Essays on Strategic Uncertainty with non-SubjectiveExpected Utility AgentsbyEvan M. CalfordB.Com(Hons), The University of New South Wales, 2008B.Sci, The University of New South Wales, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Economics)The University of British Columbia(Vancouver)July 2016c© Evan M. Calford, 2016AbstractThis thesis contains three distinct chapters that contribute to our understanding ofhow people respond, both theoretically and in controlled experimental environ-ments, to uncertainty that results from the strategic decisions of others. The stan-dard framework for studying strategic interactions involves agents with SubjectiveExpected Utility preferences (Savage, 1954) interacting in an environment where,in equilibrium, all strategies are known to all agents. This thesis studies the effectsof relaxing preferences to allow for ambiguity aversion, regret minimization, andapproximate optimization.The first chapter experimentally investigates the role of uncertainty aversionin normal form games. Theoretically, risk aversion will affect the utility valueassigned to realized outcomes while ambiguity aversion affects the evaluation ofstrategies. In practice, however, utilities over outcomes are unobservable and theeffects of risk and ambiguity are confounded. This chapter introduces a novelmethodology for identifying the effects of risk and ambiguity preferences on be-haviour in games in a laboratory environment. Furthermore, we also separate theeffects of a subject’s beliefs over her opponent’s preferences from the effects of herown preferences.The second chapter studies, experimentally, a simple dynamic entry game inboth continuous and discrete time. We introduce new laboratory methods that al-low us to eliminate natural inertia in subjects’ decisions in continuous time exper-iments. Using our novel continuous time setting and the standard discrete timesetting as benchmarks, we study the effects of inertia (caused by naturally occur-ring reaction lags) on behaviour. We demonstrate that the observed patterns of be-haviour are consistent with standard models of decision making under uncertainty,iiand that the degree of inertia affects subject responses to strategic uncertainty.The third chapter examines, theoretically, the role of mixed strategies for agentswith ambiguity averse preferences. This chapter demonstrates how a well knownresult from cooperative game theory, that a non-additive measure over a set of statescan be equivalently represented by an additive measure over the set of events, canbe used to introduce mixed strategies (in an equilibrium preserving fashion) toexisting pure strategy equilibrium concepts.iiiPrefaceChapters 2 and 4 are the unpublished, original and independent work of the author,E. Calford. Chapter 3 is the unpublished and original work of E. Calford and R.Oprea.Chapters 2 and 3 include experimental data from human subjects that wascollected at the Experimental Lab at Vancouver School of Economics (ELVSE).Chapter 2, excluding Table 2.12, is covered by UBC Ethics Certificate numberH13-02107. Table 2.12 of Chapter 2 is covered by UBC Ethics Certificate numberH15-03133. Chapter 3 is is covered by UBC Ethics Certificate number H14-00449.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Relationship to the Global Games Literature . . . . . . . . . . . . 32 Uncertainty Aversion in Game Theory: Experimental Evidence . . . 52.1 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . 102.1.1 Rationalizability . . . . . . . . . . . . . . . . . . . . . . 112.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Classification games . . . . . . . . . . . . . . . . . . . . 162.2.2 Testing game . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Experimental conditions . . . . . . . . . . . . . . . . . . 202.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . 21v2.3.2 Row player behaviour . . . . . . . . . . . . . . . . . . . 222.3.3 Column player behaviour . . . . . . . . . . . . . . . . . . 272.3.4 Discussion of results . . . . . . . . . . . . . . . . . . . . 282.4 Unpacking the Results: A New Treatment . . . . . . . . . . . . . 322.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 393 Continuity, Inertia and Strategic Uncertainty: A Test of the Theoryof Continuous Time Games. . . . . . . . . . . . . . . . . . . . . . . . 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Theoretical Background and Hypotheses . . . . . . . . . . . . . . 463.2.1 A diagnostic timing game . . . . . . . . . . . . . . . . . 463.2.2 Discrete, inertial and perfectly continuous time predictions 473.2.3 Alternative hypothesis: Inertia and ε-equilibrium . . . . . 503.3 Design and Implementation . . . . . . . . . . . . . . . . . . . . . 523.3.1 Timing protocols and experimental software . . . . . . . . 523.3.2 Treatment design and implementation . . . . . . . . . . . 543.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 Perfectly continuous and discrete time . . . . . . . . . . . 583.4.2 Inertia and continuous time . . . . . . . . . . . . . . . . . 593.5 Discussion: Strategic Uncertainty and Continuity . . . . . . . . . 603.5.1 Three decision rules . . . . . . . . . . . . . . . . . . . . 623.5.2 Validation using alternative comparative statics . . . . . . 663.5.3 Validation using other continuous time games . . . . . . . 673.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 Mental Equilibrium and Mixed Strategies for Ambiguity Averse Agents 734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.1 Mixing: Anscombe-Aumann vs. Savage interpretations ofuncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Games as Interacting Decision Problems . . . . . . . . . . . . . . 834.3.1 Translating a game into the mental space . . . . . . . . . 83vi4.3.2 Mixed strategies . . . . . . . . . . . . . . . . . . . . . . 844.4 Lo-Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 854.4.1 Formulating Lo-Nash equilibrium as a mental equilibrium 864.5 Equilibrium Under Uncertainty . . . . . . . . . . . . . . . . . . . 884.5.1 Equilibrium under uncertainty . . . . . . . . . . . . . . . 894.5.2 Mental equilibrium under uncertainty . . . . . . . . . . . 904.6 Extensive Form Games . . . . . . . . . . . . . . . . . . . . . . . 934.7 Computability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A A Brief Introduction to the Theory of Games With Ambiguity AverseAgents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.1 Theoretical Literature . . . . . . . . . . . . . . . . . . . . . . . . 112A.1.1 Lo-Nash equilibria . . . . . . . . . . . . . . . . . . . . . 115A.1.2 Nash Equilibrium under uncertainty . . . . . . . . . . . . 119B Appendix for “Uncertainty Aversion in Game Theory: ExperimentalEvidence” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122B.1 Experimental Methodology Appendix . . . . . . . . . . . . . . . 122B.1.1 Piloting and framing effects . . . . . . . . . . . . . . . . 122B.1.2 Assumptions underlying the classification procedure . . . 123B.1.3 Comprehension questions . . . . . . . . . . . . . . . . . 126B.1.4 Order of realization of randomizations . . . . . . . . . . . 128B.1.5 Related experimental literature . . . . . . . . . . . . . . . 130B.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134B.3 Results Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 135B.3.1 Demographic analysis . . . . . . . . . . . . . . . . . . . 135B.3.2 Preferences and beliefs . . . . . . . . . . . . . . . . . . . 135B.3.3 Comprehension data . . . . . . . . . . . . . . . . . . . . 138B.4 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141viiB.4.1 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . 143C Appendix for “Continuity, Inertia and Strategic Uncertainty: A Testof the Theory of Continuous Time Games” . . . . . . . . . . . . . . 153C.1 Instructions to Subjects . . . . . . . . . . . . . . . . . . . . . . . 153C.2 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 159C.2.1 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . 159C.3 ε-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168C.3.1 Characterization of ε-equilibrium sets . . . . . . . . . . . 168C.3.2 ε-equilibrium: Proofs of propositions stated in section 3.2.3173C.4 Decision Making Under Uncertainty . . . . . . . . . . . . . . . . 175C.4.1 Three decision rules . . . . . . . . . . . . . . . . . . . . 175C.4.2 Minimax regret in a broader class of games . . . . . . . . 178C.4.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183C.5 Time Series of Median Entry Times . . . . . . . . . . . . . . . . 190D Appendix for “Mental Equilibrium and Mixed Strategies for Ambi-guity Averse Agents” . . . . . . . . . . . . . . . . . . . . . . . . . . . 191D.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191D.1.1 Proof of theorem 4.1 . . . . . . . . . . . . . . . . . . . . 191D.1.2 Proof of theorem 4.2 . . . . . . . . . . . . . . . . . . . . 193D.1.3 Proof of lemma 4.3 . . . . . . . . . . . . . . . . . . . . . 193viiiList of TablesTable 2.1 The type space for preferences. . . . . . . . . . . . . . . . . . 13Table 2.2 Row player classification. . . . . . . . . . . . . . . . . . . . . 19Table 2.3 Column player classification. . . . . . . . . . . . . . . . . . . 19Table 2.4 Subjects classified by type. . . . . . . . . . . . . . . . . . . . 21Table 2.5 Row player results. . . . . . . . . . . . . . . . . . . . . . . . 23Table 2.6 Row player results as a function of preferences. . . . . . . . . 24Table 2.7 Row player choice between B and C. . . . . . . . . . . . . . . 26Table 2.8 Column player results. . . . . . . . . . . . . . . . . . . . . . . 27Table 2.9 Column player results as a function of beliefs. . . . . . . . . . 28Table 2.10 Counter factual results under self-matching. . . . . . . . . . . 31Table 2.11 Results when opponent preferences are observed. . . . . . . . 35Table 2.12 Results when opponent’s ambiguity preferences are observed. . 35Table 2.13 Results as a function of observed opponent’s preferences. . . . 36Table A.1 Summary of ambiguity averse equilibrium concepts. . . . . . . 114Table B.1 Row player results with demographics. . . . . . . . . . . . . . 136Table B.2 Column player results with demographics. . . . . . . . . . . . 137Table B.3 Summary of subject ambiguity preferences and beliefs. . . . . 138Table B.4 Summary of subject risk preferences and beliefs. . . . . . . . . 139Table B.5 Comprehension scores by game. . . . . . . . . . . . . . . . . 139Table B.6 Comprehension score summary. . . . . . . . . . . . . . . . . . 140Table B.7 Row player results including subjects that failed the compre-hension tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . 141ixTable B.8 Row player choice between B and C, including subjects thatfailed the comprehension tasks. . . . . . . . . . . . . . . . . . 151Table B.9 Column player results including subjects that failed the com-prehension tasks. . . . . . . . . . . . . . . . . . . . . . . . . . 152xList of FiguresFigure 2.1 Testing game. . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.2 Classification game 1. . . . . . . . . . . . . . . . . . . . . . 16Figure 2.3 U urn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.4 K urn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.5 Classification game 2. . . . . . . . . . . . . . . . . . . . . . 18Figure 2.6 Testing game. . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.7 The matching protocol. . . . . . . . . . . . . . . . . . . . . . 33Figure 3.1 Continuous time screen shot. . . . . . . . . . . . . . . . . . . 52Figure 3.2 Discrete time screen shot. . . . . . . . . . . . . . . . . . . . 52Figure 3.3 Entry times: PD and PC treatments. . . . . . . . . . . . . . . 57Figure 3.4 Entry times: Main treatments. . . . . . . . . . . . . . . . . . 59Figure 3.5 Basin of attraction estimates. . . . . . . . . . . . . . . . . . . 63Figure 3.6 Decision rule predictions. . . . . . . . . . . . . . . . . . . . 63Figure 3.7 Entry times: Low treatments. . . . . . . . . . . . . . . . . . . 66Figure 3.8 Cooperation rates: Friedman and Oprea (2012). . . . . . . . . 66Figure 4.1 An example with two acts and two states. . . . . . . . . . . . 79Figure 4.2 An example with two acts and two states extended into theevent space. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.3 An example game . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 4.4 An example game in the mental state space with payoffs . . . 83Figure 4.5 A normalized stag hunt game. . . . . . . . . . . . . . . . . . 92Figure A.1 Testing game . . . . . . . . . . . . . . . . . . . . . . . . . . 116xiFigure B.1 Effects of risk aversion under CARA utility. . . . . . . . . . . 125Figure B.2 Effects of risk aversion under CRRA utility. . . . . . . . . . . 126Figure B.3 U bag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure B.4 K bag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure B.5 K bag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Figure B.6 U urn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Figure B.7 K bag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Figure B.8 Example Game X. . . . . . . . . . . . . . . . . . . . . . . . 148Figure B.9 Example Game XT. . . . . . . . . . . . . . . . . . . . . . . . 149Figure B.10 K bag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure C.1 Screen shot: Nobody has entered. . . . . . . . . . . . . . . . 155Figure C.2 Screen shot: One player has entered. . . . . . . . . . . . . . . 155Figure C.3 Screen shot: The green player has entered. . . . . . . . . . . . 159Figure C.4 Screen shot: Both players have entered. . . . . . . . . . . . . 159Figure C.5 Median entry times by period, all treatments. . . . . . . . . . 190xiiAcknowledgmentsThis dissertation would not have been possible without the fantastic support ofmany people. Foremost, I would like to thank my supervisor, Yoram Halevy, forall of his support, guidance and encouragement. Yoram introduced me to decisiontheory, and his careful style of analytical thinking has improved this dissertationin numerous ways. I also thank Ryan Oprea, who complements Yoram perfectly.Ryan has been extremely generous with his time, energy and even his home (whereChapter 3 was born). If this dissertation reflects even a fraction of both Yoram’scareful rigor and Ryan’s artful storytelling then it must surely be a success. I wouldalso like to thank my other committee members, Mike Peters and Wei Li, who havechallenged and improved my thinking and writing greatly over the years. Thankyou also to Li Hao and Vitor Farinha Luz for helpful discussions.Thank you also to my family: my parents, who perhaps unwittingly set meon this path by teaching me the value of both education and incentives when theyinstituted pay-for-performance in high school, have provided me with unendingand unquestioning emotional and financial support throughout my entire journey;my sister and brother, who motivate me both with their direct support and thedrive that can only arise from having two fantastically successful younger siblingschasing after you. And, finally, my deepest gratitude must go to Kiri, who hassupported me wholeheartedly throughout my entire PhD. Without her support andlove I would likely never have reached this point and it is only fitting that she cannow share in the rewards, just as she has shared in the journey.xiiiTo the ever-loving Kiri.Chapter 1IntroductionStrategic uncertainty is a fundamental property of any strategic interaction betweenagents. Different agents, in different situations, will respond to strategic uncer-tainty in different fashions. An agent with subjective expected utility preferenceswill always behave as if they can form a precise belief regarding the behaviour ofothers. Other agents may be unable or unwilling to form such beliefs, and theseagents will necessarily have non-subjective expected utility preferences. This dis-sertation studies such agents using both theoretical and experimental methodolo-gies.Chapter 2 is an experimental study of the effects of uncertainty aversion innormal form games. The experimental design is straightforward: first risk and am-biguity preferences, as well as beliefs regarding others’ preferences, are elicited viaa series of “classification” games. Second, subjects play a normal form “testing”game. This structure allows us to directly observe the mapping between prefer-ences, and beliefs over others’ preferences, to behaviour in a strategic setting. Thepayoffs of the testing game are designed so that we can isolate the effects of pref-erences from the effects of beliefs over preferences, thereby teasing apart two keysources of strategic uncertainty.The original contributions of Chapter 2 are threefold. First, we develop andimplement a procedure for measuring subject preferences and beliefs over others’preferences in normal form games. Second, the chapter is the first paper to isolatethe effects of ambiguity from the effects of risk in a normal form game. Third, the1chapter is the first to study the role of beliefs over others’ preferences for uncer-tainty in a strategic environment.Chapter 3 is a joint work with Ryan Oprea. This chapter studies the effectsof various timing protocols on behaviour in a simple dynamic entry game. Thechapter is made possible by an innovative experimental design which allows usto implement the theoretical premises of a continuous time game directly in thelaboratory. Previous experimental studies of continuous time games have used set-tings where subjects’ natural reaction lags drive a wedge between the experimentalsetting and the theory of continuous time games (in which agents can respond in-stantaneously to changes in their environment). Our innovation is to introduce whatwe call the “freeze time protocol”, whereby the game clock is frozen at the momentof entry. During the freeze other subjects can formulate their response to the entry,and implement it, allowing for truly instantaneous responses as measured by thegame clock and satisfying key tenets of the theory.Using our freeze time protocol as well as the standard discrete time protocol asbenchmarks, we study the effects of reaction lags on behaviour in our entry game.The existence of reaction lags generates inertia in subject actions, and we manipu-late the severity of inertia by speeding up and slowing down the game clock acrosstreatments. We propose ex-ante hypotheses motivated by ε-equilibrium in con-tinuous time games (Simon and Stinchcombe, 1989; Bergin and MacLeod, 1993)which are strongly borne out by the data. In our setting ε-equilibrium leads to alarge multiplicity of equilibria in most treatments, and this multiplicity motivatedan ex-post exploration of theories that were able to explain the data with greaterprecision. The best descriptor of our data is a theory of strategic uncertainty knownas minimax regret aversion, which posits that when faced with an uncertain envi-ronment a decision maker should seek to choose that strategy that minimizes theirmaximal potential deviation from optimality. We find that changes in the degreeof inertia change the payoff effects of strategic uncertainty and that minimax re-gret aversion captures subject responses to changing strategic uncertainty acrosstreatments very well.Chapter 4 is a contribution to the theoretical literature of game theory with am-biguity agents. In games with standard, subjective expected utility, agents there aremultiple interchangeable interpretations of mixed strategies. First, we may inter-2pret agents as submitting explicitly mixed strategies to the game maker. Second,we may restrict agents to pure strategies only, and then interpret a mixed strategyequilibrium as an equilibrium in beliefs. Third, agents may use a pre-play random-ization device before reporting a pure strategy to the game maker.For games with ambiguity averse agents the different interpretations do notgenerate observationally equivalent equilibrium because the agents may have astrict preference for mixing. It has been demonstrated that the first and secondinterpretations lead to different equilibrium: compare Lo (1996) to Dow and Wer-lang (1994), for example. It is not, however, known how the third interpretationof mixed strategies relates to the first two. The third interpretation is particularlyimportant for experiments with ambiguity averse agents, such as the one presentedin Chapter 2, given that subjects must enter a pure strategy into the experimentalinterface but are free to randomize across their strategies before selecting a purestrategy. Chapter 4 fills this gap in the literature by establishing conditions un-der which the third interpretation of mixed strategies is equivalent to the secondinterpretation.1.1 Relationship to the Global Games LiteratureThe most common methodology for modeling strategic uncertainty is derived fromthe literature on global games (Carlsson and van Damme, 1993; Morris and Shin,2003). The nature of strategic uncertainty considered in this dissertation is distinctfrom the strategic uncertainty considered in the global games literature.In this dissertation, strategic uncertainty arises from agents not having a precisebelief regarding the strategic behaviour of their opponent. The strategic uncertaintyis first order strategic uncertainty (although higher order beliefs may also be im-portant), it is Knightian uncertainty, and the uncertainty may exist even in gamesof complete information. A necessary condition for such strategic uncertainty toaffect behaviour is that the agents have non-subjective expected utility preferences.The area of study in this thesis is, therefore, strategic uncertainty in games of com-plete information when agents have non-subjective expected utility preferences.In the global games literature, strategic uncertainty arises from a relaxing ofcomplete information while maintaining subjective expected utility preferences.3The key insight of the global games literature is that when complete informationis relaxed then uncertainty is generated across the entire hierarchy of beliefs, andbeliefs about beliefs, and so on. In the limiting case as first order uncertaintydisappears, the effect of strategic uncertainty over the entire belief hierarchy canstill influence behaviour. This is, however, a very different mechanism to the onestudied in this dissertation.Each approach has some advantages and disadvantages and it is important thatwe understand the effect of each mechanism on behaviour. It is well establishedthat many people do not have subjective expected utility preferences (Table 2.4confirms this to be true), and it is clearly also the case that many strategic inter-actions involve incomplete information. By isolating, and studying, each cause ofstrategic uncertainty we can generate a complete picture of the role of strategicuncertainty in human decision making.4Chapter 2Uncertainty Aversion in GameTheory: Experimental EvidenceStrategic interactions are a source of subjective uncertainty for agents. A ratio-nal agent must, necessarily, form subjective beliefs regarding their opponent’s be-haviour. But what form do these beliefs take? The standard approach to modellingstrategic interactions, Nash equilibrium, resides on a bed of expected utility the-ory: in an equilibrium each agent has a consistent and precise belief over other’sbehaviour, and the only remaining source of uncertainty is risk that stems from thepossible use of mixed strategies. This approach, however, fails to consider thatagents may retain some uncertainty regarding their opponent’s choice of (mixed)strategy. We shall refer to this uncertainty over mixed strategies as strategic uncer-tainty. Strategic uncertainty is the natural condition of real-world strategic interac-tions.There is a well-developed theoretical literature on ambiguity aversion in games(see Lo (2009), Dow and Werlang (1994), Epstein (1997) or Eichberger and Kelsey(2000), for example) that provide guidance on how agents should respond to strate-gic uncertainty. But how do people respond to strategic ambiguity? Do people be-have as if they have unique probabilistic beliefs over their opponent’s strategies, ordo they behave as if they are ambiguity averse (perhaps in a fashion consistent withevidence found in individual decision making experiments)? Can people identifywhen their opponent is facing strategic ambiguity and, if so, do they respond ra-5tionally? If not, why not? We use experimental methods to provide answers tothese questions; answers that have important implications for both applications,and development, of the theory of ambiguity aversion in games.There is, however, a fundamental problem with direct inference of ambiguityaversion from behaviour in games: risk aversion. How does a subject’s ambiguitypreference affect their behaviour in the presence of risk aversion? In theory, atleast, the separation is straightforward: for a game where subjects earn monetarypayoffs, we first take a monotonic transformation of the payoffs to move from amoney space to a utility space. Then, ambiguity aversion acts to affect the way inwhich a subject evaluates her strategies which earn utility denominated payoffs. Inpractice, however, the two affects are much more difficult to disentangle, and thischapter is the first to tackle this separation in a game theoretic setting. Becauserisk and ambiguity aversion have similar effects in games (making ‘safe’ strategiesappear relatively more attractive), and are positively correlated1, studies that focusonly on risk aversion or ambiguity aversion in games will be prone to omittedvariable bias.To disentangle these effects we use a laboratory experiment to measure ambi-guity and risk aversion in games, at the individual level, and study how behaviourin a carefully chosen normal form “testing” game is related to these measures. Thetesting game is designed so that the set of rationalizable strategies varies with pref-erences, allowing for a partial separation of behaviour as a function of preferences.The testing game also allows for a separation of the effects of ambiguity prefer-ences from the effects of beliefs over an opponents’ ambiguity preferences. Thesedual separations allow a detailed investigation of the role of uncertainty in normalform games, providing answers to our primary research questions. In a follow uptreatment we allow a group of subjects to observe their opponents’ responses to thepreference measuring tasks, allowing us to study how subjects reason conditionalon their opponents’ preferences.Subjects play 3 distinct games, and play each game as both the row player andthe column player. The first two games, the ‘classification’ games, are used tomeasure ambiguity and risk preferences, respectively. The third game, the ‘test-1Although there is some debate on this topic, risk and ambiguity aversion are correlated in thedata presented in this chapter. See footnote 22 for more details.6ing’ game, is a 3×2 normal form game used to investigate the role of preferenceson strategic behaviour and strategic reasoning. Across the three games, the be-haviour of row players is used to study the effect of preferences on behaviour innormal form games. In comparison, the behaviour of column players across thethree games is used to study the effect of beliefs over opponents’ preferences onbehaviour in normal form games. Details on the structure of the games are pro-vided in section 2.2. Importantly, preferences are measured using games, ratherthan individual decision tasks, which allows the entire experiment to be conductedin a game environment.Consider the testing game used in this experiment, presented in figure 2.1. Fora risk neutral row player with subjective expected utility (SEU) preferences, C isnever a best response. Climbing the chain of rationalizability2, once C is eliminatedthen Y is never a best response for the column player and once Y is eliminated B isnever a best response for the row player. The unique rationalizable outcome of thegame is (A,X). Naturally, (A,X) is also the unique Nash equilibrium of the game.X YA 25,20 14,12B 14,20 25,12C 18,12 18,22Figure 2.1: Testing game. Payoffs are in Canadian Dollars.In contrast, consider an agent who is ambiguity averse, with the maxmin ex-pected utility (MEU) preferences of Gilboa and Schmeidler (1989). For the rowplayer, C is now a best response for at least some feasible beliefs.3 To see this,suppose that the row player faces complete uncertainty regarding her opponent’sstrategy. Then, applying MEU preferences, she evaluates strategy A by consideringthe worst possible scenario (her opponent playing Y ) and values the strategy such2Recall that the rationalizable set can be found via iterated elimination of never-best-responsestrategies or, equivalently for two-player games, iterated elimination of strictly dominated strategies.An implication of this is that if no strategies are strictly dominated then all strategies are rationaliz-able.3Under MEU preferences an agent has a set of beliefs, and evaluates the utility of a prospectwith respect to the worst possible belief in the set. MEU preferences are not critical here, any of thestandard models of ambiguity aversion could be used with only minor modifications throughout thechapter.7that U(A) = 14. Similarly, when evaluating B she considers the worst possible sce-nario (her opponent playing X) so that U(B) = 14 as well. However, U(C) = 18and therefore C is rationalizable for the row player. Obviously A and B are alsopotential best responses (to the belief the column player is using pure X or pure Y ,respectively), and both column player strategies are also potential best responses.Therefore, under MEU preferences the rationalizable set is the full strategy set forboth players.Finally, note that if the row player has SEU preferences with a high enoughlevel of risk aversion then the rationalizable set is also the complete set of out-comes. For a subject with SEU preferences, C is never a best response to anybeliefs for the row player if and only if C is not a best response to an opponentwho mixes 50-50 over X and Y . When C is a best response to the 50-50 mix thenall strategies for both players are best responses to some beliefs and, therefore, allstrategies are rationalizable.The size of the rationalizable set hinges critically on the row player’s prefer-ences, but not the column player’s preferences: the rationalizable set is (A,X) ifand only if the row player can eliminate C in the first round. This provides the sep-aration of the role of preferences and beliefs, which is one of the key innovationsof this chapter. If the row player has SEU preferences and low risk aversion thenshe should never play C regardless of her beliefs regarding her opponent’s prefer-ences. The column player should never play Y if he believes that his opponent hasSEU preferences and low risk aversion regardless of his own preferences. The sep-aration of the role of preferences and beliefs is a novel design feature that allowsfor an investigation of the structural underpinnings of ambiguity averse solutionconcepts.Preferences organize the data: in the game in figure 2.1 ambiguity neutral sub-jects with low risk aversion play A more than twice as often as ambiguity aversesubjects with high risk aversion. On the other hand, measured beliefs over oppo-nent’s preferences are independent of column player behaviour: amongst subjectswho passed a series of incentivized comprehension tasks, there is no difference inbehaviour between subjects who believe their opponent to be ambiguity neutral orambiguity averse: both groups play X approximately three-quarters of the time.A follow-up treatment, where subjects were shown their opponents’ behaviour in8the classification games before playing the testing game, establishes that this nullresult is not driven by an inability of subjects to use preference information to pre-dict opponent behaviour. A subject that observes his opponent choosing optionsconsistent with ambiguity and risk neutrality in the preference measuring gameschooses X 100% of the time, while a subject who observes his opponent choosingoptions consistent with ambiguity and risk aversion in the preference measuringgames chooses X 30% of the time.Taken as a whole, our findings provide support for the claim that ambiguitypreferences are an important determinant of behaviour in games. However, thedata also suggest that mutual knowledge of preferences is not satisfied in the sub-ject population and an intervention that provides credible preference informationto subjects induces large behavioural changes. Nash equilibrium, therefore, is re-jected on two accounts. First, Nash fails to take account of ambiguity preferences.Second, subjects do not appear to hold a consistent set of beliefs across games. Fur-thermore, the results suggest that, empirically, both risk and ambiguity aversion areimportant factors in determining how subjects respond to the natural uncertaintythat arises in strategic interactions.There is a wealth of evidence, tracing back to Knight (1921) and Keynes (1921)via Ellsberg (1961), of ambiguity affecting decisions in individual decision mak-ing environments. The relative paucity of experimental evidence on the role ofambiguity aversion in strategic environments was a key motivation for this study.The previous literature provides a series of snapshots into how subjects behave inthe face of strategic uncertainty, and suggests that ambiguity aversion plays a keyrole in strategic decision making. Camerer and Karjalainen (1994) provides evi-dence that subjects, on average, prefer to avoid strategic uncertainty by betting onknown probability devices rather than on other subjects’ choices. Eichberger et al.(2008) establish that subjects find grannies to be a greater source of strategic am-biguity than game theorists. Kelsey and le Roux (2015a) find that subjects exhibithigher levels of ambiguity aversion in games than in a 3-colour Ellsberg urn task.This chapter is the first to give a complete picture of the role of uncertainty aver-sion in games: we document the first-order (how do subjects respond to strategicuncertainty?) and second-order (how do subjects respond to opponents who face9strategic uncertainty?) effects of both risk aversion and ambiguity aversion.4 Thischapter is also the first to provide a procedure for measuring preferences using dis-crete choice tasks in a framing that is consistent with typical normal form gameexperiments.5This chapter proceeds as follows. Section 2.1 provides a brief overview ofsome relevant theoretical considerations, while section 2.2 presents the experimen-tal design in detail. Section 2.3 presents the experimental results, and section 2.4presents an additional treatment motivated by these results. Section 2.5 provides adiscussion and conclusion. Appendix B presents additional results not included inthe main text as well as proofs and the instructions for subjects. Appendix A con-tains some additional background on the theory of games with ambiguity averseagents, a topic that is also explored in chapter 4.2.1 Theoretical ConsiderationsThe identification of the relationship between preferences and “reasonable” strate-gies in the testing game can be motivated either using rationalizability argumentsor equilibrium concepts (i.e. Nash equilibrium for SEU subjects and an appropri-ately chosen ambiguity averse equilibrium concept for ambiguity averse subjects).We use rationalizability rather than equilibrium concepts in the body of the chapterbecause of the stronger epistemic assumptions required to justify the use of equi-librium; however, we provide two equilibrium derivations (in the style of Lo (2009)and Dow and Werlang (1994)) in Appendix A for the interested reader.Throughout we assume that subjects choose from a set of pure strategies, anddo not play mixed strategies. This is consistent with the experimental implemen-tation where subjects were required to select a pure strategy choice for each game.The reason for this choice, which is also common amongst the ambiguity aversionin game theory literature, is that models of ambiguity aversion typically imply astrict preference for mixed strategies or are not able to define a utility level for4Interestingly, Ivanov (2011) asks the dual of our research question by estimating ambiguityaversion from behaviour in games. A more detailed literature review can be found in appendix B.1.5.5Heinemannn et al. (2009) also recognized the importance of using frame-consistent tasks tomeasure preferences and strategic uncertainty. In their case, they used a modified coordination gamethat was framed as a multiple price list to study strategic uncertainty through the lens of global games.10mixed strategies at all. Chapter 4 and Eichberger and Kelsey (2000) contain ex-tensive discussion on the role of mixed strategies in games with ambiguity averseagents. As Chapter 4 demonstrates it is possible to extend the theory in this sectionto allow for subjects to use mixed strategies if the mixed strategy randomizationdevice is resolved before the game is played. Therefore, a subject who rolls a diceor otherwise randomizes their choice, before clicking on a pure strategy, could alsobe accommodated by the theory described below.2.1.1 RationalizabilityIn this section we formalize the interactions between preferences and rationaliz-ability in the game in figure 2.1. We use Epstein’s (1997) notion of rationalizabil-ity, which allows for a generalization of SEU preferences to the MEU preferencesof Gilboa and Schmeidler (1989). Theorem 3.2 in Epstein (1997) establishes thatthe set of rationalizable strategies can be found by iteratively eliminating all strate-gies that are not best responses given the preferences of the agents, and thereforegeneralizes the more familiar Pearce (1984)/Bernheim (1984) rationalizability fortwo player games. The only conceptual difference between the Pearce-Bernheimframework and the Epstein framework is the treatment of mixed strategies: Epstein(1997) restricts the feasible set of strategies to consist only of pure strategies, al-though agents may still hold beliefs over the mixed strategies of their opponents(following a population or belief-based interpretation of mixed strategies).Consider Gilboa and Schmeidler’s (1989) MEU preferences (for ambiguityaverse agents), in which an agent’s beliefs regarding her opponent’s strategy isa closed and convex subset of probability measures over her opponent’s strategyset. The agent then evaluates her utility for a given prospect with respect to her‘worst case’ belief, and may use different beliefs to evaluate different prospects.For example, in our game, the set of feasible row player beliefs is a subset of theprobability simplex (ΦR ⊆ ∆({X ,Y})) and, given ΦR, the row players’ preferencescan be represented by:U(aR) = minφR∈ΦR ∑aC∈{X ,Y}uR(mR(aR,aC))φR(aC) ∀aR ∈ {A,B,C} , (2.1)11where uR(.) is the row player’s utility over monetary outcomes, mR(aR,aC) isthe monetary payoff for outcomes as shown in figure 2.1, and aR ∈ {A,B,C} andaC ∈ {X ,Y}. Given these preferences, we interpret risk aversion, in the standardmanner, as curvature of the utility function. We model ambiguity aversion in abinary fashion. For subjective expected utility subjects ΦR is constrained to bea singleton (ΦR ∈ ∆({X ,Y})), while for ambiguity averse subjects ΦR is uncon-strained (ΦR ⊆ ∆({X ,Y}). Column player preferences are defined analogously.Proposition 2.1. Consider the normal form game in figure 2.1. If the row playerhas preferences such that u(25) + u(14) > 2u(18) and ΦR is a singleton for allfeasible beliefs then the rationalizable set is {(A,X)}. If u(25)+ u(14) ≤ 2u(18)or ΦR is unrestricted then all pure strategies are rationalizable.Proof of proposition 2.1. See Appendix B.2.The condition u(25)+u(14)≤ 2u(18) provides a necessary and sufficient con-dition on the utility function of a SEU agent for C to be undominated.Notice that proposition 2.1 does not restrict the column player’s preferences:the column player’s preferences play no role in determining the size of the rational-izable set. Also note that the use of MEU preferences to model ambiguity aversionis not essential: any of the standard models of ambiguity aversion could be used.We use MEU here because it is arguably the simplest model of ambiguity aversepreferences for this game. Furthermore, Epstein (1997) provides a fully workedapplication of his rationalizability concept using MEU preferences, and it is alsothe preference structure underlying Lo (2009) (which is discussed in detail in Ap-pendix A). Alternatives such as Choquet Expected Utility (Schmeidler (1989)),which underlies Dow and Werlang (1994) (also discussed in discussed in detail inAppendix A), or Klibanoff et al.’s (2005) smooth ambiguity aversion preferenceswould work just as well. In fact, in a recent working paper Battigalli et al. (2015)independently6 establish a related result, illustrated using an example on a similar6The experimental design in this chapter was created in late 2013, and experiments were con-ducted in early 2014. The first version of this chapter that was circulated outside UBC was inNovember 2014. The earliest version of Battigalli et al. (2015) available on Google Scholar is datedSeptember 2014, and the first version that I read was dated May 2015.12game to the one studied here, for the case of smooth ambiguity aversion prefer-ences. Their example also highlights and reaffirms that “the risk and ambiguityattitudes of the (column player) are immaterial.”For the remainder of the chapter, we shall restrict ourselves to four possiblepreference types that can be held by the row player. The row player may eithersatisfy or violate u(25)+u(14)> 2u(18) and the row player may either have SEUor MEU preferences, as shown in table 2.1. Applying proposition 2.1 to this clas-sification structure, it is immediate that the rationalizable set is {A,X} if and onlyif the row player has Type 1 preferences.SEU MEUu(25)+u(14)> 2u(18) Type 1 Type 2u(25)+u(14)≤ 2u(18) Type 3 Type 4Table 2.1: The type space for preferences.Implicit in proposition 2.1, because of the use of rationalizability as the so-lution concept, is an assumption that the column player knows the row player’spreferences. Under this assumption the column player has no role to play in de-termining the rationalizable set. We can, however, relax the link between the rowplayer’s preferences and the column player’s beliefs by modeling the game as aBayesian game with one sided private information.7 Suppose there is a commonprior, α ∈ [0,1], that the row player has Type 1 preferences. The sum of the prob-abilities of Type 2, 3 or 4 row players is, therefore, 1−α .8Proposition 2.2. Consider the normal form game in figure 2.1. Suppose that thegame is a Bayesian game with one-sided private information and the row playerhas Type 1 preferences with prior probability α ∈ [0,1]. It follows that:1. if α > α¯ , then the rationalizable set is {(A,X)},7It is possible to relax the knowledge requirements even further by modeling the game as a Savagegame ala Grant et al. (Forthcoming), or by allowing both the row and column to have private prefer-ence information. These richer models will generate a similar result, replacing the α > 59 conditionin the proposition with an object that is harder to interpret.8Note that in this structure the prior is a singleton, and any ambiguity is captured through themapping from types to actions; that is, all ambiguity is ambiguity with respect to strategies whiletypes introduce only objective risk.132. if α ≤ α¯ , then all actions are part of a rationalizable profile, but a Type 1row player will never be observed to play C.where α¯ = u(22)−u(12)u(22)+u(20)−2u(12) . α¯ =59 for a risk neutral column player. Furthermore,α¯ > 12 and is decreasing in the column player’s level of risk aversion.Proof of proposition 2.2. See Appendix B.2.The key implication of proposition 2.2 is that Y is not rationalizable for thecolumn player if α > 59 .In the next section we shall discuss the procedure that was used to classifysubjects into types, and to elicit subjects beliefs regarding their opponent’s type.We close this section by noting that although both risk aversion and ambiguityaversion have identical effects on the rationalizable sets, there are some differencesin the effects of risk and ambiguity aversion in the equilibria of the game in figure2.1. It is straightforward to establish that the Nash equilibrium set for SEU subjectswith high risk aversion (preferences that satisfy u(25)+u(14)< 2u(18)) includesequilibria where the row player mixes over exactly two of their strategies. Thereare no fully mixed equilibria, and no equilibria that involve pure C for the rowplayer. However, for ambiguity averse subjects there are fully mixed strategiesand, for some equilibrium concepts, pure C can be sustained in an equilibrium.As demonstrated in Appendix A the equilibria of both Lo (2009) and Dow andWerlang (1994) can support fully mixed equilibria and the latter may also supportequilibria where the column player plays pure C.92.2 Experimental DesignThe experimental design involves 3 two-player normal form games10, with sub-jects playing each of the games as both the row player and the column player. The9I thank Simon Grant (private correspondence) for demonstrating that pure C can be supportedunder ambiguity averse equilibrium concepts that do not impose consistency of the support of beliefswith the support of the actions used, such as Eichberger and Kelsey (2000) and Dow and Werlang(1994).10Subjects were also presented with a fourth game, which was intended to be used as an additionaldiagnostic test in the case of contingencies that did not arise in the data. The fourth game is availablein an earlier working paper version of this chapter.14first two games were used to measure the risk and ambiguity preference of the rowplayer, and the column player’s beliefs regarding his opponent’s risk and ambiguitypreference; these games are referred to as the classification games. The remaininggame was then used to test whether the subject’s preferences and beliefs, cou-pled with an appropriate solution concept, are associated with behaviour in normalform games; this game is referred to as the testing game. The testing game wasconstructed so that behaviour as the row player is expected to depend only on thesubject’s preferences (and not her beliefs regarding her opponent’s preferences)and behaviour as the column player is expected to depend only on the subject’sbeliefs regarding his opponent’s preferences (and not his own preferences).The experiment was designed to mitigate a number of factors that have beenidentified as affecting the elicitation of ambiguity preferences. Recent researchhas identified that subject confusion can significantly lower measured ambiguityaversion (Chew et al. (2013)), that experiments with ambiguity can be particu-larly susceptible to violations of incentive compatibility across tasks (Baillon et al.(2014), Azrieli et al. (2014)) and that framing effects may be particularly strong(Chew et al. (2013)). The experimental design presented here mitigates these fac-tors by using extensive, incentivized, comprehension tasks to test for subject un-derstanding, realizing objective randomizations prior to realizing subjective ran-domizations, and presenting all tasks in a unified normal-form game framing. Theuse of classification games to measure preferences, rather than more traditional in-dividual preference measuring tasks, ensures that the measurement of preferencesis performed in the same domain as the testing game thereby reducing the chanceof framing effects across domains contaminating the data. This chapter is the firstto elicit ambiguity preferences in a framing that is consistent with normal formgames.A range of other important, yet technical, experimental considerations, includ-ing a description of both the random payment mechanism used and the detailed andincentivized comprehension questions, are discussed in detail in the experimentalmethodology appendix B.1.152.2.1 Classification gamesThere are two classification games. The first game elicits ambiguity preferences(for row players) and beliefs regarding the opponent’s ambiguity preferences (forcolumn players), while the second game elicits risk preferences and beliefs regard-ing the opponent’s risk preferences. In each game the row player selects betweena set of prospects, whose payoffs depend only on exogenous random events, whilethe column player earns a positive payoff if and only if she correctly predicts therow player’s choice.The first (ambiguity) classification game is shown in figure 2.2. For the rowplayer, this game is isomorphic to a simplified version of the ambiguity elicitationprocess used in Epstein and Halevy (2014).11 The game involves two ball draws,one from the U urn (figure 2.3) and one from the K urn (figure 2.4). Therefore thereare four possible states of nature, but only two payoff tables. The left payoff tablerepresents the state red ball drawn from the U urn and yellow ball drawn from the Kurn: (RU ,YK). The right payoff table represents the state (YU ,RK). The payoffs forstate (RU ,RK) are found by adding the two payoff tables together, and the payoffsin state (YU ,YK) are identically 0 for both players. The relationship between statesand payoffs was carefully explained to the subjects, and understanding was testedvia a series of comprehension questions that are discussed in detail in appendixB.1.3.S′ M′S 30.1,15 30.1,0M 0,0 0,15Red ball drawn from U urnS′ M′S 0,15 0,0M 30,0 30,15Red ball drawn from K urnFigure 2.2: Classification game 1. This game is used to measure the row player’s ambiguityaversion and the column player’s belief of the row player’s ambiguity aversion.Given that row player payoffs are independent of the column player strategychoice, we can view the row player as facing a choice between a bet that pays11Epstein and Halevy (2014) use four tasks to elicit ambiguity preferences, whereas we havesimplified this to one task. We assume symmetry of beliefs, as discussed below, which allows usto drop two of the four tasks. Furthermore, we are only interested in ambiguity aversion here (andnot ambiguity seeking behaviour), allowing us to remove one additional task.16Figure 2.3: U urn. The U urn consistsof 10 balls, each of which may beeither red or yellow. The total num-ber of red balls in the urn lies be-tween 0 and 10.Figure 2.4: K urn. The K urn contains5 red and 5 yellow balls.$30.10 if a red ball is drawn from the U urn and a bet that pays $30 if a red ball isdrawn from the K urn. We assume that subjects hold symmetric beliefs about thedistribution of balls in the U urn, which appears reasonable because red and yelloware interchangeable labels.12 If a subject has SEU preferences, then they shouldstrictly prefer strategy S (the bet on the U urn). A subject with ambiguity aversepreferences should prefer strategy M (the bet on the K urn). We note that becausethe row player is indifferent to her opponent’s strategy, the existence of the columnplayer should have no effect on the row player’s choices.Now, consider the column player in the game in figure 2.2. The column playeris tasked with predicting the row player’s action. If the outcome of the game is(S,M′) or (M,S′) then the column player receives $0 in all states. So the rationalcolumn player will play S′ if they believe that the row player is more likely tochoose S, and will play M′ if they believe that the row player is more likely tochoose M.The second classification game is shown in figure 2.5 and has a very similarstructure to the first classification game, with the key difference being that thestate is now determined by a single draw from the K urn. The row player chooseswhich risky prospect they would like to hold, and the column player attempts to12Note that this is an assumption regarding the symmetry of beliefs, and not an assumption on thesubjects preferences regarding red or yellow balls. However, even if a subject does happen to preferred balls over yellow balls, for whatever reason, there are still no confounding effects. Subjects mayonly bet on the red balls in this formulation, and a general preference for red would be equivalent toincreasing the prize paid on a red ball being drawn an equal amount for each urn.17L′ I′ H ′L 25,30 25,0 25,0I 11,0 11,30 11,0H 15,0 15,0 15,30Red ball drawn from K urnL′ I′ H ′L 10,30 10,0 10,0I 23,0 23,30 23,0H 15,0 15,0 15,30Yellow ball drawn from K urnFigure 2.5: Classification game 2. This game is used to measure the rowplayer’s risk aversion and the column player’s belief of the row player’srisk aversion.predict the row player’s preferences. A highly risk averse row player will chooseH, while a risk neutral (or low risk aversion) row player will choose L. Subjectswith intermediate levels of risk aversion will choose I. Similarly, a subject thatchooses L′ believes his opponent to have low risk aversion, a subject that chooses I′believes his opponent to have intermediate risk aversion, and a subject that choosesH ′ believes his opponent to be rather risk averse.We now consider the mapping from responses in the preference measurementtasks to the type structure that was introduced in section 2.1.1. For an extensivediscussion of the auxiliary assumptions underlying the classification procedure seeSection B.1.2. The mapping is straightforward: a subject who selects S in thegame in figure 2.2 is considered to be ambiguity neutral and therefore is eithertype 1 or type 3, and a subject who selects M is considered to be ambiguity averseand therefore is either type 2 or type 4. A subject who selects L in the game inFigure 2.5 is considered to have low risk aversion and therefore is either type 1 ortype 2, and a subject who selects H is considered to have high risk aversion andis either type 3 or type 4. Subjects who selected I are not classified, on the basisthat their risk preferences are such that it is not possible to determine whether theywould prefer to play C or a 50-50 mix between A and B in the game in Figure 2.1.Column players are classified in an analogous fashion.Table 2.2 presents a summary of the mapping from row player choices in thepreference measuring games to types and table 2.3 presents a summary of the map-ping for column players.18Row Player Strategies S MLType 1 Type 2{A} {A,B,C}I N/A N/AHType 3 Type 4{A,B,C} {A,B,C}Table 2.2: The classification procedure for row players, as a function of re-sponses to the classification games. Subjects that select I in the risk mea-surement game are not classified. The rationalizable set in the testinggame for each type is also indicated.Column Player Strategies S′ M′L′ Type 1′ Type 2′{X} {X ,Y}I′ N/A N/AH ′ Type 3′ Type 4′{X ,Y} {X ,Y}Table 2.3: The classification procedure for column players, as a function ofresponses to the classification games. Subjects that select I′ in the riskmeasurement game are not classified. A type 1′ subject believes that theiropponent is a type 1 subject. The rationalizable set in the testing gamefor each type is also indicated.2.2.2 Testing gameThe testing game, reproduced in figure 2.6, is the same game that was introduced inthe introduction. To recap, the testing game will be used to test whether the strate-gies played by subjects change with preferences (and beliefs) in the same mannerthat the rationalizable set of strategies changes with preferences (and beliefs). Forrow players, subjects who are classified as type 1 (low risk aversion and ambi-guity neutrality) have a unique rationalizable set of {A} while for other subjectsall strategies are rationalizable. For column players, subjects who are classified astype 1′ (believe their opponent to be type 1) have a unique rationalizable set of {X}while for other subjects both strategies are rationalizable.The interaction of risk and ambiguity preferences in the testing game is of aprecise nature: the row player needs only be sufficiently risk averse or ambiguity19X YA 25,20 14,12B 14,20 25,12C 18,12 18,22Figure 2.6: The testing game. The row player’s rationalizable strategy set varies with herpreferences. The column player’s rationalizable strategy set varies with his belief of therow player’s preferences.averse for the rationalizable set to be the complete action space. The hypothesisof subject level behaviour are summarized in tables 2.2 and 2.3. For row players,subjects that choose S and L in the classification games are uniquely expected toplay A, while all other subjects may play any of A, B or C. For column players,subjects that choose S′ and L′ in the classification games are uniquely expected toplay X , while all other subjects may either of X or Y .2.2.3 Experimental conditionsAll sessions were held in the ELVSE lab at the Vancouver School of Economics.There were 10 sessions held in March and April 2014, and a further 10 sessionsheld in September 2014, with between 8 and 12 subjects per session. There weresome minor changes to the instructions made between the April and Septembersessions, otherwise all sessions were identical. The September instructions areincluded in the appendix. Sessions lasted between 60 and 90 minutes, and theaverage payment was $26.60. Subjects were recruited from the ELVSE implemen-tation of the ORSEE subject pool (Greiner (2015)), which is overwhelmingly madeup of UBC undergraduate students. The experiments were run using the Redwoodexperimental software tool (Pettit et al. (2015)).2.3 ResultsThis section presents the experimental results. We split this section in two: firstwe present the results and discuss the statistical methods used to produce them.Second, we discuss the economic implications of the results. There are three keydimensions to the results presented in this section:• Risk and ambiguity preferences are positively correlated;20• Row player behaviour in the classification games is correlated with rowplayer behaviour in the testing game, as predicted by theory;• Column player behaviour in the classification games is independent of col-umn player behaviour in the testing game.2.3.1 PreferencesWe begin our tour of the results with a look at the preferences of our subjects.Ambiguity preferenceNeutral AverseType 1 Type 2Low N=85 (E=75) N=36 (E=46) 121Type 3 Type 4Risk aversionHigh N=30 (E=40) N=34 (E=24) 64115 70 185Table 2.4: Number of subjects, classified by type. Expected values, assuming independenceof risk and ambiguity preferences, are in brackets. The null of independence is rejectedat the 1% level, using Pearson’s χ2 test (p = 0.002).Table 2.4 presents the classification of subjects into types based on their re-sponses as row players in the preference measuring games. 38% of the subjectswere classified as ambiguity averse, a figure that is at the lower end of the levelof ambiguity aversion reported in previous papers, and lower than that measuredpreviously in 2-urn Ellsberg tasks.13 We find that 30% of subjects with low riskaversion are ambiguity averse, while more than half of the subjects with highrisk aversion are also ambiguity averse.14 A Pearson’s χ2 exact test rejects the13Chew et al. (2013) provide an overview of previous Ellsberg urn experimental results. For the 2-urn case, as used in this chapter, previous studies have found that between 47% and 78% of subjectsare ambiguity averse (with a weighted mean of 66%). For the 1-urn (3-colour) Ellsberg task, previousstudies have found that between 79% and 8% of subjects are ambiguity averse (with a weighted meanof 27%). Pilot sessions suggest that the framing of the decision problem in a bi-matrix format maycause lower levels of ambiguity aversion to be measured, although this effect is mitigated by the useof comprehension questions. This supports the conclusions of Chew et al. (2013), who argue thatless ambiguity aversion will be measured in more complex environments.14Female subjects are significantly more likely to be risk averse than male subjects (Pearson’s χ2p= 0.025). After conditioning on preferences, gender does not play a significant role in determiningbehaviour and additional demographic analysis is relegated to the appendix. Due to a technology21null hypothesis of independence of risk and ambiguity preference at the 1% level(p = 0.002).15Result 2.1. Risk and ambiguity preferences are positively correlated.2.3.2 Row player behaviourWe now turn our attention to the testing game, beginning with row player behav-ior in the game in figure 2.6. Recall that Type 1 subjects (low risk aversion andambiguity neutral) are expected to play the unique Bernheim/Pearce rationalizablestrategy of A, while for other subjects all strategies are rationalizable.For this analysis, we restrict our sample to those subjects who “passed” thecomprehension tasks. A subject was considered to pass the comprehension task,as a row player, if they answered the comprehension questions correctly on thefirst attempt for all three games they played as the row player. This criteria wasdetermined ex-ante16, and if a subject made any mistakes they were considered tohave failed. While this criteria may seem rather harsh, it is important to note thatthe study of ambiguity aversion in games prohibits the use of experience to trainsubjects: if the subjects were to play the game repeatedly, even using a strangermatching protocol, they would learn about the distribution of behaviour in the pop-ulation and thereby reduce the degree of strategic ambiguity in later interactions.Effectively, repeated play between ambiguity averse subjects can change the equi-librium set even when there are no supergame effects present. A strict comprehen-sion criteria is therefore used as a substitute for experiential learning to identify amishap, demographics were not collected for two sessions. The change in sample has a larger effecton the results than controlling for demographic factors does, so demographic controls are not used inthe main analysis.15The choice of statistical tests for categorical data is an oft-debated topic. Throughout this chap-ter, every p-value could be calculated in multiple different ways, with the statistical inferences al-most always being the same under all alternatives. A general preference for non-parametric testsis displayed throughout, and regression (or logit) analysis is avoided in favour of χ2 tests over ap-propriately defined sub-populations where possible. Fisher exact tests are used for cases where thesample sizes are small or severeley unbalanced. An earlier version of the chapter used Fisher exacttests throughout and generated the same conclusions.16There are many different inclusion criteria that could be constructed from the comprehensiondata. See appendices B.1.3 and B.3.3 for a discussion of the comprehension data and inclusioncriterion.22sub-population of subjects that thoroughly understand the structure of the gamesthey are playing.We present results for the full sample, as a robustness check, in appendix B.3.3.Moving to the full sample strengthens the row player results presented in this sub-section, because the subjects who failed the comprehension tests display approx-imately the same effect of preferences on behaviour in the testing game and theincreased sample size helps with statistical power. We do not combine the twosamples, however, because subjects who failed the comprehension tests show astatistically significant level effect: across all preference types, subjects who failedthe comprehension play A less often in the testing game.Table 2.5 presents an aggregate view of the row player data, detailing the per-centage of subjects that play A in the game in figure 2.6 as a function of theirpreference type, for subjects who passed the comprehension tasks.Type 1 Type 2 Type 3 Type 4 TotalLRA, AN LRA, AA HRA, AN HRA, AAPr(A) 0.74 (N=57) 0.58 (N=19) 0.61 (N=18) 0.41 (N=22) 0.63 (N=116)Table 2.5: Proportion of subjects that play A in the game in figure 2.6 by preference type, re-stricted to subjects that passed the comprehension tasks. LRA denotes low risk aversion,HR denotes high risk aversion, AN denotes ambiguity neutral and AA denotes ambiguityaverse.We now turn to analyzing the effects of preferences on the probability of asubject playing A in the testing game. We focus on the choice of A because thetheory outlined in section 2.1.1 only provides predictions of A or not A and isagnostic over the choice of B or C. We shall, however, let the data speak regardingthe choice of B or C in section 2.3.2.Table 2.6 presents the estimates of the effects of preferences on behaviour inthe testing game. The top two lines present the average, unconditional, effects ofambiguity aversion and risk aversion, respectively. The next four lines present con-ditional effects of risk and ambiguity aversion, and the final line presents the jointeffect of both risk and ambiguity aversion (the difference in behaviour betweenType 1 and Type 4 subjects). All p-values are calculated non-parametrically usingPearson’s χ2 test of independence.As we can see from table 2.6 the average, unconditional, effect of both risk and23Effect of Restricted to ∆Pr(A)Ambiguity Aversion -0.22*(0.020)Risk Aversion -0.20*(0.036)Ambiguity Aversion LRA Subjects -0.16(0.194)Ambiguity Aversion HRA Subjects -0.20(0.204)Risk Aversion AN Subjects -0.13(0.307)Risk Aversion AA Subjects -0.17(0.278)Ambiguity Aversion & Risk Aversion -0.33**(0.006)Table 2.6: Change in proportion of subjects playing A in the testing game, as a function ofsubject preferences,restricted to subjects that passed the comprehension tests. N = 116.LRA denotes low risk aversion, HR denotes high risk aversion, AN denotes ambiguityneutral and AA denotes ambiguity averse. Pearson’s χ2 p-value shown in brackets. *indicates value is significantly different from 0 using a non-directional test at the 5%level and ** indicates significance at the 1% level.ambiguity on the probability of choosing A are negative, large and statistically sig-nificant. These results are, however, biased estimates of the true effects: becauseboth effects are negative, and risk and ambiguity preferences are negatively cor-related, they suffer from omitted variable bias. Nevertheless, we report them herefor comparison to other data sets that may not include measures of both risk andambiguity preference.There is clear evidence in table 2.6 for a joint effect of risk and ambiguity pref-erences on behaviour in the game in figure 2.6: the bottom row indicates that ahighly risk averse and ambiguity averse (Type 4) subject who passed the compre-hension tests is 33 percentage points less likely to play A than a slightly risk averseand ambiguity neutral (Type 1) subject who passed the comprehension tests. Theeffect is large and highly statistically significant. Decomposing the joint effect intothe individual effect of risk and ambiguity aversion is more difficult, as the smaller24sample sizes generate imprecise estimates of the effect size.17 The effect of ambi-guity aversion conditional on low risk aversion is identified from the difference inbehaviour between Type 1 and Type 2 subjects, so that the small sample of 19 Type2 subjects who passed the comprehension test reduces the power of the χ2 test ofindependence.Result 2.2. A joint measure of risk and ambiguity aversion is associated with alower probability of playing A in the game in figure 2.6.While restricting our analysis to the choice of A or not A has strong theoreticalmotivations, it does involve throwing away information regarding the choice of Band C. The next section makes use of this information to investigate if there areany differential effects of risk and ambiguity aversion in the testing game.Analysing the choice between B and CTable 2.7 breaks down the choice of “not A” and investigates how preferencesinfluence the decisions of subjects choosing between B and C. The point estimatesprovided are mechanically equivalent to the point estimates from a multinomiallogit regression where A is specified as the base outcome.18 The p-values arecalculated non-parametrically, using either Pearson’s χ2 test for larger samples orFisher’s exact test for smaller samples (p-values that were calculated using Fisher’sexact test are denoted with a F).19The coefficients in table 2.7 are interpreted as follows. Consider the top leftestimate of 0.19: this indicates that an ambiguity averse subject is 19 percentage17One potential method for decomposing the effects of risk and ambiguity aversion is to use anANOVA. The ANOVA yields a main effect of ambiguity aversion on behaviour that is statisticallysignificant at the 10% level (p = 0.07). The interpretation of this result is difficult, however. Asdiscussed in section 2.3.4 the theoretical effect of ambiguity aversion on the rationalizable set is onlyidentified for subjects with low risk aversion, which implies that any effect of ambiguity aversion forhigh risk aversion subjects must be driven by a different mechanism. The ANOVA exercise conflatesthese two different mechanisms.18We specify A as the base outcome so that we can still interpret the decision as a choice betweenA and something that isn’t A, keeping to the spirit of the theoretical predictions.19A word of warning on sample sizes: only 11 of the 116 subjects who passed the comprehensiontests chose B in the testing game (32 chose C and the remainder chose A). However, more than aquarter of Type 2 subjects (low risk aversion, ambiguity averse) chose B, and it is this effect whichdrives the results presented below. We use Fisher’s exact test instead of Pearson’s χ2 test whenconsidering the choice of B because of its favorable small sample properties.25points more likely to choose B, conditional on not choosing C, than an ambiguityneutral subject. The results suggest that the effect of ambiguity aversion, partic-ularly amongst low risk aversion subjects, is to move subjects away from A andtowards B: a Type 2 subject is 25 percentage points more likely to choose B, con-ditional on not choosing C, than a Type 1 subject. The joint effect of risk andambiguity aversion, on the other hand, is to move subjects away from A and to-wards C (see the bottom row of the table).20Effect of Restricted to ∆ NBNB+NA ∆NCNC+NAAmbiguity Aversion 0.19* 0.16(0.033)F (0.099)Risk Aversion -0.02 0.24*(1.000)F (0.011)Ambiguity Aversion LRA Subjects 0.25* -0.01(0.024)F (1.000)FAmbiguity Aversion HRA Subjects 0.09 0.20(0.590)F (0.231)Risk Aversion AN Subjects 0.02 0.13(1.000)F (0.341)FRisk Aversion AA Subjects -0.13 0.34(0.622)F (0.050)Ambiguity Aversion & Risk Aversion 0.12 0.33**(0.251)F (0.007)Table 2.7: Change in the values of NBNB+NA andNCNC+NA in the testing game as a function ofsubject type, where NA is the number of subjects selecting A. Restricted to subjects thatpassed the comprehension tasks. N = 116. LRA denotes low risk aversion, HR denoteshigh risk aversion, AN denotes ambiguity neutral and AA denotes ambiguity averse.Figures in brackets are p-values calculated under a null hypothesis that the coefficient isequal to zero under either Pearson’s χ2 test or a Fisher exact test (denoted by F). Thecoefficients are equivalent to the probabilities implied by a saturated multinomial logitregression with a choice of A denoted as the base outcome.Result 2.3. A joint measure of risk and ambiguity aversion is associated with ahigher probability of playing C in the game in figure 2.6.20An ANOVA on the choice between A and B finds a significant main effect of ambiguity aversion(p = 0.041). An ANOVA on the choice between A and C finds a significant main effect of riskaversion (p = 0.012). Again, care should be taken in interpreting the ANOVA results, see footnote17.26Result 2.4. There is evidence that ambiguity aversion is associated with a higherprobability of playing B in the game in figure 2.6, for low risk aversion subjectswho passed the comprehension tests.2.3.3 Column player behaviourWe now turn our attention to column player behaviour in the game in figure 2.6.Recall that Type 1′ subjects have a unique rationalizable action of X , while both Xand Y are rationalizable for all other subjects. We should therefore, expect subjectswho believe their opponent to be risk and ambiguity averse to play X less oftenthan subjects that believe their opponent to be ambiguity neutral and have low riskaversion.Type 1′ Type 2′ Type 3′ Type 4′ TotalBelieve opponent to be LRA, AN LRA, AA HRA, AN HRA, AAPr(X) 0.74 (N=47) 0.83 (N=18) 0.65 (N=20) 0.59 (N=17) 0.72 (N=102)Table 2.8: Proportion of subjects that play X in the game in figure 2.6, by belief type. LRAdenotes low risk aversion, HR denotes high risk aversion, AN denotes ambiguity neutraland AA denotes ambiguity averse.Table 2.9 shows the change in the proportion of subjects that play X as a func-tion of preferences, restricted to subjects who passed the comprehension tests fortheir three games as the column player. The model structure is analogous to Table2.6. Interestingly, there is no evidence of any effect of beliefs regarding an oppo-nent’s preferences on column player behavior in the testing game. In particular, thepoint estimate of the average effect of a subject believing he is facing an ambiguityaverse opponent, relative to believing he is facing an ambiguity neutral opponent,is 0 to two decimal places. Subject beliefs regarding their opponent’s preferencesare statistically independent of their behaviour as the column player in the testinggame.Result 2.5. Neither beliefs over risk preferences nor beliefs over ambiguity pref-erences explain behaviour for the column player in the game in figure 2.6.27Effect of beliefs over opponent’s Restricted to subjects who believe their opponent is ∆Pr(X)Ambiguity Aversion -0.00(0.980)Risk Aversion -0.15(0.112)Ambiguity Aversion Low Risk Aversion 0.09(0.448)Ambiguity Aversion High Risk Aversion -0.06(0.699)Risk Aversion Ambiguity Neutral -0.10(0.431)Risk Aversion Ambiguity Averse -0.25(0.109)Ambiguity Aversion & Risk Aversion -0.16(0.226)Table 2.9: Change in proportion of subjects playing X in the testing game, as a functionof reported beliefs over opponents’ preferences, restricted to subjects that passed thecomprehension tasks. N = 102. Pearson’s χ2 p-value shown in brackets. * indicatesvalue is significantly different from 0 using a non-directional test at the 5% level and **indicates significance at the 1% level.2.3.4 Discussion of resultsResult 2.1 established a correlation between risk and ambiguity preferences.21 Theprevious experimental literature on the correlation between risk and ambiguitypreferences has produced mixed results. Some studies have found positive cor-relation, others have found no correlation, and others have found correlations un-der some circumstances but not others, although the weight of evidence tends tofavour a correlation.22 From a theoretical viewpoint, axiomatic models of pref-21The data also presents a strong relationship between a subject’s preferences and his beliefs re-garding his opponent’s preferences. Further investigation of this relationship is relegated to appendixB.3.2, however, as there is evidence (discussed in section 2.4) that subjects may have had a low levelof confidence in their predictions of their opponent’s behaviour. In this context of uncertainty, pro-jecting their own preferences onto their opponent provides a sensible focal point that was followedby most subjects.22A non-exhaustive list of papers that have found a positive correlation between risk and ambiguityaversion includes: Abdellaoui et al. (2011), Bossaerts et al. (2009) and Dean and Ortoleva (2014) . Asimilar list on the other side of the debate includes: Curley et al. (1986) and Di Mauro and Maffioletti(2004).28erences (Schmeidler (1989) and Gilboa and Schmeidler (1989), for example) aretypically agnostic regarding the relationship between risk and ambiguity aversion.There are, however, some non-axiomatic theoretical models of behaviour underuncertainty that suggest a positive correlation between risk and ambiguity prefer-ences (Halevy and Feltkamp (2005), for example). The evidence provided here isof a very base variety - once the framing of the games is stripped away, subjectswere faced with either 2 or 3 element choice sets. Given that decisions over smallchoice sets form the building blocks of decision theory under uncertainty, there is astrong case to be made that decisions over small choice sets are the correct domainfor examining preferences.In the context of measuring the effects of risk and ambiguity aversion in games,correlation between risk and ambiguity preferences can lead to an omitted variablebias if only one factor is considered. Given that the current chapter is the first toconsider the effects of both risk and ambiguity, we have evidence that previous es-timates of risk aversion in games (Goeree et al. (2003), for example) and ambiguityaversion in games (Kelsey and le Roux (2015a), for example) may overestimate ef-fect sizes. This effect is demonstrated clearly in table 2.6. The unconditional effectof risk aversion on the probability of playing A is estimated to be a statisticallysignificant −0.20. Because of the omitted variable bias, this figure is larger thanboth the estimated effect of risk aversion for ambiguity neutral subjects (−0.13)and the estimated effect of risk aversion for ambiguity averse subjects (−0.17).Result 2.2 established a joint effect of risk and ambiguity preferences on be-haviour in the testing game, but did not identify the effect of ambiguity aversionseparately from the effect of risk aversion. To understand why we do not try andpull apart average affects of risk and ambiguity, recall table 2.2 which summarizesthe mapping from our Typology to rationalizable sets. The rationalizable sets dif-fer between Type 1 subjects and subjects of any other Type. This generates a clearprediction that we should expect different behaviour between Type 1 subjects andType 2 subjects (i.e. an effect of ambiguity aversion conditional on low risk aver-sion). We do not, however, have a clear prediction of a difference in behaviourbetween Type 3 and Type 4 subjects (i.e. an effect of ambiguity aversion condi-tional on high risk aversion). This differential prediction means that many of thestandard statistical tests for categorical data of this nature, such as ANOVA, do not29identify the effects that we are theoretically interested in here. The only theoreti-cally motivated test for the effect of ambiguity aversion on behaviour is to test thebehaviour between Type 1 and Type 2 subjects (which is the test that is conductedin the third row of table 2.5).We have, so far, focused our analysis on the effect of type on behaviour in thetesting game. However, also of interest is the behaviour of Type 1 subjects (whowere measured to have low risk aversion and are consistent with ambiguity neu-trality), who have a unique rationalizable strategy of A in the testing game. Table2.5 shows that 26% of Type 1 subjects who passed the comprehension tasks, andthereby demonstrated an understanding of the payoff structure of the games, failedto play a rationalizable strategy. As discussed in appendix B.1.2 to the extent thatthere is misclassification we expect to overclassify subjects as Type 1, which mayexplain some of the non-rationalizable behaviour observed here: some of these sub-jects may perceive their opponent to be a source of ambiguity but do not perceivean Ellsberg urn to be a source of ambiguity.Results 2.3 and 2.4 established that ambiguity aversion appears to be driving amovement away from A towards B, and the joint effect of risk and ambiguity aver-sion is a movement towards C. A choice of C is the natural ‘safe’ strategy in thegame in figure 2.6, and it is not surprising that risk and ambiguity averse subjectsshow a strong affinity for choosing C. The choice of B amongst ambiguity aversesubjects, however, is more interesting. While any explanation of why ambiguityaverse subjects with low risk aversion have a tendency to choose B must be takenwith caution given the small sample sizes involved, we provide a speculative hy-pothesis. Identifying and avoiding ambiguous prospects while being comfortableto take on risks at small stakes is a sophisticated set of preferences for a subject tohold.23 For an ambiguity averse subject, a choice of C in the testing game is theobvious choice – perhaps a little too obvious. Expecting their opponent to best re-spond to C, a sophisticated ambiguity averse subject may apply 2 levels of reason-ing and best respond to the best response to C: a choice of B. There is no evidenceof such reasoning amongst high risk aversion subjects. An interesting hypothesis23Chew et al. (2013) provide evidence that, in more complicated situations, the expression ofambiguity aversion is more likely amongst sophisticated subjects precisely because sophistication isrequired to identify and avoid ambiguity.30for future work is, therefore: do ambiguity averse subjects with low levels of riskaversion engage in higher levels of strategic reasoning than ambiguity neutral orhighly risk averse subjects?Result 2.5 is perhaps the most intriguing result. Why are subjects, who demon-strated a sound understanding of the games being played by passing the compre-hension test, not behaving in a fashion that is consistent with their self reportedbeliefs regarding their opponent’s preferences? There are at least three plausibleexplanations. First, as discussed in section 2.1.1, subjects may have very impre-cise beliefs regarding their opponents preferences. Second, subjects may be unableto implement the higher order reasoning needed to understand the relationship be-tween opponent preferences and opponent behaviour. Third, subjects might believethat their opponents are not rational or do not understand the game structure.There is evidence that the first explanation may hold. A natural reaction fora subject who is very unsure about their opponent’s preferences is to imagine thattheir opponent is just like them, and best respond to what they would play as theother player. Given that we observe each subject play the game as both the rowand column player, we can check if this is the case. Table 2.10 presents the data(for all subjects that are classified a type as both the row and column player) andclearly demonstrates that column players have a tendency towards best respondingto their own row player action. Notice, however, that the effect does not appear tobe symmetric: if row players had a tendency to best respond to their own columnplayer action then we should expect to see more subjects in the {B,Y} cell, andless subjects choosing C.24X YA 85 16B 7 8C 28 30Table 2.10: The number of subjects that chose each outcome, under the counter factual thateach subject was matched with themselves. Subjects who selected either I or I′ in therisk measurement game are excluded. A Pearson’s χ2 test rejects the null hypothesis ofindependence at the 1% level (p = 0.000).24In fact, if subjects were always responding to what they would do in the other role then we wouldsee everyone playing the unique Nash equilibrium.31In the next section, we introduce an additional treatment that tests between thefirst two explanations. If subjects are unable to reason at a high enough level torespond to their opponent’s preferences, then there are serious implications for thestandard set of game theoretic tools. To conduct this test, we provide subjects witha signal about both their opponent’s preferences and comprehension scores.2.4 Unpacking the Results: A New TreatmentIn the previous section we provided evidence that subjects’ preferences affect theirbehaviour in the main testing game in the direction predicted by theory but theirbeliefs over their opponent’s preferences do not affect behaviour, thereby establish-ing partial support for the role of ambiguity aversion in game theory. The data inthe previous section does not, however, allow us to identify the reason for the fail-ure of beliefs over opponent’s preferences to affect behaviour. We hypothesize thatsubjects have a low level of confidence in their reported beliefs and are, therefore,essentially ‘guessing’ in the belief measuring games. An alternative explanationis that subjects are unable to hold a consistent mental model of their opponent,and therefore respond differently in different environments. The results presentedin this section, which are remarkably strong, support the former explanation ofbehaviour in the testing game.We propose a simple new treatment: instead of eliciting subjects’ beliefs overtheir opponent’s preferences, we simply inform the subject of the choices that theiropponent made in the preference measuring games. If subjects use the signal oftheir opponent’s preferences to inform their behaviour as the column player in thetesting game then we can conclude that subjects are able to form a coherent mentalmodel of their opponent, and that it was a lack of confidence in beliefs that wasdriving behaviour in the original treatment. The results are stark: when a subjecthas observed his opponent’s preferences, the opponent’s preferences are stronglycorrelated with the subject’s behaviour as the column player in the main testinggame.For the additional treatment, 130 new subjects were recruited and bought to theELVSE lab.25 The sessions lasted between 30 and 40 minutes and average earnings25There were actually 131 subjects, but one subject had already participated in the original treat-32Figure 2.7: The matching protocol diagram, as shown to subjects. The subject was des-ignated ”You”, and the subject’s opponent was designated ”Counterpart”. Arrows areused to represent strategic interactions: the originator’s choice influences the target’spayoff. Note that the subjects were shown a transposed version of the game in figure2.6, so the row and column labels are reversed.were C$22.56. Each subject was matched with an opponent from the original ex-perimental sessions, with the subject’s opponent being selected in a pseudo-randomfashion26.91 subjects were shown the choice made by their opponent in both the riskand ambiguity measuring games (i.e. their opponent’s choice as the row playerin the games in figures 2.2 and 2.5), while 39 subjects were shown only their op-ponent’s choice in the ambiguity measuring game. Each subject was also given asummary of his opponent’s performance on the comprehension tasks. The pool ofopponents was heavily skewed towards opponents who had performed well on thecomprehension tasks – the goal was not to investigate how a subject responded tohis opponent’s comprehension score, but rather to provide the subject with a credi-ble signal that his opponent’s choices represent reasoned decisions and not randombehaviour generated by a poor understanding of the underlying game.ment and was therefore excluded from the data. The subject had created two accounts in ORSEE,which wasn’t noticed until after the session had finished. Given that there was no learning possiblein the experiment, and no subject could observe any other subject’s choices until after all decisionswere finalized, we are still able to use the other subject’s data from the offending session without fearof contamination.26The pool of subjects was chosen in a manner that was designed to provide better statistical powerthan would have been provided using true random sampling. Subjects were informed that they wereto be matched with a previous experimental participant, but were not given any information regardinghow their opponent was selected.33After subjects had viewed their opponent’s choices in the preference measur-ing tasks, subjects were asked to play the main testing game as the column playeragainst this same opponent. The matching protocol was explained to subjects withthe aid of the diagram in figure 2.7: the subject is matched with a previous partici-pant, and therefore the subject cannot influence the payoffs that are being awardedto any other player. The subjects were also required to fill in the drop down menucomprehension questions for their role as a column player, and could earn up toa $1 bonus for completing the comprehension questions correctly (see appendixB.1.3 for details). As in the main treatment, we exclude subjects who performedpoorly on the comprehension questions: 121 of the 130 subjects answered the com-prehension questions correctly on their first attempt.27The payoff structure that was used in the preference measuring games was de-signed for the original treatment in which an unobserved variable (the subject’s truepreferences) was expected to influence behaviour in both the preference measuringgames and the testing game. As a consequence, we would expect the results inthe original treatment to persist independently of whether the subject understoodthe relationship between the preference measuring games and the testing game. Inthe new treatment, however, we will only observe the predicted effects when thesubject understands the relationship between games.It is, of course, unrealistic to think that subjects will have knowledge of theformal relationship between risk preferences, ambiguity preferences and solutionconcepts across games. There are, however, some obvious and intuitive patternsthat subjects may recognize. For example, in the risk measurement game (figure2.5) it is clear that, in laymen’s terms, strategies L and I are “dangerous”, andstrategy H is “safe”. In the testing game (figure 2.6) it is also clear that strategiesA and B are “dangerous”, and strategy C is “safe”. If an opponent is observed to bewilling to take risks in one situation, then it seems reasonable to suppose that theymight take risks in another situation.The data for this new treatment is summarized in tables 2.11 and 2.12. Table2.11 presents the data for subjects who observed both their opponent’s risk and27For the comparable game in the main treatment 179 of 206 subjects answered the comprehensionquestions correctly on the first attempt. The difference (93% vs. 87%) is not statistically significant.See appendix B.3.3 for further details.34ambiguity preferences. Remarkably, every single subject who observes a Type 1opponent plays X , which is their unique rationalizable strategy. There is also alarge drop off in the proportion of subjects that play X , conditional on observinga highly risk averse opponent (Type 3 or Type 4) which is also in concordancewith theory. Table 2.12 presents the data for subjects who observed only theiropponent’s ambiguity preferences.Opponent type Type 1 Type 2 Type 3 Type 4 TotalObserved preferences LRA, AN LRA, AA HRA, AN HRA, AAPr(X) 1.00 (N=26) 0.92 (N=12) 0.56 (N=16) 0.30 (N=23) 0.69 (N=77)Table 2.11: Percentage of subjects that play X in the game in figure 2.6, by opponents’preference type, restricted to subjects who passed the comprehension task, for subjectswho observed both their opponents’ risk and ambiguity preference. LRA denotes lowrisk aversion, HR denotes high risk aversion, AN denotes ambiguity neutral and AAdenotes ambiguity averse.Opponent type Type 1 or 3 Type 2 or 4 TotalObserved preferences AN AAPr(X) 0.95 (N=22) 0.69 (N=16) 0.84 (N=38)Table 2.12: Percentage of subjects that play X in the game in figure 2.6, by opponents’ pref-erence type, restricted to subjects who passed the comprehension task, for subjects whoonly observed their opponents’ ambiguity preference. AN denotes ambiguity neutraland AA denotes ambiguity averse.Table 2.13 presents a more detailed analysis of the results. The average effectsof observing a risk averse or ambiguity averse opponent are both large and signif-icant although, again, they may be subject to omitted variable bias. In particular,the average effect of observing an ambiguity averse opponent is overstated. Theconditional estimates demonstrate that the effect is working chiefly through the riskaversion, rather than ambiguity aversion, channel. The effect of observing a riskaverse opponent is large and statistically significant whether we restrict the sampleto only ambiguity neutral or ambiguity averse opponents. On the other hand, theeffect of observing an ambiguity averse opponent, conditional on the opponent’srisk aversion, is not significant. This may not be surprising given the difference inpayoff structures between the risk and ambiguity measurement tasks: the risk taskmeasures preferences over prospects with differences in expected values of a few35dollars, while the ambiguity task measures preferences over prospects with differ-ences in expected values of a few cents, implying that the signal of opponent’s riskpreferences is stronger than the signal of opponent’s ambiguity preferences.It could also be the case, however, that subjects are able to identify and reasonabout their opponent’s risk preferences, but are unable to identify or understand therole of ambiguity preferences. Table 2.12 establishes that this is not the case: whensubjects observe only their opponent’s ambiguity preference they are more likelyto play X when facing an ambiguity neutral than ambiguity averse opponent. Theeffect of ambiguity preferences is approximately half that of risk preferences, anda Fisher exact test with a null hypothesis of no effect of an opponent’s ambiguitypreferences on behaviour generates a two-sided p-value of 0.06 and a one-sidedp-value of 0.04.Effect of opponents Restricted to opponents who are ∆Pr(X)Ambiguity Aversion -0.32**(0.003)Risk Aversion -0.56**(0.000)Ambiguity Aversion Low Risk Aversion -0.08(0.316)FAmbiguity Aversion High Risk Aversion -0.26(0.107)Risk Aversion Ambiguity Neutrality -0.44**(0.000)FRisk Aversion Ambiguity Aversion -0.61**(0.001)Ambiguity Aversion & Risk Aversion -0.70**(0.000)Table 2.13: Change in proportion of subjects playing X in the testing game, as a functionof opponent’s preferences, restricted to subjects that passed the comprehension tests.N = 77. Pearson’s χ2 p-value shown in brackets. p-values denoted with an F arecalculated using Fisher’s exact test because of small or unbalanced samples. * indicatesvalue is significantly different from 0 using a non-directional test at the 5% level and** indicates significance at the 1% level.It is obvious that the results for subjects with Type 1 opponents are exactlyequal to the pure strategy Nash equilibrium of the game. A natural follow up ques-36tion is: are the results for subjects with Type 3 opponents also rationalizable by aNash equilibrium?28 The answer is yes: the proportion of subjects playing X whenthey observe their opponent to be ambiguity neutral but risk averse (9 out of 16subjects, see Table 2.11) forms a Nash equilibrium of the game (where row playersmix between A and C) when the row player has a CRRA utility function with riskaversion parameter β = 2.81. Given that the choice of C in the risk measurementgame implies β > 1.93 the results are consistent with potential equilibrium play atreasonable levels of risk aversion.Result 2.6. When a subject observes a signal of her opponent’s risk and ambiguitypreferences, her behaviour as the column player in the game in figure 2.6 is affectedby the signal in a fashion that is consistent with the theory of section 2.1.1.2.5 DiscussionThe data presented in sections 2.3 and 2.4 provides a clear picture of the role ofuncertainty aversion in games. A subject’s preferences over risk and ambiguityare correlated with behaviour in our normal form testing game. There is no re-lationship, however, for subjects that passed our comprehension tests, between asubject’s beliefs over her opponent’s preferences and her behaviour in the testinggame. In a follow up treatment, where subjects were shown their opponent’s be-haviour in the preference measuring tasks, a very strong relationship between theopponent’s risk preferences and the subject’s behaviour was observed.Uncertainty aversion is not the only alternative theory that has been able toexplain deviations from Nash equilibrium in previous work. Indeed, other modelssuch as the level-k model, quantal response equilibrium and other regarding pref-erences can, to various extents, explain part of the data presented in this chapter.There is, however, a key piece of the data that these other models cannot explain:the relationship between ambiguity preferences in the classification game and be-haviour in the testing game. One of the key motivations for the experimental designimplemented here was to isolate and identify the role of preferences independentlyof these ‘other’ behavioural models. It may well be the case that a level-k (or28Because of the very large range of mixed equilibrium probabilities for subjects with Type 2 orType 4 opponents, there is not much point in repeating the analyses for those subjects.37quantal response, or other regarding preferences) model that is augmented withheterogeneous uncertainty preferences will provide a highly predictive model ofstrategic behaviour across a wide range of settings but these are not the focus ofthis chapter. However, for this chapter, we invoke the principle of parsimony andfocus on isolating and identifying the role of strategic uncertainty in determiningbehaviour in strategic settings.The results presented here are in line with the results found in other recentexperimental papers that have, broadly speaking, taken an epistemic approach tobehaviour in games. The experimental epistemic games literature seeks to identifythe state of knowledge and modes of reasoning that are used by subjects in variousexperimental environments, with a goal of understanding the process of subjectdecision making via a convergence of theory and evidence.Kneeland (2015) finds, in a paper that focuses on level-k reasoning (rather thanambiguity aversion), using a novel ring-game structure, that while most subjects(94%) are rational they do not form consistent conjectures regarding each other’sbehaviour. Kneeland also finds that a large proportion of subjects believe theiropponents are rational (72%) and that 44% have second-order beliefs regarding therationality of others.The implications of the results in this chapter for subject rationality are broadlycomparable with those found by Kneeland. Amongst ambiguity and risk neutralsubjects, we find that 77% of subjects played an undominated strategy in the maintesting game, and therefore exhibit rational behaviour. While this is lower than the94% found by Kneeland, it was easier for Kneeland’s subjects to identify dom-inated strategies because they were dominated by a pure strategy (rather than amixed strategy as was the case in the game in figure 2.6). We find higher rates ofequilibrium play than Kneeland29, although the game in figure 2.6 is dominancesolvable and therefore requires weaker epistemic assumptions to justify equilib-rium behaviour than in Kneeland’s game. Furthermore, the data in the treatmentpresented in section 2.4, where subjects were able to observe their opponent’s be-haviour in the classification games, can be interpreted as providing evidence thatmost subjects believe that their opponents are rational.29The equilibrium analysis is not included in the final published version of Kneeland (2015), butcan be found in earlier working paper versions.38Healy (2013) also provides empirical evidence regarding the epistemic statusof laboratory subjects. One of his main finding is that subjects are generally poorat estimating their opponent’s preferences over outcomes. Here, we find evidencethat subjects are very much aware that they are poor at estimating their opponent’spreferences: subjects who observed their opponent’s behaviour in the classificationgames exhibited markedly different behaviour than subjects in the baseline treat-ment, and subjects responded to information about their opponent’s preferences ina highly rational manner.Esponda and Vespa (2014) find that, when subjects play a voting game againstcomputer agents with known strategies, subjects can correctly extract informationfrom the computers’ strategies much more often after observing the computers’votes than they can extract information from a strategically equivalent simultane-ous move game. Similarly, our subjects can extract information from their oppo-nent’s behaviour in the classification games if they have observed their opponent’sbehaviour. However, when a subject only predicts, but does not observe, the oppo-nent’s behaviour the subject does not behave as if her prediction of the opponent’spreferences contains relevant information.2.5.1 ConclusionsThis chapter presents evidence that both risk and ambiguity preferences play a rolein determining behaviour in normal form games, but that beliefs over an opponent’spreferences do not influence behaviour in games. As demonstrated in a followup treatment, subjects have a high level of uncertainty regarding their opponent’spreferences; when informed of their opponent’s preferences, subjects behaved ina fashion that is largely consistent with equilibrium play. The results suggest thatboth strategic uncertainty and uncertainty regarding an opponent’s preferences playan important role in determining behaviour in normal form games. Additionally,this chapter presents evidence that ambiguity and risk preference are correlated inthe context of games.The results in this chapter, and the previous research discussed above (Knee-land (2015), Healy (2013) and Esponda and Vespa (2014)), provide some strongsuggestions for empirically founded models of strategic behaviour.39Generally, assuming first order rationality of subjects appears to be a reason-able assumption (particularly when dominance relationships are strict). However,the standard game theoretic assumption of common knowledge of preferences issoundly rejected by the data: even mutual knowledge of preferences is too strong.Preference heterogeneity (across both risk and ambiguity dimensions) plays an im-portant role in strategic behaviour, but subjects do not seem to have well calibratedbeliefs regarding the preferences of others. On aggregate, the evidence suggeststhat preferences should be modeled as private information with diffuse or uncertainpriors. However, the evidence from the follow-up treatment presented in section2.4 suggests that when subjects are informed of their opponent’s preferences (i.e.when beliefs over preferences become ‘precise enough’) then it is reasonable to as-sume that subjects will respond to this preference information in a rational fashion.Grant et al. (Forthcoming) provides an important step towards synthesizing someof these ideas in a general theoretical framework.Furthermore, the results presented here suggest that studies that focus on eitherambiguity or risk preferences in strategic environments may overstate the effectsof risk or ambiguity preferences on behaviour via an omitted variable bias effect.Unpacking the relationship between preferences and strategic uncertainty requiresa careful consideration of both risk and ambiguity aversion.40Chapter 3Continuity, Inertia and StrategicUncertainty: A Test of the Theoryof Continuous Time Games.3.1 IntroductionIn game theoretic models, players usually make decisions in lock-step at a prede-termined set of dates – a timing protocol we will call “Perfectly Discrete time.”Most real world interaction, by contrast, unfolds asynchronously in unstructuredcontinuous time, perhaps with some inertia delaying mutual response. Does thisdifference between typical modeling conventions and real-world interactions mat-ter? Theoretical work on the effects of continuous time environments on behavior(developed especially in Simon and Stinchcombe (1989) and Bergin and MacLeod(1993)) focuses on what we will call “Perfectly Continuous time,” a limiting casein which players can respond instantly (that is with zero inertia) to one another,and arrives at a surprising answer: Perfectly Discrete time and Perfectly Contin-uous time can often support fundamentally different equilibria, resulting in widegaps in behavior between the two settings.In this chapter we introduce new techniques that allow us to evaluate these the-orized gaps in the laboratory directly and assess their relevance for understanding41real world behavior. We pose two main questions. First, does the gulf between Per-fectly Discrete and Perfectly Continuous time suggested by the theory describe realhuman behavior? Though equilibria exist that produce large differences in behav-ior (and authors such as Simon and Stinchcombe (1989) argue that these equilibriashould be considered highly focal), multiplicity of equilibrium in Perfectly Con-tinuous time means that the effect of continuous time is, ultimately, theoreticallyindeterminate. Second, we ask how empirically relevant these gaps are: can morerealistic, imperfectly continuous time games (games with natural response delaysthat we call “Inertial Continuous time” games) generate Perfectly Continuous-likeoutcomes? Nash equilibrium suggests not but, as Simon and Stinchcombe (1989)and Bergin and MacLeod (1993) emphasize, even slight deviations from Nash equi-librium assumptions (a´ la ε-equilibrium) allow Perfectly Continuous-like behaviorto survive as equilibria in the face of inertia, provided inertia is sufficiently small.Recent experiments have begun to investigate the relationship between contin-uous and discrete time behavior in the lab (e.g. Friedman and Oprea (2012) andBigoni et al. (2015))1 but have not yet directly tested the theory motivating thesequestions for a simple reason: natural human reaction lags in continuous time set-tings generate inertia that prevents a direct implementation of the premises of thetheory. These Inertial Continuous time settings are empirically important (and ofindependent interest) but are insufficient for a direct theory test because they gen-erate very different equilibrium behavior from the Perfectly Continuous time envi-ronments that anchor the theory (a prediction we test and find strong though highlyqualified support for in our data). In our experimental design, we introduce a newprotocol (“freeze time”) that eliminates inertia by pausing the game for several sec-onds after subjects make decisions, allowing them to respond “instantly” to actions1Both Friedman and Oprea (2012) and Bigoni et al. (2015) report evidence from prisoner’s dilem-mas played with flow payoffs in Inertial Continuous time (i.e. subjects in these experiments suffernatural reaction lags that prevent instant response to the actions of others). While the Friedmanand Oprea (2012) design varies the continuity of the environment (discrete vs. continuous time in-teraction) in deterministic horizon games, the Bigoni et al. (2015) design centers on varying thestochasticity of the horizon (deterministic vs. stochastic horizon) in continuous time games. Othermore distantly related continuous time papers include experimental work on multi-player centipedegames (Murphy et al., 2006), public-goods games (Oprea et al., 2014), network games (Berninghauset al., 2006), minimum-effort games (Deck and Nikiforakis, 2012), hawk-dove games (Oprea et al.,2011), bargaining games (Embrey et al., 2015) and the effects of public signals (Evdokimov andRahman, 2014).42of others (i.e. with no lag in game time) and thus allowing us to test PerfectlyContinuous predictions. By systematically comparing behavior in this PerfectlyContinuous setting to both Perfectly Discrete time and Inertial Continuous timesettings we are able to pose and answer our motivating questions.We apply this new methodology to a simple timing game similar to one dis-cussed in Simon and Stinchcombe (1989) that is ideally suited for a careful testof the theory.2 In this game, each of two agents decides independently when toenter a market. Joint delay is mutually beneficial up to a point, but agents benefitfrom preempting their counterparts (and suffer by being preempted). In PerfectlyDiscrete time, agents will enter the market at the very first opportunity, sacrificingsignificant potential profits in subgame perfect equilibrium. By contrast, in Per-fectly Continuous time, agents can, in equilibrium, delay entry until 40% of thegame has elapsed, thereby maximizing joint profits.3 (Simon and Stinchcombe(1989) emphasize this equilibrium and point out that it uniquely survives iteratedelimination of weakly dominated strategies, but many other equilibria – includ-ing inefficient immediate-entry equilibrium – exist in Perfectly Continuous time.)Importantly, Inertial Continuous time of the sort studied in previous experimentsleads not to Perfectly Continuous time-like multiplicity in equilibrium but only tothe inefficient instant entry predicted for Perfectly Discrete time – as Bergin andMacLeod (1993) point out even a small amount of inertia theoretically erases all ofthe efficiency enhancing potential of continuous time in Nash equilibrium.4In the first part of our experimental design we pose our main question by com-paring Perfectly Discrete and Perfectly Continuous time using a baseline set of2Compared to, for instance, the continuously repeated prisoner’s dilemma, our timing game hasseveral advantages for a diagnostic test. First, the joint profit maximizing outcome predicted bySimon and Stinchcombe (1989) is interior, meaning simple heuristics like “cooperate until the endof the game” cannot be confused with equilibrium play. Second, the strategy space is considerablysimpler than the prisoner’s dilemma, making measurement of decisions and inferences about strate-gies crisper. Finally, the prisoner’s dilemma frames the contrast between cooperation and defectionsomewhat starkly and may therefore trigger social behaviors that have little to do with the forces wedesigned our experiment to study – we speculated in designing the experiment that our timing gamewould be somewhat cleaner from this perspective.3More precisely, the agents are maximizing joint profits subject to playing non-strictly dominatedstrategies in every subgame that is reached on the path of play.4We note that all equilibria of the game we study in this chapter are subgame perfect and thatsome of our proofs rely on backwards induction. In the remainder of the chapter, for readibility, weomit the modifier “subgame perfect” when discussing equilibria of our game.43parameters and 60 second runs of the game. In the Perfectly Discrete time proto-col, we divide the 60 second game into 16 discrete grid points and allow subjectsto simultaneously choose at each grid point whether to enter the market. In thePerfectly Continuous time protocol we instead allow subjects to enter at any mo-ment but, crucially, eliminate natural human inertia by freezing the game after anyplayer enters, allowing her counterpart to enter “immediately” from a game-timeperspective if she enters during the ample window of the freeze. We find evidenceof large and extremely consistent differences in behavior across these two proto-cols. Virtually all subjects in the Perfectly Discrete time treatment suboptimallyenter at the first possible moment while virtually all subjects in Perfectly Contin-uous time enter 40% of the way into the period, forming a tight mode around thejoint profit maximizing entry time. The results thus support the conjecture of alarge - indeed, from a payoff perspective, maximally large – gap between PerfectlyContinuous and Discrete time behaviors.In the second part of the design, we study how introducing realistic inertia intocontinuous time interaction changes the nature of the results observed in our Per-fectly Continuous time treatment. Though Nash equilibrium predicts that even atiny amount of inertia will force behavior back to Perfectly Discrete-like immediateentry times, alternatives such as ε-equilibrium suggest that Perfectly Continuous-like results may survive as equilibria at low levels of inertia. In Inertial Continuoustime treatments we replicate our Perfectly Continuous time treatment but removethe freeze time protocol, thereby allowing natural human reaction lags to producea natural source of inertia. We systematically vary the severity of this inertia byvarying the speed of the game relative to subjects’ natural reaction lags and findthat when inertia is highest, entry times collapse to zero in continuous time as pre-dicted by Nash equilibrium. However, when we lower inertia to sufficiently smalllevels, we observe large entry delays that are nearly as efficient as those observedin Perfectly Continuous time. Thus, realistic Inertial Continuous Time behavior iswell approximated by the extreme of Perfectly Discrete time when inertia is largeand better approximated by the extreme of Perfectly Continuous time when inertiais small. While these patterns are inconsistent with Nash equilibrium, they are,as both Simon and Stinchcombe (1989) and Bergin and MacLeod (1993) stress,broadly consistent with ε-equilibrium.44Though ε-equilibrium is consistent with our data, it also generates imprecisepredictions. In the final part of the chapter, we consider, ex post, explanationsthat can more sharply organize our data in order to better understand how inertiaand continuity influence behavior. Recent experimental work on dynamic strategicinteraction (e.g. Dal Bo and Frechette (2011), Embrey et al. (2016), Vespa and Wil-son (2016),Dal Bo and Frechette (2016)) has emphasized the crucial role strategicuncertainty (assumed away in Nash equilibrium) plays in predicting both equilib-rium selection and non-Nash equilibrium behavior, and has focused especially onthe predictive power of the basin of attraction of defection relative to cooperativealternatives. We show that the basin of attraction becomes more hospitable to con-tinued cooperation at each moment in time as inertia falls towards the continuouslimit and that measures of risk dominance starkly organize our data, tying our re-sults directly in with this recent work.We then consider more explicit ways of modeling subjects’ responses to strate-gic uncertainty by studying simple (and parsimonious) heuristic rules (drawn fromMilnor, 1954) discussed in the literature on highly uncertain environments: max-imin ambiguity aversion (MAA), minimax regret avoidance (MRA) and Laplaciansubjective expected utility (LEU). Of these, we find that the MRA decision ruledramatically outperforms Nash equilibrium, making extremely accurate point pre-dictions for our game. We show that MRA also organizes data in an additional pairof diagnostic treatments that generate comparative statics unanticipated by Nashequilibrium. Finally, we consider the relevance of this result to other commoncontinuous time games. In the online appendix we show theoretically that MRApredicts a smooth approach to Pareto optimal, Perfectly Continuous-like play asinertia falls to zero in an important, broad class of continuous time games underempirically sensible restrictions on beliefs. We directly test this claim by showingthat MRA predictions almost exactly track data on behavior in continuous timeprisoner’s dilemma from previous work. Our analysis thus suggests that heuristicresponses to strategic uncertainty like MRA may be a productive way of organizingand interpreting data in a wide range of dynamic strategic settings.The results of our experiment suggest a role for Perfectly Continuous time the-oretical benchmarks in predicting and interpreting real-world behavior, even if theworld is never perfectly continuous. Changes in technology have recently narrowed45– and continue to narrow – the gap between many types of human interactions andthe Perfectly Continuous setting described in the theory. Constant mobile access tomarkets and social networks, the proliferation of applications that speed up searchand the advent of automated agents deployed for trade and search have the effectof reducing inertia in human interactions. Our results suggest, contra Nash equilib-rium, that such movements towards continuity may generate some of the dramaticeffects on behavior predicted for (and observed in) Perfectly Continuous time evenif inertia never falls quite to zero. Guided by these results, we conjecture that theshare of interactions that are better understood through the theoretical lens of Per-fectly Continuous time than that of Perfectly Discrete time will grow as social andeconomic activity continues to be transformed by this sort of technological change.The remainder of the chapter is organized as follows. Section 3.2 gives anoverview of the main relevant theoretical results that form hypotheses for our ex-periment and section 3.3 describes the experimental design. Section 3.4 presentsour results, Section 3.5 interprets the results in light of metrics and models ofstrategic uncertainty, and Section 3.6 concludes the chapter. Appendices collecttheoretical proofs and the instructions to subjects.3.2 Theoretical Background and HypothesesIn section 3.2.1 we introduce our timing game and in section 3.2.2 we state anddiscuss a set of propositions characterizing Nash equilibrium and providing us withour main hypotheses. In section 3.2.3 we consider alternative hypotheses motivatedby ε-equilibrium.3.2.1 A diagnostic timing gameConsider the following simple timing game, adapted from one described in Simonand Stinchcombe (1989). Two firms, i ∈ {a,b}, each choose a time ti ∈ [0,1] atwhich to enter a market, perhaps conditioning this choice on the history of thegame.5 Payoffs depend on the order of entry according to the following symmetric5To conserve notation, we normalize the length of the game to be 1 for the theoretical analysis.In our experiment, we sometimes vary the length of the game (and with it the severity of inertia andpredicted time of entry) across treatments.46function:Ua(ta, tb) =1−tb2[ΠD+(tb− ta)(1+ 21−tb )ΠF]− c(1− ta)2 if ta < tb1−ta2 ΠD− c(1− ta)2 if ta = tb1−ta2 [ΠD− (ta− tb)ΠS]− c(1− ta)2 if tb < ta(3.1)with parameters assumed to satisfy 0 < 2c < ΠS ≤ ΠD < 4c and 4c3 ≤ ΠF ≤ 4c.Though the applied setting modeled by this sort of game matters little for our rel-atively abstract experiment, we can interpret the model as one in which firms facequadratic costs for time spent in the market (parameterized by c), earn a duopolyflow profit rate of ΠD while sharing the market, earn a greater flow profit ΠF whilea monopolist and suffer a permanent reduced earnings rate (parameterized by ΠS)proportional to the time one’s counterpart has spent as a monopolist.Several characteristics of this game are particularly important for what follows.First, firms earn identical profits if they enter at the same time and this simultaneousentry payoff is strictly concave in entry time, reaching a maximum at a time t∗ =1− ΠD4c ∈ (0, 12). Second, if one of the firms instead enters earlier than the other(at time t ′), she earns a higher payoff and her counterpart a lower payoff than hadthey entered simultaneously at time t ′. The firms thus maximize joint earningsby delaying entry until an interior time t∗ but at each moment each firm has amotivation to preempt its counterpart and to avoid being preempted.3.2.2 Discrete, inertial and perfectly continuous time predictionsWhat entry times can be supported as equilibria in this game? The key observationmotivating both the theory and our experiment is that the answer depends on howtime operates in the game. In this subsection we characterize equilibrium underthree distinct protocols: Perfectly Discrete time, Perfectly Continuous time andInertial Continuous time (here we only sketch the main conceptual issues, deferringtechnical discussion to Appendix A).We begin with Perfectly Discrete time, the simplest and most familiar case.Here, time is divided into G+1 evenly spaced grid points (starting always at t = 0)47on [0,1] and players make simultaneous decisions at each of these points. Moreprecisely, each player chooses a time t ∈ {0,1/G, ...,(G−1)/G,1} at which to en-ter, possibly conditioning this choice on the history of the game, Ht at each gridpoint. Earnings are given by equation 3.1 applied to the dates on the grid at whichentry occurred.6 As in familiar dynamic discrete time games like the centipedegame and the finitely repeated prisoner’s dilemma there is a tension here betweenefficiency (which requires mutual delay until at least the grid point immediatelyprior to t∗) and individual sequential rationality (which encourages a player to pre-empt her counterpart). Applying the logic of backwards induction, strategies thatdelay entry past the first grid point unravel, leaving immediate entry at the first gridpoint, t = 0, as the unique subgame perfect equilibrium, regardless of G.Proposition 3.1. In Perfectly Discrete time, the unique subgame perfect equilib-rium is for both firms to enter at time 0, regardless of the fineness of the grid, G.Proof. : See Appendix C.2.1.At the opposite extreme, in Perfectly Continuous time players are not confinedto a grid of entry times but can instead enter at any moment ti ∈ [0,1] (again, possi-bly conditioning on the history of the game at each t, Ht). Simon and Stinchcombe(1989) emphasize the relationship between the two extremes, modeling PerfectlyContinuous Time as the limit of a Perfectly Discrete time game as G approachesinfinity. In this limit, players can respond instantly to entry choices made by oth-ers: if an agent enters the market at time t her counterpart can respond by alsoentering at t, moving in response to her counterpart but at identical dates. Since, inour game, delaying entry after a counterpart enters is strictly payoff decreasing, noplayer can expect to succeed in preempting her counterpart (or have reason to fearbeing preempted). This elimination of preemption motives also protects efficientdelayed entry from unravelling and thus makes it possible to support any entry timet ∈ [0, t∗] as an equilibrium.76For example, if firm a entered at the third grid point, and firm b entered at the fifth grid point,the payoff for firm a is given by Ua( 2G ,4G ).7Entry times greater than t∗ cannot be supported in equilibrium because they are always payoffdominated by t∗. Notice that despite the symmetry of the (joint entry) payoff function around t∗, the48Proposition 3.2. In Perfectly Continuous time, any entry time t ∈ [0, t∗] can besupported as a subgame perfect equilibrium outcome.Proof. We provide three proofs of this proposition. Appendix C.2.1 includes aself-contained heuristic proof, a more formal proof that draws directly from Simonand Stinchcombe (1989) and an alternative proof that instead follows the modelingapproach of Bergin and MacLeod (1993).Though it is possible for Perfectly Discrete and Perfectly Continuous behaviorsto radically differ in equilibrium, this is hardly guaranteed. Because of multiplicity,Perfectly Continuous behavior may be quite different or quite similar to PerfectlyDiscrete behavior in equilibrium (t = 0 and t∗ are both supportable in equilibriumin Perfectly Continuous time) depending on the principle of equilibrium selectionat work. This multiplicity is in fact a central motivation for studying these environ-ments in the laboratory. Simon and Stinchcombe (1989) emphasize that t∗ is theunique entry time to survive iterated elimination of weakly dominated strategiesin our game and they argue that this refinement is natural in the context of Per-fectly Continuous time games. Evaluating the organizing power of this refinementis another central motivation for our study.Remark 3.1. In Perfectly Continuous time, joint entry at t∗ = 1− ΠD4c is the onlyoutcome that survives iterated elimination of weakly dominated strategies.Proof. A heuristic proof is provided in appendix C.2.1. For further details, seeSimon and Stinchcombe (1989).Finally, Inertial Continuous time lies between the extremes of Perfectly Dis-crete and Perfectly Continuous time, featuring characteristics of each. Here, as inPerfectly Continuous time, players can make asynchronous decisions and are notconfined to entering at a predetermined grid of times. However, as in PerfectlyDiscrete time, players are unable to respond instantly to entry decisions by theircounterparts. In Inertial Continuous time, inability to instantly respond is due towhat Bergin and MacLeod (1993) call inertia (here, simply response lags of ex-equilibrium entry set is not symmetric around t∗ because of the temporal nature of the game.49ogenous size δ ).8, 9 With inertial reaction lags, the logic of unravelling returns asplayers once again have motives to preempt one another. As a result, the efficientdelayed entry supported in equilibrium in Perfectly Continuous time evaporateswith even an arbitrarily small amount of inertia. Theoretically then, even a tinyamount of inertia pushes continuous time behavior to Perfectly Discrete levels.Proposition 3.3. In Inertial Continuous time, only entry at time 0 can be supportedas a subgame perfect equilibrium regardless of the size of inertia, δ > 0.Proof. See Appendix C.2.1.Instead of modeling Perfectly Continuous time as a limit of Perfectly Discretetime as the grid becomes arbitrarily fine as Simon and Stinchcombe (1989) do,Bergin and MacLeod (1993) model it as the limit of Inertial Continuous time asinertia approaches zero. This alternative method for defining Perfectly Continu-ous time leads to an identical equilibrium set to the one described by Simon andStinchcombe (1989) for our game.3.2.3 Alternative hypothesis: Inertia and ε-equilibriumContinuous time can fundamentally change Nash equilibrium behavior but thiseffect is extremely fragile: even a slight amount of inertia will eliminate any pro-cooperative effects of continuous time interaction in games like ours. Since inertiais realistic, this frailty in turn calls into question the usefulness of the theory forpredicting and interpreting behavior in the real-world. Perhaps for this reason bothSimon and Stinchcombe (1989) and Bergin and MacLeod (1993) motivate the the-ory of continuous time explicitly with reference to the more forgiving alternative ofε-equilibrium, emphasizing that any Perfectly Continuous time Nash equilibriumis arbitrarily close to some ε-equilibrium of a continuous time game with inertia(and vice versa).10 If agents are willing to tolerate even very small deviations from8 Though, in the context of our experiment, inertia simply refers to natural human reaction lags,Bergin and MacLeod (1993) point out that more general types of inertia are possible.9Throughout the chapter we define inertia δ as the ratio of an agent’s reaction lag, δ0, to the totallength of the game, T (i.e. δ ≡ δ0/T ). Inertia is thus the fraction of the game that elapses before anagent can respond to her counterpart.10More precisely, Simon and Stinchcombe (1989) make this same point with respect to syn-chronous, discrete time games with very fine time grids.50best response, they can support Perfectly Continuous-like outcomes as equilibriaeven in the face of inertia, provided inertia is sufficiently small. Indeed, in ourgame, when inertia is large, ε-equilibrium coincides perfectly with Nash equilib-rium, supporting only immediate entry and mirroring Perfectly Discrete time Nashequilibrium. However when inertia falls below a threshold level (determined by ε)the equilibrium set expands to support any entry time t ∈ [0, t∗], instead mirroringPerfectly Continuous time Nash equilibrium. We formalize this in the followingproposition:Proposition 3.4. Consider a game in inertial continuous time. For any ε > 0, thereexists δˆ > 0 such that for all levels of inertia δ < δˆ any entry time in [0, t∗] can besustained in a subgame perfect ε-equilibrium.For any 0 < ε < ΠF − c there exists δ such that for all levels of inertia δ > δimmediate entry is the unique subgame perfect ε-equilibrium.Proof. See Appendix C.3.2.11This result is useful because it emphasizes that even very small deviations fromthe assumptions underlying Nash equilibrium – for instance small amounts of noisein beliefs or imprecision in payoffs specified in the game – can make either Per-fectly Discrete or Perfectly Continuous time benchmarks more predictive, depend-ing on the severity of inertia.12 For this reason (and because of the important roleε-equilibrium plays in the theory), we built our experimental design in part withthis alternative prediction in mind as an ex ante alternative hypothesis to Nashequilibrium. In section 3.5, we consider, ex post, more specific interpretations forthe sort of non-Nash equilibrium behavior ε-equilibrium is capable of sustaining,focusing on the role of strategic uncertainty in supporting non-Nash equilibriumoutcomes. By doing so we are able to more sharply organize our results and tieour findings to important themes explored in recent, closely related literature ondynamic strategic interaction.11In Appendix C.3 we prove a set of propositions fully characterizing the ε-equilibrium sets forour protocols.12Generically, δˆ < δ . However, when ΠF = 4c, as in our main treatments, δˆ = δ which implies adiscontinuity in the equilibrium set. See proposition C.4 for further details regarding the continuity(or otherwise) of the equilibrium set.51Figure 3.1: Screen shot from PerfectlyContinuous and Inertial Continu-ous time treatments (under the low-temptation parametrization).Figure 3.2: Screen shot from PerfectlyDiscrete time treatments (under thelow-temptation parametrization).3.3 Design and ImplementationIn section 3.3.1 we discuss our strategy for implementing our three timing protocolsin the lab and present the experimental software we built to carry out this strategy.In section 3.3.2 we present our treatment design.3.3.1 Timing protocols and experimental softwareWe ran our experiment using a custom piece of software programmed in Redwood(Pettit et al. (2015)). Figures 3.1 and 3.2 show screenshots. Using this software,we implemented the three timing protocols described in Section 3.2 as follows:Inertial Continuous time. Figure 3.1 shows an Inertial Continuous time screen-shot. As time elapses during the period, the payoff dots (labeled “Me” and “Other”)move along the joint payoff line (black center line) from the left to the right of thescreen. (In most treatments periods last 60 seconds meaning it takes 60 seconds forthe payoff dot to reach the right hand side of the screen.) When a subject is the firstplayer to enter the market, her payoff dot shifts from the black to the green line (thetop line), while her counterpart’s payoff dot (the dot of the second mover) shifts tothe red line (the bottom line).13 When the second player enters the market, period13Because of the nature of the payoff function, the green and red line change throughout the period52payoffs for both players are determined by the vertical location of each player’sdot at the moment of second entry. Once both players have entered, they wait untilthe remaining time in the period has elapsed before the next period begins (see theinstructions in Appendix B for more detail). Because subjects, on average, takeroughly 0.5 seconds to respond to actions by others, subjects have natural inertia intheir decision making that should theoretically generate Inertial Continuous timeequilibrium behavior.Perfectly Continuous time. The Perfectly Continuous time implementation isidentical to the Inertial Continuous time implementation (as shown in Figure 3.1)except that when either subject presses the space bar to enter the game freezes (wecall this the “Freeze Time” protocol) and the payoff dots stop moving from leftto right across the screen for five seconds. Subjects observe a countdown on thescreen and the first mover’s counterpart is allowed to enter during this time. If thecounterpart enters during this window, her response is treated as simultaneous toher counterpart’s entry time and both players earn the amount given by the currentvertical location of their payoff dot. Otherwise, the game continues as in InertialContinuous time once the window has expired. Regardless, subjects must waituntil the remaining time in the period has elapsed before the next period begins.The length of the pause was calibrated to be roughly 10 times longer than themedian reaction lag measured in Inertial Continuous time, giving subjects ampletime to respond, driving inertia to 0 and thus satisfying the premises of PerfectlyContinuous time models.14Perfectly Discrete time. Figure 3.2 shows a screen shot for the Perfectly Dis-crete treatments, which is very similar to the continuous time screen but for a fewprior to entry and stabilize once one player has entered.14An alternative way of formally implementing Perfectly Continuous time is to allow subjects topre-specify an entry time and a stationary response delay (possibly set to zero). We opted to usethe Freeze Time protocol instead of this sort of strategy method for three reasons. First, employingthe strategy method would force us to substantially constrain subjects’ strategy space, eliminatingor limiting the dependence of strategies on histories. Second, using the “Freeze Time” protocolallows us to directly compare entry decisions to inertia generated by naturally occurring reactionlags in Inertial Continuous time – a central goal of the experiment that would be impossible usingthe strategy method. Finally, for realism, we wanted subjects to actually see the unfolding of payoffsand behavior in real time. Nonetheless, see Duffy and Ochs (2012) for evidence from a distantlyrelated entry game that simultaneous choice and dynamic implementations can generate very similarresults.53changes. First, periods are divided into G = 15 subperiods, which begin at grid-points t = {0,4,8, ...,56} (measured in seconds)15, each marked by a vertical grayline on the subject’s screen. Instead of moving smoothly through time, as in thecontinuous time treatments, the payoff dots follow step functions and “jump” tothe next step on the payoff functions at the end of each subperiod. Actions areshrouded during a subperiod, so payoff dots will only move from the black to thegreen (or red) payoff lines after the subperiod in which a subject chose to enter hasended. Payoffs are determined according to equation (3.1), calculated at the gridpoint that began the subperiod in which the subject entered.163.3.2 Treatment design and implementationOur experimental design has three parts. In the first we implement 60 secondtiming games using the parameter vector (c,ΠD,ΠF ,ΠS) = (1,2.4,4,2.16) underthe extremes of Perfectly Discrete and Perfectly Continuous time.17 We call theseBaseline treatments PD (Perfectly Discrete) and PC (Perfectly Continuous).Second, we examine the effects of inertia on continuous-time decisions by run-ning a series of Inertial Continuous time treatments using the same Baseline pa-rameters. In the IC60 treatment we run periods lasting 60 seconds each (just as inthe PC and PD treatments). In the IC10 and IC280 treatments we repeat the IC60treatment but speed up or slow down the clock so that periods finish in 10 or 280seconds (respectively). By speeding up the game clock so that the game lasts only10 seconds (the IC10 treatment) we dramatically increase the severity of inertia; byslowing down the game so that it takes 280 seconds to finish (the IC280 treatment),15We used the convention that any subjects who were yet to enter at the t = 56 subperiod will beforced to enter at the t = 60 subperiod, which would result in a payoff of 0. In practice, however, nosubjects came close to waiting this long to enter.16For example, if in our Baseline treatment a subject entered in the first subperiod and her counter-part entered in the third subperiod, payoffs would be given by U(0, 215 ) for the subject and U(215 ,0)for his counterpart.17To allow the entire payoff space to be shown on a single reasonably scaled plot we truncated themaximum payment to be 75 points per period (for context U(t∗, t∗) was normalized to be 36 points).This truncation (which subjects can clearly see on their screen) only affects the payoff of the firstmover under the unusual circumstance that her opponent delays entry for a significant amount oftime. Regardless, this design choice only affects payoffs that are well off the equilibrium path anddoes not affect any of the equilibrium sets discussed in the chapter.54we substantially reduce the severity of inertia.18Finally, in the Low Temptation treatments (discussed in Section 3.5.2), we ex-amine the robustness of explanations for our main results by changing the payofffunctions in Perfectly Continuous and Discrete time. In the L-PD (Low temptation- Perfectly Discrete) and L-PC (Low temptation - Perfectly Continuous) treatmentswe replicate the PD and PC treatments but lower the premium from preemptingone’s counterpart, ΠF , from 4 to 1.4. Changing ΠF has no effect on Nash equilib-rium in either case but can have substantial effects in Perfectly Discrete time undersome alternative theories.All treatments are parameterized such that t∗ occurs 40% of the way into theperiod (the 7th subperiod in Perfectly Discrete time treatments).19We ran the PD, IC, PC, L-PD and L-PC treatments using a completely between-subjects design. In each case we ran 4 sessions with between 8 and 12 subjectsparticipating. Each session was divided into 30 periods, each a complete run ofthe 60 second game, and subjects were randomly and anonymously matched andrematched into new pairs at the beginning of each period. We ran the IC10 and IC280treatments using a within-subject design consisting of 3 blocks each composed of3 IC280 periods followed by 7 IC10 periods, for a total of 30 periods.20 Once again,subjects were randomly and anonymously rematched into new pairs each period.We conducted all sessions at the University of British Columbia in the Van-couver School of Economics’ ELVSE lab between March and May 2014. Werandomly invited undergraduate subjects to the lab via ORSEE (Greiner (2004)),assigned them to seats, read instructions (reproduced in Appendix B) out loud andgave a brief demonstration of the software. In total 274 subjects participated, werepaid based on their accumulated earnings and, on average, earned $26.68 (includ-18Recall that we define inertia δ as the ratio of an agent’s reaction lag, δ0, to the total length of thegame, T (i.e. δ ≡ δ0/T ).19In 60 second period treatments this occurs 24 seconds into the period while in the IC10 and IC280treatments this occurs after 4 or 112 seconds respectively.20We used a within design for these treatments mostly because we were concerned that the ex-treme duration of IC280 periods would cause boredom in subjects if repeated a number of times. Byinterspersing these with fast-paced IC10 we were able to reduce this concern. During the experiment,we revealed the next period’s treatment (IC280 or IC10) only after the conclusion of the previousperiod.55ing a $5 show up payment).21 Sessions (including instructions, demonstrations andpayments) lasted between 60 and 90 minutes.3.4 ResultsIn section 3.4.1 we report results from the Baseline treatments, comparing PerfectlyContinuous and Perfectly Discrete time behaviors under identical parameters. Thedata strongly supports Simon and Stinchcombe (1989)’s conjecture of a large gapbetween Perfectly Continuous and Discrete time: PD subjects nearly always inef-ficiently enter immediately while PC subjects nearly always delay entry until t∗,the joint profit maximizing entry time. In section 3.4.2, we study the relationshipbetween the relatively realistic setting of Inertial Continuous time and the extremesof Perfectly Continuous and Discrete time, varying the severity of subjects’ inertiaby varying the speed of an Inertial Continuous time game under Baseline param-eters. We find that at high levels of inertia, behavior follows Nash equilibriumpredictions, collapsing to Perfectly Discrete levels. As inertia drops towards zero,however, entry times approach Perfectly Continuous time levels, a result inconsis-tent with Nash equilibrium.Most of the distinctive predictions and comparative statics discussed in Sec-tion 3.2 concern the timing of first entry and documenting first entry times willbe our primary focus in the data analysis. Before turning to this data, however,it is useful to briefly document second mover behavior across treatments. Focus-ing attention on behavior after the first 10% of periods (after subjects have had afew periods to become comfortable with the interface), we find quite uniform andsensible behavior across treatments: subjects almost universally enter as soon aspossible (given inertia or discretization) following a counterpart’s entry, meaningthat subjects constrain themselves to playing admissible strategies (strategies thatare weakly undominated, see Brandenburger et al. (2008)). In both Perfectly Con-tinuous and Perfectly Discrete time, over 95% of second movers enter at the firstpossible opportunity after their first-moving counterparts (immediately in PC andno later than the very next sub-period in PD).22 In Inertial treatments we measure21Funds for subject payments were provided by a research grant from the Faculty of Arts at theUniversity of British Columbia.22About 5% of subjects in the PC protocol entered with a delay of exactly 0.1 seconds, which we560 10 20 30 40 500.00.10.20.30.40.5(a)% of Period ElapsedDensityPDPC0 10 20 30 40 500.00.20.40.60.81.0(b)% of Period ElapsedCDFFigure 3.3: (a) The left hand panel shows kernel density estimates of entry times (normal-ized as fraction of the period elapsed) in the PD and PC treatments. For both treatmentst∗, which generates the maximal symmetric payoff, lies 40% of the way into the pe-riod. (b) The right hand panel shows CDFs of subject-wise medians from product limitestimates (Kaplan and Meier (1958)) of intended entry times.the median subject’s reaction lag, δ0, at 0.5 seconds, closely matching reactionlags documented in previous research (e.g. Friedman and Oprea, 2012). Giventhe incentives in our game, these rapid responses strongly suggest that subjectsunderstood the structure and incentives of the game across treatments (as delay inresponse is strictly dominated in each treatment in the experiment). Unless oth-erwise noted, remaining references to entry times will refer to the timing of firstentry.573.4.1 Perfectly continuous and discrete timeFigure 3.3 (a) plots kernel density estimates of observed entry times for our PD (inred) and PC treatments (in black). Figure 3.3 (b) complements the kernel densityestimates by plotting CDFs of subject-wise median entry times using product limitestimation intended to minimize the potential downward bias introduced by firstmovers preempting – and therefore censoring – the intended entry times of secondmovers.23The results are striking. In the PD treatment, virtually all subjects choose toenter immediately as the theory predicts, generating highly inefficient outcomes.The PC treatment, by contrast, induces radically different24 behavior: entry timesare tightly clustered near t∗, with subjects maximizing joint earnings by delayingentry until about 40% of the period has elapsed. Recall that though t∗ is only one ofa continuum of equilibria in PC, it is the outcome uniquely selected by eliminationof weakly dominated strategy and is advanced as a focal prediction by Simon andStinchcombe (1989). The tightly clustered behavior in the PC treatments supportsthis conjectured focality and suggests that equilibrium selection is very uniform inPerfectly Continuous time. This pattern of behavior thus strongly supports the con-jecture that Perfectly Discrete and Perfectly Continuous time induce fundamentallydifferent behaviors in otherwise identical games.Result 3.1. Under Baseline parameters, Perfectly Continuous interaction inducesfundamentally different behavior from Perfectly Discrete interaction. While sub-jects virtually always enter immediately in the PD treatment, they virtually alwaysdelay entry until t∗ in the PC treatment.believe is due to a rounding error by the software and which we treat as a zero second lag in thiscalculation.23Specifically we use techniques introduced by Kaplan and Meier (1958) to calculate non-parametric, maximum likelihood estimates of each subject’s distribution of intended entry times inthe face of censoring bias introduced by counterpart preemption. The procedure uses observed entrytimes to partially correct censoring bias introduced in periods in which the subject is preempted byher counterpart. For each subject we estimate these distribution functions and then take the median.Figure 3.3 (b) plots distributions (across subjects) of these medians.24Mann-Whitney tests on session-wise median product-limit estimates of entry times allows us toreject the hypothesis that PC and PD distributions are the same at the five percent level.58Figure 3.4: CDFs of subject-wise product limit estimates of entry times in each of the maintreatments of the experiment.3.4.2 Inertia and continuous timePerfectly Continuous time generates a dramatic change in behavior, but environ-ments with zero inertia are probably rare in the real world. How robust are theseextreme results to a re-introduction of inertia into the game? In order to study thisquestion we ran a series of Inertial Continuous time (IC) treatments, varying theseverity of inertia from very high to very low. In the IC60 treatment we duplicatedthe PC treatment but eliminated the freeze time protocol, allowing subjects’ reac-tion lags to generate natural inertia in the game. In the IC10 and IC280 treatments,run within-subject, we sped up (IC10) or slowed down (IC280) the game clock rela-tive to the 60 second IC60 periods, generating periods that lasted 10 or 280 secondsrespectively. Speeding up the game dramatically increases the magnitude of inertia(defined, recall, as the ratio of reaction lags to game length) while slowing downthe game reduces inertia substantially.59Figure 3.4 shows the results, plotting CDFs of subject-wise median product-limit estimates of entry times for the IC10 (high inertia), IC60 (moderate inertia),IC280 (low inertia) and the PC (zero inertia) treatments (for reference we also plotthe PD treatment in red). The results reveal dramatic and quite systematic effectsof inertia on continuous time behavior as inertia drops towards zero. First, the tightoptimal entry delays observed in the PC treatment almost completely collapse inthe high inertia case, generating Perfectly Discrete-like near-immediate entry aspredicted by Nash equilibrium. However, when we reduce the severity of inertia,CDFs shift progressively to the right, with median entry times rising to t = 0.2 atmedium inertia and t = 0.3 (where subjects earn 95% of earnings available at t∗) atlow inertia and finally reach t = t∗ = 0.4 when inertia reaches zero.25 The resultsthus show that entry times rise smoothly towards Perfectly Continuous levels asinertia falls towards zero, providing us with a next result:Result 3.2. High levels of inertia cause entry delay to completely collapse as Nashequilibrium predicts. However as inertia falls towards zero, entry times approachPerfectly Continuous levels.The survival of high levels of cooperative delay in the face of small amountsof inertia is starkly inconsistent with Nash equilibrium (which predicts a com-plete collapse in cooperative delay with any inertia) but broadly consistent withε-equilibrium. Though the results are perfectly consistent with ε-equilibrium, thesmooth path of convergence is not explained by ε-equilibrium due to multiplicity(once inertia falls enough to allow entry times later than t = 0, any entry time in[0, t∗] is supportable in ε-equilibrium). In the next section we consider our resultsin light of recent findings in the literature on dynamic strategic interaction anddevelop a more satisfying, structured and precise explanation for these patterns.3.5 Discussion: Strategic Uncertainty and ContinuityWhy does inertia have a “smooth” effect on entry instead of causing the immediatecollapse in delay predicted by Nash? ε-equilibrium is broadly consistent with this25An exact Jonckheere-Terpstra test allows us to reject the hypothesis that distributions of session-wise median product limit estimates of entry times are invariant to inertia against the alternativehypothesis that they are (weakly)monotonically ordered by inertia (p < 0.001).60pattern but provides little insight into either its source or (due to multiplicity) itsprecise shape. One appealing answer is that the rich dynamic environment of acontinuous time game makes it difficult to arrive at the sort of common knowledgerequired to support Nash equilibrium, forcing subjects to grapple with unresolvedstrategic uncertainty when making their decisions. Indeed, strategic uncertaintyhas emerged as a central explanatory variable for cooperation in dynamic gamesand both finitely and infinitely repeated prisoner’s dilemmas in prominent recentwork. To measure the strategic risk of cooperating, the literature typically restrictsattention to the strategies Always Defect and Grim Trigger and calculates the basinof attraction of defection (hereafter, the BOA) – the minimal probability one mustassign to one’s counterpart playing Grim in order for Grim to be a best response.Intuitively, the greater the basin of attraction, the more risky it is to attempt to coop-erate: when the BOA is greater than 0.5 it becomes risk dominant to always defect.Both prospective experiments (e.g. Dal Bo and Frechette (2011), Embrey et al.(2016), Vespa and Wilson (2016)) and wide ranging retrospective meta-analyses(Dal Bo and Frechette (2016), Embrey et al. (2016)) reveal that this simple short-hand measure and corresponding notions of risk dominance have startlingly strongpredictive power for cooperation rates in both infinitely repeated games where co-operation is an equilibrium and – importantly for our application – finitely repeatedgames where it is not.Importantly, this simple measure of strategic uncertainty, when adapted to ourgame, also crisply organizes the large treatment effects of inertia we observe inour data. To adapt this measure to our setting, we consider the strategies “enternow at time t” and “wait to enter at t∗”– the closest analogues to Always Defectand Grim Trigger for our game – and calculate the “enter now” BOA for each tin [0, t∗], allowing us study how the strategic risk of entering immediately changesas the game progresses.26 Figure 3.5 plots the BOA at each t for each of our26Again, here and in the remainder of this section, we restrict attention to admissible strategies (astrategy is admissible if it is not weakly dominated: see Brandenburger et al. (2008) for an extensivediscussion of the role of admissibility in games.). The main implication is that when a player chooses“enter now” and the other “wait until t∗,” the second entrant responds to her counterpart by enteringas soon as possible (given the timing protocol). The strategy “wait until t∗” should therefore be readas “enter at t∗ or as soon as possible after my opponent enters, whichever is earliest.” As emphasizedabove, such admissible strategies are virtually universally employed in the data.61continuous time treatments and plots portions of the BOA at which “enter now” isrisk dominant in red. Under high inertia (IC10) the BOA is always 1 and it is alwaysrisk dominant to enter now27 while under zero inertia (PC) the BOA is always 0and delay is risk dominant until t∗. In the two intermediate treatments IC60 andIC280, the BOA changes over time, with immediate entry becoming risk dominantat a different intermediate time in each case.We make two observations. First, except where the measure reaches its bound-ary of 1, the basin of attraction is smaller, at each t, in treatments with larger inertia,suggesting that cooperation is indeed more strategically risky (relative to immedi-ate entry) at higher levels of inertia. Second, the times at which immediate entrybecomes risk dominant almost perfectly corresponds to median entry times in all ofour treatments: BOA reaches 0.5 at times (0, 0.198, 0.308 and 0.4)28 in treatments(IC10,IC60, IC280 and PC) and median product limit entry times track closely at(0.022,0.188,0.325, 0.392). Thus, subjects enter in each treatment precisely whenimmediate entry becomes risk dominant, suggesting that strategic uncertainty hasa strong role in shaping our treatment effects.3.5.1 Three decision rulesThe basin of attraction provides a convenient and easily interpretable measure ofstrategic uncertainty and suggests a strong link between strategic uncertainty andbehavior in our data. It is, however, built on fundamental simplifications that makeit better suited to benchmarking levels of strategic uncertainty in our game than tomodeling exactly how subjects make decisions in the face of strategic uncertainty.Can we put our findings on a firmer footing by considering specific heuristic re-sponses to strategic uncertainty that generate point predictions against which wecan compare our data? Because we are conducting this exercise ex post, our aim isto focus on parsimonious decision rules that put minimal structure on beliefs sub-jects hold about their counterparts’ strategies, and are therefore difficult to adjustto fit the data ex post. To achieve this parsimony we consider models that replaceNash equilibrium’s extreme assumption that agents perfectly know one another’s27The BOA is also always 1 in the PD treatment.28Strictly speaking, the BOA never rises above 0 in the PC treatment but we describe the separatrixas 0.4 to highlight the fact that even at times arbitrarily close to 0.4, entry is not risk dominant.620.0 0.1 0.2 0.3 0.40.00.20.40.60.81.0TimeBasin of AttractionIC-10IC-60IC-280PCFigure 3.5: Basin of attraction of im-mediate entry, calculated at eachtime t. Red coloring denotes pointsat which immediate entry is riskdominant. Horizontal lines at 1 (forIC-10) and 0 (for PC) signify treat-ments in which the basin of attrac-tion does not change over time.0.00.10.20.30.4Entry TimePD IC-10 IC-60 IC-280 PCDataMRAMAALEUFigure 3.6: Median product limit en-try times and predictions fromthe Minimax Regret Avoidance(MRA), Maximin Ambiguity Aver-sion (MMA) and Laplacian Ex-pected Utility (LEU) models.strategies with the opposite extreme assumption that agents are maximally uncer-tain, weighting all counterpart strategies symmetrically ex ante.29, 30In a classic paper, Milnor (1954) considers three non-parametric (and there-fore highly parsimonious) decision rules for uncertain environments like this thatwe can apply to our strategy set and compare to our data. The Laplacian31 Ex-29See Stoye (2011a) and Milnor (1954) for descriptions of the symmetry axiom we have in mind.See Arrow and Hurwicz (1972) for an argument that this type of symmetry is appropriate in modelsof fundamental uncertainty.30As in previous sections, we place only two restrictions on the set of priors: (i) we restrict toadmissible strategies and (ii) entry occurs in [0, t∗]. Both of these characteristics of strategies arevirtually universally observed in the data. In considering models of strategic uncertainty, we restrictattention to simple decision rules and beliefs that are not disciplined by equilibrium, though severalauthors have proposed equilibrium extensions of these rules (e.g. Renou and Schlag (2010), Halpernand Pass (2012), Lo (2009)). While we avoid adopting the stronger assumptions and greater structurerequired of these equilibrium concepts for this exercise, we note that many of these equilibriumconcepts generate identical predictions to those discussed below.31So named for Laplace’s (1824) argument that uniform beliefs should be applied to unknown63pected Utility (LEU) rule is the subjective expected utility response to this typeof strong uncertainty and models agents as simply choosing actions that constitutebest responses to a uniform (or “Laplacian”) distribution of entry times by theircounterparts on [0, t∗]. The Minimax Regret Avoidance (MRA) rule, first proposedin Savage (1951) and axiomatized by Milnor (1954) and Stoye (2011a), resultsfrom relaxing the independence of irrelevant alternatives axiom and specifies thatagents choose the action that minimizes the largest ex post regret (the differencebetween earnings actually generated by a strategy choice and the earnings a dif-ferent strategy choice might have generated given counterparts’ strategies) over allstrategies. Finally, the Maximin Ambiguity Avoidance32 (MAA) rule, proposedby Wald (1950) and axiomatized by Milnor (1954), Gilboa and Schmeidler (1989)and Stoye (2011b), relaxes the independence axiom in expected utility theory andspecifies that agents choose a strategy that yields the largest minimum payment(over other subjects’ strategies) an agent could achieve.33As we show in Online Appendix C.4.1, these decision rules make very differentpredictions for the way inertia shapes decisions in continuous time games like ours.The MAA rule predicts exactly what Nash equilibrium predicts: immediate entry att = 0 for any inertia greater than zero. The MRA and LEU rules, by contrast, eachpredict a smooth pattern of progressively later entry as inertia falls towards zero,terminating at t∗ when inertia is zero, though the rate of convergence is different ineach case. We calculate predictions for each of these rules for each treatment, andpresent the results in Figure 3.6 along with medians of subject-wise product limitestimates. Of the three heuristics, MAA does the worst by predicting exactly whatNash equilibrium does for each treatment. Both MRA and LEU, by contrast, do anexcellent job of tracking the data but the MRA heuristic fits point estimates fromevents due to the principle of insufficient reason (see e.g. Morris and Shin, 2003). Laplacian beliefshave an important role in the literature on global games (see Morris and Shin (2003)).32We deliberately avoid the more common MEU acronym for this decision rule to emphasize asubtle difference in interpretation between the standard MEU model and our application. In thestandard MEU model, as in Gilboa and Schmeidler (1989), the set of priors is treated as endogenousto the agent’s preferences. By contrast, we interpret MAA as a decision rule that is applied to anexogenous set of uninformative beliefs (as in Stoye (2011b)).33Milnor refers to these as the Laplace, Savage and Wald rules respectively. He also discusses afourth rule that he calls Hurwicz (commonly today called α-maxmin) which we reject because it hasa free parameter and therefore can be “tuned” to the data in an ex post exercise like this one.64the data almost perfectly and is the most accurate of the three models. We reportthis as our next result:Result 3.3. Median entry times across treatment are almost perfectly organized bypredictions made by the MRA decision rule, suggesting that reactions to strategicuncertainty are an important driver of behavior.Our results suggest that assuming subjects are highly strategically uncertainabout their counterparts’ behavior can generate significantly better predictions thanmaking the opposite assumption that strategic uncertainty is eliminated in equilib-rium. Interestingly evidence for such strategic uncertainty doesn’t seem to easemuch as subjects acquire experience in our data: median entry times in the finalperiod of play track MRA predictions across levels of inertia just as well as productlimit estimates do using data from the whole session (as visualized in Figure 3.6).There is, moreover, no evidence of movement towards Nash equilibrium over timein any of our continuous time treatments (except in IC10 where MRA actually pre-dicts Nash-like outcomes), suggesting that strategic uncertainty continues to playan essential role in determining behavior even after dozens of periods of play. As itturns out, MRA predictions are surprisingly robust to the type of feedback subjectsacquire in dynamic settings like ours because most uncensored feedback subjectsreceive via repeated play concerns whether counterparts tend to enter early in thegame. Since regret is primarily shaped by the possibility that one’s counterpartwill enter later in the game, MRA predictions change little when subjects learnthat early entry events by counterparts are unlikely. Consequently, MRA predic-tions tend to be fairly durable in the face of experience, just as later entry times arein our data.3434Consider a subject who repeatedly enters at or near the MRA predicted entry time, tMRA. She willlearn the approximate distribution of entry times over the interval [0, tMRA], but because of censoringshe will not learn anything about the distribution of (intended) entry times over the interval [tMRA, t∗].For all entry times t < tMRA the subject faces her maximal regret when her opponent enters at t∗ and,because of censoring, she cannot rule out that her opponent may intend to enter at t∗. For entry timest > tMRA she faces her maximal regret when her opponent slightly pre-empts her, which again shecannot rule out. In other words, our MRA prediction is robust to a subject learning that some, oreven all, early entry times are never used. This is because early entry times do not cause large regret:regret is maximized when opponent’s are either fully cooperative (enter at t∗) or enter immediatelyprior to the subject’s intended entry time. Given that the MRA decision rule seeks to minimizemaximal regret, it is uncertainty over these later entry times that causes MRA deviations from Nashequilibrium. See the proof in Online Appendix C.4.2 for more detail on the mapping between the set650.00.10.20.30.4TreatmentEntry TimePD L-PD PC L-PCMRA PredictionEntry TimeFigure 3.7: Median observed entrytimes (first column) and MRA pre-dictions from the diagnostic L-PDand L-PC treatments. PD and PCtreatments from the main designare also included for comparison.0 10 20 30 40 50 600.00.20.40.60.81.0GridpointsThreshold EstimateFigure 3.8: Median cooperation ratesfrom the Grid-n treatment fromFriedman and Oprea (2012) com-pared to MRA predictions.3.5.2 Validation using alternative comparative staticsWe designed and ran two additional treatments to study whether our explanationfor the comparative static effect of inertia can also explain other, distinct compar-ative statics. In the L-PD and L-PC treatments we replicate the Perfectly Discreteand Perfectly Continuous treatments but dramatically lower the preemption temp-tation parameter ΠF from 4 to 1.4. In the PC treatment, lowering this parameterhas no effect on strategic risk as measured by the basin of attraction and does notchange the MRA point prediction under Perfectly Continuous time protocols (theprediction is t = 0.4 in either case). That is, under the class of explanations we’veconsidered thus far, the PC and L-PC treatments should generate identical behav-iors.By contrast, in Perfectly Discrete time protocols, strategic uncertainty changesa great deal when we lowerΠF in the L-PD treatment: while immediate entry is al-ways risk dominant in the PD treatment (under the simple BOA measures discussedabove) it becomes risk dominant only at an interior point in the L-PD treatment, asof believed entry times and MRA predictions.66in the IC60 and IC280 treatments, suggesting lowering ΠF may generate a later en-try time in discrete time. Most importantly for our purposes, the MRA predictionrises from 0 to 0.2 when we lower ΠF in the L-PD treatment.35Figure 3.7 shows median subject-wise product limit estimates for the PD, L-PD, PC and L-PC treatments and MRA point predictions. The results nearly per-fectly track the point predictions provided by the MRA heuristic. Entry times risefrom 0 to 0.2 when we lower ΠF in the L-PD treatments but remain constant atabout 0.4 when we make the same parameter change in the L-PC treatments, justas the MRA rule suggests.36Result 3.4. Results from additional diagnostic treatments varying payoff param-eters in discrete and continuous time are well organized by measures of strategicuncertainty and point estimates are virtually identical to point predictions gener-ated by the MRA rule.3.5.3 Validation using other continuous time gamesThe MRA decision rule organizes behavior in our game remarkably well, predict-ing, in particular, the smooth approach to Perfectly Continuous-like cooperativebenchmarks we observe as inertia falls to zero. How relevant are these sorts ofresults for understanding behavior in other continuous time games? To find out,we test the MRA rule against data from the continuous prisoner’s dilemma, thesimplest game in a broad and empirically important class of games in which ef-ficient outcomes are in tension with individual incentives. The Grid-n treatmentin Friedman and Oprea (2012) studies 60 second prisoner’s dilemmas that are di-vided up into 4, 8, 16, 32 and 60 Perfectly Discrete time subperiods, within subject.This time protocol creates the equivalent of exogenous reaction lags in continuoustime lasting 50%, 25%, 12.5%, 6.6%, 3.3% and 1.6% of the game, respectively,generating a similar effect to inertia in our Inertial Continuous time games.35We originally designed these additional treatments to validate ε-equilibrium comparative staticswhich are similar to and consistent with MRA predictions but are less precise, just as with treatmentsfrom the main design.36A Mann-Whitney allows us to reject the hypothesis that session-wise median product-limit es-timates of entry times in the PD and L-PD treatments are from the same distribution (p = 0.017);the same test does not allow us to reject the same hypothesis regarding the PC and L-PC treatments(p = 0.183).67In Figure 3.8 we plot median final mutual cooperation times (measured as afraction of the period) as a function of the number of grid points.37 Over thiswe overlay MRA predictions38 for the earliest time at which mutual cooperationcan evaporate as a function of the number of grid points. (Any date after the timesplotted can be supported under the MRA). Strikingly, these earliest MRA defectiontimes nearly perfectly match median final cooperation times, converging towardsthe Perfectly Continuous time limit of 1 (cooperation until the very end of theperiod) as the number of grid points grow large and the forced reaction lag growssmall. The results thus provide strong out-of sample confirmation that the MRAheuristic organizes convergence paths to Perfectly Continuous time benchmarks.39Result 3.5. The earliest MRA defection time generates accurate point predictionsfor convergence to continuity in the continuous time prisoner’s dilemma.The prisoner’s dilemma is the simplest in a broad set of strategically similargames that include important applications like Bertrand pricing, Cournot quantitychoice, public goods, and team production problems. Though, of course, we can-not test every game in this class, the fact that MRA-predicted results like oursextend to the continuous prisoner’s dilemma is strongly suggestive that such ef-fects are relevant for a much larger set of strategically similar games. In OnlineAppendix C.4.2 we provide some support for this intuition by showing that, underplausible (and particularly empirically relevant) specifications of beliefs over thestrategy space, the MRA rule predicts similar convergence results for this broadclass of dilemma-like games. Combined, our results suggest that rules like the37As in the other analyses in this chapter we use the full dataset in making these measurements.Restricting attention to the final 2/3 of the session as Friedman and Oprea (2012) do generates similarresults.38Calculated under restrictions on beliefs discussed in Online Appendix C.4.2.39MRA also predicts the high rates of cooperation and low variation over parameters Friedmanand Oprea (2012) observe in their (Inertial) Continuous time treatments. These treatments study60 second continuous time prisoner’s dilemmas – prisoner’s dilemmas with flow payoffs realized incontinuous time (though with subject-generated inertia). Friedman and Oprea’s (2012) design setsmutual cooperation payoffs of 10 and “suckers” payoffs of 0 and varies the temptation payoff (x) anddefection payoff (y) cyclically over 32 periods over four parameterizations: Hard (x=18, y=8), Easy(x=14, y=4), Mix-a (x=18, y=4), Mix-b (x=14, y=8). The median final time of mutual cooperation (in60 second periods) are 59.6, 58.4, 58.44, 57.6 in the Easy, Mix-a, Mix-b and Hard treatments, whichare very tightly clumped near the Perfectly Continuous benchmark time of 60. This nearly perfectcooperation and minimal variation over parameters is explained by MRA, which predicts earliestcollapse of cooperation at 58.9, 58.5, 56.5 and 55.5 seconds in these four treatments.68MRA, founded in strategic uncertainty, provide an empirically plausible mecha-nism by which we might expect nearly-Perfectly Continuous levels of cooperationto emerge and persist even in the presence of realistic inertia.3.6 ConclusionPerfectly Continuous and Perfectly Discrete time are both idealizations but theyare illuminating ones, functioning as strategic analogues to vacuums in the physi-cal sciences. Like vacuums, they are environments in which theoretical forces arecast in particularly high relief and results can be crisply interpreted in the light oftheory. Although Perfectly Discrete time behavior has been exhaustively studiedin thousands of experimental investigations, Perfectly Continuous time has neverbeen studied before and for a very simple reason: natural frictions in human in-teraction that loom especially large in the relatively fast pace setting of a labora-tory experiment push strategic environments meaningfully away from the PerfectlyContinuous time setting described in the theory. Our chapter introduces a method-ological innovation that eliminates these frictions, allowing us to observe, for thefirst time, Perfectly Continuous behavior. By observing and comparing behavioracross these two “pure” environments and by comparing both to more naturalisticprotocols in between we learn some fundamental things about dynamic strategicbehavior.Results from our initial baseline parameters are nearly perfectly organized bybenchmarks proposed in the literature. Though our game suffers from multiple(indeed, a continuum of) equilibria in Perfectly Continuous Time, we observe en-try times tightly clustered at the interior joint profit maximizing entry time underthis timing protocol. This decisive equilibrium selection strongly supports a weakdominance refinement argued for by Simon and Stinchcombe (1989) for PerfectlyContinuous games. By contrast, under the exact same parameters, in Perfectly Dis-crete Time we observe almost universal, highly inefficient first-period entry that isperfectly in line with backwards induction. Thus our baseline results show strongevidence of a large and economically significant gulf between Perfectly Discreteand Perfectly Continuous time behaviors.How do results from these artificial settings relate to more realistic strategic69interactions? Most real human decisions are made neither perfectly synchronously(as in Perfectly Discrete time) nor with instant response (as in Perfectly Contin-uous time). More realistic are real time, asynchronous settings in which there issome delay in mutual responses, even if small. Nash equilibrium predicts that evena tiny amount of such inertia will be sufficient to erase all of the cooperative equi-libria generated by Perfectly Continuous time. However ε-equilibrium suggeststhat the correspondence between Inertial Continuous time behavior and the bench-marks of Perfectly Discrete and Perfectly Continuous time depends crucially onthe size of inertia. While very high levels of inertia can cause ε-equilibrium sets tocoincide with Perfectly Discrete behavior (as suggested by Nash), very low levelsof inertia can push the ε-equilibrium set to coincide with the Perfectly Continuousequilibrium set. We study such settings in our Inertial Continuous time treatmentsin which subjects interact (under Baseline parameters) in continuous time but withnatural human reaction lags (clocked at roughly 0.5 seconds in our subjects). Byvarying the speed of the game clock we are able to alter the severity of naturallyoccurring inertia in subjects’ decision making and study the robustness of PerfectlyContinuous time behavior to multiple levels of inertia. Our results show that Nashequilibrium-like collapses to Perfectly Discrete-like benchmarks occur in contin-uous time when inertia is very high. But at low levels of inertia, subject entrydelays approach the efficient levels generated in (and predicted for) the PerfectlyContinuous treatment.We close the chapter by considering sharper (psychologically) and crisper (pre-dictively) explanations for our results than ε-equilibrium can provide. Recent re-search on dynamic strategic interactions have amassed a great deal of evidence thatstrategic uncertainty faced by attempting cooperation (as measured by the basin ofattraction for defection) and related notions of risk dominance have strong predic-tive power for cooperation even in games in which cooperation is unsustainable asa Nash equilibrium. Applying similar measures to our game, we find that strategicuncertainty subjects face when attempting to cooperate rises sharply with inertia,supporting a conjecture that strategic uncertainty shapes behavior in continuoustime games in a way that Nash equilibrium cannot capture.Inspired by the organizing power of this measure in our (and other) data, weconsider a series of simple heuristics that replace Nash equilibrium’s extreme as-70sumption that subjects perfectly know their counterparts’ strategies with the op-posite extreme that subjects know very little about counterparts’ strategies. Weshow that several such decision rules strongly outperform Nash equilibrium andthat one (the Maximin Regret Avoidance model) almost perfectly matches cross-treatment point estimates from our data. To strengthen our analysis we exposethe MRA rule to an additional test using a pair of additional treatment and findsimilarly strong evidence. Finally, we show that heuristics like the MRA generateincreasingly cooperative behavior as games grow more continuous in an importantclass of empirically relevant games. We test this prediction on continuous timeprisoner’s dilemmas from past work and show, once again, that the MRA heuristicnearly perfectly organizes both point predictions and treatment effects over whichNash equilibrium makes starkly counterfactual predictions. The results suggestthat benchmarks that assume no knowledge of others’ strategies and impose littlestructure on subjects’ beliefs do a better job of anticipating behavior (even behaviorof experienced subjects) than benchmarks that assume perfect knowledge. Thesemodels also explain why, as in our data, Perfectly Continuous-like behavior persistseven in the presence of inertia.The results from our experiment – and supporting theoretical benchmarks –suggest an appealing framework for understanding the relationship between theabstractions of Perfectly Discrete and Perfectly Continuous time and real worldbehavior. Perfectly Discrete and Perfectly Continuous time predictions can bethought of as polar outcomes that each approximate realistic (Inertial Continuoustime) behavior when inertia is either very high or very low, respectively. Indeed,we can easily push real time (Inertial) behavior close to either Perfectly Discrete orPerfectly Continuous time behavior simply by varying the severity of inertia. Con-cretely, these sorts of results suggest that Perfectly Continuous time benchmarkscan, in some cases, be more empirically relevant than Discrete time benchmarks,even if agents face frictions that should be sufficient to short circuit Perfectly Con-tinuous time equilibria under standard theory. The rise of thick online global mar-kets, always-accessible mobile technology, friction reducing applications and auto-mated online agents have made strategic interactions more asynchronous and lagsin response less severe. These trends, which seem likely to intensify in the comingyears, have the effect of pushing many interactions closer to the setting of Perfectly71Continuous time. Though these technological changes may never drive inertia en-tirely to the Perfectly Continuous limit of zero, our results suggest that behaviorcan nonetheless come close to Perfectly Continuous levels as inertia falls. This de-viation from standard theory in turn suggests that we might expect Perfectly Con-tinuous time predictions to become an increasingly relevant way of understandingeconomic behavior relative to the Perfectly Discrete predictions most often used ineconomic models.72Chapter 4Mental Equilibrium and MixedStrategies for Ambiguity AverseAgents4.1 IntroductionConsider a game between two agents that is mediated by a game theorist. Theagents report their strategies to the game theorist, who then resolves the outcomeof the game and pays the agents their winnings (or collects their losses). The gametheorist may allow mixed strategy reports from the agents, and resolve the mixedstrategy herself, or she may require that the agents report a pure strategy. If thegame theorist requires pure strategy reports, as is the case in Chapter 2, then thegame theorist should be aware that the agents may still be using a mixed strategythat they are resolving privately before reporting. Under the standard formulation,where agents have expected utility preferences, the set of equilibrium under thetwo reporting requirements will be indistinguishable.However, when agents have ambiguity averse preferences then the different re-porting requirements may induce different games with different equilibrium. Equi-librium concepts such as Lo (2009), Dow and Werlang (1994) and Eichberger andKelsey (2000) enforce pure strategy reporting and generate larger equilibrium sets73than Lo (1996) or Klibanoff (1996) which allow for mixed strategy reports. Thedifference arises because ambiguity averse preferences are non-linear and agentsmay have a strict preference for mixed strategies. The equilibrium concepts thatenforce pure strategy reporting assume that only pure strategies are available toagents: they implicitly rule out private pre-play mixing. By its very nature, how-ever, private pre-play mixing will be unobservable to the game theorist and cannotreadily be prevented.How, then, does allowing for private pre-play mixing affect the equilibrium inpapers such as Lo (2009), Dow and Werlang (1994) and Eichberger and Kelsey(2000)? The answer, provided for the first time in this chapter, is that allowing forprivate pre-play mixing has no effect on the equilibrium set for agents with pref-erences that lie in the intersection of Choquet Expected Utility (Schmeidler, 1989)and Maxmin Expected Utility (Gilboa and Schmeidler, 1989). While this resultis of independent interest, it is particularly relevant for experimental tests of am-biguity averse equilibrium concepts. Recent experiments1 have used equilibriumconcepts that restrict themselves to pure strategies, and thereby implicitly assumethat their subjects are using only pure strategies and are not engaging in pre-playmixing. Given that it is not possible to actively prevent subjects from pre-play mix-ing, the results in this Chapter are essential for a direct interpretation of the data inthe previous experimental literature using pure strategy solution concepts.We establish our key result using an application of results from Gilboa andSchmeidler (1994) for agents in individual decision making problems. The applica-tion of Gilboa and Schmeidler (1994) to games yields two additional insights. First,the technique allows for an extension of the static equilibrium concepts discussedabove into dynamically consistent extensive form equilibrium concepts. Second,because the technique linearizes non-linear ambiguity averse preferences, it alsoallows us to formulate the equilibrium set of a game with ambiguity averse agentsas the solution to a linear complementarity problem – thereby providing a simplemechanical technique for solving games with ambiguity averse agents.The rest of this chapter is organized as follows. Section 4.2 introduces themathematical tools from Gilboa and Schmeidler (1994) that are used in the rest1See Section B.1.5 for an overview of the literature, that also includes Chapter 2 of this disserta-tion.74of the chapter. Section 4.3 introduces the notion of a game as a set of interactingdecision problems and presents the key idea of a “mental” state space. Sections 4.4and 4.5 apply the structure of a “mental” state space to the equilibrium conceptscontained in Lo (2009) and Dow and Werlang (1994), respectively, and establishesthe key result of this Chapter. Section 4.6 demonstrates the extension of a “mental”equilibrium to an extensive form game in a dynamically consistent fashion, usingthe equilibrium concept from Lo (2009) as an example. Section 4.7 demonstrateshow the “mental” formulation of a game gives rise to a linear complementarityprogram that can be easily solved to find the equilibrium of a normal form game.Section 4.8 concludes, and proofs are relegated to Appendix D.4.2 PreliminariesSuppose that there exists a finite2 set of states of the world, ω ∈ Ω, and that anact, f , maps each state to an outcome in R; that is f : Ω 7→ R. Choquet ExpectedUtility (CEU), first introduced by Schmeidler (1989), generalizes Subjective Ex-pected Utility by allowing a decision maker to hold non-additive beliefs which arerepresented by a capacity, ν , defined over the set of events Σ= 2Ω. Suppose, with-out loss of generality, that for a given act, f , the set of states can be ordered sothat f (ω1)≥ f (ω2)≥ . . .≥ f (ωn). A CEU agent calculates her utility of an act byevaluating the (discrete) Choquet integral:∫f dν =n∑i=1f (ωi)[ν(∪im=1ωm)−ν(∪i−1m=1ωm)](4.1)We shall assume throughout that the capacities, νi, are belief functions and weuse V to denote the space of all such capacities. That is, we assume that νi(Σi) =1 and that νi is totally monotone.3 Under these assumptions it is also possibleto represent the agent’s preferences using Maxmin Expected Utility (Gilboa andSchmeidler, 1989, 1994).Throughout this chapter, we shall rely on two basic mathematical results thatare demonstrated in Gilboa and Schmeidler (1994). Firstly, the non-additive mea-2Although the finiteness of the state space can be relaxed, we restrict attention to finite sets hereto mirror our later restriction to finite games.3A totally monotone capacity is convex, but the converse need not hold.75sure ν can be spanned by an additive measure over an appropriately defined (larger)state space. Secondly, we can represent an agent with CEU preferences over Ω as,equivalently, having SEU preferences over the larger state space with an appropri-ately transformed set of acts.Result 4.1 (Adapted from Gilboa and Schmeidler (1994)). For T,A ∈ Σ′ = Σ\{ /0},defineeT (A) ={1 T ⊆ A0 otherwiseThen the set {eT}T∈Σ′ forms a linear basis for V . The unique coefficients {ανT }satisfyingν = ∑T∈Σ′ανT eTare given byανT = ∑S⊆T(−1)|T |−|S|ν(S) (4.2)Furthermore, if ν is totally monotone then ανT ≥ 0 for all T ∈ Σ′ and if ν isnormalized then ∑T∈Σ′ ανT = 1.Result 4.1 provides the key building block for this chapter: any non-additivemeasure over a state space can be spanned by an appropriately formed set of statesconstructed from the power set of the original state space. Furthermore, whenthe non-additive measure is a belief function then the spanning coefficients canbe interpreted as probabilities over the newly constructed state space. Note therelationship between Result 4.1 and the proof of the representation theorem forCEU in Schmeidler (1989). In Result 4.1, we begin with a non-additive measureand ‘restore’ additivity by extending the state space. In Schmeidler (1989), theprimitive is a SEU representation with respect to an additive measure, which is thenextended to generate a CEU representation with respect to a non-additive measure.This tight relationship between Choquet Expected Utility and Subjective ExpectedUtility is formalized in the next result.Recall that when ν is a belief function the core of ν is simply the set of prob-76ability measures, p, such that p(A) ≥ ν(A) for all A ∈ Σ.4 We now state result4.2.Result 4.2 (Corollary 4.4 from Gilboa and Schmeidler (1994)). Suppose that ν isa belief function. Then for every f ∈ F∫f dν = ∑T∈Σ′ανT[minω∈Tf (ω)](4.3)= minp∈Core(ν) ∑ω∈Ωp({ω}) f (ω). (4.4)Result 4.2 demonstrates that an agent with CEU preferences with respect to abelief function can have their preferences represented via either MEU or SEU pref-erences. While this relation between MEU and CEU preferences is both straight-forward and well known, the representation with SEU preferences requires theformation of a new set of acts over the set Σ = 2Ω, with the outcome associatedwith each new act defined by the min function in equation 4.3. We shall call theseacts “mental” acts, and will sometimes refer to the event space as the “mental”state space. This terminology reflects that the event space may not be observableto an external observer and may, therefore, represent the mental accounting of theagent.5Definition 4.1. A mental act, f ′ , is an extension of an act, f , defined over the eventspace, Σ′, such that f ′ : Σ′ 7→ R with f ′(T ) = minω∈T f (ω) for all T ∈ Σ′.It follows from result 4.2 that the preferences of an agent, with CEU prefer-ences with respect to a belief function, can be written in the expected utility formwith respect to mental acts over the mental state space. Billot and Vergopoulos(2014) generalize this result to a broader set of preferences than just CEU with4Note that, because each p in the core is additive, the core can be equivalently defined to be therestriction of this set of probability measures to an equivalent set of probability measures definedover Ω. In some places we use core (ν) to denote this restriction, but this should be clear from thecontext.5The interpretation of the event space as a mental “state” space implies that the agent, at leastin his mental accounting, is not well calibrated about the nature of the world. In a sense, this isprecisely the trade-off that allows us to move from non-linear preferences in the observable statespace to linear preferences in the mental state space.77respect to a belief function, and explore the implications of result 4.2 for problemswith a single decision maker in some detail. It is possible to generalize the re-sults in this chapter to cover the broader class of preferences covered in Billot andVergopoulos (2014).Before continuing with the substantive heart of the chapter, we first review whythe introduction of mixed strategies may change the equilibrium set in games withambiguity averse agents relative to a game where only pure strategies are available.4.2.1 Mixing: Anscombe-Aumann vs. Savage interpretations ofuncertaintySuppose that an agent, who may choose from a finite set of acts, obtains a mix-ing device (a random number generator, say) and that they may condition theirchoice of act on the output of the mixing device. How should we define a mix overtwo acts, and how should the agent evaluate the payoffs that arise from these mixedacts? It is well established in the literature (Seo, 2009; Eichberger et al., 2016) that,for an ambiguity averse agent, the answer to these questions depends crucially onthe way that we model the mixing device. The two most common modeling strate-gies originate with Anscombe and Aumann (1963) and Savage (1954). While bothAnscombe and Aumann (1963) and Savage (1954) developed an axiomatization ofadditive expected utility preferences, our interest in these papers lies in the model-ing frameworks rather than the representation theorems. Each of these frameworkshas been extended and adapted to formulate representations of non-additive Cho-quet Expected Utility and Maxmin Expected Utility preferences; in what follows,we illustrate the role of mixing with reference to various axiomatizations of CEUpreferences.The Anscombe and Aumann (1963) model of decision making is a two stagemodel.6 In the first stage the (uncertain) state of the world is resolved, and theoutcome of an act in a state is a (risky) lottery; this timing structure is some-times referred to as a horse-roulette lottery. The second stage is the resolution6In the original formulation the Anscombe and Aumann (1963) model had three stages. Here,we refer to the cleaner two-stage adaptation of Anscombe and Aumann (1963) that was developedin Fishburn (1970) and used in later works including Schmeidler (1989). Seo (2009) also uses thethree-stage formulation and provides a brief discussion.78of the roulette lottery and, in the CEU model in an Anscombe-Aumann state space(Schmeidler (1989)), the agent is assumed to have expected utility preferences overroulette lotteries. The randomization device is therefore appropriately modeledas acting state-by-state on the roulette lotteries: consider two acts, f and g andtheir mixture β f +(1−β )g. For each state s, the mixed act is defined such that(β f +(1−β )g)(s) = β f (s)+(1−β )g(s).The Savage (1954) model of decision making is a one stage model. All uncer-tainty, whether it is ambiguous or unambiguous, is resolved simultaneously. Forexample, in the Sarin and Wakker (1992) axiomatization of CEU in the Savagemodel there is a distinction between ambiguous and unambiguous states, while inthe Gilboa (1987) axiomatization no such distinction is made. In this frameworkthere is no clear meaning to the idea of mixing two acts; the object β f +(1−β )gis simply not defined. The appropriate way to incorporate such an act in a Savageframework is to enlarge the state space in such a way that the new act can be definedin an appropriate fashion across each state. For example, each state, s, could be splitinto a state s f and a state sg and then we would define (β f +(1−β )g)(s f ) = f (s)and (β f + (1− β )g)(sg) = g(s), and if the agent understands that the states s fand sg were realized with subjective probabilities in accordance with some mix-ing device then we would expect their preferences to reflect this knowledge.7 Asdiscussed below this approach is not compatible with our interpretation, in Sec-tion 4.3, of a game as a set of interacting decision problems.As an example, consider the following setup. There are two states s = {A,B}and two acts f and g, with payoffs given in table 4.1. For simplicity, assume thatthe agent is risk neutral.s = A s = Bf 20 10g 10 20Figure 4.1: An example with two acts and two states.Suppose that the agent has CEU preferences with ν(A) = 0.4 = ν(B). Theagent therefore values both f and g such that U( f ) =U(g) = 14.7See Klibanoff et al. (2005) for a formal exposition.79Now, suppose that we are working in an Anscombe-Aumann framework, andconsider the prospect h = 12 f +12 g. In the state s = A, h will generate a fifty-fiftylottery of 20 and 10. In the state s = B, h will generate a fifty-fifty lottery of 10and 20. Therefore, in each state, h generates a lottery which the agent will valueat 15.8 h produces a constant lottery in all states, and should therefore be valuedat the value of that lottery. Notice that this implies that, for our ambiguity averseagent, h f and h g: the agent strictly prefers the mixed prospect h over theprospects that form the support of h.In the Savage framework there is only one resolution of uncertainty so wecannot assign a lottery as an outcome; the outcome that occurs when a state occursafter an act has been chosen must be real valued. So, what value should we assignh in the state s = A? It is tempting to claim that this value should be 15, as in theAnscombe-Aumann example. However, recall that our agent has CEU preferencesthat represent pessimism. It seems reasonable that our agent may be pessimisticand believe that they will hold g more often when s = A and hold f more oftenwhen s = B; in essence, our state space is not rich enough to fully capture theagent’s preferences.The standard approach in the Savage framework is to expand the state spaceby forming a new state space that is the cross product of the original state spaceand the set of possible outcomes of the available randomization device. This isprecisely the approach outlined by Eichberger and Kelsey (1996), wherein theyestablish an indifference to randomizations when the state space is expanded in thisfashion for the class of preferences considered in this chapter. We do not pursuethe Eichberger and Kelsey (1996) approach in this chapter because it is difficultto reconcile this expanded state space with the interpretation of a game as a set ofinteracting decision problems with pure strategies as the choice objects, as we seekto do in Section 4.3.As an alternative, we can use the results presented in section 4.2 to translatethe agent’s preferences into SEU preferences over the event space. To see this, weintroduce a new state s= {A,B} and form two new prospects f ′ and g′ (figure 4.2).We call this the mental state space method, as the new state is a mental construct8Recall that, in the CEU framework, the agent has standard SEU preferences over objective lot-teries.80of the agent.s = A s = B s = {A,B}f 20 10 ·g 10 20 ·f ′ 20 10 10g′ 10 20 10Figure 4.2: An example with two acts and two states extended into the eventspace.It is easy to see from Result 4.1 that ανA = ανB = 0.4 and αν{A,B} = 0.2. ThenResult 4.2 allows us to write the agents preferences in an expected utility form, andwe conclude that we can form the prospect h′ = 12 f′+ 12 g′ and that the agent musthave preferences such that U(h′) = 12U( f′)+ 12U(g′) = 14. Furthermore, becausepreferences are preserved in the mapping from the original acts to mental acts byconstruction, we conclude that U(h) = 14.It is also important to identify the timing of randomizations in each of themodels. In the Anscombe-Aumann model it is explicitly the case that the objectivemixing occurs after the subjective state is realized. In the case of a game, thiswould be consistent with the case where agents submit mixed strategies to thegame theorist, the game theorist announces the mixed strategies and then resolvesthe mixed strategies to reveal the outcome of the game. Alternatively, the gametheorist could simply announce the (pure) outcome of the game as a function of thesubmitted mixed strategies. In the Savage model mixing, and by extension, mixedstrategies are not well defined. In this case, the only reasonable interpretation isthat agents use and report pure strategies to the game theorist.The mental state space interpretation of mixed acts is that the mixing deviceis resolved (weakly) before the resolution of the state. In this interpretation wemay model the resolution of uncertainty in two stages. Ex-ante, the agent hasSEU preferences over the mental state space and mental acts and, by extension,any available mixing devices. At the interim stage, the mixing device has beenresolved and the agent is holding one of the original acts in the original state space(over which there may be subjective uncertainty). Ex-post, all uncertainty has beenresolved and the agent receives the outcome associated with the realized state and81act selected by the randomization device. In a game setting, this interpretation isconsistent with the agent using a private mixing device before reporting the purestrategy realization of the mixing device to the game theorist.The example presented above highlights the two main concerns for mixedstrategies in games with ambiguity averse agents (and also suggests a resolution).If the game theorist models the game with an Anscombe and Aumann (1963) statespace then agents will have a strict preference for mixed strategies which, in manycases, causes the behaviour of ambiguity averse agents to be indistinguishable fromstandard Nash agents (Lo, 1996; Klibanoff, 1996). Alternatively, if the game the-orist models the game with a Savage (1954) state space then the payoff for mixedstrategies is not well defined and the game theorist does best by restricting agentsto pure strategies. Each of these approaches is considered in the previous literature,which is summarized in the appendix in Table A.1.The resolution proposed in this chapter is to expand the game into the mentalstate space, where mixed strategies are well defined provided that the mixing deviceis realized before strategies are revealed.The discussion so far is consistent with, and could also have been motivatedby, more recent literature on the timing of randomization in individual decisionproblems. Seo (2009) demonstrates that, for agents who violate the reduction ofcompound lotteries, the above argument hinges critically on whether the objectiverandomization occurs before or after uncertainty is resolved. Furthermore, Halevy(2007) provides experimental evidence that subjects who exhibit ambiguity aver-sion overwhelmingly do not reduce compound lotteries. Therefore we conclude,for ambiguity averse agents, that the order of realization of randomizations is im-portant. Additionally, Eichberger et al. (2016) demonstrate that, for a dynamicallyconsistent agent, the agent will be indifferent between the randomization and theinitial prospects if the randomization occurs before the uncertainty is resolved , butthat a preference for randomization may exist if the randomization occurs after theuncertainty is resolved.824.3 Games as Interacting Decision ProblemsIt is possible to reformulate a standard game as a set of interacting decision prob-lems.9 Recall that a normal-form game is fully defined by the set of players, I, aset of strategies, {Si}, for each player and a set of utility functions, {ui}. For eachagent we can formulate their decision problem as choosing from a set of acts overstates where the states are given by Ωi = {×S j 6=i} and fi(ω) = si(ω) = ui(si,ω) =ui(si,s j 6=i).4.3.1 Translating a game into the mental spaceAny finite normal form game can be translated into a game played in the mentalstate space via an application of result 4.2 and definition 4.1. Figures 4.3 and 4.4provide an example for the 3×2 game studied in Chapter 2.10X YA 25,20 14,12B 14,20 25,12C 18,12 18,22Figure 4.3: An example gameX Y {X ,Y}A 25,20 14,12 14,·B 14,20 25,12 14,·C 18,12 18,22 18,·{A,B} ·,20 ·,12{A,C} ·,12 ·,12{B,C} ·,12 ·,12{A,B,C} ·,12 ·,12Figure 4.4: An example game in the mental state space with payoffsThe interpretation of figure 4.4 is that the row player perceives there to be an9Aumann (1987) is the classic example.10Groes et al. (1998) use a similar visual representation of extending a game into the event space,although they allow for both optimism and pessimism when extending preferences into the extrastates, and exclude explicit randomizations. In contrast, the purpose of expanding the state space inthis Chapter is precisely to allow for the inclusion of mixed strategies.83extra possible state, the state {X ,Y}. In this state, the row player is uncertain whichstrategy the column player will play and evaluates their payoffs pessimistically. Ifthe row player forms beliefs that can be represented by the belief function νR inthe game in figure 4.3 then the row players beliefs over states are given by the ανRTgiven by definition 4.1.4.3.2 Mixed strategiesThe interpretation of mixed strategies in a game with ambiguity averse agentsrequires some care. Typically, the previous literature has taken one of two ap-proaches: either use an Anscombe and Aumann (1963) interpretation of uncer-tainty in which case mixed strategies are well defined but their inclusion rendersthe behaviour of ambiguity averse agents as indistinguishable from SEU agents, oruse a Savage (1954) interpretation under which mixed strategies are not well de-fined and therefore are excluded from the feasible set of strategies (mixed equilibriaare then interpreted using either a population interpretation or as an equilibrium inbeliefs). See table A.1 for a summary of the previous literature and their treatmentof mixed strategies.Recently, the decision theory and experimental design literature has drawn adistinction between the case where subjective uncertainty is resolved before objec-tive risk and the case where risk is resolved before uncertainty (Seo (2009), Azrieliet al. (2014), Baillon et al. (2014) and Eichberger et al. (2016)). The interpre-tation of mixed strategies proposed here is closely related to this distinction: wepropose thinking of mixed strategies as a randomization over pure strategies thatoccurs in an agent’s mental state space before the game is played. The implica-tions for experimental implementations of games with ambiguity averse subjectsare two-fold. Firstly, if subjects must choose pure strategies (i.e. they must entera pure strategy choice into a computer which then matches the subject’s responsewith another subject) then we are free to interpret the subject as playing an explic-itly mixed strategy (where the subjects may resolve the randomization internally).Given that, as demonstrated below, the introduction of mixed strategies does notaffect the equilibrium set this is purely a matter of interpretation. However, if pro-viding subject’s with the ability to choose a mixed strategy (perhaps the subject is84given the ability to choose a point in a simplex to represent their strategy) then itis critically important to resolve the mixing device before the subject views theiropponent’s strategy choice.114.4 Lo-Nash EquilibriumThis section introduces Lo-Nash equilibrium, following Lo (2009) closely. Definea set of players N = {1, . . . ,n}, let each player i ∈ N have a finite set of actionsAi, and define A = ×i∈NAi and A−i = × j 6=i∈NA j. We shall endow each agent witha Von Neumann-Morgenstern utility function u : A 7→ R. Suppose that an agenthas uncertainty regarding the strategy choices of their opponents, A−i. Then wecan regard a strategy, ai, as an act over the state space A−i generating a payoffui(ai,a−i) when the state a−i is realized.In a manner consistent with Gilboa and Schmeidler (1989)’s MEU formula-tion, we suppose that an agent’s beliefs regarding their opponents strategies are aclosed and convex set of probability measures Φi ⊆ ∆(A−i). Given Φi an agentspreferences are represented byminφ∈Φ ∑a−i∈A−iui(ai,a−i)φi(a−i).Furthermore, we use σ to denote a probability measure on A. We defineσAi(ai)=∑a−i∈A−i σ(ai,a−i) as the marginal distribution of σ on Ai and σA−i(a−i)=∑ai∈Ai σ(ai,a−i) as the marginal distribution of σ on A−i. Then, in the usual fash-ion we write σ(a−i|ai) = σ(ai,a−i)σAi (ai) .Finally, we write suppσ to denote the support of the probability distribution σ ,and define suppΦi to be the union of the supports of the elements of Φi, and writeΦ to denote the profile (Φi)i∈N . We are now ready to define a Lo-Nash equilibrium.Definition 4.2 (Lo-Nash equilibrium). A pair < σ ,Φ> forms a Lo-Nash equilib-11Saito (2015) provides an alternative framework for understanding preferences for randomizationunder uncertainty. In the Saito framework, a preference (or not) for randomization is an endogenousfeature of an agent’s preferences rather than simply a function only of the timing of realizations. InSaito’s notation an agent with preference parameter δ = 0 is indifferent to randomization while anagent with δ = 1 behaves as if a randomization device completely eliminates the effects of uncer-tainty, and an agent may hold any δ ∈ [0,1].85rium if it satisfiesσ(·|ai) ∈Φi ∀ai ∈ suppσAi ,∀i ∈ N (4.5)suppΦi =× j 6=isuppσA j ∀i ∈ N (4.6)andai ∈ argmaxaˆi∈Aiminφi∈Φi ∑a−i∈A−iui(aˆi,a−i)φi(a−i) ∀ai ∈ suppσAi ,∀i ∈ N (4.7)Equation 4.7 requires that all strategies that are played in an equilibrium arebest responses, with preferences defined as MEU preferences with respect to theequilibrium conjecturesΦ. Equations 4.5 and 4.6 are the consistency requirements:equation 4.6 ensures that a strategy is played with a positive probability iff it isexpected to be played with a positive probability, and equation 4.5 forces actualstrategies to be contained in the belief sets. Note that 4.5 allows for conditioningof σ on ai - this allows for strategies to be correlated, but the realized strategymust lie within player i’s belief set for all ai. Nash equilibrium is a special caseof Lo-Nash equilibrium, thereby ensuring existence of Lo-Nash equilibrium for allfinite normal form games. Section A.1.1 presents a detailed example of Lo-Nashequilibrium.4.4.1 Formulating Lo-Nash equilibrium as a mental equilibriumTo introduce a mental version of Lo-Nash equilibrium we first need to extend thestate space into the event space and define preferences over the new state space.In a Lo-Nash equilibrium, each agent i faces a state space A−i: their opponent’sstrategy space is their state space. We denote the extended state space for agent ito be Σi = {2A−i/ /0}. Then we define an agents’ extended utility function, for allT ∈ Σi, as:u′(ai,T ) = mina−i∈Tu(ai,a−i).Notice that when T ∈ A−i then u′(ai,T ) = u(ai,a−i) so that the extended utilityfunction is consistent with the original utility function. We also define the naturalextension of σ over Σi: σ(ai,T ) = ∑a−i∈T σ(ai,a−i) for all T ∈ Σi. Finally we86introduce the probability measure αi ∈ ∆(Σi) which can be interpreted as agent i’sbelief over the event space Σi. We write α to denote the profile (αi)i∈N .We are now ready to define our Mental Lo-Nash equilibrium. In essence Men-tal Lo-Nash equilibrium is simply an application of Result 4.1 to Lo-Nash equilib-rium, as we establish in Theorem 4.1 below.Definition 4.3 (Mental Lo-Nash equilibrium). A pair < σ ,α > form a MentalLo-Nash equilibrium ifσ(T |ai)≥ ∑τ⊆Tαi(τ) ∀T ∈ Σi, ∀ai ∈ suppσAi ,∀i ∈ N (4.8)∑{T :a−i /∈T}αi(T ) = 1⇔∏j 6=iσA j(a j) = 0 ∀a−i ∈ A−i, ∀i ∈ N (4.9)andai ∈ argmaxaˆi∈Ai ∑T∈Σiu′i(aˆi,T )αi(T ) ∀ai ∈ supp σAi , ∀i ∈ N (4.10)Remark 4.1. If we restrict α such that αi(T ) = 0 for all T /∈ A−i for all i thenEquation 4.8 reduces toσA−i(T ) = αi(T ) ∀T ∈ A−i,∀i ∈ N (4.11)and Equation 4.9 is redundant.Equation 4.10 reduces to the standard Nash equilibrium best response conditionbecause ui(ai,T ) = u′i(ai,T ) for all T ∈ A−i.Evidence. When αi(T ) = 0 for all T /∈ A−i then ∑T∈A−i αi(T ) = 1. Given that∑T∈A−i σ(T |ai) = 1 because σ is a probability measure, Equation 4.8 implies thatσ(T |ai) = αi(T ) for T ∈ A−i for every ai. This ensures that the conditional distri-butions of σ are equal, and that they must also be equal to the marginal distribution– σ is a product measure. Equation 4.9 is then trivially satisfied.Remark 4.1 implies that when beliefs are additive over the strategy space, thenMental Lo-Nash equilibrium reduces to Nash equilibrium. We therefore do notneed to independently establish the existence of Mental Lo-Nash equilibrium: theexistence of Nash equilibrium guarantees the existence of Mental Lo-Nash equi-librium. It is not, however, necessarily the case that there exists a Mental Lo-Nash87equilibrium that is not a Nash equilibrium (the prisoner’s dilemma provides a sim-ple counterexample).Theorem 4.1. If < σ ,α > is a mental Lo-Nash equilibrium then there exists a Φsuch that < σ ,Φ> a Lo-Nash equilibrium. Conversely, if < σ ,Φ> is a Lo-Nashequilibrium and Φi is the core of a belief function for all i then there exists an αsuch that < σ ,α > is a mental Lo-Nash equilibrium.The interpretation of a mental Lo-Nash equilibrium is straightforward whenσ is a product measure (i.e. strategies are uncorrelated). In this case, each agentuses an independent pre-play mixing device yet may place a positive weight onbeliefs that imply correlation between their opponents, or even that their opponentsstrategies may be correlated with their own strategy. Such beliefs may be justifiedin cases where an agent believes that their opponents may be able to communicate,or where they fear that their opponents may be able to see the agent’s strategychoice (or a signal correlated with their strategy choice) before making a decision.It is feasible that some subjects in a standard game theory experiment may nottrust the experimenter and may hold such concerns. It would also be possible foran experimenter to cultivate such concerns (either deliberately or otherwise) viathe instructions given to subjects.12When σ is not a product measure, the interpretation is more complicated. Asa potential example, imagine a Professor of business strategy who suggests thattheir students use the positioning of a second hand on a clock face as a random-ization device – when students of the same Professor play against each other in asimultaneous game, this will be enough to induce correlated randomizations. It is,therefore, the case that there exists mental Lo-Nash (and, also, Lo-Nash) equilibriathat involve correlated outcomes. However, as discussed by Lo (2009), Lo-Nashequilibrium is distinct from Correlated Equilibrium.134.5 Equilibrium Under UncertaintyDow and Werlang (1994) introduce equilibrium under uncertainty, a belief based12Any experiment that intended to investigate such concerns would need to be carefully designedto avoid deceiving the subjects.13Lo actually names his equilibrium concept “Correlated Nash equilibrium” to emphasize theconnection.88equilibrium concept for two-player games that identifies only the set of (pure)strategies that an agent believes their opponent will use. While equilibrium un-der uncertainty is one of the earliest ambiguity averse equilibrium concepts (moremodern concepts allow for more sophisticated preferences and belief structures)14we focus on it here as it allows for an interesting interpretation of equilibrium whenit is translated into the mental equilibrium framework outlined above. Specifically,the equilibrium presented in Dow and Werlang’s lone theorem can be interpreted,through the lens of mental equilibrium, as an equilibrium of a normal form gamewith pre-play communication where pessimistic agents believe that their opponentmay deviate from their pre-agreed strategy. Alternatively, it may be modeled as acorrelated equilibrium where pessimistic agents believe their opponent’s may dis-regard their signal with some positive probability.4.5.1 Equilibrium under uncertaintyWe focus on the special case where the belief functions take a particular form:νi(T ) = (1− ci)pi(T ) for all T ∈ Σ′ such that T 6=Ωi where pi is an additive prob-ability distribution.15 Dow and Werlang (1994) also consider a more general classof capacities, but focus on and prove existence for the special case considered here.Belief functions of this form exhibit constant ambiguity aversion, with the level ofambiguity aversion parameterized by the variable ci, which we shall treat as anexogenous parameter of the game.Definition 4.4 (Adapted from Dow and Werlang (1994)). Consider a two playergame with N = {i, j}. Suppose that the agents have CEU preferences with respectto a belief function, and that the belief functions are such that νi(T )= (1−ci)pi j(T )for all T ∈ Σ′i/Ωi and ν j(T ) = (1− c j)pii(T ) for all T ∈ Σ′j/Ω j and νi(Ωi) =ν j(Ω j) = 1.1614In particular Eichberger and Kelsey (2000) generalizes Dow and Werlang (1994) to allow forn > 2 players.15To be explicit, ν(Ω) = pi(Ω) = 1. This formulation generates a simple set of αν : ανT = (1−ci)pi(T ) for all singleton T ∈ Σ′, ανΩ = ci and ανT = 0 otherwise. These belief functions are thespecial case of neo-additive capacities introduces in Chateauneuf et al. (2007) restricted to ambiguityaversion.16Note the switch in subscripts between the belief function νi (denoting agent i’s beliefs) and pi jwhich, in the sequel, will be interpreted as agent j’s mixed strategy.89Then {νi,ν j} form an equilibrium under uncertainty iff si ∈ argmaxs′i∈Si∫s′idνifor all si ∈ supp(pi j) and s j ∈ argmaxs′j∈S j∫s′jdν j for all s j ∈ supp(pii).Notice that this definition of equilibrium is strictly an equilibrium in beliefs;the equilibrium places no restrictions on strategies, but only restricts the beliefsthat each player may hold about their opponent’s strategies: beliefs must be mu-tually consistent in the sense that every strategy that I believe my opponent mightplay must be optimal for my opponent given their beliefs. An alternative definitionof equilibrium under uncertainty, using sub-additive probabilities instead of capac-ities, is included, along with an example, in Section A.1.2. It is possible, however,as formulated in section 4.5.2, to reformulate and reinterpret an equilibrium underuncertainty as a mixed strategy equilibrium in the mental state space.4.5.2 Mental equilibrium under uncertaintyA mental equilibrium under uncertainty is a pair of strategies and beliefs over themental state space such that the strategies are best responses given the beliefs, andthe beliefs are consistent with the strategies in the sense that no observable eventoccurs less often than the beliefs suggest it should occur.Definition 4.5 (Mental Equilibrium under uncertainty). A pair < σ ,α > form aMental equilibrium under uncertainty ifσA j(T )≥ ∑τ⊆Tαi(τ) ∀T ∈ Σi, ∀i ∈ N (4.12)andσAi ∈ arg maxσˆ∈∆(Ai) ∑T∈Σiu′i(σˆ ,T )αi(T ) ∀i ∈ N (4.13)The following theorem confirms that an equilibrium under uncertainty can berewritten as a mental equilibrium under uncertainty. We restrict attention to thecase of constant ambiguity aversion because of the interesting interpretation ofsuch equilibrium afforded by Definition 4.6 below. Dow and Werlang (1994) alsoargue that constant ambiguity is the most interesting case because it allows the90game to be parameterized by the degree of ambiguity exhibited by the agents. The-orem 4.2 can be generalized to more general belief functions, given an appropriaterewriting of Definition 4.4.Theorem 4.2. Suppose that νi = (1−ci)σA j and ν j = (1−c j)σAi . Define αi suchthatαi(T ) =(1− ci)σA j(T ) if T ∈ A jci if T = A j0 otherwiseIf < νi,ν j > form an equilibrium under uncertainty, then <σ ,α > form a men-tal equilibrium under uncertainty. Furthermore, for any σ ′A j such that σ ′A j(a j)≥αi(a j) for all a j ∈ A j and supp σ ′A j = supp σA j , < σ ′,α > is also a mental equi-librium under uncertainty.If < σ ,α > form a mental equilibrium under uncertainty, then < νi,ν j > forman equilibrium under uncertainty.A mental equilibrium under uncertainty can also be interpreted in the sameway that we can interpret a mental Lo-Nash equilibrium: agents have access to apre-play mixing device that they consult before playing a pure strategy, and theconstruction of the mixing device and the beliefs regarding the opponent’s mixingdevice must satisfy the requirements of the relevant equilibrium. There is also analternative interpretation of a mental equilibrium under uncertainty.Consider the following story that supports a mental equilibrium under uncer-tainty. There are two agents, who are about to play a normal form game with eachother. Before the game commences the agents are allowed to communicate witheach other, and reach an agreement on which (potentially mixed) strategy they willplay.17 After the pre-play communication each agent retires to their own room todecide their strategy in private. Once in their room, the agents may question thesincerity of their opponents commitment; these doubts may be objectively justified(as in Aumann (1990)) or may be purely subjective. The agent concludes that theycan trust their opponent to follow the agreement with probability 1− c and thattheir opponent will deviate from the agreement with probability c.18 Furthermore,17In the case of a mixed strategy, the agents are agreeing on the mixing device that they will consultprior to selecting their pure strategy.18We treat the parameter c as exogenous, but a richer model may seek to endogenize it.91the agent is pessimistic in the sense of Maxmin or Choquet Expected Utility.Some natural questions arise: Under what conditions will the agent be willingto play their agreed strategy? Which pre-play commitments are self-enforcing inthis environment? The answers are straightforward: if the pre-play commitmentsform a mental equilibrium under uncertainty then they will be self-enforcing andthe agents will be willing to play their agreed strategy.Definition 4.6. If < σ ,α > form a mental equilibrium under uncertainty andαi(T ) =(1− ci)σA j(T ) if T ∈ A jci if T = A j0 otherwisethen < σ ,α > form a c-robust agreement.When c= 0, we recover a common justification for Nash equilibrium: if agentsbelieve that their opponent will follow their pre-play commitment with probability1, then the strategies are self enforcing iff the strategies form a Nash equilibrium.We illustrate c-robust agreements by presenting an example.X YA 3,3 0,2B 2,0 1,1Figure 4.5: A normalized stag hunt game.Consider the stag hunt game presented in figure 4.5 and the associated sym-metric c-robust agreements:• If c > 12 there is a unique robust agreement: {B,Y}.• If c ≤ 12 there are three robust agreements: {B,Y}, {A,X} and the mixedrobust agreement σ(X) = σ(A) = 12(1−c) .Notice that in the mixed robust agreement the weight placed on the ‘cooper-ative’ strategy increases as the parameter c increases. Intuitively, in order for anagent to be willing to mix (i.e. be indifferent between their strategies) against arelatively untrustworthy opponent the agreement must be weighted more heavily92towards ‘cooperation’. As the level of trust decreases, we find a point (c= 12 ) wherecooperation is no longer sustainable.4.6 Extensive Form GamesModeling extensive form games with ambiguity averse agents is a difficult taskgiven that ambiguity averse agents do not, in general, have dynamically consistentpreferences. One key implication of this is that Kuhn’s theorem (establishing theequivalence of behavioral and mixed strategies) breaks down in games with ambi-guity averse agents, thereby forcing the modeler to make difficult choices regardingthe structure of allowable strategies (Aryal and Stauber, 2014). The natural struc-ture of a mental equilibrium provides a possible solution: because preferences havea SEU representation in the mental state space Bayesian updating will produce dy-namically consistent preferences in the mental state space. The only remainingchallenge is to determine when dynamically consistent preferences in the mentalstate space map into dynamically consistent preferences in the observable game.We begin by stating the formal requirements for dynamically consistent ob-servable preferences, and then discuss the intuition behind the result.Consider the standard formulation for an extensive form game< I,A,X ,n,H, ι ,u>,where I is the finite set of players, X is the (partially ordered) finite set of nodes(with x0 denoting the initial node), A(x) denotes the actions available at each node,n(x,a) denotes the successor to node x when action a is played, ι(x) denotes theplayer to move at node x, H(x) is an information partition such that if x′ ∈ H(x)then ι(x) = ι(x′), A(x) = A(x′) and H(x′) =H(x), and u is a set of preferences overoutcomes. We write Ai,Hi and ui to denote the actions, histories and (Bernoulli)utility function for player i. Xi denotes the nodes at which player i has the move.We map the extensive form game into a mental extensive form game as follows.As is standard, si : Hi 7→ Ai denotes a strategy for player i. The set of player istrategies is Si, S =×i∈ISi and S−i =× j 6=iS j. Player i’s mental state space is givenby Σi = 2S−i/ /0.19 Beliefs over the mental state space are denoted by αi that areformulated from beliefs over the observable game according to Result 4.1. Result4.2 therefore implies that preferences have a SEU representation over the mental19As for a normal form game, the mental state space is equivalent to the observable event space.93state space. By assumption, beliefs over the mental state space are updated viaBayesian updating.Given X ,n and Hi we write ψi : Xi⋃x0 7→ S−i to denote the set of opponentsstrategies that are consistent with play reaching node x∈ Xi. Note that ψi(x0) = S−ifor all i because the initial node is consistent with any feasible opponent strategy.Assuming perfect recall, it is possible to build an ordered filtration of S−i usingψi. First, note that the successor operator n defines a partial order over all nodes;call this partial ordern. Define the total orderψi to be an arbitrarily chosen totalorder such that xn x′ implies xψi x′ for all x,x′ ∈ {Xi⋃x0}. The total order ψiallows us to order the set {Xi⋃x0} as {x0,x1, . . .xn}. We are now ready to definethe filtration Ψi as follows:Definition 4.7. Define Ψi = {Ψx0 ,Ψx1 , . . . ,Ψxn} to be a set of partitions. Thefirst partition is defined to be Ψx0 = Σi. Subsequent partitions are defined recur-sively. Suppose that the m-th partition has k elements, so that it may be writ-ten as Ψxm = {Ψ1xm , ...,Ψkxm}. Then the m+ 1-th partition is given by Ψxm+1 ={Ψ1xm/ψ(xm+1), ...,Ψkxm/ψ(xm+1),ψ(xm+1)}, after discarding any Ψ jxm/ψ(xm+1)that are empty. Finally, if Ψx = Ψx′ for any x,x′ ∈ {Xi⋃x0} then delete eitherΨx or Ψx′ from the set.20Definition 4.8 (Filtration). Ψi = {Ψx0 ,Ψx1 , . . . ,Ψxn} is a filtration if each Ψxm isa partition of A−i and for each Ψkxm ∈ Ψxm there exists a Ψ jx′m ∈ Ψx′m such thatΨkxm ⊆Ψ jx′m for all m,m′ with m > m′ . 21Lemma 4.3. If the extensive form game satisfies perfect recall thenΨi is a filtrationfor all i.We are now ready to state the main result of this section, which provides con-ditions on the agents’ beliefs such that the agent will exhibit dynamically consis-tent preferences. The proof is borrowed directly from Eichberger et al. (2005).The novelty of the result rests on using Result 4.1 as the vehicle that allows us20Note that, by construction, Ψx =Ψx′ whenever x′ ∈ h(x).21Note that the term filtration is used in slightly different forms in different literature. It is possibleto recover the most common definition (typically used in the mathematics literature) by completingeach Ψx j into a σ -algebra.94to translate the Eichberger et al. (2005) result from an individual decision makingframework into a game theoretic setting.Theorem 4.4. Consider an extensive form game in the mental state space. Foreach player, write the final stage of their filtration as Ψixn = {Ψi1, ...,Ψik}. Ifk∑κ=1∑s⊆Ψiκαi(s) = 1 (4.14)and the agent uses Bayesian updating to update α then agent i has dynamicallyconsistent preferences.Furthermore, ifΨixn is non-trivial (i.e. |Ψiκ | ≥ 2 for all κ) and the agent believesall nodes are reached with positive probability thenk∑κ=1∑s⊆Ψiκαi(s) = 1if and only if agent i has dynamically consistent preferences.Proof. See Theroem 2.1 in Eichberger et al. (2005). Note that the condition∑kκ=1∑s⊆Ψiκ αi(s)=1 is equivalent to their condition ∑E∈E ν(E) = 1.The practical implication of equation 4.14 is that the agent may only be un-certain (i.e. have non-additive beliefs) regarding actions occur after the agents lastdecision node on that branch, or regarding which node they are at within an infor-mation set when there are no future nodes at which the agent has the move wherehe learns which node he was previously at within the information set.22 This isa reasonably restrictive condition, although as the theorem states this condition istight in a fully mixed equilibrium for dynamically consistent agents who are neverthe last agent to act.In a general framework, dynamic consistency will imply ambiguity neutrality(via the tight connection between dynamic consistency and Savage’s P2 postulate).That is, if we wish to impose dynamic consistency across all possible extensiveform games at the same time then the agents must necessarily be expected utility22We conjecture that this condition implies that the agent’s beliefs must satisfy the rectangularitycondition of Epstein and Schneider (2003).95maximizers. Here, however, we take a weaker stance of enforcing dynamic con-sistency for a single, predefined, game tree. Any mental equilibrium of the normalform game derived from the game tree such that equilibrium beliefs satisfy Equa-tion 4.14 will be dynamically consistent. Checking whether equation 4.14 holdsfor a given equilibrium is straightforward: the mental state space is already speci-fied as part of the equilibrium so all that remains is to find the finest partition of thefiltration for each subject, which can usually be read directly off the game tree.It is also straightforward to define a sub-game perfect refinement of mentalLo-Nash equilibrium.Definition 4.9 (Sub-game perfect mental Lo-Nash equilibrium). A pair <σ ,α > isa sub-game perfect mental Lo-Nash equilibrium if it forms a Lo-Nash equilibriumon every sub-game when restrictions of both σ and α to any sub-game are foundvia Bayesian updating.4.7 ComputabilityOne natural concern with the use of ambiguity averse equilibrium concepts is thatthey are difficult to calculate. In this section we demonstrate that the computationalcomplexity of the mental equilibrium concepts presented above are similar to thecomputational complexity of finding a Nash equilibrium. The fact that mentalequilibrium consist of linear utility functions is the key feature that makes thispossible. Importantly, the mental equilibrium approach provides an algorithmicmethodology for solving for the equilibrium of either Dow and Werlang (1994) orLo (2009): first, find the mental state space and associated payoffs and then applythe same techniques that are used for computing Nash equilibrium.The textbook approach, which we present here following Shoham and Leyton-Brown (2009), for computationally searching for Nash equilibrium involves for-mulating the problem as a linear complementarity problem (LCP). The LCP canthen be solved as a feasibility program. The LCP for a two player game with finiteaction sets is given by:96∑ai∈Aiu j(a j,ai)σ(ai)+ r(a j) =U j ∀a j ∈ A j (4.15)∑a j∈A jui(ai,a j)σ(a j)+ r(ai) =Ui ∀ai ∈ Ai (4.16)∑ai∈Aiσ(ai) = 1, ∑a j∈A jσ(ai) = 1 (4.17)σ(ai)≥ 0,σ(a j)≥ 0 ∀ai ∈ Ai,a j ∈ A j (4.18)r(ai)≥ 0,r(a j)≥ 0 ∀ai ∈ Ai,a j ∈ A j (4.19)r(ai)σ(ai) = 0,r(a j)σ(a j) = 0 ∀ai ∈ Ai,a j ∈ A j (4.20)A solution to the LCP consists of a set of σ ,r and U such that the σ form a Nashequilibrium, the U denote the equilibrium utility for each player and the r a set ofslack parameters that are set to 0 for any strategy that is used in the equilibrium.The LCP can be solved using, for example, the Lemke-Howson algorithm.We can also write a mental equilibrium under uncertainty as a LCP. We presenthere the case considered in Theorem 4.2:∑a j∈A ju′i(ai,a j)αi(a j)+u′i(ai,A j)α(A j)+ r(ai) =Ui ∀ai ∈ Ai (4.21)∑ai∈Aiu′j(a j,ai)α j(ai)+u′j(a j,Ai)α(Ai)+ r(a j) =U j ∀a j ∈ A j (4.22)∑T∈A jαi(T ) = ci, ∑T∈Aiα j(T ) = c j (4.23)αi(A j) = 1− ci,α j(Ai) = 1− c j (4.24)αi(Tj)≥ 0,α j(Ti)≥ 0 ∀Tj ∈ A j,∀Ti ∈ Ai (4.25)r(ai)≥ 0,r(a j)≥ 0 ∀ai ∈ Ai,a j ∈ A j (4.26)α(ai)r(ai) = 0,α(a j)r(a j) = 0 ∀ai ∈ Ai,a j ∈ A j (4.27)This LCP solves for the beliefs of the agents (the α), but does not directlyprovide us with the strategies (the σ ). However, the α bound a convex hull offeasible σ and it is possible to write down a feasible σ directly from the output ofthe LCP.97When we view the values (ci,c j) as fixed parameters of the game, then the(ci,c j) parameters and equation 4.24 drop out and the mental equilibrium underuncertainty LCP is exactly as computationally difficult as the Nash equilibriumLCP. If, on the other hand, we treat (ci,c j) as variables then we have 1 additionalparameter, and no additional equations, to calculate for each agent. Given thatincreasing the strategy set for an agent in the Nash equilibrium LCP introducesboth an additional parameter and an additional equation, we conclude that solvinga k× k mental equilibrium under uncertainty is harder than solving a k× k Nashequilibrium but easier than solving a (k+1)× (k+1) Nash equilibrium.Mental Lo-Nash equilibrium is more computationally intensive because of thestricter requirements on the supports of the strategies and the larger mental statespace. The mental Lo-Nash LCP is given by:∑T∈Σiu′i(ai,T )αi(T )+ r(ai) =Ui ∀ai ∈ Ai (4.28)∑T∈Σ ju′j(a j,T )α j(T )+ r(a j) =U j ∀a j ∈ A j (4.29)∑T∈Σiαi(T ) = 1, ∑T∈Σ jα j(T ) = 1 (4.30)∑T :a j∈Tαi(T ) = I(a j) ∀a j ∈ A j (4.31)∑T :ai∈Tα j(T ) = I(ai) ∀ai ∈ Ai (4.32)I(a j)≤ 1 ∀a j ∈ A j (4.33)I(ai)≤ 1 ∀ai ∈ Ai (4.34)αi(Tj)≥ 0,α j(Ti)≥ 0 ∀Tj ∈ Σi,∀Ti ∈ Σ j (4.35)r(ai)≥ 0,r(a j)≥ 0 ∀ai ∈ Ai,a j ∈ A j (4.36)(1− I(ai))r(ai) = 0,(1− I(a j))r(a j) = 0 ∀ai ∈ Ai,a j ∈ A j (4.37)The first two equations contain 2k α parameters compared to k parametersin the Nash case, and we have also introduced the I parameters which impose asub-constraint on the α parameters in addition to the standard requirements that98the α form a probability distribution. While the problem has substantially moreparameters, it is not is any sense fundamentally different from the Nash equilibriumLCP.4.8 ConclusionsThis chapter presents a methodology that extends static ambiguity averse equilib-rium concepts that only allows for pure strategies to allow for mixed strategies,as well as be extended into a dynamically consistent ambiguity averse equilibriumconcept for extensive form games. The technique is restricted to preference struc-tures which allow only ambiguity aversion.23 Furthermore, an application of thetechnique to Dow and Werlang’s (1994) equilibrium under uncertainty produces amodel of partially robust agreements.The theoretical interest in the structure and interpretation of mixed strategyequilibrium for agents with uncertainty averse preferences is readily apparent. Un-like standard SEU agents, ambiguity averse agents are not necessarily indiffer-ent between a mixed strategy and the pure strategy supports of the mixed strat-egy. Considering this, some of the standard interpretations of mixed strategies(via purification arguments or population interpretations of mixed strategies) maynot be appropriate in the context of ambiguity averse agents; indeed, a majorityof ambiguity averse equilibrium concepts explicitly restrict their analyses to purestrategies. In this context, the methodology introduced here can be viewed as anequilibrium-preserving interpretation of mixed strategies: when agents are ambi-guity averse and have access to a mixing device that resolves before the strategicinteraction occurs then the agents will, in equilibrium, be indifferent between usingthe mixing device or not. In this fashion the population interpretation of mixingcan be restored without the need to explicitly assume away the existence of mixingdevices.A commonly held piece of folk wisdom is that people play pure strategies,and do not use mixed strategies. Experimental evidence appears to contradict this,23Preference structures that allow only for ambiguity seeking can also easily be accommodated bydefining the dual of eT (A) in theorem 4.1. Dealing with preferences structures that allow for eitherambiguity aversion or ambiguity seeking is a significantly more difficult proposition, although it maybe tractable under some (possibly unpalatable) restrictions.99however, and indicates that a majority of subjects make use of randomization de-vices when available in zero-sum games (see Schachat (2002) for example). Giventhis evidence that mixed strategies are used when available, extending an equilib-rium concept to allow for mixed strategies while preserving other features of theequilibrium represents an improvement in the descriptive power of the equilibriumconcept.The ubiquitous rise of the internet and digital communications has lead to aproliferation of pre-programmed bots that are increasingly interacting in strategicsituations.24 In some cases bots that play pure strategies are exploitable, therebynecessitating the use of randomization. In cases where the programmer has ex-pected utility preferences over the outcomes produced by their bot there is no con-ceptual problem with evaluating the (ex-ante) expected performance of differentlyspecified programs. However, in cases where the programmer has ambiguity aversepreferences the evaluation of such programs is a conceptually challenging task.The theory presented in this chapter presents one potential way forward. Giventhat current ambiguity averse equilibrium concepts that allow for mixed strategiesare observationally equivalent to Nash equilibrium for 2-player games, a theorythat allows for mixed strategy equilibrium that is distinct from Nash equilibrium isnecessary for ambiguity averse preferences to be a positive descriptor of behaviourin games.24Online pricing algorithms for concerts and sporting events interacting with automated purchas-ing algorithms owned by scalpers are but one example.100Chapter 5ConclusionIn this dissertation, I have contributed to the literature on strategic uncertainty ingames with non-Subjective Expected Utility agents.Chapter 2 provides the most thorough and convincing evidence to date thatambiguity preferences influence behaviour in normal form games. It also demon-strates that the usual assumption of common knowledge of preferences does nothold among a typical experimental population; even the much weaker mutual knowl-edge of preferences fails to obtain. However, in an experimental intervention thatrestored mutual knowledge of preferences, solution concepts that allow for ambi-guity aversion perform remarkably well.Chapter 2 introduces three experimental innovations. First, it introduces amethodology for measuring risk and ambiguity preferences, and beliefs over op-ponent’s preferences, using normal form games. Second, it identifies and separatesthe effects of risk aversion and ambiguity aversion in a normal form game. Third,it separates the effect of preferences on behaviour from the effects of beliefs overopponent’s preferences in a normal form game.Chapter 3 investigates the role of naturally occurring reaction lags in determin-ing how subjects perceive and respond to strategic uncertainty. One key aspectof the results is that when reaction lags are either very small or very large thenbehaviour approximates predictions that are formed from models without strate-gic uncertainty. This implies that in some settings with very large reaction lagsdiscrete time Nash equilibrium is a good approximation of behaviour and in other101settings with very small reaction lags continuous time Nash equilibrium is a goodapproximation of behaviour. We speculate that as technology improves and com-munication and reaction lags shrink, continuous time models of behaviour willbecome increasingly important in applications.The original contributions of Chapter 3 have two main dimensions. First, weintroduce a new experimental technique that allows us to implement the premises ofthe theory of continuous time games directly in the laboratory. Second, we identifythat theories of strategic uncertainty, particularly minimax regret theory, explainthe data collected in our experiment remarkably well. We further demonstrate thatminimax regret theory can also explain data from similar previous experiments,and that a theoretical application of minimax regret theory generates plausible pre-dictions across a broad class of dilemma-like games in inertial continuous time.Chapter 4 provides a theoretical motivation for interpreting experiments withambiguity averse agents using solution concepts that restrict the action space topure strategies. The Chapter establishes that, for two key equilibrium concepts,allowing pre-play mixing is equivalent to restricting to pure strategies. The theo-retical results of Chapter 4 are essential for interpreting the experimental results ofChapter 2. 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Experimenting with equilibrium selectionin dynamic games. Mimeo, 2016. → pages 45, 61Abraham Wald. Statistical Decision Functions. Wiley, 1950. → pages 64, 177111Appendix AA Brief Introduction to theTheory of Games WithAmbiguity Averse AgentsThis appendix provides a brief background on the theory of games with ambiguityaverse agents; it is particularly relevant for readers of both Chapters 2 and 4. Sec-tion A.1 surveys the existing solution concepts for ambiguity averse agents. Sec-tion A.1.1 re-introduces Lo-Nash equilibrium (originally previewed in Chapter 4)and uses the game studied in Chapter 2 as an illustrative example. Section A.1.2presents a slightly different formulation of equilibrium under uncertainty than thatpresented in Chapter 4. A comparison of the differences in the equilibrium sets be-tween Lo-Nash equilibrium and equilibrium under uncertainty highlights the rolethat the different consistency conditions play across the two equilibria; in partic-ular, (C,Y ) is an equilibrium of the testing game from Chapter 2 for equilibriumunder uncertainty but not for Lo-Nash equilibrium.A.1 Theoretical LiteratureThere is a growing literature that models the extension of game theory to subjectswith non-neutral ambiguity preference. There is, however, surprisingly little agree-ment on how to proceed. Should an ambiguity averse equilibrium concept maintain112mutual knowledge of rationality? Should stochastic independence be preserved, orshould we allow for correlation between agent’s strategies? How much (if at all)should consistency of beliefs be relaxed? Each of these questions could be used tocategorize the literature in various ways. We focus, instead, on the role of mixedstrategies.To reiterate, there are three ways for the game theorist to deal with mixed strate-gies. Either allow agents to hold strict preferences for mixed strategies, restrict thestrategy space to contain only pure strategies (and thereby avoid the issue alto-gether), or to place restrictions on the model such that the agent does not have astrict preference for mixed strategies. Models that fit the third category requirecareful attention to be paid regarding the timing of realizations of randomizationdevices. Currently, the only two papers that have used the third approach have notbeen sufficiently careful (Rothe (2010) and Lehrer (2011)). Table A.1 provides asummary of the state of the literature.As can be seen in table A.1 a majority of the literature simply excludes mixedstrategies. The experimental design in Chapter 2 could be reinterpreted using anyof the equilibrium concepts that exclude mixed strategies, and this appendix in-cludes the equilibrium derivations for Dow and Werlang (1994) and Lo (2009). Itis not, however, compatible with models that allow a strict preference for random-ization, such as Lo (1996) or Bade (2011).1 For example, propositions 4 and 5 inLo (1996) imply that for two-player games with a unique strict pure strategy Nashequilibrium, such as the game in figure 2.6 here, the only ambiguity averse equi-librium is the Nash equilibrium – a conclusion that is rejected in our data. Bade(2011) gives agents access to an ambiguous randomization device (i.e. agents cancondition their strategy on a draw from an Ellsberg urn, for example), but finds ob-servational equivalence between Nash equilibria and ambiguity averse equilibria intwo-player games.Returning to the questions raised in the opening paragraph of this section, inthe opinion of this author, the most convincing answers have been provided by Lo1The experimental results presented in Chapter 2 could be interpreted as providing some evidenceagainst models such as Lo (1996) or Klibanoff (1996). An alternative viewpoint would be to arguethat a proper test of these papers would require an explicit mixing device be provided to subjects.Similarly, a proper test of Bade (2011) would require subjects to have access to an ambiguous mixingdevice.113Paper Equilibrium concept Treatment of mixed strategiesDow and Werlang (1994) Nash equilibrium under uncertainty Pure strategies onlyLo (1996) Belief equilibrium Strict preference for mixed strate-gies allowedKlibanoff (1996) Equilibrium with uncertainty aver-sionStrict preference for mixed strate-gies allowedEpstein (1997) Rationalizability Pure strategies onlyGroes et al. (1998) Nash equilibrium with lower prob-abilitiesPure strategies onlyMarinacci (2000) Belief equilibrium Pure strategies onlyEichberger and Kelsey (2000) Equilibrium under uncertainty Pure strategies onlyLo (2009) Correlated Nash equilibrium Pure strategies onlyRothe (2010) Choquet Nash equilibrium Mixed strategies have payoffs thatare assumed to be linear in utilityBade (2011) Ambiguous act equilibrium Strict preference for mixed strate-gies allowed. Allows for ambigu-ous strategies (using an ambiguousmixing device)Azrieli and Teper (2011) Ex-ante J equilibrium (games withincomplete information)Strict preference for mixed strate-gies allowed.Lehrer (2011) Partially specified equilibrium Mixed strategies have payoffs thatare assumed to be linear in utilityGrant et al. (Forthcoming) Savage games Pure strategies onlyEichberger and Kelsey (2014) Equilibrium under ambiguity Pure strategies onlyRiedel and Sass (2014) Ellsberg equilibrium Strict preference for mixed strate-gies allowed. Allows for ambigu-ous strategies (using an ambiguousmixing device)Table A.1: This table contains summary of the treatment of mixed strategies in the literatureof ambiguity averse equilibrium concepts. Only papers that define and introduce equilib-rium concepts are included, applications are not included. Note that the approach takenby Lehrer (2011) and Rothe (2010), of assuming that the payoffs of mixed strategies arelinear in utility, raises some difficulties that are not addressed in either paper.114(2009). Lo posits an epistemic approach to ambiguity averse game theory; begin-ning with the epistemic conditions of Nash equilibrium, what are the consequencesof relaxing rationality to allow for ambiguity averse preferences? The resultantequilibrium concept substantially relaxes stochastic independence2, allows agentsto hold set valued beliefs regarding their opponent’s strategies and requires actualplay to lie within the set of beliefs, and maintains mutual knowledge of rationality.The only desiderata missing from Lo (2009) is that it does not allow for mixedstrategies; the strategy space is restricted to pure strategies, and a population inter-pretation of mixing is invoked. Chapter 4 extends Lo (2009) to allow for mixedstrategies.A.1.1 Lo-Nash equilibriaThis section introduces Lo-Nash equilibrium, following Lo (2009) closely. Definea set of players N = {1, . . . ,n}, let each player i ∈ N have a finite set of actions Ai,and define A = ×i∈NAi and A−i = × j 6=i∈NA j. We shall endow each agent with aBernoulli utility function u : A 7→R. Suppose that an agent has uncertainty regard-ing the strategy choices of their opponents, A−i. Then we can regard a strategy, ai,as an act over the state space A−i generating a payoff ui(ai,a−i) when the state a−iis realized.In a manner consistent with Gilboa and Schmeidler (1989)’s MEU formula-tion, we suppose that an agent’s beliefs regarding their opponents strategies are aclosed and convex set of probability measures Φi ⊆ ∆(A−i). Given Φi an agentspreferences are represented byminφ∈Φ ∑a−i∈A−iui(ai,a−i)φi(a−i).Furthermore, we use σ to denote a probability measure on A. We defineσAi(ai)=∑a−i∈A−i σ(ai,a−i) as the marginal distribution of σ on Ai and σA−i(a−i)=∑ai∈Ai σ(ai,a−i) as the marginal distribution of σ on A−i. Then, in the usual fash-ion we write σ(a−i|ai) = σ(ai,a−i)σAi (ai) .2In fact, Lo refers to his equilibrium as Correlated Nash equilibrium to emphasize the similar-ities with Aumann’s Correlated equilibrium. We avoid using this name, and instead refer to theequilibrium as Lo-Nash equilibrium, to avoid confusion between the two concepts.115Finally, we write suppσ to denote the support of the probability distribution σ ,and define suppΦ to be the union of the supports of the elements of Φ. We are nowready to define a Lo-Nash equilibrium.Definition A.1 (Lo-Nash equilibrium). A pair < σ ,Φ> forms a Lo-Nash equilib-rium if it satisfiesσ(·|ai) ∈Φi ∀ai ∈ suppσAi ,∀i ∈ N (A.1)suppΦi =× j 6=isuppσA j ∀i ∈ N (A.2)andai ∈ argmaxaˆi∈Aiminφi∈Φi ∑a−i∈A−iui(aˆi,a−i)φi(a−i) ∀ai ∈ suppσAi ,∀i ∈ N (A.3)Equation A.3 requires that all strategies that are played in an equilibrium arebest responses, with preferences defined as MEU preferences with respect to theequilibrium conjectures Φ. Equations A.1 and A.2 are the consistency require-ments: equation A.2 ensures that a strategy is played with a positive probability iffit is expected to be played with a positive probability, and equation A.1 forces ac-tual strategies to be contained in the belief sets. Note that equation A.1 allows forconditioning of σ on ai - this allows for strategies to be correlated, but the realizedstrategy must lie within player i’s belief set for all ai.Lo-Nash equilibrium: an exampleConsider the normal form game presented in figure A.1. For this section, we shallfollow the standard game theoretic approach and assume that the payoffs in figureA.1 are utility values, thereby abstracting from issues of risk aversion. Recall thatthe game has a unique Nash equilibrium (A,X).X YA 25,20 14,12B 14,20 25,12C 18,12 18,22Figure A.1: Testing game116In contrast, there are fully mixed Lo-Nash equilibria in this game. Intuitively,this is easy to see. If the row player is uncertain about the column player’s strategychoice, and the row player is ambiguity averse, then there are set-valued beliefs forwhich C is a best response. There does not, however, exist a Lo-Nash equilibriumwhere the row player plays C with probability 1. The consistency requirementsfor Lo-Nash equilibria would then imply that the column player would know withcertainty that the row player will play C, so the unique best response for the columnplayer is Y . In this case, C is no longer the best response for the row player andthe equilibrium breaks down. It is the case, however, that C can be played with aprobability that is arbitrarily close to one. The full set of Lo-Nash equilibria forthis game is large.Lemma A.1. The set of distributions, Σ, that can be supported as Lo-Nash equilib-ria for the game in figure A.1 is Σ= {σ(A,X) = 1}⋃{σ : 0< σ(A), 0< σ(B), 0<σ(C), 411 ≤ σ(X |·)≤ 711 }⋃{σ : σ(A) = 0, 0 < σ(C)< 1, σ(X |·)≤ 711 , σ(Y )<1 }⋃{σ : σ(B) = 0, 0 < σ(C)< 1, 411 ≤ σ(X |·), σ(Y )> 0 }.Proof of lemma A.1. We proceed case-by-case, considering each possible set ofpure strategy supports for the row player in turn. It follows directly from the def-inition of Lo-Nash equilibrium that we can find the full set of equilibrium distri-butions for each case by first identifying the largest feasible equilibrium sets ofbeliefs.It is straightforward to see that the only pure strategy Lo-Nash equilibrium is{A,X}.Next we consider equilibria that are fully mixed for the row player. The rowplayer will be indifferent between all three of their strategies only when ΦR = {φ :411 ≤ φ(X)≤ 711}. This implies that the column player can play any mixed strategywithin this range. The largest set of beliefs such that the column player is indif-ferent between both X and Y is beliefs of complete uncertainty. The consistencyrequirements on the row player are therefore very weak, such that they need to useeach strategy with a strictly positive probability. Note that the equilibrium allowsfor correlation between strategies.Next, we consider equilibria where the row player does not play A. In suchcases, the largest feasible set of row player beliefs are given by ΦR = {φ : φ(X)≤117711}. Consistency requirements restrict the column player’s beliefs to be ΦC = {φ :φ(A) = 0,φ(B)> 0,φ(C)> 0}, which is also the largest feasible set of beliefs thatensures column player indifference. Therefore the row player can play any mixthat places positive weight on both B and C. The correlation requirements allowfor equilibria that are arbitrarily close to pure {C,Y}. Consider the equilibriumwith σ(B,X) = 7ε , σ(B,Y ) = 4ε and σ(C,Y ) = 1− 11ε with all other outcomesbeing assigned a probability of 0, and take ε to be arbitrarily small. The equilibriawhere the row player does not play B are built in a similar fashion.There are no equilibria where the row player mixes between only A and B. Insuch a case, the column player must believe that φ(C) = 0 and therefore they havea unique best response of X .This example illustrates a key feature of Lo-Nash equilibrium. The source ofambiguity is not from doubts over which Nash equilibrium is to be played, nor isit over which strategy will be realized in a mixed strategy Nash equilibrium. Thesource of ambiguity in this game is endogenous to the game structure and agent’spreferences. The only requirement is that the row player has ambiguity aversepreferences, so that C is a best response for some potential row player beliefs andthereby breaking the rationalizability chain required to produce the unique (A,X)equilibrium.As is always the case, the unique Nash equilibrium is also a Lo-Nash equi-librium. Ambiguity averse preferences, on their own, need not necessarily affectthe play of a game if the agents still form point-valued conjectures. If an agentbelieves, with certainty, that their opponent will play a particular (possibly mixed)strategy then there is no role for their ambiguity to play as, subjectively, the agentfaces no ambiguity.The other key feature of the game in figure A.1 is that equilibria other than(A,X) exist iff the row player is ambiguity averse (lemmas A.2 and A.3). If the rowplayer has SEU preferences, then the column player’s ambiguity preference has noeffect on the equilibrium set. Using this result, we can differentiate the effect of aplayers own ambiguity preference (which should affect row player behaviour only)from the effect of a players beliefs about their opponent’s ambiguity preference(which should affect column player behaviour only).118Definition A.2. A Lo-Nash equilibrium is proper for agent i iffΦi is not a singleton.A Lo-Nash equilibrium is proper iff it is proper for at least one agent.Lemma A.2. All proper Lo-Nash equilibrium of the game in figure A.1 are properfor the row player.Proof of lemma A.2. From lemma A.1, all proper Lo-Nash equilibria involve therow player mixing between C and at least one of A or B. If the row player hassingleton beliefs, then their set of best responses can never include C. Therefore,any proper Lo-Nash equilibrium must be proper for the row player.Lemma A.3. There exist proper Lo-Nash equilibrium of the game in figure A.1that are not proper for the column player.Proof. Consider the equilibrium where σ(A,X)=σ(A,Y )= 16 , σ(B,X)=σ(B,Y )=19 , σ(C,X) = σ(C,Y ) =29 , ΦR = {φ : 411 ≤ φ(X) ≤ 711} and ΦC = {φ : φ(C) =49 ,φ(A) =13 ,φ(B) =29}.A.1.2 Nash Equilibrium under uncertaintyIn this section we describe Dow and Werlang’s (1994) Nash Equilibrium underuncercainty. The analysis will follow Dow and Werlang (1994) very closely. Webegin by introducing the notion of a sub-additive probability, P, being a probabilitywhich satisfiesP(A)+P(B)≤ P(A∩B)+P(A∪B).The expected utility of an agent with respect to a sub-additive probability overa non-negative random variable, X , is given by the Choquet integralE(X) =∫R+P(X ≥ x)dx.In many cases this will be equivalent to calculating the Maxmin Expected Utilitywith respect to the core of the sub-additive probability (see Gilboa and Schmeidler(1994) for details).Dow and Werlang define the support of a sub-additive probability to be anevent, A, such that P(Ac) = 0 and P(Bc)> 0 for all B⊂ A. We note that the support119need not be unique, and that there are other reasonable definitions of the support ofa sub-additive probability that are not used here.A Nash equilibrium under uncertainty is then simply the requirement that allstrategies that are in the support of an opponent’s beliefs are best responses for anagent given the agent’s beliefs. Label the set of available strategies for player i asAi.Definition A.3 (From Dow and Werlang (1994)). A pair (P1,P2) of non-additiveprobabilities P1 over A1 and P2 over A2 is a Nash Equilibrium under Uncertaintyif there exist a support of P1 and a support of P2 such that(i) for all a1 in the support of P1, a1 maximizes the expected utility of player1, given that P2 represents player 1’s beliefs about the strategies of player 2; andconversely,(ii) for all a2 in the support of P2, a2 maximizes the expected utility of player 2,given that P1 represents player 2’s beliefs about the strategies of player 1.We refrain from giving the full set of equilibrium sub-additive probabilities, asmany of them produce observationally equivalent outcomes, but instead establishtwo facts: that a fully mixed equilibrium can be supported, and that pure {C,Y}can be supported, in a Nash Equilibrium under Uncertainty.Lemma A.4. The pure strategy equilibrium {C,Y} can be supported by the equi-librium sub-additive probability PC with PC(Y ) = 311 and PC(X) = 0 and additiveprobability PR with PR(C) = 1.Proof of lemma A.4. The support of PC is {Y} and the support of PR is {C}. It isobvious that Y is the unique best response for the column player. The row playerevaluates their strategies as follows:U(A) = 1∗14+0∗ (25−14) = 14U(B) = 1∗14+ 311∗ (25−14) = 17U(C) = 18C is the unique best response.120Lemma A.5. A fully mixed equilibrium can be supported by the equilibrium sub-additive probability PC with PC(X) = PC(Y ) = 411 and additive probability PR withPR(C) = 49 , PR(A) =39 , PR(B) =29 .Proof of lemma A.5. The support of PR is {A,B,C} and the support of PC is {X ,Y}.Given that PR is additive we can calculate the column player’s utility in the standardfashion, and verify that U(X) =U(Y ) = 1489 ∼ 16.4. The row player evaluates theirstrategies as follows:U(A) = 1∗14+ 411∗ (25−14) = 18U(B) = 1∗14+ 411∗ (25−14) = 18U(C) = 18Therefore, both players are indifferent between all strategies in the supports of thedistributions.121Appendix BAppendix for “UncertaintyAversion in Game Theory:Experimental Evidence”B.1 Experimental Methodology AppendixIn every experimental design there are a myriad of decisions that the experimentermust make. This section discusses some of the more important choices that weremade in the design of this experiment. The focus is on issues that are likely to beless familiar to readers, rather than standard experimental practices such as ensur-ing payments are anonymous.B.1.1 Piloting and framing effectsThis experiment has two distinct sections. First, we measure the subjects prefer-ences and beliefs over their opponent’s preferences. Second, we give the subjectsa variety of normal form games. Normal form games are normally presented tosubjects in a bi-matrix format. In contrast, decision theorists use a range of pro-cedures to measure preferences, none of which are bi-matrices. A natural concernis that the framing of the task might affect the measurement (and it turns out thatthis concern is well founded). In a series of pilot experiments, we observed that the122proportion of subjects that display ambiguity aversion is significantly higher whenthe task is presented as a worded decision problem (when compared to a bi-matrixgame).1The existence of a framing effect precludes presenting the preference measure-ment tasks and normal form games in differing formats. To resolve this problem,we presented all tasks as games. As we are interested in measuring both a sub-ject’s own preferences and their beliefs regarding their opponent’s preferences, weformulate the decision problems as a game where the row player reveals their ownpreferences and the column player attempts to predict the row player’s behaviour.A natural concern is that, by moving the preference measurement tasks intoa game format, our preference measures will be distorted relative to previous re-search on ambiguity aversion. The pilot experiments suggested that adding a seriesof comprehension questions, as described in the next section, does a reasonable jobof reducing the gap between worded decision problems and bi-matrix games.2B.1.2 Assumptions underlying the classification procedureThere are several assumptions underlying the classification procedure used in sec-tion Section 2.2.1, which are discussed in this section. Briefly, the assumptionsare: source independence of ambiguity preferences; CRRA utility; and, for col-umn players, a strength of beliefs assumption. The results are robust to varying theCRRA utility assumption. The strength of beliefs assumption may be violated inthe data, and the additional treatments in Section 2.4 investigate this possibility.Assumptions on ambiguity preferencesUnderlying the classification mapping is an assumption that a subject will (sub-jectively) view an Ellsberg urn as a source of ambiguity if and only if they view1Similar evidence of framing effects for ambiguity aversion measurement can be found in Chewet al. (2013).2Despite the inclusion of the comprehension questions, the measured level of ambiguity aversionis still slightly lower than that measured in previous 2-colour Ellsberg urn experiments. The resultsreported here do appear to be consistent with recent evidence from Chew et al. (2013), who arguethat we should expect to measure less ambiguity aversion in more complex environments.123their opponent’s strategy as a source of ambiguity.3 Using an Ellsberg urn to pro-duce the classification provides a stable point of comparison to the vast literatureon ambiguity aversion in individual decision problems.Furthermore, it is possible that some subjects who chose S in in the game infigure 2.2 would switch to M if the payoff for S was lowered to, say, $30.01. Such asubject has ambiguity averse preferences but would be classified as ambiguity neu-tral given the payoff structure of my game. However, the results in previous work,such as Epstein and Halevy (2014), suggest that this is not a significant source ofmisclassification even when the payoff for S is as high as $31.00. Therefore, forthese two reasons, we conclude that our classification provides a lower bound onthe proportion of subjects who may exhibit ambiguity aversion with respect to theiropponent’s preferences.Assumptions on risk preferencesThe mapping from choices in the risk measurement game to types will depend onthe parameterization of the utility function. We consider two common utility func-tions: constant relative risk aversion (CRRA, equation B.1) and constant absoluterisk aversion (CARA, equation B.2).4 Figures B.1 and B.2 present the implications3Heath and Tversky (1991) were the first to provide evidence of differing source preferences forambiguity, comparing bets on political outcomes with bets on football matches. They found evidencethat subjects prefer betting on events for which they have a greater familiarity. While it is unlikelythat the subjects in this experiment were substantially more familiar with either the playing of normalform games or betting on Ellsberg urns, it is still possible that source dependent preferences couldbe present in the current environment. In particular, some subjects may find the Ellsberg urn to beless ambiguous than their opponent’s behaviour because an Ellsberg urn is symmetric and determinedsolely by ‘nature’, while an opponent has their own agency – an argument first put forward by Kelseyand le Roux (2015a). To the extent that we mis-classify subjects, we should expect there to be somesubjects who are measured as ambiguity neutral with respect to the urn yet are ambiguity averse withrespect to their opponent’s action.4The CRRA utility function is given by the equationu(c) =c1−β −11−β (B.1)and the CARA utility function byu(c) = 1− e−ρc. (B.2)Note the implicit assumption that subjects treat the experiment as an independent event, and do notintegrate their outside wealth into their utility functions. This is the standard approach in experi-mental economics, although recently some authors have begun to consider partial wealth integration124of the CARA and CRRA parameterizations, respectively.Figure B.1: The role of risk aversion in the risk preference measuring game and the testinggame assuming SEU preferences with CARA utility: u(c) = 1− e−ρc. Subjects whoselect L in the game in figure 2.5 have type 1 or 2 preferences, while subjects whoselect H have type 3 or 4 preferences. Subjects who select I are of an indeterminatetype.As can be seen from figure B.1, the CARA parameterization provides a clearclassification for subjects that choose either L or H, but is indeterminate for sub-jects who choose I. Meanwhile, figure B.2 indicates that, under the CRRA param-eterization, subjects that choose L and I should be pooled into a low risk categorywhile subjects who choose H should be considered high risk (although there is asmall region of misclassification for subjects that choose H). Given that both mod-els essentially agree on the treatment of subjects that choose L and H, we shallimplement the CARA parameterization and remove subjects that selected I fromthe data analysis. Robustness checks indicate that moving to a CRRA parame-terization has no effect on the statistical inferences implied by the results, whichis unsurprising given that only 10% of subjects choose I in the risk measurementgame.Additional assumptions on beliefsThe mapping for the column players is calculated analogously to the mapping forrow players, although it requires an additional assumption regarding the strength ofbeliefs. The process for eliciting beliefs regarding opponent’s preferences elicitsthe subject’s ‘best guess’ regarding his opponent. In the worst case scenario asubject may be completely unsure about their opponent’s preferences, implying(Holt and Laury (2014)). A theoretically more robust formulation would use u(c+w) where w is thesubjects initial wealth level, but we follow the experimental standard here. As suggested by SimonGrant, readers who are troubled by this formulation should focus on the CARA utility parameteriza-tion (and not the CRRA parameterization).125Figure B.2: The role of risk aversion in the risk preference measuring game and the testinggame assuming SEU preferences with CRRA utility: u(c) = c1−β−11−β . Subjects whoselect L or I in the game in figure 2.5 have type 1 or 2 preferences, while subjects whoselect H are likely to have type 3 or 4 preferences.that their choices as the column player in the classification games may contain noinformation. The hypothesis of a relationship between column player behaviourin the classification games and the testing game can, therefore, be seen as a jointtest of the theory outlined in section 2.1 and the assumption that beliefs are strongenough to be informative. Section 2.4 introduces an additional treatment in whichsubjects are shown a signal of their opponent’s preferences, allowing us to separatethis joint hypothesis into its component parts.B.1.3 Comprehension questionsOn the experimental screen, underneath each of the normal form games, a series ofdynamic drop down menus were included for each game. Before a subject couldconfirm their strategy choice in a game, they were required to fill in the drop downmenus correctly. To ensure that subjects took the drop down menus seriously theywere paid a bonus of $1 for each game where they filled in the drop down menuscorrectly on their first attempt. Each incorrect attempt reduced the bonus payment,for that game, by $0.25.The drop down menus were designed in such a way that the subjects recreatedthe worded decision problem that describes the relevant game. For example, for thegame in figure 2.2, the worded problem for option S would read “Your earnings forthis choice [are/are not] affected by your counterpart’s strategy. Your earnings forthis choice will be [$30.10/$30/$15/$0] if a red ball is drawn from the [U urn/Kurn] and nothing otherwise.” For each set of terms that are square bracketed, thesubjects were required to select the correct term (shown in bold here) from a drop126down menu.Subjects were required to fill in drop down menus that described the possibleoutcomes for each of their strategies.5 The process of interpreting the bi-matrixgame and converting it into a worded decision problem helped to ameliorate theframing effects induced by the bi-matrix game.In addition, the drop down menus also served as a comprehension check. Thistest of subject comprehension was quite stringent. There were a handful of subjectswho had extreme difficulty with the drop down menus, and only 25% of subjectsearned the maximum possible bonus payment of $7. We used the level of bonuspayment earned by the subject as a measure of comprehension, and removed sub-jects who performed poorly from our sample.Given the extensive piloting process and experimenter degrees of freedom in-volved in using the comprehension data to exclude subjects, the possibility of datamining for statistically significant results needs to be addressed. Two approacheswere taken to alleviate these concerns. First, pilot sessions were run using only theclassification games of section 2.2.1, and not the testing game of section 2.2.2.6Second, the rules for excluding subjects from the data, based on comprehensionscores, were determined ex-ante rather than ex-post.Two rules for excluding subjects were determined ex-ante. A strict rule, and arelaxed rule. There was an expectation that the strict rule may be too strict, so therelaxed rule was designed as a back up. The expectations were correct, and afterapplying the strict rule only 8 ambiguity averse and risk neutral subjects remained(and only 4 risk averse and ambiguity neutral subjects); therefore, the relaxed rulewas used. The comprehension data is reported in further detail in appendix B.3.3.5In one set of pilot sessions subjects were required to fill in the drop down menu only for thestrategy that they selected. This approach was soon abandoned, as it turned out that subjects weremaking their decisions without referring to the drop down menus, and then only filling in the dropdown menus after they had already come to a decision. The drop down menus thereby failed to breakdown the framing effect of presenting the choices in a bi-matrix format. In the final experimentaldesign subjects were required to fill in the drop down menus before they could select a strategy.6The only exceptions to this were the first two pilot sessions which used a 3-colour Ellsberg urninstead of the 2-colour Ellsberg urn used in the final design. Only 1 out of 14 subjects was found tobe ambiguity averse, meaning that it was not possible to perform any tests on the effects of ambiguityaversion on behaviour, and the 3-colour Ellsberg urn was removed from the design.127B.1.4 Order of realization of randomizationsAs discussed in section 2.1, many models of ambiguity aversion induce preferenceswith a strict preference for randomization. In an experiment, such as this one,where we are asking subjects to respond to multiple games there are two ways inwhich a preference for randomization could undermine the incentive structure. Inboth cases, however, the results of Eichberger et al. (2016) (and similar ideas foundin earlier papers, dating back to at least Kreps (1988)) can be used to overcome anyconcerns: if objective randomizations occur before subjective states are realizedthen dynamic consistency implies no preference for randomization.The first difficulty, as discussed in Chapter 4, occurs within a game. If subjectshave a strict preference for randomization then the availability of mixed strategiescan change the equilibrium set (relative to the game where only pure strategiesare available). The strict preference for randomization can be negated via twoarguments. First, subjects were required to select pure strategies in each of thegames (i.e. the computer interface did not allow for mixed strategies to be chosenexplicitly). In order to play a mixed strategy subjects would need to provide theirown randomization device, whether it be a mental mixing, tossing of a coin orrolling of a die.7 Second, following the arguments of Eichberger et al. (2016)there should be no preference for randomization if a subjects own randomizationis resolved before they view their opponents strategy choice. This requirementwas satisfied during the experiment as a consequence of requiring subjects to enterpure strategy choices into the computer, and not revealing their opponent’s strategychoices until the end of the experiment.The second difficulty occurs between games. If subjects have a strict prefer-ence for mixing between games, then using a random payment mechanism8 willprovide the subject with an opportunity to hedge their ambiguity across games andthey will not treat each game as an independent decision problem. On the otherhand, paying subjects for their decisions in all games will obviously give rise tohedging opportunities across games, particularly given that several of the games7No subjects were observed to flip coins or roll dice during the experiment.8A random payment mechanism chooses, at random, a subset of the tasks in the experiment to bethe tasks for which subjects are actually paid. Usually, the number of tasks that are selected is one,and this is sometimes referred to as a “pay-one” incentive scheme.128include a draw from an Ellsberg urn that determines the ‘play’ of nature. Baillonet al. (2014) provides some solace, and a joint reading of Azrieli et al. (2014) andEichberger et al. (2016) reinforces this view. Baillon et al. (2014) demonstrate thatif objective randomizations are resolved after subjective randomizations then theelicitation is not incentive compatible; if objective randomizations are realized firstthen incentive compatibility may be restored. Azrieli et al. (2014) suggest that,absent an assumption that subjects are indifferent between ex-ante and ex-post ran-domizations, there are no strong objections to be made against using a random pay-ment mechanism in an experiment of this nature. Following this, it is clear fromEichberger et al. (2016) that the random payment mechanism should be resolvedprior to the resolution of play in the games. Furthermore, Johnson et al. (2014)demonstrate experimentally that an elicitation process which they call PRINCE,including this order of realizations, can improve the quality of elicited preferences.Resolving the random payment mechanism prior to the resolution of play in thegames requires some care. It is obviously not appropriate to allow the subjects toknow which game will be chosen for payment before the subjects play the games.The solution implemented here was as follows: when subjects entered the experi-mental lab there were 7 flash cards stuck to one of the walls. On the front (visibleto subjects) side of the flash cards were the letters A through G, one letter per flashcard. On the back (not visible to subjects) were the numbers 1 through 7, one num-ber per flash card. The matching of letters to numbers was randomly determined bythe experimenter.9 During the instructions, a subject was asked to choose a letterfrom A to G, and the number was recorded. Subjects were informed that, at theend of the experiment, the number on the back of the chosen letter would determinewhich game would be paid. From the subjects perspective the choice of game wasthen fixed, but unknown. At the end of the experiment, the letter was flipped overto reveal the game to be paid before any balls were drawn from the urns.All randomizations and ball draws were conducted using physical devices thatwere as procedurally transparent as possible.9There were actually two sets of flash cards, one with letters and one with numbers. Letterflashcards were then randomly blu-tacked to number flashcards. This enabled the flash cards to beeasily re-randomized between sessions.129B.1.5 Related experimental literatureThere is very little experimental evidence regarding ambiguity aversion in gametheory. There are, as far as I am aware, only a handful of papers that have collectedexperimental data that was designed to investigate ambiguity aversion in game the-ory (Kelsey and le Roux (2015a), Kelsey and le Roux (2015b), Ivanov (2011),Eichberger et al. (2008), Pulford and Colman (2007), and Camerer and Karjalainen(1994)).10 Additionally, Eichberger and Kelsey (2011) re-analyses previous exper-imental data and Heinemannn et al. (2009) study strategic uncertainty through thelens of global games. We proceed chronologically.Camerer and Karjalainen (1994) present evidence from 4 games. The first twogames are designed to replicate the standard Ellsberg tasks, but replacing the sub-jective ball draw with a choice by an opponent. The third game requires subjectsto predict the outcome in a co-ordination game that had previously been played byother subjects. In each of these games Camerer and Karjalainen found, on average,a consistent yet small amount of ambiguity aversion along with a large amount ofheterogeneity between subjects. These games, however, do not really have strate-gic interaction in a meaningful sense; in each game the strategic uncertainty onlyaffects one of the players.The fourth game that Camerer and Karjalainen study is a matching penniesgame with an extra risky option for the row player. This is a very clever designthat allows the row player to, effectively, choose between a bet on their opponent’sstrategy choice or a bet on a random draw. Again, the average subject was mea-sured to be ambiguity averse.Pulford and Colman (2007) took an approach that is somewhat opposed tothat of Camerer and Karjalainen (1994). In Pulford and Colman (2007) subjectsfaced either a complete information game where their opponent was completelyindifferent between all outcomes, or a game of incomplete information where theiropponent had one of two types and each type of opponent had a differing dominantstrategy. Pulford and Colman argue that the complete information game is a riskygame and the incomplete information game is an ambiguous game (when subjects10Di Mauro and Castro (2011) study a voluntary contribution game where subjects play againstvirtual agents. It is debatable whether this should be viewed as a game, or simply a specially struc-tured decision problem.130were not informed about the probability distribution of types).Pulford and Colman argue, using the principle of insufficient reason, that theonly reasonable beliefs are that an opponent who is indifferent everywhere shouldbe expected to choose each strategy precisely half the time. This assumption is verystrong. A weaker, more reasonable, assumption would be that we require beliefsin this case to be symmetric (but possibly non-additive). Under this alternativeassumption it becomes very difficult to interpret the results of Pulford and Colman(2007).Eichberger et al. (2008) is a very entertaining paper. Subjects played a varietyof normal form games against either a granny, a game theorist, or another sub-ject. Eichberger et al. hypothesized, and it was confirmed in the data, that subjectswould view playing against the game theorist as being less ambiguous than play-ing against the granny.11 There was no evidence of any differences in behaviourbetween subjects playing against the granny or against another subject.Eichberger et al. structured their games in a manner which allowed the am-biguity involved in a game to be quantified. Fixing the identity of the opponent,it was usually the case that subjects played the Nash equilibrium strategy less ingames with higher levels of ambiguity.12The results in Eichberger et al. (2008) are clearly consistent with ambiguityaverse behaviour, although without observing individual level ambiguity measuresit is difficult to rule out other potential explanations of the data. Eichberger andKelsey (2011) revisit the data from Goeree and Holt (2001), and establish that thedata is also broadly consistent with ambiguity averse behaviour. The analysis inboth papers is, however, at the aggregate level, and given the large amount of be-tween subject heterogeneity that has been observed in decision theory ambiguityaversion experiments (see Halevy (2007) for example) it seems that tracking indi-vidual subject level behaviour across games would be a natural next step.Ivanov (2011) takes this next step. He uses a considerably more complicateddesign, and makes extensive use of stated beliefs in his identification process.1311Personally, I find this result to be surprising. My prior was that undergraduates would feel morecomfortable predicting the behaviour of a granny than a game theorist, given that most undergradu-ates would presumably have spent more time interacting with grannies than game theorists.12The exceptions occurred for subjects that were playing the game theorist.13Eichberger et al. (2008) also asked for stated beliefs, but they did not put them to work in the131First, subjects played a series of normal form games. Next, subjects were asked tostate their beliefs regarding their opponent’s behaviour in the normal form games.14Finally, subjects were asked to choose between lotteries that were constructed fromtheir beliefs in part two and the payoffs from the games in part one.The central question posed in Ivanov (2011) is quite different from the ques-tions raised in this paper. Ivanov seeks to classify subjects as ambiguity averse/am-biguity neutral/ambiguity loving using only their responses to normal form games;the headline result is that 22/46/32 percent of subjects fall into each category, re-spectively. Unlike Ivanov (2011), this paper does not consider ambiguity seekingbehaviour (ambiguity seeking subjects will be indistinguishable form ambiguityneutral subjects given the experimental design in the current paper). On the otherhand, Ivanov (2011) requires an assumption that stated beliefs are equal to truebeliefs (troublingly, if we model ambiguity aversion using the MEU model, thenthe notion of ‘true beliefs’ may not even be be well defined) in order to identifyambiguity preference, whereas the current paper measures ambiguity preferencedirectly.Kelsey and le Roux (2015a) is the only other papers that compares behaviour inEllsberg urn tasks to behaviour in normal form games at the individual level. Theresults reported in Kelsey and le Roux (2015a) are mostly negative, however. Theyfind significant levels of ambiguity aversion in their normal form game (a battleof the sexes game augmented with a safe option for the column player), but findmuch less ambiguity aversion (and even some ambiguity seeking) in their Ellsbergurn tasks. This contrasts with the results here, where Ellsberg urn behaviour wascorrelated with behaviour in the game in figure 2.6.There are three potential reasons for the differences in results between the twopapers. Firstly, Kelsey and le Roux (2015a) do not control for risk aversion. Thebehaviour in their normal form game might be driven by risk aversion, rather thanambiguity aversion. Secondly, Kelsey and le Roux (2015a) used a three-colourEllsberg urn, rather than the two-colour Ellsberg urn used here. Chew et al. (2013)report that measured levels of ambiguity aversion are often much lower in two-fashion of Ivanov.14In one treatment beliefs were elicited at the same time as the games were played.132colour rather than 3-colour Ellsberg tasks.15 The reasons for this difference are notexactly clear, although Chew et al. argue that it is a confusion effect (that subjectsdo not recognise the 3-colour urns as being ambiguous).The third difference between Kelsey and le Roux (2015a) and the current paperis the use of comprehension questions. The results of Chew et al. (2013) suggestthat, in some situations, subjects will have trouble recognizing and responding toambiguity. To ensure that this does not affect results, we used a series of drop downmenu comprehension questions to screen for understanding. Kelsey and le Roux(2015a) did not use any tests for comprehension, and this may make it harder toidentify a relationship between ambiguity averse behaviour in the games and am-biguity averse behaviour in the Ellsberg tasks.The identification of the link between individual choice tasks and normal formgame behaviour is always going to be difficult because the Nash equilibrium re-mains an equilibrium for ambiguity averse subjects. This implies that even in idealcircumstances the power of a test to reject the null hypothesis of independence maybe low (particularly if the Nash equilibrium is focal). Although, as demonstratedin this paper, the effect can be recovered with a clean experimental design. Onbalance, the evidence strongly suggests that ambiguity aversion has an influenceon behaviour in normal form games.Kelsey and le Roux (2015b) study games of public good provision. In the‘best shot’ formulation of the game ambiguity averse subjects are expected toover-contribute to the public good and in the ‘weakest link’ formulation ambiguityaverse subjects are expected to under-contribute. On aggregate the data supportsthe hypothesis that ambiguity aversion is prevalent in the subject population, al-though there is significant heterogeneity and no direct measures of ambiguity pref-erences are reported. Kelsey and le Roux also hypothesized that British subjectswould behave in a more ambiguity averse fashion when paired with Indian subjects(when compared to British subjects paired with other British subjects) althoughthey do not find evidence to support this in the data.15In piloting for this experiment, two sessions were run using a 3-colour Ellsberg urn. Only 1 outof 14 subjects was classified as being ambiguity averse. Given this extremely low observed rate ofambiguity aversion, 3-colour Ellsberg urns were not used in any other sessions.133B.2 ProofsThe proofs in this section use the method of iterated elimination of strategies thatare never a best response. This method produces the set of Pearce/Bernheim ratio-nalizable strategies under SEU preferences, and produces the set of Epstein (1997)rationalizable strategies under MEU preferences.Proof of proposition 2.1. Suppose that the row player has preferences such thatu(25)+ u(14) > 2u(18) and ΦR is restricted to be a singleton. Then the best re-sponse for the row player is A whenΦ(X)R > 12 , is B whenΦ(X)R <12 and is eitherA or B when Φ(X)R = 12 . C is never a best response.Using the procedure of iterated elimination of never best response strategies,we eliminate C. Now, in the reduced game, Y is never a best response for thecolumn player. Eliminate Y . Now B is never a best response.Therefore {A,X} is the unique rationalizable strategy.Now, consider the case where u(25)+u(14)≤ 2u(18). C is now a best responseto the belief set Φ(X) = {φ : φ(X) = 12}. A and B are both clearly best responses,as are X and Y . Therefore all strategies are rationalizable.Now, consider the case where Φ(X) may be set valued. C is now a best re-sponse to the belief set Φ(X) = {φ : 0≤ φ(X)≤ 1}. A and B are both clearly bestresponses, as are X and Y . Therefore all strategies are rationalizable.Proof of proposition 2.2. We use iterated elimination of never best response strate-gies.In the first round, the type 1 row player eliminates C. All other strategies forall other players are best responses.In the second round, the payoff to the column from playing Y is increasingin the probability that the row player plays C, and the payoff from playing X isdecreasing the in the probability that the row player plays C. Therefore, becausethe largest probability that the column player can assign to the row player playingC is α , Y can only be sustained as a best response whenever U(Y )−U(X) =u(22)(1−α)+u(12)α−(u(12)(1−α)+u(20)α)> 0. Therefore, X dominates Ywhenever α > u(22)−u(12)u(22)+u(20)−2u(12) = α¯ , so that Y is eliminated as never being a bestresponse iff α > α¯ .134A consequence of this is that no more strategies can be eliminated if α ≤ α¯ . Inthis case, both column player strategies enter the rationalizable set, as do all threerow player strategies, as all three strategies may be played by the non-type 1 rowplayer. The Type 1 row player may play either A or B.If α > α¯ , then we must consider the third round. In this case, Y has beeneliminated in the second round, and both types of row players now have a uniquebest response of A. Therefore, {A,X} is the only rationalizable strategy whenα > α¯ .If we normalize the utility function so that u(22) = 1 and u(12) = 0 then wehave that α¯ = 11+u(20) . It is clear that α¯ is bounded below by12 and is decreasing inthe degree of risk aversion.B.3 Results AppendixB.3.1 Demographic analysisTable B.1 presents an analysis of the demographic data for row players. Speci-fications are equivalent to those presented in table 2.6 in the main text, with andwithout demographic controls: an indicator for female subjects, and an indicatorfor STEM (plus economics) majors. The sample size is reduced from that in themain text for two reasons: first, there were two sessions where software errors pre-vented the recording of demographic information and, second, for some subjectsthe self-reported major of study was not identifiable. As the table makes clear, nei-ther gender nor area of study affects row player decisions in the testing game andthe inclusion of demographic controls does not change the estimates of the effectsof preferences on behaviour.Table B.2 presents an analysis of the demographic data for column players.As for the row player data, there is no effect of either gender or area of study onbehaviour in the testing game.B.3.2 Preferences and beliefsTable B.3 presents the relationship between a subject’s own ambiguity aversion andtheir beliefs regarding their opponent’s ambiguity aversion. Again, the sample is135Effect of Conditional on ∆Pr(A) ∆Pr(A) ∆Pr(A) ∆Pr(A)Ambiguity -0.14 -0.15Aversion (0.181) (0.165)[0.10] [0.11]Risk -0.13 -0.14Aversion (0.224) (0.211)[0.11] [0.11]Ambiguity Aversion Low Risk Aversion -0.13 -0.13(0.332) (0.332)[0.14] [0.14]Ambiguity Aversion High Risk Aversion -0.16 -0.17(0.358) (0.313)[0.17] [0.17]Risk Aversion Ambiguity Neutrality -0.12 -0.12(0.399) (0.413)[0.14] [0.14]Risk Aversion Ambiguity Neutrality -0.14 -0.16(0.379) (0.336)[0.16] [0.17]Risk Aversion & Ambiguity Aversion -0.28* -0.29*(0.040) (0.033)[0.13] [0.14]Female 0.02 0.02(0.869) (0.876)[0.10] [0.10]STEM -0.08 -0.08(0.454) (0.448)[0.10] [0.10]Table B.1: Change in proportion of subjects playing A in the testing game, restricted tosubjects that passed the comprehension tests and that have demographic informationrecorded. N=100. All estimates calculated using a linear probability model, with p-values in brackets and standard errors in square brackets. * indicates value is signif-icantly different from 0 using a non-directional test at the 5% level and ** indicatessignificance at the 1% level.136Effect of beliefs over Conditional on ∆Pr(X) ∆Pr(X) ∆Pr(X) ∆Pr(X)Ambiguity 0.03 0.04Aversion (0.740) (0.737)[0.10] [0.10]Risk -0.17 -0.17Aversion (0.112) (0.104)[0.10] [0.10]Ambiguity Aversion Low Risk Aversion 0.12 0.11(0.393) (0.422)[0.14] [0.14]Ambiguity Aversion High Risk Aversion -0.08 -0.07(0.623) (0.672)[0.16] [0.16]Risk Aversion Ambiguity Neutrality -0.08 -0.09(0.541) (0.496)[0.14] [0.14]Risk Aversion Ambiguity Aversion -0.28 -0.27(0.083) (0.091)[0.16] [0.16]Risk Aversion & Ambiguity Aversion -0.16 -0.16(0.216) (0.217)[0.13] [0.13]Female 0.09 0.08(0.350) (0.410)[0.10] [0.10]STEM 0.00 0.01(0.974) (0.903)[0.10] [0.10]Table B.2: Change in proportion of subjects playing X in the testing game, as a function ofbeliefs over opponent’s preferences, restricted to subjects that passed the comprehensiontests and have demographic data recorded. N = 87. All estimates calculated using a lin-ear probability model, with p-values in brackets and standard errors in square brackets.* indicates value is significantly different from 0 using a non-directional test at the 5%level and ** indicates significance at the 1% level.137restricted to subjects who filled in the comprehension drop down menus correctlyon the first attempt; 95 of the 206 subjects failed this test, with most of those sub-jects failing comprehension for the game that measured their beliefs. Table B.4presents the relationship between a subjects own risk aversion and their beliefs re-garding their opponents risk aversion. Only 35 subjects failed the comprehensionstests for the risk games, leaving a sample size of 171 subjects.The most striking feature of tables B.3 and B.4 is the extremely strong rela-tionship between the subject’s own preferences and their beliefs regarding theiropponents preferences. On both the risk and ambiguity dimension the Fisher exacttest rejects the null hypothesis that preferences and beliefs are independent at allreasonable significance levels.The results in the main text suggest that subjects were not very confident intheir predictions of their opponent’s preferences, while the results here demon-strate that subjects predicted their opponent to behave like themselves. A reconcil-iation of these facts is that when faced with predicting behaviour in an uncertainenvironment the subject’s own behaviour acts as a very strong focal point.Predicted ambiguity preferenceNeutral AverseNeutral 63 4 67Own ambiguity preferenceAverse 12 32 4475 36 111Table B.3: Subject’s own ambiguity preferences crossed with subject’s predictions of theircounterpart’s ambiguity preferences, restricted to subjects who passed the comprehen-sion test for the relevant games. Non-directional Fisher’s exact test p < 1×10−10.Result B.1. Subjects believe that their opponents preferences are the same as theirown preferences.B.3.3 Comprehension dataThis section provides an analysis of the comprehension data, and a discussion ofthe robustness of the results presented in the main body of the paper. Table B.5presents the subjects comprehension scores broken down by game number, and ta-ble B.6 presents the aggregated data. As the tables demonstrate, only about a quar-138Predicted risk aversionLow Med HighLow 73 6 19 98Med 5 9 2 16Own risk aversionHigh 9 2 46 5787 17 67 171Table B.4: Subject’s own risk aversion crossed with subject’s predictions of their counter-part’s risk aversion, restricted to subjects who passed the comprehension test for therelevant games. Non-directional Fisher’s exact test p < 1×10−10.ter of the subjects answered all of the comprehension questions correctly on theirfirst attempt. Subjects performed better on the risk measurement game than theambiguity measurement game, and also performed better as the row player (mea-suring their own preferences) than as the column player (measuring their beliefsregarding their opponent’s preferences) in these games.Figure number of gameComp Score 2.2R 2.2C 2.5R 2.5C 2.6R 2.6C1 159 131 196 177 168 1790.75 26 39 5 18 32 210.5 7 13 2 5 2 40.25 9 4 0 1 1 00 5 19 3 5 3 2Total 206 206 206 206 206 206Table B.5: Number of subjects attaining each comprehension scores for each of the sevengames played. Subjects were awarded a comprehension score of 1 for answering thecomprehension questions correctly on the first attempt, and were penalized 0.25 for eachincorrect attempt. The column heading 2.2R (resp. 2.2C) indicates the responses to thegame in figure 2.2 as the row (resp. column) player.There are obviously many different inclusion criteria that could be constructedusing the comprehension data. The results in this paper were not data-mined bychoosing the most convenient inclusion criteria. Ex-ante, two candidate inclusioncriterion were identified. One was the criterion used in the body of the paper.The other criterion was to only use subjects who score a perfect 7 out of 7 on thecomprehension questions. The latter criterion was unable to be used because ofthe small and unbalanced sample sizes that it produced. For example, of the 52139Total Comp Score Number of Subjects7 526.75 596.5 356.25 126 155.75 85.5 65.25 25 34.75 54.5 14.25 14 13.75 13.5 23.25 11.25 2Total 206Table B.6: Total comprehension score by subject. A score of 7 indicates that the subjectanswered the comprehension questions correctly on the first attempt for each of the 7games.subjects who satisfied the criterion only 8 were risk neutral and ambiguity averse,while only 4 were risk averse and ambiguity neutral. Therefore, this criterion wasrejected for not providing enough power, and the criterion in the main text wasadopted.We also include, as an additional robustness check, the estimates of the maintables on the full sample of subjects, including those who failed the comprehensiontests. Generally, the results do not differ markedly from those presented in the maintext. The largest differences occur for column player behaviour in the originaltreatment in the main testing game, where subjects who failed the comprehensiontests show a much larger effect of beliefs over their opponents’ preferences onbehaviour than was presented in the main text. It is unwise to ascribe intentionalityto these subjects, given that they failed to demonstrate a sound knowledge of thepayoffs. Nevertheless, we report the estimates here for completeness.Table B.7 is the full sample analogue of Table 2.6. The only difference in statis-140tical inference between the two samples is a stronger effect of ambiguity aversionconditional on high risk aversion.Effect of Conditional on ∆Pr(A)Ambiguity Aversion -0.21**(0.005)Risk Aversion -0.18*(0.017)Ambiguity Aversion Low Risk Aversion -0.12(0.229)Ambiguity Aversion High Risk Aversion -0.28*(0.027)Risk Aversion Ambiguity Neutrality -0.07(0.485)Risk Aversion Ambiguity Aversion -0.23(0.051)Ambiguity Aversion & Risk Aversion -0.35**(0.001)Table B.7: Change in proportion of subjects playing A in the testing game, as a functionof subject preferences. N = 185. Pearson’s χ2 p-value shown in brackets. * indicatesvalue is significantly different from 0 using a non-directional test at the 5% level and **indicates significance at the 1% level.Table B.8 is the full sample analogue of table 2.7. The effect of ambiguityaversion being associated with a move from A to C amongst subjects with low riskaversion is no longer apparent in the full sample. This change is caused by subjectswho failed the comprehension tests playing B with an essentially equal probabilityacross all subject Types.Table B.9 is the full sample analogue of table 2.9. We see a strong effectdifference in behaviour between Type 1′ and Type 4′ subjects (bottom row of thetable) in the full sample. This effect is being driven almost entirely by subjectswho failed the comprehension tests, and therefore should be treated with caution.B.4 InstructionsThe instructions presented to subjects in the September sessions subjects are re-produced below. The instructions in the March and April sessions had a different141example game presented as “Game X”. Otherwise, the two sets of instructions wereidentical. The instructions were originally written in HTML, CSS and Javascriptand were presented to students on their computer in an Internet browser. The in-structions were interactive, and subjects could highlight strategies in the examplegames and practice filling in the drop down menus for “Game X”. The instructionspresented below have been reformatted to meet the formating requirements of thisdissertation, but are materially the same as those presented to the subjects.142B.4.1 InstructionsThis is a research experiment designed to understand how people make economicdecisions. To assist with our research, we would greatly appreciate your full atten-tion during the experiment. Please do not communicate with other participants inany way and please raise your hand if you have a question.You will participate in a series of 7 games. In each game you will make onedecision, and a subject you will be paired with, called your counterpart, will makeone decision. One, and only one of the games will be chosen, in a random fashiondescribed below, as the game for which you will be paid. The amount that youearn will depend on a combination of your decision, your counterpart’s decisionand, for some games, the colour of a ball drawn from a bag. Your counterpart willbe one of the other participants in this room and the identity of your counterparthas been randomly pre-determined.How your payment will be determinedAs mentioned above, only one game will be chosen as the game for which you willbe paid. Each of the 7 games is equally likely to be chosen. You will notice thatthere are 7 pieces of paper labelled from ’A’ to ’G’ stuck to the wall. On the backof each of them is a number. At this point, I would like to ask one of you to chooseone of the pieces of paper labelled from ’A’ to ’G’. The number that is written onthe back of the chosen piece of paper will determine which of the 7 games is chosenfor payment. Although the choice of game has already been fixed by the choice ofletter, we will not reveal the choice to you until the end of the experiment. Thisprocedure suggests that you should treat each game independently (i.e. as if it wasthe only game in the experiment). After the experiment you may come and checkthat each of the games actually is represented on the back of one of the cards.Your payment may also depend on the colour of a ball drawn from a bag. Insome games there will no balls drawn, in some other games there will be only oneball drawn from a bag, and in some games there will be two balls drawn (each froma different bag). The ’Unknown’ bag (denoted by the letter U) will contain onlyRED and YELLOW balls in an unknown ratio, while the ’Known’ (denoted by theletter K) bag will contain RED and YELLOW balls in an equal ratio. The total143number of balls in each bag will always be 10. A graphical representation of thetwo bags is shown below.Figure B.3: U bag. Figure B.4: K bag.When you are playing the games there will be a picture of the relevant bag(s)on the screen to remind you of the composition of the bag(s).Before the experiment begun, I asked a graduate student in economics, whohas no knowledge of this experiment, to place 10 balls into the U bag. I instructedthe student to place exactly 10 balls in the bag, and that only RED and YELLOWballs are to be placed in the bag. I have no knowledge of the composition of theballs in the bag, other than what is described above. There might be 10 RED ballsand 0 YELLOW balls, or 9 RED balls and 1 YELLOW ball, or 8 RED balls and2 YELLOW balls, and so on up to 0 RED balls and 10 YELLOW balls. Afterthe experiment you may come and check that the bag satisfies the requirementsoutlined above.I shall now create the K bag, by placing 5 RED balls and 5 YELLOW balls intoa second bag.At the end of the experiment, after the chosen game has been revealed, I willask one of you to draw a ball from the K bag (if required), and another one of youto draw a ball from the U bag (if required).The combination of the game chosen, your choices, your counterpart’s choicesand the colour of the ball(s) will determine how much money you make during theexperiment. The amount that you make during the experiment will be added toyour $5 show up fee and will be paid to you, in cash, at the end of the experiment.One of the conditions of the ethics approval for this experiment is that I do notdeceive the subjects (i.e. you). If you feel that I have deceived you in any way, youmay contact either my thesis supervisor or the UBC Behavioural Ethics Review144Board to lodge a complaint. Their contact details are included on the consent formthat you have read and signed.The structure of the gamesIn each game, you will need to make a choice of either A or B or C (in some gamesyou will only have 2 options, A and B). You may only ever choose one option pergame. Your counterpart will make a choice of either X or Y or Z (in some gamesyour counterpart will only have 2 options, X and Y). Your counterpart may alsoonly ever choose one option per game. The amounts that each of you can earn willbe presented to you in a table format, as seen below.X Y ZA 20,10 8,0 0,0B 30,0 6,10 4,0C 30,0 15,0 9,10Your options will always be shown on the rows of the table. Your counterpart’soptions will always be shown on the columns of the table (the computer flips thegame so that everyone can look at the table from the same perspective). Withineach cell of the table your earnings will always be shown first, ¡b¿in bold¡/b¿, andyour counterpart’s earnings will always be shown second. Values shown are alwaysshown dollar amounts. For example, in the above game, if you chose action ’A’ andyour counterpart chose action ’Y’ then you would receive a $8 and your counterpartwould receive $0. On the other hand, if you chose ’A’ and your counterpart chose’Z’ then you would each earn $0. Games that have only a single payoff table, likethe above example, do not require a ball to be drawn.In other games, your earnings will also depend on the colour of a ball whichwill be drawn from a bag after you have both made your decisions. These gameswill have two tables, and the colour of the ball(s) will determine which table isused to calculate your earnings. In the example given below, the left table will beused if a RED ball is drawn and the right table will be used if a YELLOW ball isdrawn.Suppose, in the above example, that you chose option ’B’ and your counterpartchose option ’Y’. Then your payment will be $9 if the ball drawn is RED, and $4145L′ I′L 12,10 18,0I 17,0 9,10Red ball drawn from K bagL′ I′L 12,10 19,0I 8,0 4,10Yellow ball drawn from K bagFigure B.5: K bag.if the ball drawn is YELLOW. Your counterpart’s payment will be $10 if the balldrawn is RED, and $10 if the ball drawn is YELLOW.The game shown above was a K game. In a K game there is only one balldrawn, from the K bag, and the colour of the ball will determine which table willbe used. There will also be U games, where there is also only one ball drawn, fromthe U bag. The other type of game, a U and K game, is shown below. It will alwaysbe clear which type of game you are playing from the labels on the tables and thepictures of the bags underneath the game.L′ I′L 12,10 18,0I 17,0 9,10Red ball drawn from U bagL′ I′L 12,10 19,0I 8,0 4,10Red ball drawn from K bagIn a U and K game there will be two balls drawn - one from each of the bags.If a RED ball is drawn from the U bag then you will be paid according to the lefthand table. If a RED ball is drawn from the K bag then you will be paid accordingto the right hand table. It is also possible that a RED ball will be drawn from bothbags. In this case, you would be paid according to both tables (the payments willbe added together). However, it is also possible that a RED ball will not be drawnfrom either bag. In this case you would receive no payment (other than the $5 showup fee and any bonus payments you may earn).146Figure B.6: U urn. Figure B.7: K bag.Bonus payments and drop down menusBefore you can confirm your choices in the game itself, you will need to fill in aseries of dynamic drop down menus to confirm that you have understood the game.You should pay careful attention to the drop down menus, becuase you will earnbonus payments that depend on whether you have filled the drop down menus incorrectly.For each game, you should fill in the drop down menus first. Once you arehappy that your choices adequately describe the game, you should click on the”Check Description” button. If you fill in the drop down menus correctly on yourfirst attempt you will earn a $1 bonus for that game. As there are 7 games, youmay earn up to $7 in bonus payments. If you click ”Check Description”, but havefilled in the dropdown menus incorrectly or have not filled the dropdown menus inat all, your bonus payment will decrease by $0.25. After four incorrect attemptsyour bonus payment for only that game will reach zero.WARNING: The bonus payment system relies on ’alerts’ that will pop upon your screen. Sometimes the Chrome browser will give you option of turningthe alerts off. Do not do this; if you do then the computer may not record yourbonus payments. If you accidentally turn the alerts off then please raise yourhand and we will reset your browser.Before we continue shall work through an example of the drop down menustogether.Note that most of the games appear in pairs - each game has two players, andyou will play each game in each role. (Recall that there are seven games - theseventh game does not have a pair). Below is an example of Game X (with thedrop down menus removed), viewed from the other role. Notice that while in147Figure B.8: Game X.Game X your earnings were not affected by your counterpart’s choices, in GameXT your counterpart’s earnings are not affected by your choices.How to use the game interfaceOnce your description of the game is correct (and you have verified this by clickingthe “check description” button) the drop down menus will inactivate, and the “lock”button will activate. At this point you can enter your choice by clicking on thedesired option in the table. When you click on a choice, the computer will highlightthe row that you have chosen (you may try this in Game X above). If you want to148X Y ZA 10,9 0,4 0,12B 0,9 10,4 0,12C 0,9 0,4 10,12Red ball drawn from K bagX Y ZA 10,9 0,15 0,6B 0,9 10,15 0,6C 0,9 0,15 10,6Yellow ball drawn from K bagFigure B.9: Game XT.Figure B.10: K bag.change your mind, you can simply click on a different choice. Once you are happywith your choice you should click on the “lock” button before moving on to thenext game. If, later, you wish to change your mind you can always click on the“unlock” button and then change your choice.You may also highlight any of your counterpart’s options by clicking on thelabel for that choice (as long as the game is unlocked) - this feature is providedto you as a visual aid to assist in your decision making process, but will have noimpact on the earnings received by either you or your counterpart. There is anotherdecision making aid that has been provided to you: a textbox. In the past, somesubjects have found it useful to write down the reasoning behind their decisions.You do not have to write anything in the textbox, and nothing you do write in thetextbox will be used in the determination of your earnings. Nothing you write willbe shown to your counterpart. Likewise, nothing your counterpart writes will beshown to you. When you lock a game, the textbox associated with that game willlock as well.Once you have entered your choices for each of the 7 games, and locked each ofthe 7 games, you can click the button to submit all of your answers. Once you havepressed the “submit” button you can no longer go back and change your answers.Once everyone has clicked submit, the computer will match your responses with149your counterpart’s choices. The computer will not match you with your counterpartuntil everyone has finished the experiment, so there is no advantage to rushing.Take your time and make sure you are happy with your choices.Summary of the order of events• Read instructions, and choose a letter on the wall that will determine whichgame will be paid.• Play the 7 games. For each game, begin by filling in the drop down menus,and then make your choice in the game. Lock each game after you havemade your choice. If you want to change your mind, you can always unlockany game and change your choice.• Once you are happy with all of your choices, press the ’submit’ button. Onceyou have pressed the ’submit’ button your choices are final.• Enter some demographic information into the computer.• Reveal the game that will be paid.• Draw ball(s) from the bag(s).• Calculate your payment, based on the chosen game, colour of ball, youraction and your counterpart’s action.• Receive payment and leave the experiment.150Effect of Conditional on ∆ NBNB+NA ∆NCNC+NAAmbiguity Aversion 0.13 0.19*(0.053) (0.013)Risk Aversion -0.02 0.23**(1.000)F (0.003)Ambiguity Aversion Low Risk Aversion 0.12 0.06(0.186)F (0.510)Ambiguity Aversion High Risk Aversion 0.16 0.27*(0.288)F (0.039)Risk Aversion Ambiguity Neutrality -0.06 0.11(0.675)F (0.269)Risk Aversion Ambiguity Aversion -0.02 0.31*(1.000)F (0.015)Ambiguity Aversion & Risk Aversion 0.10 0.38**(0.373)F (0.000)Table B.8: Change in the values of NBNB+NA andNCNC+NA in the testing game as a functionof subject type, where NA is the number of subjects selecting A. N = 185 Figures inbrackets are p-values calculated under a null hypothesis that the coefficient is equal tozero under either Person’s χ2 test or a Fisher exact test (denoted by F). The coefficientsare equivalent to the probabilities implied by a saturated multinomial logit regressionwith a choice of A denoted as the base outcome.151Effect of beliefs over opponent’s Conditional on ∆Pr(X)Ambiguity Aversion -0.13(0.069)Risk Aversion -0.15*(0.033)Ambiguity Aversion Low Risk Aversion -0.06(0.491)Ambiguity Aversion High Risk Aversion -0.18(0.111)Risk Aversion Ambiguity Neutrality -0.09(0.268)Risk Aversion Ambiguity Aversion -0.21(0.083)Ambiguity Aversion & Risk Aversion -0.28**(0.006)Table B.9: Change in proportion of subjects playing X in the testing game, as a function ofreported beliefs over opponents’ preferences. N = 184. Pearson’s χ2 p-value shown inbrackets. * indicates value is significantly different from 0 using a non-directional testat the 5% level and ** indicates significance at the 1% level.152Appendix CAppendix for “Continuity, Inertiaand Strategic Uncertainty: A Testof the Theory of Continuous TimeGames”C.1 Instructions to SubjectsInstructionsYou are about to participate in an experiment in the economics of decision-making.If you follow these instructions carefully and make good decisions, you can earna CONSIDERABLE AMOUNT OF MONEY, which will be PAID TO YOU INCASH at the end of the experiment.Your computer screen will display useful information. Remember that the in-formation on your computer screen is PRIVATE. To insure best results for yourselfand accurate data for the experimenters, please DO NOT COMMUNICATE withthe other participants at any point during the experiment. If you have any ques-tions, or need assistance of any kind, raise your hand and the experimenter willcome and help you.153In the experiment you will make decisions over several periods. At the end ofthe last period you will be paid, in cash, the sum of your earnings over all periods.The Basic Idea.In each of several periods, you will be randomly matched with another partic-ipant for 60 seconds and you will each decide when to Enter the market. If youboth enter at the same time, you will both earn the same amount, which will dependon the time at which you both entered. If one player enters earlier than the other,she will earn more money while her counterpart will earn less. The longer the sec-ond player waits to enter after the first player enters, the greater the difference intheir earnings.Screen Information.A vertical dashed line marks the passage of time, moving from left to right overthe course of the period until both players have chosen to Enter the Market. Adot moving left to right labeled ‘Me’ and another labeled ‘Other’ shows paymentinformation for each player, although before either player has entered the marketthe two dots will be precisely on top of each other. The dots show the amountof money each subject will earn if both players have entered now If nobody hasentered yet the dots (which are on top each other) show what will happen if bothenter now. If one player has entered already the dots (which are now separated)will show the amount each player will earn if the second player enters the marketnow.You can choose to Enter the market at any time by pressing the space bar. Thetime of entry will be shown on the screen as a dashed vertical line.The screen gives you information on your potential earnings under three pos-sible scenarios:If you both enter in the same sub-periodIf you and your counterpart enter at the same time, you will earn exactly thesame amount as your counterpart. The black line that looks like a hill shows154Figure C.1: Nobody has entered. Figure C.2: One player has decided toenter and the clock has frozen.exactly what you and your counterpart would both earn if you both entered in eachmoment of the period.Notice that your joint earnings depend on when you both choose to enter in.For example, if you and your counterpart both entered at time 0, you would bothearn 20 points. However, if you both entered at time 24, you would both earnapproximately 36 points.If you enter firstIf your counterpart enters later than you, she will earn less and you will earnmore than if she had entered at the same time you did. At every moment the screentells you what would happen if you entered now, and your counterpart entered at alater time than you:• The green line shows you what you would earn if you entered now and yourcounterpart entered in each of the remaining moments in the period.• The red line shows you what your counterpart would earn if you entered nowand she entered in each of the remaining moments in the period.155Notice that the longer your counterpart waits to enter after you, the less sheearns. For instance in the example in Figure 1, if you entered now and your counter-part entered 5 seconds later, you would earn approximately 38 points (the amounton the green line 5 seconds later) and your counterpart would earn approximately20 points (the amount on the red line 5 seconds later). If, instead, your counter-part waited 10 seconds to enter, you would earn approximately 50 points and yourcounterpart approximately 18 points.Note that these lines will change as you move along: the green line will alwaysbe above the current point on the black hill and the red line will always be be-low, reflecting the fact that you earn more (and your counterpart less) than if yourcounterpart entered when you did.If you enter secondIf you enter at a later time than your counterpart, you will earn less and yourcounterpart more than if you had moved at the same time as your counterpart. Im-portantly, these graphs and payoffs are symmetric: your counterpart sees the samescreen (at least prior to anyone entering) and faces the same payoff consequencesas you do. Thus it is also true that:• The green line shows you what your counterpart would earn if (s)he enterednow and you entered later at a future time.• The red line shows you what you would earn if your counterpart entered nowand you entered at a later time.For instance in the example in Figure 1, if your counterpart entered now andyou entered in 5 seconds, your counterpart would earn approximately 38 points (theamount on the green line 5 seconds to the right) and you would earn approximately20 points (the amount on the red line 5 seconds to the right). If, instead, you waited10 seconds to enter, you would earn approximately 18 points and your counterpartapproximately 50 points.Time Freeze156After a player first enters, the game will freeze for 5 seconds. During these 5seconds the player’s counterpart can choose whether to enter too by pressing thespacebar. If (s)he does, the software will treat both entry decisions as occurringat the same time and both players will earn the exact same amount (the amountshown on the black line at the moment of entry). If she does not choose to enterduring the time freeze, the clock will resume and her earnings will drop, followingthe red line. The time freeze is demonstrated in Figure 2.After Entry OccursIf one player enters before the other, you will see a dotted green line mark thetime of first entry and you will see the dots separate, one following the green lineand the other red line. A label next to each dot will tell you which corresponds toyou (labeled “Me”) and which to your counterpart (labeled “Other”). A messageat the top of the screen will also remind you whether you were the first to enter(in green) or the second (in red). If you were not the first to enter, the timing ofyour entry decision will now determine both of your earnings. Figure 3 shows anexample.After both players enter, horizontal lines will appear showing your earnings.At the end of these lines (on the right side of the screen) you will see your andyour counterparts’ exact earnings (if you both entered at the same time, you willboth earn the same amount). Figure C.4 shows an example in which one player hasentered at time 10 and the other later at time 14. As a consequence the first to enter(in green) earns 45.01 points and the second (in red) player earns 26.23 points.Other Information and Earnings.At the top of the screen you will see the current period number, the time re-maining in the current period, the number of points you will earn this period basedon current decisions and the total number of points you have accumulated over allperiods so far. You will be paid cash for each total point you have earned at the endof the experiment, at a rate given by the experimenter.157At the beginning of each new period, you will be randomly matched with anew participant from the room to play again. All matchings are anonymous – youwill never know who you were matched with and neither will they.Summary• Your and your counterparts’ earnings depend on the time you each decide toEnter the market (by pressing the spacebar).• When either player enters, the clock will freeze and her counterpart will havethe opportunity to enter too, at the same time.– If you both enter at the same time, you will earn the same amount,shown on the black hill-like line. As the black hill line shows, yourjoint earnings depend on the time at which you both enter.– If your counterpart enters at a later time than you, you will earn moreand she will earn less than if she had entered when you did. Theseamounts are shown via the green and red lines.– Likewise, if you enter at a later time than your counterpart, you willearn less and your counterpart more than if you had entered when shedid.• You will be paid based on the total number of points you earn, over all peri-ods.158Figure C.3: The green player has en-tered but the red player has not.Figure C.4: Both have entered.C.2 Nash EquilibriumIn section C.2.1 we provide proofs for propositions stated in section 3.2.2 of themain paper. In section C.3 we prove a set of propositions characterizing ε-equilibriumfor our games and provide proofs for propositions stated in section 3.2.3 of the mainpaper.C.2.1 Nash equilibriumIn subsection C.2.1 we prove proposition 3.1, which relies exclusively on standardtheoretical tools.In subsection C.2.1 we provide more details on the theoretical schema of Simonand Stinchcombe (1989) and provide a self contained heuristic proof of proposition3.2 following the intuition of Simon and Stinchcombe (1989). We close the subsec-tion by providing a full proof of proposition 3.2 that draws directly on theoreticalmachinery developed in Simon and Stinchcombe (1989). Because this machin-ery is substantial, this full proof is not self contained and relies on definitions andlemmas developed in Simon and Stinchcombe (1989).In subsection C.2.1 we provide some details on the theoretical apparatus ofBergin and MacLeod (1993), an alternative way of modeling continuous timegames, and specialize it to our setting. We then re-prove proposition 3.2 using159this method. Again, because the required tools are substantial, this proof is not selfcontained but draws on definitions and results from Bergin and MacLeod (1993).Finally, we use ideas from Bergin and MacLeod (1993) to prove proposition 3.3.Proof of proposition 3.1Proof of Proposition 3.1. We prove by contradiction. First, consider the case ofpure strategies: Suppose that there is an equilibrium where player i enters at aweakly later grid point than player j, and that this is not the first grid point. Playeri has a profitable deviation, which is to enter one grid point before player j or at thefirst grid point if there are no grid points before player j’s entry time. Contradiction.The modification to allow for mixed strategies is straightforward. Call thelast grid point that player i places positive weight on ki. Suppose that there isan equilibrium with ki ≥ k j and ki 6= 0 (where we are labeling t = 0 as the zerothgrid point). Player i can improve their payoff by moving weight from entry at ki toentry at the grid point identified by max{k j−1,0}.Simon and Stinchcombe (1989)In this section, following Simon and Stinchcombe (1989), we model our gameby considering a set of strategies that are unambiguously defined in the limit of adiscrete time grid as the grid becomes infinitely fine. Together, the two player gamegenerated by the utility function in equation 3.1, the histories Ht and strategies sigiven below define what we call the Perfectly Continuous time game.1 We beginby defining a history, and use the history to define a set of strategies.In our timing game, a history consists of two pieces of information: the currenttime, and a record of players that have already entered the market. Therefore, ahistory at time t, Ht , is an object in [[0, t]∪{1}]2, where the first element denotesthe entry time for firm i, and the second element the entry time for firm j, with 1indicating that a firm has not yet entered. For example, a history at time t = 0.5 canbe written H0.5 = {0.4,1}0.5, indicating that the first firm entered at time t = 0.4,the second firm is yet to enter, and the subscript denotes the current time.1The strategies also implicitly define the action sets and the allowable order of moves, and thegame is of complete and perfect information.160Given this definition of histories, a strategy is a mapping si : Ht 7→ {0,1}, where1 indicates an action of “enter” at that history. We fix si({t ′, ·}t)= 1 for all t ′< t (af-ter a firm has entered they cannot leave the market again), require that si is a piece-wise continuous function, and also require that if si({1, t ′}t) = 1 then si({1, tˆ}t) = 1for all tˆ ≤ t ′ (if, at time t, I wish to enter in response to an opponent who entered attime t ′ then I must also enter in response to any opponent that entered prior to t ′).Notice that these strategies satisfy assumptions F1 - F3 of Simon and Stinchcombe(1989).2Most of the technical complications that arise from modeling games in contin-uous time are related to the “well-ordering paradox” that, in continuous time, thenotions of “the next moment” or “the previous moment” have no meaning. As Si-mon and Stinchcombe (1989) point out this means that sensible sounding strategiessuch as “I play left at time zero; at every subsequent time, play the action I playedlast period” are not well defined. The approach that Simon and Stinchcombe takeis to restrict attention to strategies that are uniquely defined as the limit point forany discrete time grid as the grid becomes arbitrarily fine; their conditions F1-F3are designed to generate such strategies.We present two proofs of proposition 3.2 in this section that follow this model-ing approach. The first proof, which we call the “heuristic proof” does not directlyaddress many of the complexities associated with continuous time, other than to re-strict the strategy set to the strategies defined above. The question of whether thisrestriction is appropriate is not undertaken here. For this reason we also providean alternative proof, resting almost entirely on results in Simon and Stinchcombe(1989), which demonstrates that our equilibria are sound. We also provide a heuris-tic proof of remark 3.1.Heuristic proof of proposition 3.2. Consider, the strategies:st′i (Ht) ={0 if Ht = {1,1}t with 0≤ t < t ′ ≤ t∗1 otherwise(C.1)2The conditions require a bounded number of changes in action for each player, piecewise con-tinuity with respect to time and strong right continuity with respect to histories. The first two areobviously satisfied here, and the requirement that si({1, t ′}t) = 1 then si({1, tˆ}t) = 1 for all tˆ ≤ tensures strong right continuity with respect to histories.161Each of these strategies generates simultaneous entry at t ′. To aid intuition,we focus on these strategies precisely because they are strategies such that thereis no delay in responding to an opponent’s entry in any history. Such a delay is(weakly) payoff decreasing, and certainly cannot be observed in an equilibrium.More formally, this implies ruling out all strategies si(Ht) = 0 where Ht = {1, t ′}for t ′ < t.We claim that the strategies given in equation C.1 form a set of (symmetric)equilibria. That is, the pair of strategies (st′i (Ht),st ′j (Ht)) are an equilibrium of ourPerfectly Continuous time game. To see that this is true, consider two cases.In the first case, consider a strategy that deviates from st′i (Ht) by replacingst′i = 1 with s′t ′i = 0 for some set of histories. Given our restriction that a firmcannot re-enter after it has exited, this can only occur for histories in which the firmhas not yet entered. This implies that these must be histories in which either ouropponent has entered, or no one has entered. Histories in which no one has enteredat times after t ′ are off the equilibrium path, so the change in strategy profile hasno effect on payoffs. Histories in which the opponent has entered will either be offthe equilibrium path, or on a path for which U(st′i ,st ′j )>U(s′t ′i ,st ′j ) (this is becausethe payoff for the second entrant is a decreasing function of their entry time).In the second case, consider a strategy that deviates from st′i (Ht) by replacingst′i = 0 with s′t ′i = 1 for some set of histories. It must be the case that at least one ofthese histories is on the equilibrium path: s′ will cause the firm to enter before t ′.Given the equilibrium strategies this deviation will generate an instantaneous replyfrom the firm’s opponent. We shall now observe joint entry at some time prior tot ′. Given that the payoff to joint entry is increasing on the interval [0, t∗], such adeviation reduces the firms payoff.Heuristic proof of remark 3.1. We proceed somewhat unusually and consider thesecond round of iterated elimination of weakly dominated strategies first.Consider the strategies used to build the equilibrium in the heuristic proof ofproposition 3.2. Given that we restrict attention to only these strategies, each of thestrategies of the type given in equation C.1 is weakly dominated by the strategy162si(Ht) ={0 if Ht = {1,1} with 0≤ t < t∗1 otherwise(C.2)which generates joint equilibrium entry at t∗.We now show that any strategy not in equation C.1 is weakly dominated by astrategy that is in equation C.1.Consider the class of strategies that has si({1,1}t)= 0 for all t < t ′ and si({1,1}t ′)=1. Each strategy in this class is weakly dominated by st′i as defined in equation C.1.3If their opponent is yet to enter by t = t ′ then all strategies in the class provide thesame utility. The only way that strategies in the same class can differ is in how theyrespond to entry from their opponent: because st′i responds instantly in all cases,each other strategy in the class must have a delay in response for at least someopponent entry time. Given that payoffs are decreasing in the entry time of thesecond firm, the dominance relation is established.All strategies must fall into one of the above classes.The strategy in equation C.2 is therefore the unique strategy that survives iter-ated elimination of weakly dominated strategies.We conclude this section with a more formal (but less self contained) proof ofproposition 3.2, drawing more directly on definitions and results from Simon andStinchcombe (1989).Formal proof of Proposition 3.2. The proof consists entirely of demonstrating thatTheorem 3 of Simon and Stinchcombe (1989) is applicable to our environment.We begin by stating this theorem. The definitions of all relevant notation can befound in Simon and Stinchcombe (1989).Theorem C.1 (Theorem 3 from Simon and Stinchcombe (1989).). Consider acontinuous-time game with a dH-continuous valuation function. Let f be a continuous-time strategy profile satisfying F1-F3. Suppose that there exists a sequence of δ n-fine grids, (Rn), where δ n→n 0 and a sequence (gn, εn) such that εn→n 0 and foreach n, gn is an εn-SGP equilibrium for the game played on Rn. Further suppose3Or, if t ′ > t∗ in violation of equation C.1, then the strategy is weakly dominated by st∗i .163that gn is defined by further restricting f|Rn to the R-admissible decision nodes.Then f is an SGP equilibrium for the continuous time game.We proceed in 5 steps:1. The game has a dH-continuous valuation function because all games with in-tegrable flow payoff functions, such as ours, have a dH-continuous valuationfunction.2. Label the strategy profile that induces entry at time t in equation C.1 as f (t).f (t) satisfies Simon and Stinchcombe (1989)’s conditions F1-F3.3. Take the sequence of grids to be grids with G = n uniformly spaced points.4. Define g(t)n to be the strategy profile where each agent plays “enter” at allgrid points that occur (weakly) after t. At all grid points (strictly) before t,each agent plays “wait” if both agents have played “wait” at all previous gridpoints and otherwise they play “enter”. This strategy is the restriction f|Rn tothe R-admissible decision nodes.5. It follows immediately from the proof of proposition C.3 that g(t)n formsan ε(t)n-equilibrium where ε(t)n is defined by replacing the inequality inequation C.4 with equality. Furthermore, ε(t)n→ 0 as G = n→ ∞.We therefore conclude that f (t) is an equilibrium of the continuous time game.We finish by noting that the strategy f (t) induces an outcome where both agentsenter at time t, and that the definition of f (t) allows 0 < t ≤ t∗. Joint entry at t = 0is obviously also supported by an equilibrium (if your opponent enters at t = 0 thenyour best response is also to enter at t = 0). Therefore, we can support entry at anytime t ∈ [0, t∗] as an equilibrium in the continuous time game.Bergin and MacLeod (1993)Bergin and MacLeod (1993) take an alternative approach to modeling continuoustime games by introducing a general notion of inertia and modeling continuous164time as the limit as inertia disappears.4 In this subsection, we first formally de-fine a narrower notion of reaction-lag-based inertia appropriate for the setting ofour experiment. We then prove proposition 3.3 which claims that short of the con-dition limit (i.e. when inertia is greater than zero) firms must enter immediatelyin equilibrium. Finally, we provide an alternative proof of proposition 3.2 usingproposition 3.3 and Bergin and MacLeod (1993)’s Theorem 3. By doing so, weshow that in our game the two approaches to modeling Perfectly Continuous time– e.g. as the limit of Perfectly Discrete time a´ la Simon and Stinchcombe (1989)or as the limit of Inertial Continuous time a´ la Bergin and MacLeod (1993) – makeidentical predictions.5We introduce a simplified version of Bergin and MacLeod (1993)’s definitionof inertia that is appropriate for our game, capturing the key idea of inertia as areaction lag.6Definition C.1 (Inertia). Fix a time tˆ ∈ [0, t∗]. Suppose that si({1,1}t) = 0 for allt < tˆ and that si({1,1}tˆ) = 1. A strategy satisfies inertia if there exists a δ > 0 suchthat si({1, t ′}t) = 0 for all t, t ′ such that t ′ ≤ t < min{tˆ, t ′+δ}.Our inertia condition prevents firms from responding immediately to the entrydecisions of their opponents - responses are delayed by at least δ > 0 of the game.7We can think of tˆ as a firm’s planned entry time – if their opponent has not enteredyet, then the firm plans to enter at tˆ. The inertial requirement then states that thefirm cannot enter within δ of their opponent’s entry time, unless they are enteringat their planned entry time.We define Inertial Continuous time simply as Perfectly Continuous time re-stricted to strategies that satisfy the inertia condition of definition C.1 and can use4A key technical innovation of Bergin and MacLeod (1993) is that it allows one to model contin-uous time in infinitely repeated games. Infinitely repeated games violate the Simon and Stinchcombe(1989) assumption that an agent will change their behaviour a finite number of times.5We conjecture that this is also true more generally, whenever the relevant conditions of bothBergin and MacLeod (1993) and Simon and Stinchcombe (1989) are satisfied.6Bergin and MacLeod (1993) assume that the action space is constant across time, violating acondition in our experiment: we do not allow firms to exit the market once they have entered (i.e.once a firm has exited their action space shrinks from two actions to only one action). This technicalproblem can be neatly resolved by imposing an arbitrarily large amount of inertia to all histories inwhich the firm has already entered.7Recall that we define δ ≡ δ0T , where δ0 is a subject’s natural reaction lag in real time and T isthe game length in real time.165this to provide a proof of proposition 3.3.Proof of proposition 3.3. The pure strategy case provides the intuition for the fullproof, so we first prove that there are no pure strategy equilibria with delayed entry.We proceed by contradiction. Suppose that there is a pure strategy equilibriumwhere the earliest entry is not at time 0. Without loss of generality assume thatfirm i enters weakly before firm j at time t > 0. It is easy to verify that firm j hasa profitable deviation: entering at a point that is δ before t (or at 0 if t < δ ). Thiscontradicts the assumption that the strategies form an equilibrium.To deal with mixed strategies, the proof by contradiction needs only minimalchanges. Write t j > 0 and ti for the latest entry time used by firms j and i in amixed strategy equilibrium for histories where their opponent has yet to enter. Weproceed by demonstrating that there is a strategy in the support of player j’s mixthat is not optimal.Suppose that t j ≥ ti. Consider the history where we arrive at time max{ti−δ ,0}and neither firm has entered yet. Clearly firm j’s strategy is not optimal: given firmi’s strategy, and the fact that we reach the current subgame with positive probability,firm j should enter immediately with certainty. This contradicts the assumptionthat the strategies form an equilibrium.The logic behind the proofs of propositions 3.1 and 3.3 does not apply to propo-sition 3.2. The reason is that in continuous time a firm can respond instantaneouslyto its opponent’s entry. The logic of the proof by contradiction thus does not hold:firm j’s best response is no longer to preempt firm i, but to enter simultaneouslywith firm i. We conclude the section with an alternative proof of proposition 3.2.Alternative proof of Proposition 3.2 using Bergin and MacLeod (1993). The proofconsists entirely of demonstrating that Theorem 3 of Bergin and MacLeod (1993) isapplicable to our environment. There are some technical hurdles to be surmountedas Bergin and MacLeod (1993) define strategies as being mappings from outcomesto actions, rather than the more standard mappings from histories to actions. Fortu-nately, in our game, there is a straightforward mapping from histories to outcomesthat simplifies matters greatly.We first state Bergin and MacLeod’s Theorem 3:166Theorem C.2 (Theorem 3 from Bergin and MacLeod (1993)). A strategy x ∈ S∗is a subgame perfect equilibrium if and only for any Cauchy sequence, {xn} con-verging to x, there is a sequence εn → 0, such that xn is an εn-subgame perfectequilibrium.To start, we note that although the theorem implies that all convergent Cauchysequences need to form a convergent sequence of ε-equilibrium it is clear fromBergin and MacLeod’s proof that it is sufficient to find a single satisfactory se-quence. Once a single sequence is found, part (b) of their proof implies that x isan equilibrium and then part (a) of their proof implies that all other convergentsequences must also have associated convergent ε paths.We note again that Bergin and MacLeod (1993) use a special domain for theirstrategies. The domain of x is the object (A,T ) where A = {ti, t j} is a set of out-comes (one for each player) and T = [0,1] is the set of feasible entry times. Clearly,we can translate each of our strategies into the Bergin and MacLeod (1993) formu-lation. For example, the strategy that induces player i to enter at the earlier of timeτ and immediately after their opponent enters could be written as:x1((·, t j), t) ={0 if t < τ and t < t j1 otherwise(C.3)The set of strategies that satisfy our inertia condition (definition C.1) maps intoa subset of Bergin and MacLeod’s set of strategies S. If we add the set of strategiesthat are formed in the limit as δ → 0, then we have a set of strategies that are asubset of Bergin and MacLeod’s set of strategies S∗.8Now, for each time τ ∈ [0, t∗], let x(τ)n be a sequence of strategies for bothplayers that satisfy definition C.1 with τ = tˆ and δ = δ n such that δ n→ 0. Thenx(τ)n Cauchy converges to the strategy given in equation C.3. Furthermore, fromthe proof of proposition C.4, for each x(τ)n we can identify an εn, which is foundby replacing the inequality in equation C.7 with an equality, such that the strategiesform an εn-equilibrium. Clearly, the sequence εn→ 0 as δ n→ 0.Therefore, we conclude that the strategy x(τ), which generates joint entry att = τ , forms an equilibrium of the continuous time game. Given that this is true for8Note that the strategy presented in equation C.3 is in S∗ but not in S because it does not satisfythe inertia condition.167all τ ∈ [0, t∗], we can sustain all such entry times in equilibrium.C.3 ε-equilibriumIn subsection C.3.1 we prove three propositions that completely characterize the ε-equilibrium sets for each of our timing protocols. In appendix C.3.2 we use thesecharacterizations to prove comparative static propositions stated in section 3.2.3that form the basis of alternative hypotheses for the experiment.Following Radner (1980) we assume that players are willing to tolerate a pay-off deviation from best response of size ε and treat their counterparts as havingthe same tolerance. We can then form a set of propositions that provide alterna-tive predictions to Nash equilibrium as a function of ε for each of our three mainprotocols.C.3.1 Characterization of ε-equilibrium setsAgain, we begin by considering Perfectly Discrete Time. Recall that for a gamewith grid size G, dates begin at t ={0, 1G ,2G , ...,1}(a total of G+ 1 dates). Fornotational simplicity, assume that the grid is such that t∗ = 1− ΠD4c lies exactlyon some grid point (this assumption simplifies the following expressions and isimposed in the parameterization of our experiment), and label this gridpoint kG .Proposition C.3. Suppose that all agents enter at or before t∗.9 In a PerfectlyDiscrete time game with G+ 1 periods, t∗ = kG and tolerance ε the set of entrytimes that can be sustained in a pure strategy sub-game perfect ε-equilibrium isgiven by{0, 1G , ...,κG}where κ is the largest integer that satisfies 0≤ κ ≤ k and1G[3ΠF2− c(2+ 1G)]+κG2[2c− ΠF2]≤ ε. (C.4)If no non-negative integer satisfies equation C.4 then κ = 0 (i.e. the unique equi-librium is immediate entry).9If the ε-equilibrium set includes t∗, it is possible for entry times after t∗ to also be supported as ε-equilibria. Such entry times never arise in the data and violate the conceptual spirit of ε-equilibriumoutlined by Radner and we assume them away purely to simplify the discussion and notation.168Proof of proposition C.3. We begin with two assumptions. Firstly, we assume thatall firms enter at or before t∗. Secondly, when one firm enters the other firm shallenter as soon as possible afterwards (in the next sub period). We shall demonstrateat the end of the proof that the second assumption can be disposed of, but imposingit simplifies the proof.We proceed by backwards induction. Suppose that firms arrive at period k−1and neither firm has entered yet. We are interested in determining whether cooper-ation can be sustained for one more period. If it can, then the payoff to each firmwill be U( kG ,kG). If, however, a firm defects and enters immediately then they willearn U( k−1G ,kG).After some algebra, we see that U( k−1G ,kG)−U( kG , kG) = 1G[3ΠF2 − c(2+ 1G)]+kG2[2c− ΠF2].Therefore, we conclude that if neither firm has entered when firms arrive atperiod k−1, then joint entry at period k can be sustained as an ε-equilibrium ifε ≥U(k−1G,kG)−U( kG,kG). (C.5)Now, rollback to period k− 2. Suppose that firms arrive at period k− 2 andneither firm has entered yet. Can cooperation be sustained for one more period?There are two cases.In the first case equation C.5 holds, so that if both firms wait at period k−2 thenthere is an equilibrium continuation where each firm earns U( kG ,kG). If, however,a firm enters at period k− 2 they will earn U( k−2G , k−1G ). Waiting can thereforebe sustained as an ε-equilibrium if ε ≥ U( k−2G , k−1G )−U( kG , kG). This inequalitymust hold whenever equation C.5 holds because the utility function is increasingin (joint delay of) entry times. We therefore conclude that if both firms may waitin an ε-equilibrium at period k−1 then they may also wait in an ε-equilibrium atperiod k−2 (and the same logic implies this must be true for any earlier period aswell).In the second case equation C.5 does not hold. Therefore, if both firms waitat period k−2 they must both enter at period k−1 and the continuation payoff isU( k−1G ,k−1G ) for both firms. Defecting to immediate entry will earn U(k−2G ,k−1G ).After some algebra, we see that U( k−2G ,k−1G )−U( k−1G , k−1G )= 1G[3ΠF2 − c(2+ 1G)]+169k−1G2[2c− ΠF2]<U( k−1G ,kG)−U( kG , kG).Therefore, we conclude that if neither firm has entered when firms arrive atperiod k−2, then joint entry at period k−1 can be sustained as an ε-equilibrium ifε ≥U( k−2G , k−1G )−U( k−1G , k−1G ). Notice that the ε cutoff at period k−2 is less thanthe cutoff at k− 1. Therefore, there are some ε values where co-operation can besustained at period k−2, but not at period k−1.The proof continues by induction, rolling back one period at a time and notingthat U( k−n−1G ,k−nG )−U( k−nG , k−nG ) is decreasing in n, and substituting n= κ deliversthe equation in the proposition.We still need to demonstrate that relaxing our initial assumption that entry byone firm would be immediately followed by entry from the other will not changethe equilibrium set found above. There are two concerns: firstly, that allowinglengthier delays might destroy some equilibria described above and, secondly, thatallowing lengthier delay might create new equilibria.10No equilibria can be destroyed. Take an equilibrium found above, and allowan agent to unilaterally deviate to a strategy that involves lengthy delays after theiropponent’s entry (in at least some subgames). This deviation must weakly decreasethe agent’s payoff, so it is not a profitable deviation and the original equilibriumsurvives.New equilibria can be created, but none of these equilibria will involve aninitial entry time that is later than the set of entry times described in the proposition.We demonstrate this with a proof by contradiction. For a given ε , define κ as givenby equation C.4. Now, suppose that there is an equilibrium with entry at the pair ofnodes (κ+ jG ,κ+ j+iG )with j≥ 1 and i≥ 2. Then, from the definition of ε-equilibriumit must be the case that U(κ+ j−1G ,κ+ jG )−U(κ+ j+iG , κ+ jG )≤ ε (otherwise the secondentrant would have a profitable deviation).Now, it is also true that U(κ+ j−1G ,κ+ jG )−U(κ+ j+iG , κ+ jG ) > U(κ+ j−1G , κ+ jG )−U(κ+ j+1G ,κ+ jG )≥U( κG , κ+1G )−U(κ+2G , κ+1G )>U( κG , κ+1G )−U(κ+1G , κ+1G )> ε .The second inequality is demonstrated by writing out U( k−1G ,kG)−U( k+1G , kG),collecting terms and noting that the coefficient on k is 8c−(ΠF+ΠS)2G2 > 0 (intuitively,10Intuitively, one way to view this is to think of longer delays as being analogous to a gamewhere the grid becomes coarser after one player has entered. This increases returns to defection andtherefore shrinks the equilibrium set.170this is demonstrating that the payoff difference between preempting rather than be-ing preempted is increasing in time). The final equality follows from the definitionof κ . This establishes the contradiction.We now characterize the ε-equilibrium set for Inertial Continuous time.Proposition C.4. Assume that both firms enter at some time t ≤ t∗.11 In an inertialcontinuous time game with inertial delay δ , and tolerance ε , all entry times t ∈(0, τ¯] can be sustained in a subgame perfect ε-equilibrium where τ¯ is the solutionto:arg maxτ∈[δ ,t∗]τ (C.6)s.t. δ[(1− τ2+1)ΠF − c(2−2τ+δ )]≤ ε. (C.7)If no such τ exists then the unique equilibrium is immediate entry.12Proof of proposition C.4. The proof technique is the same as for proposition C.3.Again, we begin by assuming immediate response (or as soon as possible given thereaction lag) to an entry, and establish that it can be discarded ex-post.Consider a strategy such that each firm enters at t = τ if their opponent has yetto enter at τ , and they enter as soon as possible if their opponent enters before τ . Abest response to such a strategy is to to enter at τ−δ , and enter as soon as possibleif their opponent enters before τ−δ . Notice that entering in range (τ−δ ,τ) is nota best response because of the assumption implicit in definition C.1 which tells usthat the opponent could still enter at τ in this case.The payoff from waiting both firms waiting until τ is given by U(τ,τ), and thebest response strategy of entering at τ−δ earns a payoff of U(τ−δ ,τ).11See Footnote 9.12In our experimental implementation of inertial time immediate entry typically involved subjectsentering at δ , as it took subjects a reaction lag to respond to the start of the period.171Observe that U(τ − δ ,τ)−U(τ,τ) = δ [(1−τ2 +1)ΠF − c(2−2τ+δ )], andthat this expression is increasing in τ .We therefore conclude that the strategy considered above can be sustained asan ε-equilibrium if ε ≥ δ [(1−τ2 +1)ΠF − c(2−2τ+δ )].This establishes the proposition.As dealt with in the proof of proposition C.3, it is possible to discard the as-sumption of immediate response. Suppose that there is an ε-equilibrium wherethere is a delay δˆ > δ between entry times and that the first entry time occurs att > τ¯ where τ¯ is the largest solution to equation C.7. The payoff for the secondmover is then given by U(t+ δˆ , t). The assumption that this constitutes a subgameperfect ε-equilibrium implies that U(t−δ , t)−U(t + δˆ , t)≤ ε . But, we also haveU(t−δ , t)−U(t+ δˆ , t)>U(t−δ , t)−U(t+δ , t)>U(τ¯−δ , τ¯)−U(τ¯+δ , τ¯)>U(τ¯ − δ , τ¯)−U(τ¯ + δ , τ¯) > U(τ¯ − δ , τ¯)−U(τ¯, τ¯) = ε . Note that the second in-equality follows from the derivative d(U(t−δ ,t)−U(t+δ ,t))dt =δ2 [8c− (ΠF +ΠS)]> 0,and that the final equality follows from the definition of τ¯ . We have reached acontradiction, and reject the existence of such equilibria.Finally, we note that the central motivation for ε-equilibrium – that agents arewilling to tolerate small deviations from best response in order to achieve high,cooperative payouts – loses its bite in Perfectly Continuous time where agents canachieve these same payouts without deviating from best response at all. For com-pleteness we state this as a final proposition.Proposition C.5. Assume that all agents enter at or before t∗.13 In Perfectly Con-tinuous Time, the set of first entry times that can be supported in Nash equilibriumand the set of first entry times that can be supported in ε-equilibrium are identicalfor any ε: any entry time t ∈ [0, t∗] can be supported in either case.Proof of proposition C.5. The Nash equilibrium set is given by proposition 3.2. AllNash equilibria are ε-equilibria, so all entry times t ∈ [0, t∗] can also be supported13See Footnote 9.172in an ε-equilibrium. The proposition rules out entry times after t∗ by assumption,so the two sets are identical over the allowable range of entry times.14C.3.2 ε-equilibrium: Proofs of propositions stated in section 3.2.3In this subsection we use the preceding characterizations to prove propositionsfrom section 3.2.3.Proof of proposition 3.4. Proposition 3.4 is an application of proposition C.4.Substituting t∗ into equation C.7, we see that t∗ is an equilibrium for any ε > 0when δ satisfies:δ[(1− t∗2+1)ΠF − c(2−2t∗+δ )]≤ ε. (C.8)Equation C.8 bounds a quadratic equation in δ with negative leading coeffi-cient, so that any δ less than the smaller root will satisfy our requirements. Denotethis root by δ˜ and write b = (1−t∗2 +1)ΠF − c(2−2t∗) for convenience. The ever-handy quadratic formula delivers:δ˜ =b2c−√b24c2− εcNoting that δ˜ is always positive whenever ε > 0 completes the proof of the firstpart of the proposition.For the second part of the proposition, note that because inertia cannot exceedthe total game length δ ≤ 1. Substituting τ = δ into equation C.7 we find that therewill exist no ε-equilibrium with delay if:δ[32ΠF −2c+δ (c− ΠF2 )]> ε.The left hand side of this equation is increasing in δ over the interval [0,1] andclearly less than ε at δ = 0. We can therefore find a δ that satisfies the inequality14Again, in Perfectly Continuous time, it is possible to support entry times after t∗ in an ε-equilibrium. All such equilibria would, however, be ruled out by applying Simon and Stinchcombe’s1989 iterated elimination of weakly dominated strategies argument.173only if the inequality holds when δ = 1. When δ = 1 the inequality simplifies toΠF − c > ε .The upper bound on ε in the second part of the proposition is a natural conse-quence of the “thick” indifference curves that are associated with large values of ε .For any game with bounded payoffs there will exist a ε large enough that an agentis indifferent between all outcomes, and all outcomes may therefore be sustainedin a ε-equilibrium. The restriction on ε can therefore be viewed as a non-trivialityrequirement.174C.4 Decision Making Under UncertaintyThis appendix contains further information on the decision making under uncer-tainty rules that are discussed in the main text. In section C.4.1 we give moredetails on the three decision rules considered in section 3.5.1. Section C.4.2 pro-vides a proposition showing that the asymptotic effect inertia has on cooperationin our game under the MRA heuristic holds for a broader class of dilemma-likegames.C.4.1 Three decision rulesIn this section we give more details on the three non-parametric heuristics dis-cussed in Milnor (1954) and examined in 3.5.1 of the body of the paper. Supposeagents play trigger strategies15 where a trigger time t is the strategy that enters atthe earliest of time t and the soonest available entry time after their opponent enters.We shall write U¯(t, t ′) to identify the payoff associated with agents using triggerstrategies with entry times t and t ′ so that, for example, U¯(t, t ′) = U(t, t + δ ) inInertial Continuous Time when t +δ < t ′. Notice that the payoffs associated witha particular pair of trigger times will therefore vary with the continuity and inertiaof the game.Our key assumption on beliefs is that agents assume that their opponent willuse a trigger strategy with t ∈ [0, t∗] but have complete uncertainty regarding whichtrigger strategy within the set will be played (this uncertainty is imposed in Milnor(1954) and (Stoye, 2011a) via a symmetry axiom argued for in Arrow and Hurwicz(1972)).16 In order to maintain parsimony (particularly important here because weare conducting this analysis ex post), we focus on simple applications of threedecisions rules without adding additional potentially ad hoc structure (for example15For our game, a restriction to admissible strategies implies trigger strategies.16We interpret the term “complete uncertainty” under a “no priors” interpretation (Stoye, 2011a).In the context of our game this implies that the agent believes that any pure strategy in the set [0, t∗]may be used, but the agent has no additional information. Alternatively, it would be possible to use an“exogenous priors” interpretation (Stoye, 2011a) which allows for any probability distribution overthe set of states – a multiple priors framework. For our game, this interpretation requires the inclusionof mixed strategy beliefs. It can be demonstrated, however, that the results are robust to introducingbeliefs over mixed strategies (e.g. see proposition 3.19 in Halpern and Pass (2012) for the MRAcase). As a consequence our predictions do not change under an exogenous priors interpretation ofbeliefs.175equilibrium structure).17In a first model, which we call Laplacian Expected Utility, agents respond tostrategic uncertainty as if they are expected utility maximizers with uniform beliefsover counterpart strategies. Milnor (1954) provides an axiomatization of this de-cision model, showing that this representation is the result of combining expectedutility axioms with the symmetry axiom that underlies the type of extreme strategicuncertainty we are considering here. In our case, the utility of an LEU agent canbe written asULEU(t) =∫ t∗0U¯(t,s)ds,and finding the LEU maximizing entry time is then straightforward.A second model, Maximin Ambiguity Aversion (or Maxmin Expected Utility)has been axiomatized for endogenous priors by Gilboa and Schmeidler (1989) andfor exogenous priors by Stoye (2011b). The model provides a straightforwardconservative heuristic for decision making under uncertainty. As with our otherdecision rules, we assume complete uncertainty over the interval [0, t∗]. Adaptedto our setting, the maximin expected utility of an agent entering at time t can bewritten as:UMAA(t) = mint ′∈[0,t∗]U¯(t, t ′).It is straightforward to see that the argument of the minimization can be takento be t ′ = 0 for all possible entry times t: having your opponent enter immediatelyis (weakly) the worst possible thing that can happen for any strategy in every treat-ment. The best response to t ′ = 0 is t = 0 so that the maximum of the minimum17Though we do not impose equilibrium structure on our decision rules, we note that there existequilibrium concepts that produce predictions that are equivalent to our MRA and MAA decisionrules. For ambiguity averse agents, Lo (2009) introduces an equilibrium concept that generates iden-tical predictions to our MAA decision rule. For agents with minimax regret preferences, the Halpernand Pass (2012) equilibrium notion of “iterated regret minimization with prior beliefs” generatesthe same predictions as our MRA decision rule when prior beliefs are restricted to trigger strate-gies. While we elect to maintain parsimonious decision rules with exogenous beliefs, it is possibleto build equilibrium frameworks (with varying degrees of endogenous belief formation) that producethe same behavioral predictions.176payoffs is given by UMAA(0) =U(0,0).18Finally, the third (and main) model discussed in the paper, Minimax RegretAvoidance, can be traced back to Wald (1950) and Savage (1951) and was axiom-atized by Milnor (1954) for the case of discrete states and by Stoye (2011a) forthe case of continuos states.19 Applied to a strategic setting, it is a decision rulewhereby agents choose a strategy that minimizes the worst case regret with respectto the range of possible counterpart strategies. More formally, the regret betweentwo strategies, R(t, t ′), is the difference between the best response payoff and therealized payoff, so thatR(t, t ′) = ( maxtˆ∈[0,t∗]U¯(tˆ, t ′))−U¯(t, t ′).The maximal regret associated with a strategy is then defined asR(t) = maxt ′∈[0,t∗]R(t, t ′).Finally, the minimax regret strategy is the strategy that minimizes R(t). Writing tfor the minimax regret entry time20 we havet = arg mint∈[0,t∗]R(t) = arg mint∈[0,t∗]maxt ′∈[0,t∗]R(t, t ′). (C.9)18Again, while we focus on a very simple application of maximin to the strategy space, predictionsreported in the text are unchanged in equilibrium variations. For example, applying the epistemicallyfounded equilibrium model of Lo (2009) we find that the unique equilibrium entry time is t = 0 forall of our treatments (with the exception of PC where it also mirrors the Nash equilibrium), just as inour simpler, non-equilibrium approach.19Interest in minimax regret has increased in recent years with general theories of minimax regretequilibrium provided by Renou and Schlag (2010) and Halpern and Pass (2012), and specific appli-cations to bargaining in Linhart and Radner (1989) and monopoly pricing are found in Bergemannand Schlag (2008) and Bergemann and Schlag (2011).20Notice that there are two places in equation C.9 that we have implicitly restricted attention topure strategies. The first is in the max operator, where we consider only beliefs over pure strategies.This restriction is without loss of generality because the regret maximizing strategy will always be apure strategy in our game (see Halpern and Pass (2012), particularly proposition 3.19, for a detailedand general discussion). The second is in the argmin operator, where we require agents to selecta pure strategy. This deliberate modeling choice reflects the experimental design, where subjectsmust implement a pure strategy. While the introduction of mixed strategies can affect the set ofregret minimizing strategies, it is not appropriate to allow for mixing in that fashion when modelingexperimental behavior with non-expected utility preferences.177We calculate specific MRA predictions in the body of the paper numerically.21 Insection C.4.2, below, we show that for a broad class of games in trigger strategies(including ours) the main effect of inertia observed in these calculations (and inour experiments) must hold under the MRA decision rule.C.4.2 Minimax regret in a broader class of gamesIn our experiment, behavior grows more efficient as inertia falls to zero, a resultthat we show is consistent with subjects using a minimax regret avoidance ruleto choose among strategies. In this section we show that there is a general classof dilemma-like inertial time games for which the minimax regret decision rulegenerates very similar predictions: when inertia is large, minimax regret predictsbehavior that is bounded away from the socially optimal outcome and as inertiashrinks towards 0 behavior approaches the socially optimal outcome. Our experi-mental design can be rewritten as a special case of this result, as can the prisoner’sdilemma, Bertrand competition, Cournot competition and public goods games.We begin by defining the structure of the game and the assumptions on thepayoff functions that are necessary for the result to hold, and then define our classof strategies. We restrict attention to trigger strategies, following a suggestion inHalpern and Pass (2012), for both normative and positive reasons. On the pos-itive side, trigger strategies are typically used by subjects in dilemma games, apoint emphasized in Friedman and Oprea (2012). On the normative side, allowingfor fully general strategies leads to agents using minimax regret decision rules to21 Specifically, we define two types of regret: type 1 regret R1 from entering too early, and thetype 2 regret R2 from entering too late:R1(t, t ′) ={R(t, t ′) if t < t ′−δ0 if t ′−δ ≤ t (C.10)andR2(t, t ′) ={R(t, t ′) if t ′−δ < t0 if t ≤ t ′−δ . (C.11)We note that R1(t) = R(t, t∗) =U(t∗−δ , t∗)−U(t, t∗) and R2(t) = R(t, t−δ ) =U(t−2δ , t−δ )−U(t, t−δ ), giving us each type of regret as a function of δ . Then, by noting that R1(t) is decreasingand R2(t) is increasing, and noting that R(t) = max{R1(t),R2(t)}, we can find R(t) by solving theequation R1(t) = R2(t). The resultant solution is a function of δ and it is then straightforward tonumerically compute minimax regret predictions.178“believe” that their opponent is using strategies that are both non-admissible andinvolve making large sacrifices with no hope of reciprocation.22Suppose that we have a game being played in inertial continuous time betweentwo players on the interval [0,1]. Payoffs are defined via a symmetric flow utilityfunction. The (instantaneous) action space for both players is constant with rep-resentative elements ai,a j ∈ A, and the flow utility is denoted by uti(ai,a j). Weidentify some key action profiles, which shall be assumed to exist:• argmaxa∈A ut(a,a) = aˆt , with aˆt unique for all t. We shall call aˆt the cooper-ative action at time t. Furthermore, assume that aˆt is constant so that aˆt = aˆfor all t.23• The instantaneous game at t has a unique Nash Equilibrium that is denotedby a∗t . We shall call a∗t the Nash equilibrium action at time t. Furthermore,assume that a∗t is constant so that a∗t = a∗ for all t.24• If ut(a˜, aˆ) > ut(aˆ, aˆ), then we shall call a˜t an exploitative action at time t.25We shall assume that there exists at least one exploitative strategy at every t.It will typically be the case that a∗ is an exploitative strategy.We now define a class of generalized trigger strategies. Our trigger strategiescollapse to the equivalent of grim trigger in games with only two strategies. Weshall require that each trigger strategy uses a fixed, pre-determined and constantexploitation strategy a˜.26 A trigger strategy with trigger time t1 and a constant a˜satisfies the following conditions:22As an example, consider a repeated prisoners dilemma in continuous time. Consider an agentwho believes that their opponent will respond to any defection with permanent defection unlessthe initial defection occurs precisely√epi seconds into the game, in which case the opponent willcontinue to cooperate. Such an agent will experience extremely large regret unless they defect atprecisely√epi . Restricting beliefs to trigger strategies removes such strategies from the considerationset of agents when assessing regret.23This final assumption is without loss of generality. For example, if we identify actions withchoosing a real number, we can simply relabel the cooperative action to be 1 at every instant.24When a∗t 6= aˆt for all t, then this is also without loss of generality.25Where it will not result in confusion, we will sometimes write a˜ instead of a˜t .26While this assumption seems quite restrictive it turns out not to affect outcomes very much.The results presented below hold for any pair of exploitation strategies, and the difference in regretminimizing trigger times for differing exploitation strategies is arbitrarily small for small δ .1791. At time 0, play aˆ.2. At each time t < t1 if the history is such that both players have always playedaˆ, or if the agent has always played aˆ and the opponent has played aˆ at all t ′such that t ′ ≤ t−δ , then play aˆ.3. At each time t < t1 if the agent has played anything other than aˆ then play a∗at all remaining moments.4. At each time t < t1, if the opponent has played anything other than aˆ at anytime t ′ such that t ′ < t−δ then play a∗ at all remaining moments.5. At time t1, if the history is such that both players have always played aˆ atall t < t1 then play a˜ over the period t ∈ [t1, t1 + δ ) followed by a∗ at allt ≥ t1+δ .6. At time t1, if the history is such that the agent has always played aˆ at allt < t1 and the opponent played aˆ at all t < t ′ with t1− δ ≤ t ′ and anythingother than aˆ at t ′ then play a˜ over the period t ∈ [t1, t ′+δ ) and play a∗ at allt ≥ t ′+δ .We can therefore identify each trigger strategy by its trigger time and the ex-ploitative strategy that it implements: a trigger strategy is a pair (t1, a˜), althoughwe note again that because a˜ is fixed it is not a choice variable, and for this reasonwe shall often refer to a trigger strategy solely by its trigger time. We will use theexpression U((t1, a˜),(t ′1, a˜′)) or U(t1, t ′1) to denote the payoff induced by the use oftrigger strategies. We also interpret the reaction lag, δ , as being a fixed parameterof the game rather than a strategic choice of the agents.We shall impose the following assumptions on the instantaneous flow payofffunctions. While the assumptions may appear to be onerous, we shall demonstratethat several standard applications satisfy them.• ut(aˆ, a˜)≤ ut(a∗,a∗)< ut(aˆ, aˆ)< ut(a˜, aˆ) and ut(aˆ, a˜)≤ ut(a˜, a˜)< ut(aˆ, aˆ) forall t and all a˜, and that each of these flow payoff functions are continuouslydifferentiable in t. This can be interpreted as the game having a ‘dilemma’structure at every instant.180• We also have two conditions on the rate of change of the flow payoffs.ut+t′(a∗,a∗) < ut(aˆ, aˆ) and ut+t ′(a˜, a˜) ≥ ut(aˆ, a˜) for all t and all t ′ ≤ δ andall a˜. These rather weak conditions ensure that the ‘dilemma’ nature of thegame persists across every stretch of time that is less than or equal to onereaction lag. A game with constant flow payoffs will trivially satisfy thiscondition if it satisfies the previous dilemma conditions.• dut(a˜,aˆ)dt ≥ dut(a˜,a˜)dt ≥ 0, dut(a∗,a∗)dt ≥ 0, dut(aˆ,a˜)dt ≤ 0 and dut(aˆ,aˆ)dt ≤ 0 for all t and alla˜. Jointly, these conditions imply that playing the socially optimal strategybecomes (weakly) less attractive as the game progresses.We now proceed to our main result via a series of lemmas. Proofs are collectedin appendix C.4.3. Our first lemma and corollary clarify behaviour in the limitingcase as inertia approaches zero.Lemma C.6. When δ = 0, trigger strategies are identified only by their triggertime, and the class of trigger strategies with t = 1 is weakly dominant among theclass of all trigger strategies. Furthermore, a trigger strategy with a trigger timet < 1 is not weakly dominant.Corollary C.7. When δ = 0 any trigger strategy with a trigger time of t = 1 gen-erates a regret of 0. Any trigger strategy with a trigger time of t < 1 generatespositive regret.Next, we establish some useful properties of the best response function.Lemma C.8. The best response trigger strategy t ′ to a trigger strategy t satisfiest−δ ≤ t ′ ≤ t. Furthermore, the best response payoff, U(t ′, t) is increasing in t.We shall now reintroduce our two types of regret. Type 1 regret is regret fromusing a trigger time that is earlier than optimal, and type 2 regret is regret fromusing a trigger time that is later than optimal. In our current setup, we define thetype 1 and type 2 regret between two strategies as:R1(t, t) =U(t ′, t)−U(t, t) if t < t ′0 if t ′ ≤ t (C.12)181andR2(t, t) =U(t ′, t)−U(t, t) if t ′ < t0 if t ≤ t ′. (C.13)We can also define the type 1 and type 2 regret of a trigger strategy as R1(t) =maxt R1(t, t) and R2(t) = maxt R2(t, t), noting that typically the arguments of themaxima will be different for R1(t) and R2(t). Finally, we can write R(t)=max{R1(t),R2(t)}.Our next two propositions establish some facts about type 1 and type 2 regret.Lemma C.9. Type 1 regret is strictly decreasing in trigger time over the intervalt ∈ [0,1−δ ) and is equal to 0 in the interval t ∈ [1−δ ,1]. Furthermore, R1(t) iscontinuous.Lemma C.10. Type 2 regret is weakly increasing in trigger time. Furthermore,R2(t) is continuous and limδ→0 R2(t) = 0 for all t.Finally, we arrive at the proposition that establishes that MRA behaviour sup-ports greater cooperation as δ decreases, and that MRA behaviour converges tofully cooperative play in the limit. The proof is a straightforward application of theintermediate value theorem.Proposition C.11. When δ > 0 the earliest entry time that minimizes regret, tδ , isstrictly less than 1. Furthermore, tδ is decreasing in δ and limδ→0 tδ = 1.ExamplesThe following are examples that satisfy the requirements of our setup above.1. Prisoner’s dilemma in inertial continuous time, such as Friedman and Oprea(2012).2. Oligopoly game with a constant flow rate of customers arriving every in-stant. To illustrate, in the Bertrand case, suppose that the two firms havezero marginal costs and each instant a mass 1 of customers arrives and de-mand the quantity Q = 1− P from the lowest priced firm every instant.In this case a∗ = 0 and aˆ = 12 and take the exploitation strategy to be a˜ =1820.49. The associated flow payoffs are u(aˆ, a˜1) = 0,u(a∗,a∗) = 0,u(a˜, a˜) =0.12495,u(aˆ, aˆ) = 18 ,u(a˜, aˆ) = 0.2499.3. Cournot oligopoly game. To illustrate suppose that the two firms have zeromarginal costs and each instant a mass 1 of customers arrives and demandthe quantity Q= 1−P from the lowest priced firm every instant. In this casea∗ = 13 , aˆ =14 and an example exploitation strategy is a˜ =38 . The associatedpayoffs are u(aˆ, a˜) = 332 ,u(a∗,a∗) = 19 ,u(a˜, a˜) =332 ,u(aˆ, aˆ) =18 ,u(a˜, aˆ) =964 .4. Public goods game in continuous time. At each instant each player may con-tribute a ∈ [0,1] to the public good. The public good provides instantaneousutility to each player that is equal to three-quarters of the total instantaneouscontribution. In this case a∗ = 0, aˆ = 1 and an example exploitation strategyis a˜= 0. The associated payoffs are u(aˆ, a˜) =−0.25,u(a∗,a∗) = 0,u(a˜, a˜) =0,u(aˆ, aˆ) = 0.5,u(a˜, aˆ) = 0.75.C.4.3 ProofsProof of lemma C.6. When δ = 0, the class of trigger strategies becomes muchsimpler. The trigger strategy with trigger time t ′ implements the following strategy.At all t ≥ t ′ play a∗. At all t < t ′ play aˆ if your opponent has played aˆ at all s≤ t,otherwise play a∗. Because there is no exploitation period trigger strategies can beidentified by their trigger time alone.Compare two strategies with trigger times t1 and t2 with t1 < t2. Denote theopponent’s trigger time by t3. If t3 ≤ t1 then both t1 and t2 earn the same payoff forall t. If t3 > t1 then both t1 and t2 earn the same payoff on [0, t1] and [min{t3, t2},1]but on the interval [t1,min{t3, t2}] then t1 earns flow payoffs ut(a∗,a∗) while t2earns flow payoffs ut(aˆ, aˆ). Therefore t2 earns a strictly higher payoff against sometrigger strategies but the converse does not hold, so that t2 weakly dominates t1.Proof of corollary C.7. A weakly dominant strategy generates 0 regret, and non-weakly dominant strategies generate positive regret.Proof of C.8. In the range t ′ > t+δ the payoff U(t ′, t) is independent of t ′.183In the range t < t ′ ≤ t+δ we haveU(t ′, t) =∫ t0us(aˆ, aˆ)ds+∫ t ′tus(aˆ, a˜)ds+∫ t+δt ′us(a˜, a˜)ds+∫ 1t+δus(a∗,a∗)ds(C.14)which is decreasing in t ′ because ut(aˆ, a˜)< ut(a˜, a˜).When t ′ < t−δ we haveU(t ′, t) =∫ t ′0us(aˆ, aˆ)ds +∫ t ′+δt ′us(a˜, aˆ)ds +∫ 1t ′+δus(a∗,a∗)ds (C.15)Taking the derivative with respect to t ′ yieldsut′(aˆ, aˆ)+ut′+δ (a˜′, aˆ)−ut ′(a˜′, aˆ)−ut ′(a∗,a∗)which is strictly positive.This establishes that, for any a˜ and any (t, a˜), we must have t−δ ≤ t ′ ≤ t.We can now, therefore, write down the functional form of the best responsepayoff:maxt ′U(t ′, t)=maxt ′∫ t ′0us(aˆ, aˆ)ds+∫ tt ′us(a˜, aˆ)ds+∫ t ′+δtus(a˜, a˜)ds+∫ 1t ′+δus(a∗,a∗)ds(C.16)Now, applying the envelope theorem, we haved maxt ′U(t ′, t)dt= ut(a˜, aˆ) − ut(a˜, a˜) > 0 (C.17)if the constraints t−δ ≤ t ′ ≤ t are not binding.If the constraints are binding we have either184d maxt ′U(t ′, t)dt= ut(aˆ, aˆ) − ut(a˜, a˜) + ut+δ (a˜, a˜) − ut(a∗,a∗) > 0 (C.18)ord maxt ′U(t ′, t)dt= ut−δ (aˆ, aˆ) − ut−δ (a˜, aˆ) + ut(a˜, aˆ) − ut(a∗,a∗) > 0 (C.19)We have therefore established that the best response payoff is increasing in theopponent’s trigger time.Proof of lemma C.9. We shall use t ′ to denote the best response trigger strategy tothe trigger t. The type 1 regret between two strategies can then be written asR1(t, t) =U(t ′, t)−U(t, t) if t < t ′0 if t ′ ≤ t (C.20)The type 1 regret of an individual trigger strategy is defined by the maximumof the regret between two strategies:R1(t) = max(t)R1(t, t)Now, we wish to establish that R1(t, t) is maximized when t = 1. When t > t+δit is clear that R1(t, t) is increasing in t as the positive part of equation C.20 isincreasing in t (as established in the proof of the previous lemma), and the negativepart of equation C.20 is independent of t. When t < t ′ ≤ t ≤ t+δ we haveU(t ′, t) =∫ t ′0us(aˆ, aˆ)ds+∫ tt ′us(a˜, aˆ)ds+∫ t ′+δtus(a˜, a˜)ds+∫ 1t ′+δus(a∗,a∗)dsandU(t, t) =∫ t0us(aˆ, aˆ)ds+∫ ttus(a˜, aˆ)ds+∫ t+δtus(a˜, a˜)ds+∫ 1t+δus(a∗,a∗)ds185so thatU(t ′, t)−U(t, t)=∫ t ′tus(aˆ, aˆ)ds+∫ t ′tus(a˜, aˆ)ds+∫ t ′+δt+δus(a˜, a˜)ds+∫ t ′+δt+δus(a∗,a∗)dsis independent of t. Therefore R1(t, t) is weakly increasing in t so that type 1 regretis maximized when the opponent has a trigger time of 1. We can therefore writethe type 1 regret associated with a trigger strategy t asR1(t) =(maxt ′U(t ′,1))−U(t,1)whenever t < t ′. Because the maximization problem is independent of t, thederivative with respect to t is simply∂R1(t)∂ t=∂ −U(t,1)∂ t=−ut(aˆ, aˆ)+ut(a˜, aˆ)−ut+δ (a˜, aˆ)+ut+δ (a∗,a∗)< 0.We have therefore established that type 1 regret is decreasing in trigger timewhen t < t ′.We know from Lemma C.8 that U(t ′, t) must be maximized when 1−δ ≤ t ′ ≤1. It is easily seen that t ′ = 1− δ because u(aˆ, aˆ) < u(a˜, aˆ). Therefore, t ≥ 1− δimplies that t ≥ t ′, so that R1(t) = 0 whenever t ≥ 1−δ .We now demonstrate the continuity of R1(t). U(t, t) is continuous because theunderlying flow payoff functions are assumed to be continuous, implying that thederivatives of U(t, t) exist via the Leibniz integral rule which also implies continu-ity. An application of Berge’s maximum theorem then establishes that maxt ′U(t ′, t)is continuous in t, and a second application establishes that maxt(maxt ′(U(t ′, t))−U(t, t)) is continuous in t.Proof of lemma C.10. Type 2 regret between two strategies can be written asR2(t, t) =U(t ′, t)−U(t, t) if t ′ < t0 if t ≤ t ′ (C.21)We shall address three cases.186Case 1 (t ′ = t < t):We start by establishing that R2(t, t) is maximized at t = t−δ .There are two sub cases. First, t < t ≤ t+δ .In this case, R2(t, t) =∫ tt us(a˜, a˜)ds− ∫ tt us(aˆ, a˜)ds. This is decreasing in t be-cause ut(a˜, a˜)> ut(aˆ, a˜).In the second sub case t+δ ≤ t.In this case, R2(t, t) =∫ t+δt us(a˜, a˜)−us(aˆ, a˜)ds which is weakly increasing int because ut(a˜, a˜) is weakly increasing in t and ut(aˆ, a˜) is weakly decreasing in t.Therefore, the type 2 regret of trigger t is maximized when t = t−δ , so thatR2(t) =U(t−δ , t−δ )−U(t, t−δ ) (C.22)=∫ tt−δus(a˜, a˜)−us(aˆ, a˜)ds (C.23)which is weakly increasing in t.Case 2 (t ′ = t−δ ):We begin with two sub cases. In the first, t−δ < t ≤ t and R2(t, t) is decreasingin t because, given the constraints, an increase in t pushes the actual entry time (t)closer to the optimal entry time (t ′) which reduces regret.27In the second sub case, t+δ ≤ t. In this case, R2(t, t) =U(t−δ , t)−U(t, t) =∫ tt−δ us(a˜, aˆ)−us(aˆ, aˆ)ds+ ∫ t+δt us(a∗,a∗)−us(aˆ, a˜)ds which is weakly increasingin t because the positive integrands are weakly increasing in time and the negativeintegrands are weakly decreasing in time.We have thus established that the t that maximizes R2(t, t) must satisfy t ≤ t ≤t + δ . In this range we have dU(t,t)dt ≤ 0. It is also the case that dU(t−δ ,t)dt ≥ 0. Itmust, therefore, be the case thatR2(t) =U(t−δ , t)−U(t, t) (C.24)is weakly increasing in t.27Alternatively, we could write out the payoff functions and note that the conditions on the payofffunction that cause t ′ = t−δ also imply that type 2 regret is decreasing in t.187Case 3 (t−δ < t ′ < t)In this case we can apply the unconstrained envelope theorem, which allows usto write:dR2(t)dt=∂ −U(t, t)∂ t(C.25)When t > t it is clear that U(t, t) is decreasing in t: the agent is being pre-empted and benefits from reducing the period in which they are being exploited.When t ≤ t the case for U(t, t) being decreasing is more subtle. Recall that t ′is the best response to t and that t ′ < t. Furthermore, t ′ is strictly interior to theinterval t− δ < t ′ < t. Taken together, these imply that t lies in a region of the Ufunction that is decreasing.28It is therefore true that in the third case we also have dR2(t)dt ≥ 0.The continuity of R2 can be established in the same fashion as the continuityof R1.From the above arguments it is clear that limδ→0 t = t. From Lemma C.8 wesee that limδ→0 t ′ = t so that limδ→0 t ′ = limδ→0 t = t. It is therefore true thatlimδ→0U(t ′, t)−U(t, t) =U(t, t)−U(t, t) = 0.Proof of proposition C.11. The proof is simply a matter of stitching together theprevious results.Write T (t) = R1(t)− R2(t). T (0) > 0 and T (1− δ ) ≤ 0 and dTdt < 0. Bythe intermediate value function, there is a unique time, tδ ∈ [0,1− δ ], such thatT (tδ ) = 0. We argue that this is the earliest regret minimizing time. When t < tδwe have R1(t)> R2(t) so that R(t) = R1(t). From Lemma C.9 we know that R1(t)is decreasing, so that R(t) is decreasing as well. When t > tδ we have R2(t)> R1(t)so that R(t) = R2(t). From Lemma C.10 we know that R2(t) is non-increasing, sothat R(t) is non-increasing as well. Therefore tδ must be a regret minimizing triggertime, and must also be the smallest regret minimizing trigger time.28 The first order condition for t ′ is that ut ′(a˜, aˆ)+ ut ′+δ (a∗,a∗) = ut ′(aˆ, aˆ)+ ut ′+δ (a˜, a˜) and theassumptions on the rate of change of the flow payoff function assure that the second order conditionis satisfied globally.188From Lemma C.10, we have that limδ→0 R2(t)= 0 for all t. Therefore limδ→0 T (t)=R1(t) for all t. Furthermore, we have limδ→0 tδ = limδ→0 1− δ = 1 as R1(t) > 0for all t < 1−δ and R1(1−δ ) = 0.189010203040PeriodPercent of Period ElapsedPD IC10(High Inertia)IC60(Medium Inertia)IC280(Low Inertia)PC(Zero Inertia)L −PD L −PC(Zero Inertia)10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30Figure C.5: Median entry time (normalized to the percent of the period elapsed), by period,across all treatments. IC60 and PC, PD, L−PD and L−PC treatments ran for 30periods, while the IC10 and IC280 treatments were run within-session with three blocksof 3 periods of IC280 followed by 7 periods of IC10. Grey horizontal lines for eachtreatment mark the median final period entry time.C.5 Time Series of Median Entry TimesFigure C.5 plots period-to-period time series of median raw, observed entry timesfrom all periods in each treatment. Note that unlike the analysis in the main text,these period-to-period results have not been corrected for censoring bias usingproduct limit estimates (since such estimates, at the subject level, require multi-ple observations per subject and therefore cannot be conducted for each period ofthe dataset).190Appendix DAppendix for “MentalEquilibrium and MixedStrategies for Ambiguity AverseAgents”D.1 ProofsD.1.1 Proof of theorem 4.1Lemma D.1. Suppose that ν is a belief function and Φ = core (ν). Then A ∈supp Φ if and only if ν(Ac)< 1.Proof of lemma D.1. Suppose that A ∈ supp Φ. Therefore there exists a φ ∈ Φsuch that φ(A)> 0 and φ(Ac)< 1. Therefore there exists a p ∈ core (ν) such thatp(Ac)< 1. Therefore ν(Ac)< 1.Suppose that A /∈ supp Φ. Therefore φ(A) = 0 for all φ ∈Φ. Therefore p(A) =0 for all p ∈ core (ν). Therefore p(Ac) = 1 for all p ∈ core (ν). Note that therealways exists a p ∈ core (ν) such that p(Ac) = ν(Ac). Therefore ν(Ac) = 1.191Lemma D.2 (Lo-Nash equilibrium). Consider a pair < σ ,Φ > such that Φi =core νi for all i, where νi is a belief function. If σ and νi satisfyσ(·|ai) ∈ core νi ∀ai ∈ suppσAi ,∀i ∈ N (D.1)νi(ac−i) = 1⇔∏j 6=iσA j(a j) = 0 ∀a−i ∈ A−i, ∀i ∈ N (D.2)andai ∈ argmaxaˆi∈Ai∫aˆidνi ∀ai ∈ suppσAi ,∀i ∈ N (D.3)then the pair < σ ,Φ > form a Lo-Nash equilibrium. Conversely, if < σ ,Φ > area Lo-Nash equilibrium then σ and νi satisfy equations D.1, D.2 and D.3.Proof of Lemma D.2. We need to demonstrate that < σ ,Φ> is a Lo-Nash equilib-rium.Equation D.1 is equivalent to equation 4.5.The equivalence of equation 4.7 and equation D.3 follows directly from Corol-lary 4.4 of Gilboa and Schmeidler (1994).∏ j 6=iσA j(a j)= 0⇔∃ j 6= i s.t. σA j(a j)= 0⇔ a−i /∈× j 6=i supp σA j and LemmaD.1 establish that equation 4.6 and equation D.2 are contrapositives.Proof of theorem 4.1. Given the above lemmas, it is sufficient to show an equiv-alence between equations D.1 and 4.8, equations D.2 and 4.9, and equations D.3and 4.10. We begin by noting that result 4.1 ensures that each αi can be associatedwith a belief function νi (and vice-versa) and that Φi can be defined as the core ofνi.The equivalence of equations D.3 and 4.10 follows from results 4.1 and 4.2.The equivalence of equations D.1 and 4.8 is a consequence of the definition ofthe core of a capacity and the fact that νi(B) =∑T⊆Bαi(T ) (which follows directlyfrom the definition of αi).The equivalence of equations D.2 and 4.9 follows immediately from νi(ac−i) =∑T :a−i /∈T αi(T ) which again is an immediate consequence of the definition of αi.192D.1.2 Proof of theorem 4.2Proof of Theorem 4.2. Suppose that < νi,ν j > form an equilibrium. The αi de-fined in the theorem is the αi derived from result 4.1. Given this, Result 4.2 andDefinition 4.4 implies thatsi ∈ argmaxs′i∑T∈Σiu′i(s′i,T )αi(T )for all si ∈ supp σAi . Linearity of u′ then implies equation 4.13.To establish equation 4.12, notice that νi is additive for all pairs of events thatdo not form a partition of the state space. Therefore, σA j(T ) ≥ νi(T ) for all T ∈Σ/Ω, and σA j(Ω) = νi(Ω) = 1. Recalling that νi(T ) = ∑τ⊆T αi(τ) establishes theresult.For the converse, the linearity of preferences in the mental state space andEquation 4.13 implies that every strategy in the support of σi is optimal for agent iwith respect to αi. Then Result 4.2 implies that every strategy in the support of σiis also optimal with respect to νi.D.1.3 Proof of lemma 4.3Proof of Lemma 4.3. By construction each Ψxm is a partition.Suppose that Ψi is not a filtration. Then there exists a Ψkxm ∈Ψxm with at leasttwo elements s,s′ ∈Ψkxm such that s ∈Ψκxm′ , s′ ∈Ψκ′xm′ , s /∈Ψκ′xm′ and s′ /∈Ψκxm′ .Furthermore we can, without loss of generality, choose m,m′ such that m =m′ + 1. To see this, note that by construction Ψx0 and Ψx1 are admissible in afiltration. Suppose that Ψxm is the first partition that is not admissible. Then itmust be that there is an event in Ψxm that is not a subset of any events in a previouspartition. Suppose that this previous partition is Ψxm−n with n > 1. But, all eventsin Ψxm−1 are subsets of events in Ψxm−n , implying that there must also be an eventin Ψxm that is not a subset of any event in Ψxm−1 .Therefore there exists an information set that precedes xm through which ex-actly one of s or s′ passes (if s and s′ always passed through the same collection ofinformation sets then they would share an event in theΨxm′ partition) but both s and193s′ passes through xm. This violates perfect recall.194
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Essays on strategic uncertainty with non-subjective expected utility agents Calford, Evan M. 2016
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Title | Essays on strategic uncertainty with non-subjective expected utility agents |
Creator |
Calford, Evan M. |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | This thesis contains three distinct chapters that contribute to our understanding of how people respond, both theoretically and in controlled experimental environments, to uncertainty that results from the strategic decisions of others. The standard framework for studying strategic interactions involves agents with Subjective Expected Utility preferences (Savage, 1954) interacting in an environment where, in equilibrium, all strategies are known to all agents. This thesis studies the effects of relaxing preferences to allow for ambiguity aversion, regret minimization, and approximate optimization. The first chapter experimentally investigates the role of uncertainty aversion in normal form games. Theoretically, risk aversion will affect the utility value assigned to realized outcomes while ambiguity aversion affects the evaluation of strategies. In practice, however, utilities over outcomes are unobservable and the effects of risk and ambiguity are confounded. This chapter introduces a novel methodology for identifying the effects of risk and ambiguity preferences on behaviour in games in a laboratory environment. Furthermore, we also separate the effects of a subject's beliefs over her opponent's preferences from the effects of her own preferences. The second chapter studies, experimentally, a simple dynamic entry game in both continuous and discrete time. We introduce new laboratory methods that allow us to eliminate natural inertia in subjects' decisions in continuous time experiments. Using our novel continuous time setting and the standard discrete time setting as benchmarks, we study the effects of inertia (caused by naturally occurring reaction lags) on behaviour. We demonstrate that the observed patterns of behaviour are consistent with standard models of decision making under uncertainty, and that the degree of inertia affects subject responses to strategic uncertainty. The third chapter examines, theoretically, the role of mixed strategies for agents with ambiguity averse preferences. This chapter demonstrates how a well known result from cooperative game theory, that a non-additive measure over a set of states can be equivalently represented by an additive measure over the set of events, can be used to introduce mixed strategies (in an equilibrium preserving fashion) to existing pure strategy equilibrium concepts. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2016-07-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0305872 |
URI | http://hdl.handle.net/2429/58445 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Graduation Date | 2016-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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