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Bounded rationality in games : theory, experiments, and applications Kneeland, Terri 2013

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Bounded Rationality in Games: Theory, Experiments, and Applications by Terri Kneeland  B.A Economics and Mathematics, University of Victoria, 2005 M.A Economics, The University of British Columbia, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Economics)  The University of British Columbia (Vancouver) June 2013 c �Terri Kneeland 2013  Abstract This dissertation combines three contributions to the literature on bounded rationality in games. The aim of this thesis is to improve our understanding of how individuals make decisions in games, improve our ability to model this behavior and increase our understanding of how bounded rationality affects predictions, policy and optimal mechanisms. The first paper is an application of a boundedly rational model to explain behavior in coordinated attack games. I demonstrate that the main experimental results, such as threshold strategies, comparative statics, and the differences in behavior under public and private information, are robust predictions of limited depth of reasoning models. This is in contrast to equilibrium, which mispredicts the coordinating roles of the different types of information. The analysis has implications for macroeconomic phenomena, like currency attacks and debt crises, which are commonly modeled using incomplete information coordinated attack games. The second paper explores policy and optimal mechanism design under bounded rationality. Level-k implementation is contrasted with the more standard Bayesian implementation concept. I show that the revelation principle holds with an augmented message space and that level-k implementation is a weaker solution concept. In addition, level-k implementation is possible in a mechanism that is robust to different specifications of beliefs about depths of reasoning or to any specification of beliefs about payoffs. The third paper takes a step back from assuming a particular solution concept and investigates empirical features of strategic reasoning in the lab. I employ strategic choice data from a carefully chosen set of ring games to obtain individual-level estimates of the following three epistemic conditions: rationality, beliefs about the rationality of others, and consistent beliefs. I find that not a single subject satisfies all three of the epistemic conditions sufficient for Nash equilibrium and that consistent beliefs, rather than rationality, is the more likely source for the failure of Nash equilibrium. The design allows us to weight the relative plausibility of alternative solution concepts used to explain laboratory results.  ii  Preface This dissertation is original, unpublished, independent work by the author, Terri Kneeland. The experimental work reported in Chapter 4 is covered by UBC Behavioral Research Ethics Board Certificate number H11-02207.  iii  Table of contents Abstract Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii  Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of figures  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 Coordination under limited depths of reasoning  . . . . . . . . . . . . . . .  5  2.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5  2.2  Coordinated attack game . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  2.2.1  Common knowledge of payoffs . . . . . . . . . . . . . . . . . . . . . .  9  2.2.2  Incomplete information of payoffs . . . . . . . . . . . . . . . . . . . . .  9  2.3 2.4  2.5  2.6  2.7  Background evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Level-k thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.1  Sophisticated types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15  2.4.2  Consistency with experimental evidence . . . . . . . . . . . . . . . . . 16  2.4.3  Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18  Experimental analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.1  The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19  2.5.2  Econometric analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20  2.5.3  Qualitative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22  2.5.4  Type classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24  Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.1  Policy analysis: speculative attack . . . . . . . . . . . . . . . . . . . . 24  2.6.2  Global games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26  2.6.3  Other macroeconomic applications . . . . . . . . . . . . . . . . . . . . 28  Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 iv  3 Mechanism design with level-k types  . . . . . . . . . . . . . . . . . . . . . . 30  3.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  3.2  Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.1  Payoff environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  3.2.2  Type spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  3.2.3 3.2.4  Solution concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  3.3  Revelation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  3.4  Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  3.5  Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  3.6  Bayesian equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44  3.7  Robustness of level-k mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.7.1  Relaxing beliefs about levels . . . . . . . . . . . . . . . . . . . . . . . . 47  3.7.2  Relaxing beliefs about payoffs . . . . . . . . . . . . . . . . . . . . . . . 49  3.7.3  Relaxing truth-telling of level 0 types . . . . . . . . . . . . . . . . . . 51  4 Rationality and consistent beliefs: theory and experimental evidence . . 54 4.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.1.1  Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57  4.2  Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59  4.3  The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.1  Identifying assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 63  4.3.2  Epistemic conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65  4.3.3 4.4  4.5  Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65  4.3.2.2  Consistent beliefs . . . . . . . . . . . . . . . . . . . . . . . . 66  Separating rationality and consistent beliefs . . . . . . . . . . . . . . . 68  Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.4.1  Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70  4.4.2  Consistent beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72  4.4.3  Laboratory implementation . . . . . . . . . . . . . . . . . . . . . . . . 73  Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.1  4.6  4.3.2.1  Secondary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.6.1  Beliefs about payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82  5 Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 A Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 v  A.1 Omitted proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.2 Limited depth of reasoning model . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.3 General model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.4 Discrete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.5 Round 1 estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 B Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.1 Omitted proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.2 Relaxing belief to p-belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.3 Assignment Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 B.4 Instructions and quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 B.5 Raw data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118  vi  List of Tables 2.1  Qualitative experimental regularities . . . . . . . . . . . . . . . . . . . . . . . 10  2.2  Qualitative results under level-k thinking . . . . . . . . . . . . . . . . . . . . 18  2.3  Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20  2.4  Threshold predictions by type . . . . . . . . . . . . . . . . . . . . . . . . . . . 22  2.5  HNO’s qualitative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23  2.6  Estimated mean threshold cutoffs  2.7  Aggregate type classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25  4.1  Predicted actions under rationality and assumptions A1-A3 . . . . . . . . . . 71  4.2  Subjects assigned by category . . . . . . . . . . . . . . . . . . . . . . . . . . . 72  4.3  Subjects who failed quiz, classified by order of rationality . . . . . . . . . . . 77  4.4  Order of rationality determined by initial versus final choices . . . . . . . . . 77  4.5  Sample distributions under different game orders . . . . . . . . . . . . . . . . 78  . . . . . . . . . . . . . . . . . . . . . . . . 23  A.1 Aggregate type classification (round 1) . . . . . . . . . . . . . . . . . . . . . . 104 B.1 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119  vii  List of Figures 3.1  Payoff-direct mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  3.2  Bayesian mechanism for Example 3.1 . . . . . . . . . . . . . . . . . . . . . . . 40  3.3  Level-k mechanism for Example 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 40  3.4  Structure of a Bayesian mechanism for Example 3.2 . . . . . . . . . . . . . . 41  3.5  Level-k mechanism for Example 3.2 . . . . . . . . . . . . . . . . . . . . . . . . 42  3.6  Payoff-direct mechanism for the buyer . . . . . . . . . . . . . . . . . . . . . . 43  3.7  Payoff-direct mechanism for the seller . . . . . . . . . . . . . . . . . . . . . . 43  3.8  Ex post level-k mechanism for Example 3.2 . . . . . . . . . . . . . . . . . . . 51  4.1  3-player ring game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55  4.2  B1: rationalizable bimatrix game . . . . . . . . . . . . . . . . . . . . . . . . . 59  4.3  B2: rationalizable bimatrix game . . . . . . . . . . . . . . . . . . . . . . . . . 60  4.4  R1: rationalizable ring game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61  4.5  R2: rationalizable ring game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62  4.6 4.7  A type structure for a 3-player ring game . . . . . . . . . . . . . . . . . . . . 66 A consistent type structure for a 3-player ring game . . . . . . . . . . . . . . 68  4.8  A consistent type structure for a 3-player ring game . . . . . . . . . . . . . . 69  4.9  G1: 4-player dominance solvable ring game . . . . . . . . . . . . . . . . . . . 70  4.10 G2: 4-player dominance solvable ring game . . . . . . . . . . . . . . . . . . . 70 4.11 G3: 2-player ring game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.12 Subjects classified by order of rationality . . . . . . . . . . . . . . . . . . . . . 75 4.13 Subjects classified by epistemic type . . . . . . . . . . . . . . . . . . . . . . . 76 4.14 Proportion of subjects playing each action by game . . . . . . . . . . . . . . . 76 4.15 Proportion of subjects who play rationalizable action in G1 and G2 . . . . . . 81 4.16 QRE predictions for games G1 and G2 . . . . . . . . . . . . . . . . . . . . . . 81 B.1 Best response correspondences for game G3 . . . . . . . . . . . . . . . . . . . 111  viii  Chapter 1  Introduction In a game, a player’s payoff depends upon the action she takes and the actions of her opponents. In order to choose an action, a player must reason about the actions her opponents are taking. This will depend upon the action she believes her opponents believe she is taking, and the actions she believes her opponents believe she believes her opponents are taking, and so on. The standard economic analyses of games assume players are both capable of reasoning about these infinite hierarchies and have the ability to correctly guess what their opponents are doing. However, departures from these ’highly rational’ decisions are increasingly common in economic theory and are empirically well motivated by a wide range of behavioral data from lab and field experiments and econometric inference. This dissertation investigates the implications of relaxing these assumptions in different environments and the empirical foundations for doing so. There is an extensive existing literature that examines bounded rationality in games. This literature has put forth a number of alternative solution concepts that relax some of the reasoning requirements imposed in standard solution concepts like Nash equilibrium, Bayesian equilibrium, and rationalizability. Two popular alternative solution concepts are: limited depth of reasoning (LDoR) models introduced by Stahl and Wilson (1994), Stahl and Wilson (1995), and Nagel (1995) and formalized as the level-k model by Costa-Gomes et al. (2001), Costa-Gomes and Crawford (2006), and Costa-Gomes et al. (2009) and as the cognitive hierarchy model by Camerer et al. (2004). This literature is summarized in Costa-Gomes et al. (2013); and quantal response models (QRE) formalized by McKelvey and Palfrey (1995). This dissertation expands on this literature both theoretically through applications and experimentally. The first two chapters investigate limited depth of reasoning models. Behavior is determined by a player’s depth of reasoning. L0 types are nonstrategic - their behavior is specified  1  outside of the model. All higher types anchor their beliefs in the nonstrategic L0 type and adjust them through thought experiments with iterated best responses. L1 types have one depth of reasoning. They best respond to L0 types. L2 types have two depths of reasoning. They best respond to some prespecified belief over L0 and L1 types. And, so on, with Lk types best responding to some prespecified belief over lower types {L0,L1,. . .,Lk-1}. All Lk types, k≥1, have accurate models of the game and are rational in the sense that they best respond to their beliefs about others. Limited depth of reasoning models depart from standard solution concepts in two ways. First, these models only require players to use a finite order of beliefs to determine their optimal actions, regardless of the game. And, secondly they do not require players to hold correct beliefs about the actions of their opponents. Chapter 2 provides the first unified explanation of behavior in coordinated attack games under both private  and public information. It demonstrates that the main  experimental results, such as threshold  strategies, comparative statics, and the  differences in behavior under public and private information are robust predictions of limited depth of reasoning models. This is in contrast to  equilibrium, which  mispredicts the coordinating roles of the different information structures. Chapter 3 investigates optimal mechanism design under the level-k solution concept. The mechanism design literature typically focuses on standard equilibrium solution concepts like Bayesian Nash equilibrium. This chapter explores optimal mechanism design under a solution concept based on level-k and cognitive hierarchy models, which have strong experimental support. A version of the revelation principle is established for this solution concept which requires the message space be augmented to allow agents to reveal both their payoff type and their depths of reasoning. We provide necessary conditions for level-k implementation and demonstrate that it is a weaker solution concept than Bayesian implementation. And, we give several robustness results. Under weak assumptions, implementation is possible in a mechanism that is robust to different specifications of beliefs about the depths of reasoning of others or to any specification of beliefs about payoffs. Results are illustrated with a bilateral trade example. Chapter 4 investigates the behavioral and experimental support for the foundations of different boundedly rational models. It employs strategic choice data from a carefully chosen set of ring-network games to obtain individual-level estimates of the following three epistemic conditions: rationality, beliefs about the rationality of others, and consistent beliefs about strategies. I find that no subject satisfies all three of the sufficient epistemic conditions for Nash equilibrium. The data suggests that a failure to play Nash equilibrium stems, more often, from a failure of consistent beliefs than from a failure in rationality or beliefs about the rationality of others. The unique design allows us to weigh the relative plausibility of alternative solution concepts like QRE and LDoR models used to account for laboratory results. The data tend to support limited depth of reasoning models which  2  maintain the rationality assumptions imposed by Nash equilibrium but relax the consistent beliefs assumption. Below are outlined the main contributions of this dissertation. Chapter 2 contributes to the literature in three ways. First, it provides an explanation for existing experimental evidence. Second, it provides an example of a macroeconomic application in which the limited depth of reasoning analysis has different policy implications than equilibrium. The  analysis has implications for understanding macroeconomic  phenomena, like currency attacks  and debt crises, which are commonly modeled  using incomplete information coordinated attack games. And third, it illustrates robust predictions of limited depth of reasoning models. The main drawback of limited depth of reasoning models are that their specification depends upon structural parameters that must be specified outside of the model. However, we show that these models make robust predictions with respect to these parameters in the coordinated attack environment. Chapter 3 contributes to both the mechanism design literature and the literature on bounded rationality in games by providing formal results for level-k implementation in general environments. In addition, it contributes to the belief-free implementation literature by demonstrating robust features of level-k implementation: the mechanism does not depend upon the exact specification of beliefs over depths of reasoning and under very weak assumptions, can be implemented independent of the specification of beliefs about payoffs. Further, it is shown that a level-k mechanism that is independent of the specification of beliefs about payoffs is easier to implement than a Bayesian mechanism (which is sensitive to the exact specification of beliefs about payoffs). Chapter 4 makes two main contributions. First, it introduces a novel experimental design that allows us to separately identify features of beliefs from strategic choices. And secondly, the experiment cleanly distinguishes between two main boundedly rational solution concepts: QRE and LDoR models. In previous work, models are compared by jointly testing the epistemic assumptions and additional structural assumption that each model imposes. This approach separately identifies epistemic assumptions independently of the structural assumptions. Both Chapters 2 and 4 contribute to the discussion on the role of higher-order uncertainty in behavior. Chapter 2 provides an indirect test of responsiveness to higher-order beliefs by comparing the predictions of two models: one model that assumes players respond to small perturbations of higher-order beliefs (standard Bayesian equilibrium model) and one that is based on limited depths of reasoning. Chapter 4 provides a direct test of sensitivity to higher-order beliefs. The experiment finds that people are largely unresponsive to changes in (even fairly low) higher-order beliefs: 80 percent of the subject pool fails to respond to changes in beliefs at 3rd-order or lower. Lastly, all three chapters illustrate robust features of limited depth of reasoning models. 3  These models impose specific assumptions about beliefs over L0 behavior and the depths of reasoning of others that are specified outside of the model. This dissertation illustrates that there are macroeconomic applications in which limited depth of reasoning models make robust predictions. We can design mechanisms with robust features. And, we can provide experimental tests of the limited depth of reasoning models which are independent of the structural assumptions. Chapter 2 shows that limited depth of reasoning predictions are independent of assumptions about L0 behavior and particular specifications for beliefs about the depths of reasoning in certain classes of coordinated attack games. Chapter 3 illustrates the robust features of optimal mechanisms under limited depths of reasoning. And, Chapter 4 illustrates an experimental design that correctly identifies a subject’s depth of reasoning independently of assumptions about L0 behavior.  4  Chapter 2  Coordination under limited depths of reasoning 2.1  Introduction  Consider a simple coordination game used to model a speculative attack. Players have the option to attack a currency peg. If enough players attack, the attack is successful, and the peg collapses. If not enough players attack, the attack fails, and the peg holds. The threshold for a successful attack depends on the fundamentals of the economy. If fundamentals are strong, a large proportion of players must attack in order for the attack to be successful. If fundamentals are weak, a small proportion of players can attack and collapse the peg. Intuitively, we might expect players to attack a currency peg when fundamentals are weak, but restrain from attacking when fundamentals are strong. This is a robust outcome of coordinated attack experiments. Heinemann et al. (2004), Duffy and Ochs (2009), and Cornand (2006) test coordinated attack games in the laboratory. There are two main qualitative results. (1) The probability of an attack depends on the fundamentals. Players attack when fundamentals are weak and do not attack when fundamentals are strong. This holds regardless of whether players receive public or private information about the fundamentals. And, (2) the degree of coordination depends upon the information structure. Players are more likely to coordinate on the same strategy in public information treatments than in private information treatments. These results provide a challenge for equilibrium theory. When fundamentals are public information, there are multiple equilibria driven by self-fulfilling beliefs. Hence, equilibrium does not explain why behavior is tied to fundamentals under public information. Further, equilibrium mispredicts the coordinating roles of public and private information. Private information generates disperse higher-order beliefs which weaken complementarities in ac5  tions generating a unique equilibrium. Public information strengthens complementarities between actions generating multiplicity. In other words, there is perfect coordination under private information and in contrast, public information has destabilizing effects. This is inconsistent with existing experimental evidence that finds that public information increases coordination relative to private information. Equilibrium selection criteria like risk-dominance, payoff-dominance and global games can explain the first qualitative feature - they predict that players will play a threshold equilibrium under public information. And, if all players coordinate using the same selection criteria, public information need not be destabilizing. However, no existing equilibrium analysis explains why private information decreases the degree of coordination relative to public information. This paper shows that an alternative solution concept, based on limited depth of reasoning, can explain the intuitive behavior and is consistent with the existing experimental evidence. Limited depth of reasoning models, like level-k thinking and cognitive hierarchy, are a behaviorally-motivated approach to reasoning in games.1 Each player has a bounded depth of reasoning determined by her cognitive type. L0 types are nonstrategic - their behavior is specified outside of the model. All higher types anchor their beliefs in the nonstrategic L0 type and adjust them through thought experiments with iterated best responses. L1 types have one depth of reasoning. They best respond to L0 types. L2 types have two depths of reasoning. They best respond to some belief over L0 and L1 types. And, so on, with Lk types best responding to some belief over lower types {L0,L1,. . .,Lk-1}. All Lk types, k≥1, have accurate models of the game and are rational in the sense that they best respond to their beliefs about others. They depart from equilibrium only in basing their beliefs on simplified models of others. This yields a workable model of players’ decisions while avoiding much of the cognitive complexity of equilibrium analysis. This paper applies the limited depth of reasoning solution concept to incomplete information coordinated attack games. Limited depth of reasoning provides a unified explanation of the pattern of behavior in both public and private information games. To the best of my knowledge, this is the first paper to provide a consistent explanation of the existing experimental literature on coordinated attack games. Limited depth of reasoning provides a unified explanation of behavior under the different information conditions. Behavior is tied to fundamentals under any information structure with players attacking only when fundamentals are weak. Moreover, coordination is greater under public information than private information. Players with different depths of reasoning treat public information similarly but private information differently. Public information coordinates higher- and lower-order beliefs about fundamentals. This coordinates the behavior of players with different types of reasoning. Private information decreases coordination 1 See  Crawford et al. (forthcoming) for a recent survey of this literature.  6  by creating differences between higher- and lower-order beliefs and hence differences in the behavior of players with different depths of reasoning. The qualitative results are robust predictions of limited depth of reasoning models. The results hold under weak assumptions on the nonstrategic type L0. And, they hold under any limited depth of reasoning model. That is, they hold for any specification of beliefs that an Lk type might hold over lower types and for any distribution of types in the model. The results even hold if there is some proportion of sophisticated types in the model (types that have correct beliefs about the types of others, infinite depths of reasoning, and realize that others may also have infinite depths of reasoning). In addition, this paper analyzes the experimental data from Heinemann et al. (2004) and classifies subjects as either level-k or equilibrium types using a finite mixture model. Allowing for level-k types significantly improves the models ability to explain the experimental data. Approximately, 70 percent of subjects can be consistently classified as level-k types and 30 percent of subjects as equilibrium types. Limited depth of reasoning provides a consistent explanation for experimental behavior at both the individual and aggregate level. This study contributes to the limited depth of reasoning literature in a number of ways. First, it demonstrates that limited depth of reasoning models produce robust predictions. Part of the difficulty with models like level-k thinking and cognitive hierarchy is that L0 behavior and the type distribution must be specified outside of the model. This is problematic because both L0 behavior and the type distribution do not appear to be stable across games. However, the theoretical predictions of this paper are largely independent of these specifications. Even without knowing L0 behavior or the type distribution, limited depth of reasoning models can make robust, testable predictions. In addition, this paper highlights a novel aspect of limited depth of reasoning models. Differences between equilibrium and limited depth of reasoning may stem from nonequilibrium beliefs. This occurs under limited depth of reasoning if L0 behavior is specified different from equilibrium behavior. This is an important source of equilibrium deviations in laboratory experiments.2 However, the results in this paper do not stem from a specific specification of L0. The results hold because of player’s beliefs about the boundedness of others. Because players base optimal actions on simplified models of others, players’ beliefs are always linked to fundamentals, actions are never driven by self-fulfilling beliefs, and public information increases coordination over private information. The analysis in this paper also has relevance for macroeconomics. Phenomena, like bank runs, currency crises, debt crises, and coordinated investment, are commonly modeled using incomplete information coordinated attack games. Both equilibrium and limited depth of reasoning analyses suggest there is an asymmetry between behavior under public and private 2 See  Nagel (1995) or Costa-Gomes and Crawford (2006) for examples.  7  information. Under equilibrium, public information increases strategic complementarities, potentially decreasing coordination because of self-fulfilling beliefs. Under limited depth of reasoning, public information coordinates the beliefs of players with different depths of reasoning, increasing coordination. Public information plays a different coordinating role under the two solution concepts and will have different policy implications. However, the role of public information under equilibrium is inconsistent with the existing experimental evidence. The limited depth of reasoning analysis provides a consistent explanation of the coordinating roles of private and public information. This paper proceeds as follows. The next section describes the coordinated attack game. Section 2.2 discusses the existing experimental evidence and the inconsistency of the equilibrium analysis. Section 2.3 establishes the limited depth of reasoning results. Section 2.5 analyzes the experimental data and discusses the consistency of level-k with the experimental literature. Section 2.6 discusses the relevance of limited depth of reasoning for macroeconomics and presents a policy analysis of public information. Section 2.7 discusses related literature. Omitted proofs can be found in Appendix A.1.  2.2  Coordinated attack game  A simple model of a coordinated attack is presented. The model follows Morris and Shin (2004) and Bannier (2002). The coordinated attack game is interpreted as a speculative attack throughout the rest of this paper. This model can be applied to describe other phenomena such as bank runs, debt crises, coordinated investment, and political change. There is a continuum of players indexed by i and uniformly distributed on [0, 1]. Players may either attack the exchange rate peg or do nothing. There is a cost t of attacking. If a player attacks and the peg is abandoned, the player receives a positive payoff D on top of the cost paid for attacking. It is assumed D > t. The payoff from not attacking is zero. The exchange rate peg is abandoned if and only if the proportion of players attacking, denoted by l, is no less than a critical value θ ∈ R.  The critical value θ parameterizes the strength of the status quo. It is often referred to  as the fundamentals of the economy. Under the speculative attack interpretation, θ can be interpreted as foreign exchange reserves held by the Central Bank. A higher θ represents better fundamentals and raises the threshold for a successful attack. The payoffs for each player can be summarized by the function π : [0, 1]xR → R, the payoff gain from attacking. It is defined as  π(l, θ) =  �  D−t −t  8  if l ≥ θ  otherwise  2.2.1  Common knowledge of payoffs  There are three different cases to consider when payoffs are common knowledge: θ>1  Even if all players attack, the fundamentals are sufficiently strong to maintain the exchange rate peg. There is a unique equilibrium in which none of the players attack.  θ≤0  The fundamentals are too weak for the peg to be maintained. The unique equilibrium is one in which all players attack.  0 < θ ≤ 1 The currency regime is ripe for attack. There exist two equilibria - one in which all players attack and the exchange rate peg is abandoned and another in which no players attack and the peg is upheld. This game has multiple equilibria whenever 0 < θ ≤ 1. Attacking is only worthwhile if a player expects others to attack. As beliefs are self-fulfilling, they are not tied to fundamentals and there is no way to predict an attack.  2.2.2  Incomplete information of payoffs  The above game can be converted into an incomplete information game by letting θ be unknown and having each player receive signals about θ. The distribution of θ and the signal processes are assumed to be common knowledge. Each player receives a private signal xi and a public signal y about θ, where xi = θ + �i with �i ∼ N (0, β1 ) and y = θ + η with η ∼ N (0, α1 ). The fundamental θ is distributed uniformly on the real line.  Under this information structure, players form expectations of θ based on the public and  private information they receive. Conditional on the information received, θ is normally distributed with a mean formed by a weighted average of the public and private signals. Given signals x and y, θ is conditionally distributed according to θ|x, y ∼ N where µ =  β α+β .  �  1 (1 − µ)y + µx, α+β  �  (2.1)  A player’s strategy in the incomplete information game is a function  s : R → {0, 1}, which lists an action for any private signal she might receive (0 represents not attack and 1 represents attack.  There is an unique equilibrium in the incomplete information game provided private √ information is sufficiently precise relative to public information (i.e. √αβ ≤ 2π). In this case, there will be a unique equilibrium where players play threshold strategies. Players  attack if and only if their private signal is below some threshold cutoff. However, if public information is relatively precise then there may exist multiple equilibria driven by selffulfilling beliefs. 9  The main proposition in Morris and Shin (2004) states that if  √α β  ≤  √  2π, there exists  a unique equilibrium in the speculative attack game where players attack if and only if x≤x ¯E , for some unique x ¯E .3  2.3  Background evidence  Heinemann et al. (2004), Duffy and Ochs (2009), and Cornand (2006) test coordinated attack games in the laboratory. They test three information conditions between them. A private information condition (PI) where players receive only private information about fundamentals. A complete, common information condition (CI) where players receive complete, public information about fundamentals. And, a mixed information condition (MI) where players receive both private and public information. Heinemann et al. (2004) tests both PI and CI conditions. Duffy and Ochs (2009) test CI. And, Cornand (2006) tests MI. There are 4 main regularities that emerge from these experiments. Regularity  Experimental Findings  1  Players play threshold strategies in all three information conditions  2  Increasing t decreases the likelihood of a successful attack in all three information conditions  3  The standard deviation of thresholds in CI is smaller than the standard deviation of thresholds in PI  4  The likelihood of a successful attack is greater in CI than in PI  Table 2.1: Qualitative experimental regularities Both Heinemann et al. (2004) and Cornand (2006) find that on average over 70 percent of all strategies are consistent with undominated thresholds in the first period of play. This increases to close to 90 percent by the last round of play. Duffy and Ochs (2009) find that on average 83 percent of players never play a dominated strategy (in any round of play). All three studies find that increasing the cost of attacking results in a statistically significant lower likelihood of attacking. Regularities 1 and 2 are equilibrium predictions only in PI and MI conditions. Equilibrium does not predict this behavior when payoffs are public information. While equilibrium 3 Notice that the threshold cutoff x ¯E (y) will depend on the public signal y. Generally, we will suppress that dependence and just write x ¯E , taking the public signal y as fixed.  10  makes no predictions in the CI condition, Regularities 1 and 2 are not necessarily inconsistent with equilibrium as anything can happen with multiple equilibria. However, equilibrium is inconsistent with Regularity 3. Regularity 3 says that the standard deviation of thresholds in CI is smaller than the standard deviation of thresholds in PI. In other words, public information increases coordination. This is a finding of Heinemann et al. (2004) which is the only paper that jointly tests both CI and PI information conditions. The experimental evidence suggests that equilibrium mispredicts the coordinating roles of public and private information. Under private information, all players coordinate on the same threshold strategy. Under public information, players may or may not coordinate on the same threshold strategy. Theoretically we can define the degree of coordination by the proportion of subjects who coordinate on attack or not attack at a given signal. The degree of coordination under private information is 1 for all x ∈ R. The degree of coordination  under public information is always less than or equal to 1 for all y ∈ R. Thus, private information increases coordination under equilibrium.  Definition. Consider a strategy profile si : R → {0, 1} for each player i ∈ [0, 1] under an  information system I. The degree of coordination at a given signal z, where z is either a private signal or a public�signal, is the largest � proportion of players who successfully ´1 ´1 C coordinate, deg (I) = max 0 si di, 1 − 0 si di .  One measure of the degree of coordination empirically is the standard deviation of cutoff  thresholds. Equilibrium predicts that the standard deviation of thresholds under public information should be greater than or equal to those under private information. In other words, public information may have destabilizing effects. This is the opposite of the experimental finding that finds that public information increases coordination in the laboratory.4 Regularities 1-3 are the main qualitative results from the existing coordinated attack experiments. Regularity 4 is a quantitative result that refers to the point estimates of the threshold cutoffs. Heinemann et al. (2004) finds that the likelihood of a successful attack is significantly higher under public information (CI) than under private information (PI). Equilibrium does not predict this regularity. All four regularities are consistent with limited depth of reasoning models. The next section provides the level-k thinking results for the speculative attack game, establishes consistency with the existing experimental evidence, and demonstrates the robustness of the level-k predictions. 4 The global games literature establishes that a unique equilibrium in these types of coordinated attack games depends on the degree of payoff uncertainty in the model (e.g. Morris and Shin, 2003). Typically, this is modeled by introducing private information over fundamentals. However, we can think about introducing payoff uncertainty in other ways, such as uncertainty about the risk preferences of other players. Under some preference specifications, equilibrium will predict Regularities 1 and 2 under public information, as a high degree of payoff uncertainty will induce a unique threshold equilibrium (Hellwig, 2002). However, I believe that Regularity 3 is unlikely to hold under equilibrium analysis. Introducing other forms of payoff heterogeneity will not increase predictability under public information while decreasing predictability under private information.  11  2.4  Level-k thinking  In a limited depth of reasoning model each player’s behavior is determined by her cognitive type, which is drawn from a discrete distribution over a particular hierarchy of types {L0, L1, . . . , Lk, . . .}. L0 captures a player’s intuitive reactions to the game and is defined  via nonstrategic, behaviorally-plausible decision rules. L0 types act as the starting point for players’ strategic thinking. Type Lk, k > 0, anchors her beliefs in the nonstrategic L0 type and adjusts them through thought experiments with iterated best responses: L1 best responds to L0, L2 to a distribution over L0 and L1, and so on. Lk types have accurate models of the game and are rational in the sense that they best respond to their beliefs about others. They depart from equilibrium only in basing their beliefs on simplified models of others. This yields a workable model of players’ decisions while avoiding much of the cognitive complexity of equilibrium analysis. In doing so, these models explicitly incorporate the observation that players reason to a finite level of beliefs over beliefs.5 In a general limited depth of reasoning model, Lk types best respond to some distribution over lower types, where each type’s beliefs about others are defined according to some prespecified rule. Under cognitive hierarchy, Lk types best respond to a mixture of L0, L1, . . . , Lk − 1 types with weights determined by the Poisson distribution (e.g. Camerer et al., 2004). Under level-k thinking, Lk types best respond to Lk-1 types (e.g. Costa-Gomes and Crawford, 2006). The main portion of this paper works within a level-k thinking model. This is for simplicity only. The results hold for any limited depth of reasoning model with Lk types best responding to some distribution over lower types (see Appendix A.2). Both limited depth of reasoning and equilibrium require that players best respond given their beliefs about others. This requirement alone is not enough to generate precise predictions. Equilibrium adds the additional assumption that players’ actions and beliefs must be mutually consistent. Limited depth of reasoning models do not impose mutual consistency. Instead, they impose the assumption that players follow decision rules based on an iterated process of strategic thinking; players recursively calculate optimal behavior based on the anchoring L0 type. This defines a procedural model of player’s decisions that avoids the circular logic of equilibrium imposed by the assumption of mutual consistency. The estimated distribution of cognitive types tends to put most of the weight on L1 and L2 types and negligible weight on L0 types (Costa-Gomes and Crawford, 2006; Costa-Gomes et al., 2001). The anchoring L0 type exists mainly in the minds of others. This paper takes this position and assumes there is no support on L0 in the type distribution.6 In most applications, the specification of L0 is the key to the model’s explanatory power. However, the predictive power in this paper comes from the recursive nature of the level-k model. 5 The ability of players to reason with arbitrarily high levels of beliefs over beliefs is not supported by experimental evidence (Stahl and Wilson, 1994; Nagel, 1995; Kübler and Weizsäcker, 2004). 6 Allowing for a proportion of L0 types that played randomly would simply add noise into the analysis and not change the qualitative results in this paper.  12  The main results hold under only weak restrictions on L0 behavior. Let the behavior of L0 types be described by the cumulative distribution function Q(l|x, y) on [0, 1] and its associated density function q(l|x, y). The distribution Q represents beliefs about the proportion of L0 types attacking. An L1 type who receives private information x and public information y believes that others are behaving according to Q(l|x, y). The perceived behavior of L0 may be influenced by information.7 This is reasonable, as L0 types exist mainly in the minds of others. Two restrictions are placed on the behavior of L0. A1 Q(l|x, y) is weakly increasing in x and y for a given l. A2 q(l|x, y) is continuous in x and y for a given l. Assumption A1 assumes that the perceived behavior of L0 types varies monotonically as information varies. This captures the likelihood that an L1 type with a low signal believes that others are more likely to attack than an L1 type with a higher signal. This is the natural specification for L0 as L0 is meant to capture players’ intuitive responses to the game. Assumption A2 requires q(l|x, y) to be continuous with respect to x and y. This restriction is included for convenience to ensure continuity in the payoff functions. In previous level-k applications with incomplete information, L0 behavior was specified to be independent of information (e.g. Brocas et al. (2009) and Crawford and Iriberri (2005)). Assumptions A1 and A2 are satisfied automatically in this case. An L1 type who observes signal x and y believes that the behavior of others can be described by the density function q(l|x, y). Given this, an L1 type knows that if she attacks, she will receive positive payoff D whenever θ ≤ l and pay a fixed cost t with certainty. For a given l, the expected gain from attacking can be written as D · P r(θ ≤ l|x, y) − t. Given the conditional distribution of θ from (2.1), the probability of a successful attack is P r(θ ≤ l|x, y) = 1 − Φ  �� � α + β(µx + (1 − µ)y − l) .  Averaging over l according to density q gives the expected payoff gain for an L1 type who receives signals x and y. This can be written as  π  L1  (x) = D  ˆ  0  1  �  1−Φ  �� �� α + β(µx + (1 − µ)y − l) q(l|x, y)dl − t.  Note that π L1 is continuous and strictly decreasing in x by A1 and A2. There is a unique point κ1 such that π L1 (κ1 ) = 0. The cutoff κ1 is determined implicitly by 7 L0’s behavior may also depend upon other variables like D or t. The results presented below are robust to their inclusion.  13  ˆ  0  1  Φ  �� � D−t α + β(µκ1 + (1 − µ)y − l) q(l|κ1 , y)dl = . D  (2.2)  L1 types play according to a threshold strategy with cutoff κ1 . They attack if and only if their private signal x is below the threshold cutoff κ1 .8 L2 types best respond to the belief that all others are playing a threshold strategy with cutoff κ1 . L2 types believe that players attack only if their private signal is below the threshold cutoff κ1 . From the perspective of L2 types, there is a successful attack if and only if  � Φ( β(κ1 − θ)) ≥ θ. √ Let θ¯2 be determined uniquely by the solution to Φ( β(κ1 − θ¯2 )) = θ¯2 . As a result, an L2 type expects an attack to be successful whenever θ ≤ θ¯2 .  The expected payoff gain for an L2 type who observes signals x and y is given by D · P r(θ ≤ θ¯2 |x, y) − t. Using the conditional distribution of θ from (2.1), the expected payoff gain for an L2 type is given by  � � � π L2 (x, θ¯2 ) = D 1 − Φ( α + β(µx + (1 − µ)y − θ¯2 )) − t.  Since π L2 is continuous and strictly decreasing in x, there is a unique point κ2 such that π L2 (κ2 , θ¯2 ) = 0. Therefore, L2 types play according to a threshold strategy with cutoff κ2 . They attack if and only if x ≤ κ2 .  The behavior of higher types is similar to that of L2 types since they also believe that  others are playing threshold strategies. Lk types will play a threshold strategy with cutoff κk , where κk is determined implicitly by � D−t Φ( α + β(µκk + (1 − µ)y − θ¯k )) = D  (2.3)  � Φ( β(κk−1 − θ¯k )) = θ¯k .  (2.4)  with θ¯Lk determined by the solution to  Proposition 2.1 summarizes these results.  Proposition 2.1. Let A1 and A2 hold. In the speculative attack game, an Lk type attacks if and only if x ≤ κk for all k ≥ 1, where κ1 is determined by equation (2.2) and κk (k > 1) is determined by equations (2.3) and (2.4).  Define the likelihood of a successful attack to be proportional to the size of the interval 8 Assumption A2 guarantees continuity in expected payoffs. However, without this assumption there still exists a unique point κ such that π L1 > 0 if x < κ and π L1 < 0 if x > κ. As a result, L1 types still play threshold strategies, although they may no longer be indifferent between attacking and not attacking at the threshold signal.  14  ¯ where θ¯ is the aggregate threshold for a successful attack under level-k thinking (i.e. [−∞, θ], there will be a currency crisis if θ ≤ θ¯ and no crisis otherwise).9 The intuitive comparative static results hold under level-k thinking. If the payoff to attacking (D) decreases, the  likelihood of a successful attack decreases. If the cost of attacking (t) increases, the likelihood of a successful attack decreases. If public information (y) increases, then the likelihood of a successful attack crisis decreases. Proposition 2.2 formalizes the comparative static results. Proposition 2.2. Let A1 and A2 hold. The likelihood of a successful attack falls whenever D decreases, t increases or y increases.  2.4.1  Sophisticated types  Limited depth of reasoning models avoid the cognitive complexity of equilibrium analysis by allowing players to determine their optimal strategies recursively. However, they impose the constraint that Lk types do not believe that others reason similarly to them. Including sophisticated types relaxes this constraint. A sophisticated type is aware that there are other sophisticated types in the population. They best respond to the actual distribution over Lk and sophisticated types. The inclusion of sophisticated types allows for players who have sophisticated reasoning abilities but understand that not all players do. The existence of boundedly-rational types influences the beliefs of sophisticated players so that multiple equilibria driven by selffulfilling beliefs do not arise as readily as they do under common knowledge of rationality. As a result, behavior is unique and public information is not destabilizing even when there are sophisticated players in the level-k thinking model. Multiple equilibria may arise for the sophisticated types depending on the relative precision of public and private information. However, the range of information structures over which there is unique play is much larger in the level-k model with sophisticated players than under equilibrium. If there is a proportion γ of sophisticated players, then sophisticated √ types play according to a unique threshold strategy provided γ √αβ ≤ 2π. As a result, as long as the proportion of sophisticated types is sufficiently small, uniqueness holds. In  this way, the existence of boundedly rational types act like a coordination mechanism for sophisticated players - they drive out equilibria driven by self-fulfilling beliefs. Proposition 2.3. Let γ be the proportion of sophisticated players and let A1 and A2 hold. If √ γ √αβ ≤ 2π, there is a unique equilibrium in the speculative attack game where sophisticated types attack if and only if x ≤ κs , where κs is uniquely determined. √ The condition, γ √αβ ≤ 2π, is a sufficient condition. It holds when there is only private √ information. It fails when there is only public information. However, γ √αβ ≤ 2π, is not a 9 The  proof of Proposition 2.2 establishes that there exists a unique θ¯ such that given the distribution of ¯ cognitive types there is a currency crisis if and only if θ ≤ θ.  15  necessary condition. Consider the case where there is precise public information. Let the level-k cutoff be given by κ. This means all level k types attack whenever θ ≤ κ. If the  proportion of level-k types is greater than the cutoff κ and the proportion of sophisticated types is less than the cutoff κ (i.e. γ < κ < 1 − γ), then sophisticated types will have a unique equilibrium strategy to attack whenever θ ≤ κ.  2.4.2  Consistency with experimental evidence  This section establishes that level-k thinking is consistent with all four experimental regularities. Propositions 2.1 and 2.2 establish that level-k thinking is consistent with Regularities 1 and 2 under all information conditions. To see that Regularity 3 is consistent, consider Corollary 2.1, which restates the results of Proposition 2.1 when there is either only private information or only public information. Corollary 2.1. Let A1 and A2 hold. Suppose players receive a private signal xi or a public signal y with xi , y ∼ N (θ, β1 ). Let θ be distributed uniformly on the real line. There exists a unique cutoff, κY , and unique cutoffs, κX k , for each k ≥ 1, such that  (i) Lk types attack if and only if xi ≤ κX k , for k ≥ 1 when information is private (ii) Lk types attack if and only if y ≤ κY , for k ≥ 1 when information is public  When there is only public information, all types coordinate on the same threshold cutoff. However, when information is private, different types use different threshold signals. This results from the interaction of bounded depths of reasoning with differences in the nature of public and private information. Under public information, all types share the same information, hence all types share the same beliefs, regardless of the order of beliefs. When information is private, higher-order beliefs are more disperse than lower-order beliefs. This causes differences in behavior for players with different depths of reasoning and leads to greater coordination under public information than under private information. The degree of coordination under public information is 1 for all y ∈ R. The degree of coordination under  private information is always less than or equal to 1 for all x ∈ R (with strict inequality whenever there are at least two different depths of reasoning). Thus, public information increases the degree of coordination under level-k thinking. Regularity 4 refers to point estimates of the threshold cutoffs. If we place reasonable additional restrictions on the behavior of L0, the level-k model will be consistent with this additional regularity. In order to see this, we establish two additional results. First, we show X that we can ensure that L1 types are more likely to attack than Lk types (i.e. κX 1 > κk for all k > 1). Next, we show that when the precision of private and public information is  equal, L1 thresholds will be the same (i.e. κY = κX 1 ). And, finally, we show that increasing the precision of public information will increase cutoff thresholds, ensuring Regularity 4. 16  Proposition 2.4 establishes that the threshold cutoffs for each level-k type converge monotonically to the equilibrium threshold as k tends to infinity. Given the specification for L0, the cutoffs either monotonically increase or decrease towards x ¯E depending on whether κ1 < x ¯E or κ1 > x ¯E , respectively. Proposition 2.4. Let A1 and A2 hold. Let x ¯E be the equilibrium cutoff and {κk } be the set  of level-k cutoffs in the private information game. Then the threshold cutoffs {κk } converge monotonically towards the equilibrium cutoff x ¯E as k → ∞. In addition, (i) if κ1 < x ¯E , then {κk } is a strictly increasing sequence  (ii) if κ1 > x ¯E , then {κk } is a strictly decreasing sequence If the L1 cutoff is above the equilibrium cutoff under private information, then the L1 cutoff will be higher than the cutoffs of all other Lk types (i.e. κ1 > κ2 > κ3 > · · · ). For  this to hold we require the behavior of L0 types to be biased towards the payoff dominant equilibrium. This is a natural specification for L0. Many experiments find that players tend to play payoff-dominant actions in coordination games, at least in initial periods (e.g. Costa-Gomes et al., 2009). Let P = { 12 , 1; 12 , 0} be the distribution where half of the players attack with probability one. P represents the beliefs of the equilibrium threshold player when there is only private  information. Because information is symmetric, each player believes that half the players should receive a signal above his own. Therefore, the threshold player believes that half the players will attack. If Q first-order stochastically dominates P , L0 behavior is biased X towards the payoff-dominant equilibrium and we are guaranteed that κX ¯E for all 1 > κk > x  k > 1. Now, suppose the L0 specification does not differentiate between private and public information (i.e. Q(l|x) = Q(l|y)). Then the L1 thresholds will be equivalent under both private and public information when the precision of the signals is the same. Therefore, as long as the L1 threshold, κY , increases as the precision of public information increases, Regularity 4 will hold. We can show this happens when Q = {ρ, 1; 1 − ρ, 0} for some ρ ≥  large. The requirement that  D−t D  1 2  and  D−t D  is not too  not be too large is necessary to ensure that the cutoff is  not too close to one. To see why, notice that if t is very small, the threshold cutoffs under private information may actually be above one because the payoff to a successful attack is large compared to the cost of attacking. Thresholds will never be greater than one under precise public information. These conditions will guarantee the level-k model is consistent with Regularity 4. A3 Q(l) = {ρ, 1; 1 − ρ, 0} for some ρ > 0.5 and  D−t D  < 0.5.  Proposition 2.5. Let A1, A2, and A3 hold. Let κY be the level-k threshold in CI and let Y X {κX k } be the threshold for the Lk type in PI. Then κ > κk for all k ≥ 1.  17  Assumption A3 is not necessary but is sufficient to guarantee Regularity 4. Many other specifications for L0 behavior will also guarantee this result. For example, let Q be the distribution {1, p; 0, 1 − p} where everyone attacks with probability p and no one attacks with probability 1 − p. Then if p ≥ .55 and hold as well.  D−t D  is sufficiently small, then Regularity 4 will  Table 2.2 summarizes the empirical regularities and the corresponding level-k theory. Regularity  Experimental Findings  1  Proposition 2.1: All Lk types, k ≥ 1, play threshold strategies  2  Proposition 2.2: The likelihood of a successful attack decreases when the cost of attacking rises  3  Corollary 2.1: If there are at least 2 types, the standard deviation of thresholds in CI is less than the standard deviation of thresholds in PI  4  Proposition 2.5: Under an additional L0 assumption, the likelihood of a successful attack is greater in CI than in PI  Table 2.2: Qualitative results under level-k thinking  2.4.3  Robustness  Regularities 1-3 are robust predictions of limited depth of reasoning models. The regularities hold under only weak assumptions on the behavior of L0. And, they hold for any limited depth of reasoning model. More specifically, Regularities 1-3 only rely on assumption A1. A1 is a very weak requirement and holds trivially under the specification that L0 behavior does not depend on information. The level-k thinking model also places specific assumptions on the beliefs of each cognitive type about the cognitive types of others. Specifically, an Lk type believes there are only Lk-1 types. Cognitive hierarchy, another popular limited depth of reasoning model, assumes that a Lk type believes there are all lower types with weights determined by a conditional Poisson distribution. In general, we could think that an Lk type may hold any distribution of beliefs over the lower types. The exact belief structure does not matter. Moreover, the exact type distribution does not matter. The regularities hold for any distribution of types in the population.10 In other words, Regularities 1-3 hold for any limited depth of reasoning model. This is shown in Appendix A.2. 10 Regularity  3 holds only if there are at least two different types in the model.  18  The regularities even hold for L∞ types. L∞ types have infinite depths of reasoning but believe others have bounded depths of reasoning. Further, the regularities hold for some proportion of sophisticated types. These are types that have infinite depths of reasoning and take into account the fact that others may also have infinite depths of reasoning. As long as the proportion of sophisticated types is not too large, limited depth of reasoning with sophisticated types is consistent with the existing experimental evidence. Regularities 1-3 are not merely a consequence of players having bounded depths of reasoning, but of them basing their beliefs on simplified models of others. One of the main criticisms of the limited depth of reasoning literature is that predictions are not robust to model specification. Predictions depend upon the specification for L0 and the distribution of types. This is not true for Regularities 1-3. They are robust predictions of limited depth of reasoning models.  2.5  Experimental analysis  This section analyses the experimental data from Heinemann et al. (2004) (HNO hereafter). The distribution of level-k and equilibrium types are estimated using a finite mixture model. Allowing for level-k types significantly improves our ability to explain the experimental data.  2.5.1  The experiment  Subjects play a finite player game similar to the above game. In each session, there are 15 subjects who simultaneously decide whether to attack or not attack. An attack is associated with an opportunity cost t (which is modeled as the safe payoff to not attacking). An attacking subject earns the amount θ if the attack is successful. An attack is successful if and only if a sufficient number of traders decide to attack. The threshold to a successful attack is determined by a(θ) which is a non-increasing function in θ. Low θ represents good fundamentals and a high threshold to a successful attack. High θ represents poor fundamentals and a low threshold to a successful attack. This game differs from the game analyzed in previous sections in a number of ways: it is a discrete player game, the payoff from a successful attack depends upon the fundamentals, and a lower θ corresponds to better fundamentals. However, none of these changes alter the previous analysis in a substantial way.11 The only change to note is that the attack/nonattack regions are flipped. Players attack if and only if their signals are above the cutoff threshold. There are two information treatments in the experiment. A private information treatment (PI) and a common information treatment (CI). In PI, players do not know the fun11 See Appendix A.4 for an analysis of the level-k results. See HNO for an analysis of the equilibrium results.  19  Treatment PI20 CI20 PI50 CI50  Information Private Public Private Public  Safe Payoff (t) 20 20 50 50  Sessions 5 5 5 6  Table 2.3: Treatments damental, but know that θ is distributed uniformly on [10, 90] and receive private signals xi randomly drawn with independent and uniform conditional distributions on [θ − 10, θ + 10]. In CI, players learn the fundamental θ when they receive a precise public signal y = θ.  There are also two payoff treatments where the payoff to the safe (not attack) option varies from t = 20 to t = 50. The treatments are summarized in Table 2.3. Subjects play 16 rounds in the experiment. Because we are interested in initial play, we analyze only the first 8 rounds of play.12 We also drop 4 high stakes/extended length sessions and 4 sessions with an alternative payoff specification. Therefore, we analyze the data from a total of 21 sessions and 315 subjects. Each subject makes 10 binary choices (attack or not-attack), each round, for 8 rounds.  2.5.2  13  Econometric analysis  The analysis uses two main econometric methods. First, we follow Heinemann et al. (2004) and Duffy and Ochs (2009) and estimate aggregate mean thresholds by estimating a logit response model in which the binary attack decision depends on a constant and the signal (Z=x or y). That is, we use maximum likelihood estimation to find the coefficient estimates a ˆ and ˆb, that are a best fit to the logit response function: −1  P r(attack|Z) = [1 + exp (−a − bZ)]  The attack threshold can be viewed as the critical value, Z ∗ , for which a player is indifferent between attacking and not attacking, which obtains when P r(attack|Z ∗ ) = 0.5. Using this, we can obtain the estimated attack threshold Zˆ ∗ = − aˆ . The standard deviation of a logistic distribution with mean − aˆˆb is given by  π √ . ˆ b 3  ˆ b  We take this to be a measure of  the coordination of subjects around the estimated attack threshold. Second, we follow Costa-Gomes et al. (2001) and Costa-Gomes and Crawford (2006) and estimate a finite mixture model that allows for different behavioral types. This approach 12 Ideally, we would only consider Round 1 data, before subjects receive feedback. But, on average, each subject only makes 1 choice in the first round that separates the predictions of level-k and equilibrium types. So we analyze the 8 rounds of data in order to get better type separation. However, we could still estimate the type distribution off of these approximately 315 choices that separate types. As we would hope, the level-k model does even better at explaining the initial play than it does at explaining the full 8 rounds of data. These results are listed in Appendix E. 13 Our aggregate results are similar to HN0’s aggregate full sample results (see Table 2.6).  20  assumes that each subject’s type is drawn from a fixed common prior distribution over all types. The types we allow include: an equilibrium type (E) who believes all other types are equilibrium types, an L1 type who believes all other types are L0 types and whose behavior is fixed (and specified below), an L2 type who believes all other types are L1 types, and an L3 type who believes all other types are L2 types. We assume that each player follows the predictions of a particular type with error. Because the subjects often make a type’s exact choices, we use a simple spike-logit error structure. Index types k = 1, . . . , 4 and choices by q = 1, . . . , 80. In each choice, a subject has a given probability 1 − �k of making his type’s choice exactly, and with error-rate, �k , makes choices that follow a logistic distribution with error density, dkq (aiq , λ).14 The parameter λk represents the logistic precision parameter. For subject i, Let Qik represent the set of choices where subject i’s action is consistent with type k’s predicted action. Subject i’s log-likelihood of following action profile ai is given by:   lnLi (p, �, λ|ai ) = ln   4 �  k=1    pk   � �  q∈Qik    � � 1 − �k + �k dk (aiq , λ)   �k dkq (aiq , λk )  (2.5)  q ∈Q / ik  The aggregate log-likelihood function is given by  lnL(p, �, λ|a) =  S � i=1    ln   4 �  k=1    pk   � �  q∈Qi  where S is the number of subjects.    � � 1 − �k + �k dk (aiq , λ)   �k dkq (aiq , λk ) (2.6) q∈QiC  With four types, this model has 11 independent parameters: 3 independent type probabilities pk , 4 independent precision parameters λk , and 4 independent error-rates �k . In order to fully specify this model, we must specify the behavior of L0 types. In most level-k applications, L0 behavior is chosen to be either uniformly random (i.e. Q(l) = l) or some focal behavior. However, we know from previous coordination experiments that players tend to focus on payoff-dominant outcomes (at least during initial play) and Crawford et al. (2008) find that level-k models with L0 players biased towards payoff-dominant outcomes explain behavior in coordination games fairly well. For these reasons, we assume that the behavior of L0 types is given by the cumulative distribution function Q(l) = l2 . This biases the behavior of L0 types towards the payoff-dominant outcome (attack) relative to 14 The  error density, dkq (aiq , λk ), is defined dkq (aiq , λk ) =  h i exp λk Sqk (aiq ) h i h i. exp λk Sqk (attack) +exp λk Sqk (not−attack)  The  term, Sqk (aiq ), is type k� s expected payoff from playing action aiq in choice q, given type k’s beliefs about the actions of others.  21  the uniformly random specification Q(l) = l. Notice that this specification of L0 behavior satisfies assumptions A1 and A2. Each type’s behavior is determined by a threshold cutoff. A type attacks if and only if his signal is above the threshold cutoff. Given the behavior of the L0 types we can determine the threshold cutoffs for L1, L2 and L3 types. The threshold cutoffs for equilibrium types are given by the unique equilibrium threshold in the private information treatment and by the global games threshold in the public information treatments.15 Thresholds vary across both information treatments and safe-payoff treatments for all four types. The type’s threshold predictions are given in Table 2.4. These cutoffs fully specify a type’s predicted actions. A type k attacks if and only if her signal in choice q is greater than or equal to her cutoff threshold. CI t 20 50  E 44 64  L1+ 36 56  PI E 41.8 66.0  L1 36.9 57.1  L2 38.8 60.8  L3 40.0 62.9  Table 2.4: Threshold predictions by type  2.5.3  Qualitative results  This section demonstrates that the qualitative results exist in our restricted sample of HNO’s experimental data. Table 2.5 specifies the experiment-specific analog of each of the 4 regularities discussed in Section 2.3. The mean threshold cutoffs are estimated for each treatment using the binary response logit model specified in the previous section. Table 2.6 lists the estimated mean cutoff thresholds for each treatment and the associated standard deviations for all subjects who do not play a dominated strategy. 15 HNO find that the global games equilibrium explains this data better than any other known equilibrium selection method, i.e. payoff-dominance, risk dominance or max-min.  22  Regularity  Experimental Findings  1  80% of subjects’ play is consistent with (undominated) threshold strategies  2  The estimated mean threshold when t=50 is greater than the estimated mean threshold when t=20  3  The estimated standard deviation in CI treatments is less than the estimated standard deviation in PI treatments  4  The estimated mean threshold in CI treatments is less than the estimated mean threshold in PI treatments Table 2.5: HNO’s qualitative results  Treatment (n=25200)  PI20  CI20  PI50  CI50  Estimated mean threshold  44.96  39.04  57.75  51.14  Estimated standard deviation  14.16  13.81  13.01  11.06  Table 2.6: Estimated mean threshold cutoffs We can see from Table 2.6 that mean cutoffs are lower in the P120 and CI20 treatments relative to the PI50 and CI50 treatments. Subjects respond to changes in the safe payoff t as predicted in both the level-k and equilibrium models. Mean and individual thresholds are also lower in the CI treatments relative to the PI treatments in both t = 20 and t = 50 treatments. This is consistent with the level-k model. However, equilibrium predicts that the mean threshold should increase in the CI50 treatment relative to PI50. Lastly, notice that the standard deviation of estimated thresholds is lower in the CI treatments relative to the PI treatments. This is inconsistent with an equilibrium analysis, as discussed above, but consistent with level-k. Regularity 3 is consistent with the level-k model because of the coordinating role public information plays in this model. All types play according to the same threshold in the CI treatment and different thresholds in the PI treatment.  23  2.5.4  Type classification  This section classifies subjects into types by estimating a finite mixture model. Table 2.7 estimates equation (2.6) using maximum likelihood. Level-k types make up 70 percent of the estimated type distribution. This is compared to 30 percent for equilibrium types. L1 types are most frequent, making up 54 percent of the type distribution. The likelihood ratio test rejects the equilibrium model in favor of a model that includes all four types at all standard significance levels. In addition, a t-test rejects the null hypothesis that the proportion of L1 types are less than or equal to the proportion of equilibrium types at all standard significance levels. Both the precision parameters and the error-rate parameters combine to determine the rate of deviation from each type’s predicted play. L1 and E follow their type’s predicated action 95 percent of the time on average. L2 types follow type’s actions 80 percent of the time on average. And, L3 types follow their type’s predicted action about 56 percent of the time. In light of Proposition 2.3, we can ask what proportion of sophisticated types will be sufficiently small enough to ensure that the behavior of these types is anchored by the behavior of the level-k types?16 A proportion of sophisticated types of 30 percent is small enough to ensure that sophisticated types have a unique equilibrium in the public information treatments. In the high cost of attacking treatment (t=50), there is a unique equilibrium in the public information treatment even if the proportion of sophisticated types is as large as 45 percent. In the private information treatment, there is always a unique equilibrium for sophisticated types. As a check on our type specifications, we can check whether subjects should have been assigned as L0 types rather than as E, L1, L2 or L3 types. We can do this by comparing the likelihood of their assigned type with the likelihood of a L0 type. The aggregate L0 behavior characterized by q(l) = l2 can be supported by a variety of individual behavior. To account for this we specify that the likelihood of an L0 type is determined by randomly attacking with probability p, for any p ∈ [0, 1]. We find that 9 of the 315 subjects should be assignment as L0 types, which is about 2.8 percent of subjects. The behavior of these subjects is more consistent with random play than with threshold strategies.  2.6 2.6.1  Macroeconomics Policy analysis: speculative attack  The coordinated attack game analyzed in this paper is commonly used to determine the optimal transparency policy for central banks wishing to decrease the probability of a currency 16 We  cannot separately identify both sophisticated types and equilibrium types in this data set. Therefore, we consider the proportion of equilibrium types to be a proxy for the proportion of sophisticated types.  24  MODEL:  All Types  Level-k  Equilibrium  Log-Likelihood  -6324.8  -6614.1  -9890.7  L1 L2 L3 E  .5430  .5569  (.0264)  (.0282)  .1380  .1575  (.0224)  (.0379)  .0277  .2856  (.0355)  (.0346)  .2911  1  (.0198)  λ1 λ2 λ3 λE �1 �2 �3 �E  n  .1440  .1435  (.0100)  (.0139)  .0345  .0219  (.0408)  (.0204)  .000  .1111  (.0086)  (.0204)  .1045  .0290  (.0060)  (.0017)  .6414  .8000  (.0588)  (.0659)  .8624  .7870  (.0883)  (.0723)  .8982  1.0000  (.0854)  (.0470)  .8684  .5245  (.0604)  (.0380)  25200  25200  25200  *Notes: bracketed numbers are bootstrapped standard errors clustered at the session level.  Table 2.7: Aggregate type classification  25  crises. Central banks know the fundamentals and have control over the precision of the public information they release. Both level-k and equilibrium models make similar qualitative predictions. If the fundamentals are low, then increasing the precision of public information will increase the probability of a currency crisis. If fundamentals are strong, then increasing the precision of public information will decrease the probability of a currency crisis. This is summarized in Proposition 2.6. Proposition 2.6. There exists a y¯E , y, y¯ ∈ R, such that: √ (i) Let √αβ ≤ 2π. Under equilibrium, increasing the precision of public information  decreases the probability of a currency crisis if y > y¯E and increases the probability of a currency crisis if y < y¯E . (ii) Let A1 and A2 hold. Under level-k thinking, increasing the precision of public information decreases the probability of a currency crisis if y > y¯ and increases the probability of a currency crisis if y < y. Under limited depth of reasoning, there is some region where the public information about the fundamental is sufficiently strong such that increasing the precision of public information would result in decreasing the probability of a speculative attack. The optimal transparency policy would be to release complete information about fundamentals. However, under equilibrium there is a tradeoff between increasing the precision of public information and leaving private information sufficiently precise as to not go back to the case of multiplicity. There is no such trade-off under level-k thinking. If fundamentals are relatively strong, providing the most precise public information possible is the optimal transparency policy. Further, if L0 satisfies the assumption A3, then the public information cutoff will be higher under level-k thinking than under equilibrium. There will be a range of intermediate values where level-k thinking and equilibrium make the opposite policy prescriptions. If public information is in this intermediate range, then the equilibrium prescription is to increase the precision of public information and the level-k prescription is to decrease the precision of public information.  2.6.2  Global games  This paper is closely related to the global games literature. The speculative attack game analyzed in this paper is one of the workhorse models of global games. Because of this, the results in the current paper have implications for this literature. Global games, initiated by Carlsson and van Damme (1993b,a) and furthered by Morris and Shin (1998, 2003), has shown that multiplicity in coordination games stems from common knowledge of payoffs. The concern is that increased precision of public information may be destabilizing; it could lead to multiplicity driven by self-fulfilling beliefs. However, both the existing experimental 26  literature and a boundedly rational approach to strategic reasoning suggests that public information has no destabilizing effects. The global games literature has developed substantially beyond that of the simple model analyzed in this paper. In this section, we discuss whether the level-k insights from this simple model carry over to the second-generation global games models. Equilibrium and level-k make similar comparative static predictions in terms of how changing underlying payoff parameters, like the cost or payoff of attacking, affects the likelihood of an attack. However, comparative static predictions from changing the information structure are very different in the two models. Under equilibrium, public information is potentially destabilizing relative to private information as it leads to multiplicity. Under level-k, public information has a coordinating role over private information. The global games literature, at its core, is concerned with the destabilizing effects of different information structures. Second-generation models are built on the same premise; some information structures may enhance the coordination motive and lead to multiplicity. Thus, the secondgeneration literature is subject to the same criticism as the first-generation models; they predict that public information has a destabilizing effect. This is not found in the laboratory and, it is not an inherent concern in limited depth of reasoning models. We consider a few examples below. Angeletos and Werning (2006) look at the role of public information in coordinated attack games by introducing a financial market into the global game environment. Financial markets aggregate exogenous private information to create endogenous public information. Therefore, increasing the precision of private information automatically increases the precision of public information. This means that multiplicity remains even when we introduce private information into the model. The authors make an important point that private and public information are not independent policy tools. However, the insights are still subject to the same experimental criticism as the simple coordinated attack model analyzed in this paper. In the laboratory, exogenous public information does not have destabilizing effects. Angeletos et al. (2007) analyze coordinated attack games in a dynamic context. They allow agents to take actions in many periods and learn about the underlying fundamental over time. Agents essentially play a repeated static global game similar to the game analyzed in this paper, where information about fundamentals is different every period. They find that multiplicity may obtain under the same conditions on the information structure that generated uniqueness in the static game. In each period, multiplicity arises when the precision of public information (either exogenous public information or endogenous public information formed from private information) is too precise relative to private information. Private information is destabilizing in the dynamic game because public information is destabilizing in the static game and all private information becomes public information over time. However, experiments find that public information does not have destabilizing  27  effects in the static game. Hellwig et al. (2006) analyze the role of private information in generating currency crises in an environment where the domestic asset markets are modeled explicitly. They show that multiplicity remains even after introducing noisy private signals about interest rates. In this environment, multiplicity does not originate from a coordination problem but rather from market fundamentals. Thus, private information does not resolve the issue of multiple equilibria in this environment. Regardless, a level-k analysis would allow you to generate unique predictions (given a fixed L0) and more importantly generate comparative static predictions (which would be largely independent of L0 behavior).  2.6.3  Other macroeconomic applications  Given the results in this paper, a limited depth of reasoning analysis is likely to have interesting implications in other applications that distinguish between private and public information. For example, coordination games with weak strategic complementarities have been used to model price setting behavior in firms under monopolistic competition (Woodford, 2002). In this setup a firm’s optimal price depends on higher-order beliefs about some fundamental that describes the state of the monetary supply. Empirically, firms adjust prices slowly in response to monetary supply shocks. Under equilibrium, firms over-react to public information and under-react to private information about the fundamental relative to the optimal price setting behavior if the fundamental was known. Woodford explains inertia in price setting by arguing that firms process public information differently and hence should be treated like private information. If we apply a limited depth of reasoning model to this game, firms will under-react to both public and private information. Limited depth of reasoning provides an explanation for price inertia that provides a consistent treatment of private and public information.  2.7  Related literature  The motivation for this paper is similar to Strzalecki (2010). Both papers study behavior under limited depth of reasoning in environments where higher-order uncertainty plays a role in equilibrium behavior. Strzalecki uses limited depth of reasoning models to explain observed behavior in Rubinstein’s (1989) email game. He shows that if players have finite depths of reasoning then both the attack and no attack outcomes can be supported (where equilibrium supports only the no attack outcome). The two papers ask different questions using similar coordinated attack models. Strzalecki investigates whether the limited depth of reasoning model is consistent with players playing the attack outcome under a particular information structure. This paper is interested in the coordinating roles of public and private 28  information and investigates behavior under different information structures. Moreover, the limited depth of reasoning solution concept is applied differently in the two papers. The analysis of Strzalecki (2010) applied to the coordinated attack game in this paper would suggest that any undominated action could be played. This approach would be equivalent to fixing the solution concept as the set of rational actions or mth -order rationalizable actions (for some fixed and small m). However, the analysis in the current paper closely follows the approach of the existing level-k and cognitive hierarchy literature and focuses on the outcomes that result from a fixed L0. As a result, we show that a level-k model is consistent with the existing experimental literature under some weak restrictions on L0. The mechanism underlying this paper is similar to Crawford (2003) which studies strategic communication in cheap talk games. Under equilibrium with common knowledge of rationality, communication is not strategic and there are only babbling equilibria. Crawford essentially shows that as long as the proportion of sophisticated types is not too large there will be strategic communication. This is similar to Proposition 2.3 in the current paper which shows that as long as the proportion of sophisticated types is small, multiple equilibria will not arise. In both papers, allowing for the existence of boundedly rational types allows us to talk meaningfully about intuitive phenomena - strategic communication and the coordinating role of public information. Neither result stems solely from nonequilibrium beliefs or bounded depths of reasoning, but from the belief that others might have bounded depths of reasoning. Shapiro et al. (2010) experimentally test the level-k model in a related incomplete information coordination game with weak complementarities. They consider games where players have both an information motive (incentive to match fundamentals) and a coordination motive (incentive to match behavior of others). They study a range of games where the strength of the information and coordination motives vary. They find that the level-k model is consistent with behavior in coordination games when the coordination motive outweighs the information motive. The game in the current paper is one where players have a strong coordination motive. In fact, players care about the fundamentals only to the extent that the fundamentals may inform them about the behavior of others. Therefore, both the current paper and Shapiro et al. (2010) find evidence that the level-k model is consistent with experimental evidence in (at least some) classes of incomplete information coordination games.  29  Chapter 3  Mechanism design with level-k types 3.1  Introduction  Laboratory experiments frequently find that behavior deviates from standard equilibrium predictions when players interact in novel environments. Models that relax the belief consistency assumptions of equilibrium, like level-k and cognitive hierarchy models, have been increasingly used to explain this behavior.1 This paper contributes to this line of research by studying the following question: how is optimal mechanism design influenced by the inclusion of these boundedly rational agents? Level-k models have been successful at explaining initial play in laboratory experiments relative to standard equilibrium solution concepts. Thus, studying mechanism design under the level-k model is relevant given the central role of equilibrium assumptions in mechanism design. This is particularly true, given that mechanism design inherently deals with the creation of new mechanisms where the learning justification of equilibrium is weak. Further, equlibrium-optimal mechanisms are often not robust in that even small deviations from the assumed equilibrium behavior can make the mechanism yield poor results. A formal approach to analyze robustness of equilibrium-optimal mechanisms is lacking (with the exception of the “robust mechanism design” literature by initiated by Bergemann and Morris (2005)). As a result little is known about the practical usefulness of these mechanisms. This paper takes an evidence-based approach to the robustness question and explores optimal mechanism design under the empirically supported solution concept of level-k thinking. This paper studies standard Bayesian implementation problems, replacing the hypothesis that agents will play a Bayesian Nash equilibrium in the game created by the mechanism 1 See  Crawford et al. (forthcoming) for a review of this literature.  30  with the alternative assumptions that agents follows the level-k model. In the level-k model, agents anchor their beliefs in a naive model (level-0) of others’ likely responses, and then adjust their beliefs by a small number of iterated best responses. This yields a tractable model of strategic behavior in which all players determine their optimal actions in only a finite number of steps. This solution concept relaxes the mutual consistency assumption of Bayesian equilibrium by allowing players to anchor their beliefs in the non-strategic level-0 behavior. This paper asks three main questions. What does the revelation principle look like in this context? What is the relationship between level-k and Bayesian implementation? How robust are level-k mechanisms to alternative specifications of beliefs? We establish that the revelation principle holds under the level-k model. However, a direct mechanism in this environment is different than a direct mechanism in the Bayesian implementation environment. Under Bayesian implementation, the revelation principle ensures that we need only consider mechanisms where players truthfully reveal their payoff types. In the level-k environment, a direct mechanism must allow players to reveal both their payoff type and their level of reasoning. Since a player with a low level of reasoning may view the game very differently from a player with a high depth of reasoning, we need to allow players with the same payoff type but different levels to send different messages. Augmenting the message space (to allow more than just truthful revelation of payoff types) will expand the set of social choice correspondences that are level-k implementable relative to Bayesian implementation. We establish a set of necessary conditions for level-k implementation and show that level-k implementation is a weaker implementation concept than Bayesian implementation. Relaxing the mutual consistency assumptions of Bayesian equilibrium, allows the planner to relax cross-player consistency requirements when designing a level-k mechanism. In general, there is scope to improve efficiency under level-k implementation relative to Bayesian implementation. However, in a certain class of environments, relaxing mutual consistency gives level-k implementation no more flexibility relative to Bayesian implementation. In separable environments, Bayesian and level-k implementation are equivalent in terms of which social choice functions can implemented. Lastly, we discuss several different aspects of robustness. Optimal mechanisms typically depend strongly on specific aspects of players’ beliefs: beliefs about payoffs, beliefs about levels of reasoning, and beliefs about the equilibrium being played. Finding mechanisms that are robust to relaxing these strong common knowledge assumptions, typically known as the Wilson doctrine, can insure that a social choice correspondence will be implemented even if the designer does not know players’ beliefs.2 This paper explores three aspects of 2 Much  of this literature is due to Bergemann and Morris (2005), who investigate aspects of robust mechanism design (relaxing common knowledge of payoff assumptions) while maintaining the assumption the about common knowledge of rationality. The limited depth of reasoning literature implicitly relaxes the  31  mechanism robustness to beliefs: beliefs about players’ levels of reasoning, beliefs about payoffs, and level-k equilibrium selection. Throughout the main body of this paper we use the level-k model, a type of limited depth of reasoning model, which makes specific assumptions about what a player with a certain level of reasoning believes about the levels of others. We show that whenever a social choice correspondence is implementable, it is possible to find a mechanism that will implement the social choice correspondence under any relaxation of level-k beliefs that maintain the spirit of the ‘finite depths of reasoning’ assumption.3 Additionally, we show that under relatively weak assumptions, level-k implementation under a particular assumption about players’ beliefs about the payoff types of others guarantees the existence of a mechanism which will implement the social choice correspondence under level-k implementation and any specification of beliefs about the payoff types of others. We demonstrate the weakness of the assumptions by showing that this version of robust level-k implementation is weaker than Bayesian implementation. Lastly, the main theoretical results in this paper follow from assuming implementation in a truth-telling equilibrium. However, this assumption can be relaxed. Truthful level-k implementation ensures that we can find a mechanism that implements the social choice correspondence in both the truthful level-k equilibrium and in the level-k equilibrium determined by all (nonstrategic) level 0 players playing uniformly, which is the common assumption in the experimental and applied level-k literatures. Results are illustrated throughout this paper with simple, discrete type, bilateral trade examples. These examples illustrate the role of the augmented message space in the revelation principle, differences between level-k and Bayesian mechanisms, and adaptations of the mechanism to relax common knowledge assumptions. There is a growing literature that focuses on behavioral mechanism design.4 This paper adds to the mechanism design literature by studying implementation under relaxations of mutual consistency. To the best of my knowledge, only one other paper deals with implementation under the level-k model. Crawford et al. (2009) look at setting optimal reserve prices in first and second price auctions when players are level-k types. This paper expands on that work establishing standard mechanism design results for general environments. This paper proceeds as follows. Section 3.2 sets up the payoff environment and defines level-k implementation. Section 3.3 establishes the revelation principle. Section 3.4 discusses some bilateral trade examples. Section 3.5 establishes necessary conditions for level-k imassumption of common knowledge of rationality. In this paper, we investigate aspects of robust mechanism design that relaxes both common knowledge of payoff and rationality assumptions. 3 In the level k model, each level k type believes there are only level k-1 types. However, it is possible to find a mechanism that will be incentive compatible for a level k types that hold any beliefs over lower levels, such as in the cognitive hierarchy model. 4 See Healy and Mathevet (forthcoming), Severinov and Deneckere (2006), Eliaz (2002), and Eliaz and Spiegler (2006) for examples.  32  plementation. Section 3.6 compares level-k and Bayesian implementation. Relaxations of the common knowledge assumptions are discussed in Section 3.7.  3.2 3.2.1  Setup Payoff environment  We consider a finite set of agents 1, 2, . . . , I. Agent i’s payoff type θi ∈ Θi , where Θi is a  finite set. We write θ ∈ Θ = Θ1 × · · · × ΘI . There is a set of outcomes Y . Each agent  has a utility function ui : Y × Θ → R. Each agent cares about the payoff type profile and the outcome. There is a social planner who is concerned with implementing a social choice correspondence F : Θ → 2Y \∅. The planner would like the outcome to be an element of F (θ) whenever the true payoff type profile is θ. The environment is fixed throughout this paper.  3.2.2  Type spaces  We use the framework of a type space in order to formally define each agent’s beliefs about the payoff types of others. The standard way to do this is to use a Bayesian type space. The set of payoff types along with a common prior over the set of payoff types constitute a Bayesian type space. Definition. An Bayesian type space B is a structure B = �Θ1 , . . . , ΘI ; p�, where p ∈ �(Θ).  Given the common prior p, each payoff type forms her beliefs by conditioning on the common prior according to Bayes’ rule. The belief of payoff type θi about the payoff types of others is given by p(θ−i |θi ) =  P  p(θ) p(θi ,θ−i ) .  θ−i ∈Θ−i  Level-k models are designed to capture the idea that players are often capable of perform-  ing only a finite number of levels of reasoning in order to figure out their optimal behavior. We use a type space approach to define the level-k model based on Strzalecki (2010) who developed the framework for games of complete information. We expand the framework here to account for games of incomplete information. ¯ where Definition. A B-based level-k type space C is a structure C = �C1 , . . . , CI ; B; k�, ¯ ¯ B is a Bayesian type space B = �Θ1 , . . . , ΘI ; p�, k ≥ 1 ∈ N and Ci = Θi × {0, 1, . . . , k}. A player’s cognitive type ci = (θi , ki ) is a 2-dimensional type representing both her payoff type θi and her level ki . A player’s level represents her depth of reasoning. A player with  33  a level k uses only k steps of reasoning in order to figure out her optimal behavior in any ¯5 game. The type space contains all levels of reasoning from 0 to k. A player’s beliefs about the cognitive types of others are determined both by her payoff type and her level. The beliefs of a cognitive type ci = (θi , ki ) about the cognitive types of others c−i = θ−i × v −i is determined by the function bi (c−i |ci ) :   p(θ |θ ) if v = k − 1 ∀j �= i −i i j i bi (c−i |ci ) = . 0 otherwise  The notation θ × v is used to represent a payoff type/level profile (θ1 , v1 ) × · · · × (θn , vn ), ¯ n. for some v ∈ {0, . . . , k} A player with a level k puts weight only on other types that have a level exactly equal  to k − 1. This captures the core assumption of the level-k and cognitive hierarchy literature which is that a player of level k believes that the other players have levels strictly less than  k. This assumption ensures that players can calculate their optimal actions in a recursive fashion with a finite number of steps given the behavior of level 0 types. A player’s beliefs about the payoff types of others are determined by the common prior p. A player with payoff type θi and level k believes that the payoff types of others are determined by p(θ−i |θi ) and that others have level k − 1. We formally call this type space a Bayesian-based level-k type space because the beliefs about the payoff types of other players  are derived from a common prior. We drop this formalism throughout the rest of this paper and refer to these type spaces as simply level-k type spaces.  3.2.3  Solution concepts  Fix a payoff environment and a type space. A mechanism specifies an action set for each player and a mapping between action profiles and outcomes. Definition. A mechanism �M, f � consists of a set of actions M = M1 × · · · × MI and a function f : M → Y .  Given the payoff environment and (Bayesian or level-k) type space, a mechanism defines an I-player incomplete information game with action set Mi for player i and payoffs defined by ui (f (mi , m−i ), θi ).  For a given level-k type space, we can define the concept of a level-k equilibrium. The  level-k equilibrium imposes that all types are rational (that is, they play a best response given their beliefs about the actions of other players) and for beliefs about those actions to 5 The bound on the level of reasoning is not necessary, the results in this paper go through if C = i Θi × {0, 1, 2, . . .}, however bounding the depths of reasoning maintains the finiteness of the type space for simplicity.  34  be consistent with what other types are actually doing in equilibrium. Level 0 types are not required to be rational.6 Definition. For a given game defined by a mechanism �M, f � and type space C, a strategy profile s : C → M is a level-k equilibrium if and only if for all i ∈ I, for all ci ∈ Ci with ki > 0 and all mi ∈ Mi ´ ´ ui (f (si (ci ), s−i (c−i )), θ)dbi (c−i |ci ) ≥ ui (f (mi , s−i (c−i )), θ)dbi (c−i |ci )  Each level-k equilibrium can be calculated recursively given the behavior of level 0 types.  Level 1’s actions are a best response to level 0’s actions. Level 2’s actions are a best response to level 1’s actions, and so on. We depart from the standard application of the level-k solution concept in this paper. Typically, level-k thinking is applied to yield a precise prediction. This is done by specifying a specific behavior for level-0 types: usually that level-0 types play each action with equal probability (uniform level-0 equilibrium). However, in this paper we think about the levelk solution concept as generating a set of equilibria (each equilibrium can be determined by a behavioral specification for level-0 types) and then focus on partial implementation, whether there is an level-k equilibrium consistent with the social choice correspondence. This approach is mostly for theoretical convenience, we show in later sections that, (almost) without loss of generality, our results for implementation in ‘some’ equilibrium extend to implementation in the uniform level-0 equilibrium. Notice also, that our notion of level-k implementation does not require knowledge of the actual distribution of types (and hence levels). This is because implementation requires that the outcome be consistent with the social choice correspondence for all type profiles and hence does not depend upon the distribution of types. We will be interested in comparing our level-k implementation results with the more standard solution concept of Bayesian implementation. The two solution concepts differ in that the level-k equilibrium relaxes the constraint of mutual consistency imposed under Bayesian equilibrium by not requiring the incentive constraints to hold for level 0 types. Definition. For a given game defined by a mechanism �M, f � and type space B, a strategy profile s : Θ → M is a Bayesian equilibrium if and only if for all i ∈ I, for all θi ∈ Θi and all mi ∈ Mi ´ ´ ui (f (si (θi ), s−i (θ−i )), θ)dp(θ−i |θi ) ≥ ui (f (mi , s−i (θ−i )), θ)dp(θ−i |θi )  3.2.4  Implementation  A social choice correspondence F is level-k implementable on C if there exists a mechanism and a level-k equilibrium such that outcomes achieve F for every type profile in C with  6 Level 0 types do not have to play a best response to their beliefs (and may play actions that are not a best response to any beliefs i.e. play dominated actions). In fact, level 0’s beliefs are not formally defined in a level-k type space.  35  levels greater than 0.7 Definition. A mechanism �M, f � and a message profile m : C → M achieves F on C if ¯ × · · · × ΘI × {1, . . . , k} ¯ for all θ × k ∈ Θ1 × {1, . . . , k} f (m(θ × k)) ∈ F (θ) The notation θ × k will often be used to represent a payoff type/level profile (θ1 , k) ×  ¯ Note that for a mechanism to achieve F , we require · · · × (θI , k), for some k ∈ {0, . . . , k}.  that only the messages sent by cognitive types with levels at least one (k ≥ 1) be consistent with outcomes F (θ).8  Definition. A social choice correspondence F is level-k implementable on C if there  exists a mechanism �M, f � and a message profile m : C → M , such that m is a level-k equilibrium and m achieves F on C.  For completeness, we give the definition of Bayesian implementation below. Definition. A mechanism �M, f � and a message profile m : Θ → M achieves F on B if for all θ ∈ Θ  f (m(θ)) ∈ F (θ) The definition of level-k and Bayesian implementation differ only in that we require a level-k equilibrium rather than a Bayesian equilibrium that achieves the social choice correspondence. Definition. A social choice correspondence F is Bayesian implementable on B if there exists a mechanism �M, f � and a message profile m : Θ → M , such that m is a Bayesian equilibrium and m achieves F on B.  7 Partial level-k implementation ensures that the social choice correspondence can be implemented for some level-k equilibrium. Since the level-k equilibrium definition places no incentive constraints on level 0 types, a designer then has complete freedom in specifying the behavior of level 0 types (which then determines the behavior of the higher levels in the level-k equilibrium). In fact, the proof of the revelation principle in Section 3.3 considers the level-k equilibrium where level 0 types truthfully report their types. The strength of this assumption is discussed in Section 3.7.3 and we show that level-k implementation (for any level-k equilibrium) always implies implementation in the level-k equilibrium where level 0 types play all actions with equal probability (the common assumption made on level 0 behavior in the applied level-k literature). Thus, little applicability is lost under the assumption that the designer has a choice in the level-k equilibrium. 8 We only require the actions of cognitive types with levels k ≥ 1 to be consistent with the social choice correspondence as it is often found that level 0 types do not exist in experimental data. In addition, there is no way to control for the behavior of level 0 players if they are irrational and do not respond to incentives. Thus, this approach assumes rationality as a minimum requirement.  36  3.3  Revelation principle  It is well known that the revelation principle holds for Bayesian implementation.9 This means that we can restrict attention to payoff-direct mechanisms and equilibrium where players truthfully reveal their payoff types. In this section, we establish a version of the revelation principle for level-k implementation. However, a direct mechanism looks different in this environment. A type must truthfully reveal both his payoff type and level. We differentiate between the direct mechanisms required under Bayesian and level-k implementation by referring to the former as a payoff-direct mechanism. Definition. A payoff-direct mechanism is a mechanism �Θ, f � where the message set is equal to the set of payoff types, i.e. Mi = Θi ∀i.  A payoff-direct mechanism only allows players to reveal their payoff type. A direct mechanism allows players to reveal both their payoff types and their levels. Definition. A direct mechanism is a mechanism �C, f � where the message set is equal ¯ ∀i. to the set of cognitive types, i.e. Mi = Ci = Θi × {0, 1, . . . , k} A social choice correspondence F will be directly implementable if there exists a direct mechanism where truthfully revealing your cognitive type is a level-k equilibrium that achieves F. Definition. A mechanism �C, f � directly level-k implements F on type space C if the identity mapping I : C → C is a level-k equilibrium and I achieves F .  Proposition 3.1 states the revelation principle for level-k implementation. Proposition 3.1. (Revelation Principle) Suppose there exists a mechanism M = �M, f � that level-k implements the social choice correspondence F on the type space C. Then there exists a direct mechanism that directly level-k implements F on C. Proof If the mechanism M implements F , then there exists a strategy profile mi : Ci → Mi  for all i such that m = m1 × · · · × mI is a level-k equilibrium and m achieves F . Now consider the direct mechanism �C, g� where g = f ◦ m : C → Y .  Consider the level-k equilibrium that is determined by all level 0 types truth-telling. Consider a player with cognitive type (θi , 1). He chooses action c according to:  argmax c∈Ci 9 See  ˆ  ui (g(c, (θ−i , 0)), θ) dp(θ−i |θi )  =  argmax c∈Ci  Myerson (1991, Chapter 6).  37  ˆ  ui (f (m(c, (θ−i , 0))), θ) dp(θ−i |θi )  And, so mi (θi , 1) ∈ argmax m∈Mi  ˆ  ui (f (m, m−i (θ−i , 0)), θ) dp(θ−i |θi )  ⇒ (θi , 1) ∈ argmax c∈Ci  ˆ  ui (g(c, (θ−i , 0)), θ) dp(θ−i |θi )  Now we prove the result by induction: every cognitive type of level k ≥ 1 has an incentive  to truthfully report their type.  Suppose, that is true for all levels j < k. Now consider a cognitive type (θi , k).  argmax c∈Ci  ˆ  ui (g(c, (θ−i , k − 1)), θ) dp(θ−i |θi )  =  argmax c∈Ci  ˆ  ui (f (m(c, (θ−i , k − 1))), θ) dp(θ−i |θi )  And, so mi (θi , k) ∈ argmax m∈Mi  ˆ  ui (f (m, m−i ((θ−i , k − 1))), θ) dp(θ−i |θi )  ⇒ (θi , k) ∈ argmax c∈Ci  ˆ  ui (g(c, (θ−i , k − 1)), θ) dp(θ−i |θi )  Proof follows by induction.� The revelation principle implies that we can confine our choice of mechanisms to ones where each cognitive type truthfully reveals her own payoff type and level. Therefore we can ¯ confine our analysis to mechanisms with the following message space, Mi = Θi ×{0, 1, . . . , k}. The proof of the revelation principle relies on a truth-telling argument. Cognitive types  must truthfully report their level. However, in the context of the direct level-k mechanism, the designer cannot simply ask players to report their type (i.e. payoff type and level of reasoning), as one might do when eliciting willingness to pay, for example, as a player’s level of reasoning is not a concept that is intelligible to her. This however, does not mean that the revelation principle does not apply. Direct level-k implementation simply means that ¯ such that a player we can find a mechanism with the message space Mi = Θi × {0, 1, . . . , k} with a payoff type θi and level of reasoning k has an incentive to send the message that happens to be labelled (θi , k). However, a direct mechanism also supposes that level 0 types must truthfully report their level (i.e. partial implementation in the level-k equilibrium determined by level 0 types truthtelling). This is more problematic as level 0 types are not incentivised to do so. Typically, the level 0 behavior is thought to be some nonstrategic reaction to the game: possibly a salient action or to play all actions with equal probability. Thus, in practice we need to design mechanisms with a more refined view of level 0 behavior. Generally, level 0 behavior is defined as uniformly random in applications and experiments. These types play all actions with equal probability. In the following examples, we design optimal mechanisms  38  with this level 0 behavior in mind. However, the revelation principle is still useful for theoretical analysis, as it allows us to confine our attention to direct mechanisms. Further, it is shown in Section 3.7.3 that, under very weak assumptions, whenever a social choice correspondence is level-k implementable, it is also implementable in the level-k equilibrium where level 0 types play uniformly randomly. Therefore, almost no applicability is lost by assuming that level 0 behavior is defined by truth-telling in the theoretical results.  3.4  Examples  In this section we look at two simple bilateral trade examples. There are 2 agents, a buyer and a seller, each with two types. The social planner wants to implement the ex post efficient outcome with budget balance. In the first example, ex post efficiency is both Bayesian and level-k implementable, however we show that the two mechanisms may look quite different. In the second example, ex post efficiency is not achievable under Bayesian implementation but is achievable under level-k implementation. There is one seller with one unit of a good to sell and one buyer. Both the buyer and seller have private values for the good. The seller has two possible types {sl , sh } which represent the minimum value that the seller is willing to sell the good for. The buyer has  two possible types {bl , bh } which represent the maximum value that the buyer is willing to  pay for the good. We will assume that sellers’ types are drawn with probability � for sh and (1 − �) for sl . And, buyers’ types are drawn from the distribution with probability δ for bl and (1 − δ) for bh .  An ex post efficient, budget balanced, payoff-direct mechanism takes the following struc-  ture Seller  Buyer bh bl  sl trade at price p1 trade at price p3  sh trade at price p2 no trade with transfer p4  Figure 3.1: Payoff-direct mechanism  If there is a trade at price p, then the the buyer has utility b − p and the seller has utility  p − s. If there is no trade and p4 = 0 then the buyer’s and seller’s utility is 0. When p4 > 0, then a transfer of money is made from the buyer to the seller.  Example 3.1. Consider the following value distribution for the buyer {bh , bl } = {7, 3} and the seller {sh , sl } = {6, 2} with payoff types drawn from the common prior q, 39  sl  sh  q = bh  1/8  3/8  bl  3/8  1/8  The ex post efficient outcome requires trade at all payoff type profiles except when the buyer is low valued (bl = 3) and the seller is higher valued (sl = 6). The mechanism in Figure 3.2 implements the ex post efficient outcome with budget balance under Bayesian implementation. Buyer’s actions are the row actions {bh , bl } and seller’s actions are the column actions {sl , sh }.  bh = 7 bl = 3  sl = 2 4.5 3  sh = 6 6 no trade  Figure 3.2: Bayesian mechanism for Example 3.1  Now, suppose we wish to implement the ex post efficient outcome under level-k implementation. Consider a level-k type space with k¯ = 2 and beliefs about payoffs determined by the above common prior. The mechanism in Figure 3.3 implements the ex post efficient outcome under level-k for the uniform level 0 equilibrium.  L0 bh = 7 bl = 3  L0 3 1 no trade  sl = 2 8 4.5 3  sh = 6 no trade 6 no trade  Figure 3.3: Level-k mechanism for Example 3.1  In this mechanism, there are three actions for both the buyer and seller. The action L0 is never played in equilibrium (except by level 0 types) and all higher levels truthfully reveal their payoff type. A level 1 buyer believes that the seller is playing each action with equal probability (regardless of his payoff type). Given this belief, a level 1 buyer has an incentive to truthfully reveal his payoff type. The same goes for a level 1 seller who believes the buyer is playing each action with equal probability. Given that level 1 buyers and sellers are truthfully reporting their payoff types, it is in the best interest for level 2 buyers and sellers to truthfully report their payoff type (notice that the level 2 types have the same beliefs as a Bayesian type in the above Bayesian mechanism). The same will then hold for all higher levels.  40  Example 3.2. For the second example, consider the same set of payoff types for both the buyer and the seller but assume that payoff types are drawn from the uniform prior p.  p = bh  sl 1/4  sh 1/4  bl  1/4  1/4  Matsuo (1989) showed that whenever, bh > sh > bl > sl and �(1 − δ)bh + (1 − �)bl <  (1 − δ)sh + δ(1 − �)sl , the ex post efficient outcome will not be Bayesian implementable. To see the intuition behind this notice that the low valued buyer and the high valued seller should never have any incentive to misreport and thus must make zero rents in equilibrium. Thus, our direct mechanism should look like the mechanism in Figure 3.4.  bh = 7 bl = 3  sl = 2 p 3  sh = 6 6 no trade  Figure 3.4: Structure of a Bayesian mechanism for Example 3.2  Now, for a high valued buyer to truthfully report his payoff type the trading price (at (bh , sl )) must be less than or equal to 4 (i.e. p ≤ 4). For a low valued seller to truthfully report his payoff type the trading price must be greater than or equal to 5 (i.e. p ≥ 5). These two conditions are incompatible. There is no mechanism which will implement the ex post efficient outcome under Bayesian implementation. Can the ex post efficient outcome be level k implementable? First, notice that the ex post efficient outcome cannot be achieved with a payoff-direct mechanism under level-k implementation. We know that there is no set of prices {p1 , p2 , p3 , p4 } such that the ex post efficient outcome is Bayesian implementable. Suppose we could however find such prices for  level-k implementation. Now, consider a level 2 buyer and level 2 seller. The level 2 buyer knows that level 1 sellers are truthfully revealing their payoff types and level 2 sellers know that level 1 buyers are truthfully revealing their payoff types. Given this, level 2 sellers and buyers must want to truthfully reveal their own payoff types. But, if that is so, since level 2 types have correct beliefs about the distribution of payoff types in the population, we can find a set of prices {p1 , p2 , p3 , p4 } such that the ex post outcome is Bayesian implementable. This is a contradiction.  Therefore, it is impossible to implement the ex post efficient outcome with a payoffdirect mechanism under level-k implementation. This statement can actually be generalized, which we will show in Section 3.6: whenever a social choice correspondence is not Bayesian implementable, then it will not be level-k implementable with a payoff-direct mechanism. 41  However, we can implement the ex post efficient outcome by augmenting the message space to allow types to reveal both their payoff types and levels. The mechanism in Figure 3.5 level-k implements the ex post efficient outcome for the level-k type space with k¯ = 2 and the same uniform prior.  L0 bh = 7 L1 bh = 7 L2 bh = 7 bl = 3  L0 sl = 2 2 1 5 no trade  L1 sl = 2 8 4 3.5 3  L2 sl = 2 7 5.5 3 3  sh = 6 no trade 6 6 no trade  Figure 3.5: Level-k mechanism for Example 3.2  To see this consider the level-k equilibrium where level 0 types play uniformly randomly. That is, level 0 buyers play actions {L0, L1, L2, bl } with equal probability and level 0 sellers play actions {L0, L1, L2, sh } with equal probability.  A level 1 type believes others are playing uniformly randomly. Therefore, playing L1 is a  best response for the high valued buyer and playing bl is a best response for the low valued buyer. And, playing L1 is a best response for the low valued seller and playing sh is a best response for the high valued seller. A level 2 buyer believes that all level 1 sellers truthfully report their types. So she believes that L1 and sh are being played with equal probability. Therefore, playing L2 is a best response for the high valued buyer and playing bl is a best response for the low valued buyer. A level 2 seller believes that all level 1 buyers truthfully report their types. She believes that L1 and bl are being played with equal probability. Therefore, playing L2 is a best response for the low valued seller and playing sh is a best response for the high valued seller.  3.5  Necessary conditions  Example 3.2 demonstrates that it is possible to implement a social choice correspondence that is not Bayesian implementable under level-k implementation. This section establishes the formal link between level-k implementation and Bayesian implementation by establishing a set of necessary conditions for level-k implementation. Level-k implementation is a weaker implementation requirement than Bayesian implementation. First, consider Example 3.2 again. Any direct mechanism that level-k implements the ex post efficient outcome must provide incentives for level 2 types to truthfully report their payoff types given that level 1 types are truthfully reporting their payoff types. Therefore there must exist some payoff-direct mechanism A such that a buyer will truthfully reveal 42  his payoff type given that he believes the seller is truthfully revealing his payoff type (and given that his beliefs are determined by the common prior). A bh = 7 bl = 3  sl = 2 3.5 3  sh = 6 6 no trade  Figure 3.6: Payoff-direct mechanism for the buyer  The payoff-direct mechanism A is actually a subset of the level-k mechanism in the last section. It is the component of the level-k mechanism that a level 2 buyer takes into account when choosing his action. There also must exist some payoff-direct mechanism B such that a seller will truthfully reveal his payoff type given that he believes the buyer is truthfully revealing his payoff type (and given his beliefs are determined by the common prior). B bh = 7 bl = 3  sl = 2 5.5 3  sh = 6 6 no trade  Figure 3.7: Payoff-direct mechanism for the seller  The payoff-direct mechanism B is a subset of the level-k mechanism in the last section. It is the component of the level-k mechanism that a level 2 seller takes into account when choosing his action. The existence of these payoff-direct mechanisms are necessary conditions for level-k implementation. A level 2 type must have an incentive to truthfully report his payoff type given that he believes other payoff types are truthfully reporting their own payoff types. The existence of these payoff-direct mechanisms are also necessary conditions for Bayesian implementation. However, level-k implementation has weaker implementation requirements than Bayesian implementation because you have a freedom across players that you do not have under Bayesian implementation. To see this, notice that a necessary condition for Bayesian implementation is that the payoff-direct mechanisms A and B exist and incentivize both the buyer and seller to truthfully report their payoff types respectively. However, for a social choice correspondence to be Bayesian implementable, it must be that A=B. In this way, level-k implementation relaxes the cross-player implementation requirements imposed by the mutual consistency assumption of Bayesian implementation. The following proposition gives the necessary conditions for level-k implementation for the general environment. 43  Proposition 3.2. (Necessary Conditions) Consider a level-k type space with beliefs about payoff types determined by the common prior p and k¯ ≥ 2. If the social choice correspondence  F is level-k implementable, then there exists a function f i : Θ → Y for each i ∈ I such that the following conditions hold:  (i) f i (θ) ∈ F (θ) ∀θ ∈ Θ and ∀i ∈ I � � (ii) ui (f i (θ), θ)p(θ−i |θi ) ≥ θ−i ∈Θ−i  θ−i ∈Θ−i  ui (f i (θ� , θ−i ), θ)p(θ−i |θi ) ∀θ� ∈ Θi and ∀i ∈ I  Proof:  Suppose that the social choice correspondence F is level-k implementable. Then it ¯ is implementable by the direct mechanism �c, ×i {Θi × {0, 1, . . . , k}}� where each player truthfully reveals her type (θi , k).  Consider the behavior of a L2 type with payoff type θi . L2 types believe all other players are L1 types and have beliefs about θ−i given by the  common prior p ∈ �(Θ) ˆ  ui (c((θi , 2), (θ−i , 1)), θ)dp(θ−i |θi ) ≥  ˆ  ui (c(m, (θ−i , 1)), θ)dp(θ−i |θi )  ¯ ∀m ∈ Θi × {0, 1 . . . , k}  Define f i (θi , θ−j ) = c((θi , 2), (θ−i , 1))  Then, the above truth-telling condition for cognitive type (θi , 2) ensures that that condition (ii) holds for i. Condition (i) holds by definition of c achieving F. � The social choice correspondence F will also be Bayesian implementable as long as (i) and (ii) hold with f i = f j for all i, j ∈ I.  3.6  Bayesian equivalence  In the bilateral trade example, we saw that restricting our attention to payoff-direct mechanisms would not permit implementation of the ex post efficient outcome under level-k when it was not also Bayesian implementable. This is actually a general statement. If we restrict attention to payoff-direct mechanisms, level-k implementation will not buy us anything on top of Bayesian implementation. That is, when we restrict the message space to the set of payoff types, any social choice function that is level-k implementable will be Bayesian implementable as well. The following proposition gives this result. Proposition 3.3. Consider a social choice correspondence F. If F is not Bayesian Implementable on a type space B, then it is not implementable with a payoff-direct mechanism on any (B-based) level-k type space C with k¯ ≥ 2. 44  Proof: Suppose not. Then there exists a payoff-direct mechanism that is cognitive implementable on C.  Therefore, we have a mechanism �c, Θ� where each player truthfully reveals her payoff  type θi  Consider the behavior of type (θi , 2) Level 2 types believe all other players are level 1 types and have beliefs about θ−i given by the common prior p ∈ �(Θ). Each level 1 type truthfully reveals his payoff type. Therefore the incentive condition for type (θi , 2) is: ˆ  ui (c(θi , θ−i ), θ)dp(θ−i |θi ) ≥  ˆ  ui (c(θ� , θ−i ), θ)dp(θ−i |θi )  ∀θ� ∈ Θi , ∀θi , ∀i  where, c(θi , θ−i ) ∈ F (θi , θ−i )  But, if this is true, then �c, Θ� would implement F under Bayesian implementation. A  contradiction. �  Level-k implementation is weaker than Bayesian implementation, whenever a social choice correspondence is Bayesian implementable it will be level-k implementable. However, as Example 3.2 demonstrates the reverse does not hold. We saw in the previous section that level-k implementation relaxes the cross-player consistency conditions that Bayesian implementation imposes. Thus, when these conditions are binding level-k implementation may be possible when Bayesian implementation is not. Given this, it might be interesting to know in what environments this condition will or will not bind. It can be shown that Bayesian and level-k implementation are equivalent whenever the environment is separable. Common separable environments include the cases of quasilinear utility and social choice functions (the social choice correspondence specifies only one desirable outcome for each type profile). In these environments, either the social choice correspondence is too restrictive (in the case of a social choice function) that the binding cross-player conditions are also imposed for level-k implementation. Or, the social choice correspondence is flexible enough to ensure cross-player restrictions never bind (i.e. quasilinear utility with no budget balance conditions). The following proposition gives this result. Definition. The environment is separable if the following hold: (i) Y = Y0 × Y1 × · · · × YI  (ii) ui ((y0 , y1 , . . . , yI ), θ) = u ˜i (y0 , yi , θ) (iii)F (θ) = f0 (θ) × F1 (θ) × · · · × FI (θ)  45  Proposition 3.4. Let F be a social choice correspondence in a separable environment. Let B be a Bayesian type space. Let C be a (B-based) level-k type space with k¯ ≥ 2. Then, the following are equivalent:  (i) F is level-k implementable (ii) F is Bayesian implementable Proof (ii) ⇒ (i) Straightforward. Any Bayesian equilibrium can be replicated as a level-k equilibrium  (consider the level-k equilibrium where the cognitive type (θi , 0) plays the action that the payoff type θi plays in the Bayesian equilibrium). (i) ⇒ (ii)  By the revelation principle, level-k implementation means that F can be implemented with a direct mechanism.  ¯ Therefore, we have a mechanism �c, ×i {Θi ×{0, 1, 2, . . . k}}� where each player truthfully  reveals her type (θi , k)  Consider the behavior of a cognitive type (θi , 2). In the direct mechanism, each player truthfully reveals his cognitive type ˆ  ui (c((θi , 2), (θ−i , 1)), θ)dp(θ−i |θi ) ≥  ˆ  ui (c(m, (θ−i , 1)), θ)dp(θ−i |θi )  ∀m ∈ Θi × {0, 1, 2}  where, c((θi , 2), (θ−i , 1)) ∈ F (θ) .  Define g i (θ) = c((θi , 2), (θ−i , 1)) for all θ ∈ Θ and all i ∈ I. Then, it must true that ˆ  ui (g i (θ), θ)dp(θ−i |θi ) ≥  ˆ  ui (g i (θ� , θ−i ), θ)dp(θ−i |θi )  ∀θ� ∈ Θi , ∀θi ∈ Θi ∀i ∈ I  where, g i (θ) ∈ F (θ).  But, because we have separability, we can rewrite  g0i (θ)  =  f0 (θ)  gji (θ)  ∈  Fj (θ)  ` ´ Let h(θ) = f0 (θ), g11 (θ), . . . , gii (θ), . . . , gII (θ) . Then, under the mechanism �Θ, h�: ˆ  u ˜i (f0 (θ), gii (θ), θ)dp(θ−i |θi ) ≥  ˆ  u ˜i (f0 (θ� , θ−i ), gii (θ� , θ−i ), θ)dp(θ−i |θi )  46  ∀θ� ∈ Θi , ∀θi ∈ Θi ∀i ∈ I  ⇒  ˆ  u ˜i (h(θ), θ)dp(θ−i |θi ) ≥  ˆ  u ˜i (h(θ� , θ−i ), θ)dp(θ−i |θi )  ∀θ� ∈ Θi , ∀θi ∈ Θi ∀i ∈ I  ⇒ F is Bayesian implementable�  3.7 3.7.1  Robustness of level-k mechanisms Relaxing beliefs about levels  The analysis of this paper uses the level-k model. This is a type of limited depth of reasoning model where players have particular beliefs about the levels of others. Specifically, if a player is of level k, he believes that others have levels exactly equal to k-1. In general, we might allow a player with level k to hold beliefs over all lower levels.10 The following definition of limited depth of reasoning models generalize the level-k type space in exactly that way. Definition. A limited depth of reasoning type space (LDoR type space) is a type ¯ λ1 , . . . , λk¯ � with λk ∈ �({0, . . . , k − 1}). space C LDoR = �C1 , . . . , CI ; B, k; As in the level-k type space, a player’s beliefs about the cognitive types of others are determined both by her payoff type and her level. The beliefs of a cognitive type ci = (θi , ki ) about the cognitive types of others, c−i = (θ−i , v −i ), is determined by the function bi (c−i |ci ) :  � � bi (c−i |ci ) = λkv1 × · · · × λkvI−1 p(θ−i |θi )  where the notation θ × v represents a payoff type-level profile (θ1 , v1 ) × · · · × (θI , vI ), where ¯ I. v ∈ {0, . . . , k} A player with a level k puts weight only on other types that have a level less than or equal  to k−1. This captures the core assumption of the limited depth of reasoning literature which is that a player of level k believes that other players have levels strictly less than k. This assumption ensures that players can calculate their optimal actions in a recursive fashion with a finite number of steps given the behavior of level 0 types. As in the level-k model, a player’s beliefs about the payoff types of others are determined by the common prior p. Given the definition of an LDoR type space, we can apply the solution concepts of levelk equilibrium and level-k implementation to this generalized type space, by simply using the belief function bi (c−i |ci ) defined for the LDoR type space in the definition of a level-k equilibrium.  10 Cognitive hierarchy models relax the level-k belief structure in this way. In the cognitive hierarchy model, a level k type has beliefs over all lower levels determined by a conditional Poisson distribution. See Camerer et al. (2004) for specifics.  47  Given that the LDoR type space generalizes level-k type spaces it could be more difficult to implement a particular social choice correspondence under this generalization. But, this is actually not the case. Both level-k mechanisms used to implement the ex post efficient outcomes in Example 3.1 and Example 3.2 will be robust to replacing the level-k type space with any LDoR type space (with k¯ = 2). This can be done generally. Whenever a social choice correspondence is level-k implementable it will also be implementable for any LDoR type space. In fact, this can be done robustly, we can find one mechanism that will implement the social choice correspondence for any LDoR type space. This result is formalized in the following proposition. Proposition 3.5. Let F be a social choice correspondence. Let B be a Bayesian type space. Let C be a (B-based) level-k type space. If F is implementable in the (B-based) level-k type space then it is implementable in any (B-based) LDoR type space. Proof By the revelation principle, level-k implementable means that F can be implemented with a direct mechanism.  ¯ Therefore, we have a mechanism M = �c, Mi = Θi × {0, 1, 2, . . . k}}� where each type in  the level-k type space truthfully reveals her type (θi , k).  Statement: for each k ≥ 1, we can find a mechanism M � = �c� , Mi� = Θi × {0, . . . k}}�  such that, given any (B-based) LDoR type space, each type (θi , j) truthfully reports her type ∀θi ∈ Θi , j ≤ k, i ∈ I.  The statement is true for k = 1: let c� ((θ, v)) = c((θ, v)) ∀ (θ, v) ∈ Θ × {0, 1}I .  Now suppose the statement is true for k − 1. Let the mechanism �h, Θi × {0, 1, 2, . . . k −  1}}� be the mechanism which makes the statement hold true for k − 1.  Define a new mechanism M � = �c� , Mi� = Θi × {0, . . . k}}� where c� ((θ, v)) = h((θ, v))  for all (θ, v) ∈ Θ × {0, 1, . . . , k − 1}I , c� ((θi , k), (θ−i , v)) = h((θi , k − 1), (θ−i , v)) for all  (θ−i , v) ∈ Θ−i ×{0, 1, . . . , k−1}I−1 . And, redefine c� ((θi , k), (θ−i , v)) = c((θi , k), (θ−i , v)) for  all (θ−i , v) ∈ Θ−i ×{0, 1, . . . , k−1}I−1 such that vj = k−1 for at least some j ∈ {0, . . . , I−1}. Consider a type for player i with level k and payoff type θi .  ¯ I−1 . It must be For simplicity, define λk (v) = λkv1 × · · · × λkvI−1 for any v ∈ {0, . . . , k}  that  (θi , k) ∈  argmax  8 <  X  k  m∈Θi ×{0,1,2,...,k} : v∈{0,...,k−1}I−1  λ (v)  This is true because  48  ˆ  9 = ui (c (m, (θ−i , v)), θ)dp(θ−i |θi ) ; �  8 <  1 P k (θi , k − 1) ∈ argmax λ (v) m∈Θi ×{0,1,2,...,k−1} : v  X  k  λ (v)  v∈{0,...,k−2}I−1  ˆ  9 = ui (c (m, (θ−i , v)), θ)dp(θ−i |θi ) ; �  which ensures (θi , k) is also incentive compatible since (θi , k − 1) and (θi , k) have the same payoffs over this support. And, (θi , k) ∈  argmax m∈Θi ×{0,1,2,...,k}  ˆ  ff ui (c� (m, (θ−i , v)), θ)dp(θ−i |θi )  for any v ∈ {0, 1, . . . , k − 1}I−1 such that vj = k − 1 for at least some j ∈ {0, . . . , I − 1}. Thus, (θi , k) must be incentive compatible for type (θi , k).  In addition, adding a strategy for player i does not affect the strategic choices of any of the other players. This is because none of the types of these players put any weight on their opponent playing the strategy (θi , k) for any θi ∈ Θi .  We can augment the mechanism analogously for each of the other players j �= i. Thus,  the proposition holds by induction. �  3.7.2  Relaxing beliefs about payoffs  This subsection shows that we can design a level-k mechanism that is robust to relaxing the assumption that beliefs about payoffs are determined by a specific common prior. To do so we define a version of ex post implementation for level-k implementation. Definition. A social choice correspondence F is ex post level-k implementable on C if there exists a mechanism �M, f � and a message profile m : C → M , such that for all θi ∈ Θi ¯ and ki ∈ {1, 2, . . . , k}  ui (f (mi ((θi , k)), m−i (θ−i × k − 1)), θ) ≥ ui (f (c� , m−i (θ−i × k − 1)), θ) ∀c� ∈ Ci , θ−i ∈  Θ−i , i ∈ I  and m achieves F on C. Proposition 6 shows that under some relatively weak assumptions, a social choice corre-  spondence is ex post level-k implementable. The assumptions ensure that we can separate each payoff type for each individual in an ex post sense. In other words must be possible to design a mechanism (possibly different ones for each agent) such that each payoff type wants to truthfully report her payoff type regardless of her beliefs about the payoff types of others. Level-k implementation relaxes the cross-player consistency requirements of ex post implementation (analogously to the relationship between level-k and Bayesian implementation). Thus ex post level-k implementation is a weaker implementation concept than ex post  49  implementation. We then illustrate with an example that this a social choice correspondence may be ex post level-k implementable when it is not Bayesian implementable. Proposition 3.6. Let F be a social choice correspondence. If there exists an f i : Θ → Y for each i such that the following hold  (i) ui (f i (θ), (θi , θ−i )) ≥ ui (f j (θ� , θ−i ), (θi , θ−i )) ∀θ� ∈ Θi , θ ∈ Θ, j ∈ I, and (ii) f i (θ) ∈ F (θ) ∀i ∈ I  then F is ex post level-k implementable. Proof ¯ I → R by c((θ, v)) = f 1 (θ) for all θ ∈ Θ and v ∈ {0, . . . , k} ¯ Define c : {Θi × {0, . . . . , k}}  and redefine c((θi , k), (θ−i , k − 1)) = f i (θ) for all θ ∈ Θ.  Consider the level-k equilibrium where level 0 types truth-tell. By induction on k: Consider type (θi , 1) and θ−i ∈ Θ−i ui (c((θi , 1), (θ−i , 0)), θ)  =  ui (f i (θ), θ)  ≥  ui (f i (θ� , θ−i ), θ) ui (f j (θ� , θ−i ), θ)  ≥  ∀θ� ∀j ∈ I  And, since ui (c((θ� , j), (θ−i , 0)), θ) = ui (f j (θ� , θ−i ), θ) for some j ∈ I for all θ� ∈ Θi , type  (θi , 1) has an incentive to truthfully reveal his type for all θ−i . Now, suppose any types with level k − 1 truth-tell. Consider type (θi , k) and θ−i ∈ Θ−i  ui (c((θi , k), (θ−i , k − 1)), θ)  =  ui (f i (θ), θ)  ≥  ui (f i (θ� , θ−i ), θ)  ≥  ui (f j (θ� , θ−i ), θ)  ∀θ� ∀j ∈ I  And, since ui (c((θ� , j), (θ−i , k − 1)), θ) = ui (f j (θ� , θ−i ), θ) for some j ∈ I for all θ� ∈ Θi ,  type (θi , k) has an incentive to truthfully reveal his type for all θ−i . �  The assumptions (i)-(ii) are relatively weak. Consider Example 3.2, the ex post outcome is not ex post implementable, but this environment satisfies the conditions of Proposition 3.6. Thus the ex post outcome is implementable for any (B-based) level-k type space for any Bayesian type space B.  Define the functions f b and f s is the following way:  50     2 if (b, s) = (bh , sl ), (bl , sl )   b f (b, s) = 5 if (b, s) = (bh , sh )    no trade if (b, s) = (b , s ) l h    6 if (b, s) = (bh , sl ), (bh , sh )   f s (b, s) = 4 . if (b, s) = (bl , sl )    no trade if (b, s) = (b , s ) l h These functions satisfy the requirements of Proposition 3.6 and the following mechanism  level-k implements the ex post outcome for any Bayesian type space.  L0 bh = 7 L0 bl = 3 L1 bh = 7 L1bl = 3 L2bh = 7 L2bl = 3  L0 sl = 2 2 2 2 2 2 2  L0 sh = 6 5 no trade 5 no trade 5 no trade  L1 sl = 2 6 4 2 2 2 2  L1sh = 6 6 no trade 5 no trade 5 no trade  L2sl = 2 2 2 6 4 2 2  L2sh = 6 6 no trade 5 no trade 5 no trade  Figure 3.8: Ex post level-k mechanism for Example 3.2  In the level-k equilibrium where level 0 types truthfully reveal their types, all other types have any incentive to truthfully reveal their payoff type, regardless of their beliefs about the payoff types of their opponent.  3.7.3  Relaxing truth-telling of level 0 types  The proof of the revelation principle in this paper relies on the assumption that the designer can choose the level-k equilibrium that will be played. This assumption is similar to partial Bayesian implementation where it is assumed that the truth-telling equilibrium will be played. However, this is especially problematic under level-k implementation because level 0 types are never incentivized to truthfully reveal their type. In the applied level-k literature, it is often assumed that level 0 players play uniformly randomly (uniform level 0 equilibrium) across the action space. This assumption has reasonable empirical support. Given this, we might want to design a mechanism that implements the uniform level 0 equilibrium (plus higher types truth-telling) rather than the level-k equilibrium where level 0 types truth-tell. Often we will be able to design a mechanism to implement the uniform equilibrium with no more difficulty than designing a mechanism to implement the level 0 truth-telling equilibrium. And, in fact both mechanisms in Example 3.1 and 3.2 are designed around the uniform level 0 equilibrium. But, the revelation principle does not guarantee that we can do so. 51  However, under very weak assumptions, whenever a social choice function is implementable, it will also be implementable in the uniform level 0 equilibrium. The following Proposition gives this result. The assumption required is that a designer must be able to separate the types for each player ex post, and for all players (although, the mechanisms may differ across players). This requirement ensures that separability in payoff types does not come solely from different beliefs. Additionally, separability does not have to be consistent with the social choice correspondence. That is, f i (θi ) does not need to be related to F (θ) in any way. This is because separability without beliefs is only required for level 1 types and since level 0 types do not exist in the population, the outcome promised to level 1 types when their opponents play as level 0 do not need to coincide with the social choice correspondence. Proposition 3.7. Let F be a social choice correspondence. Let B be a Bayesian type space  and let C be a (B-based) level-k type space. If there exists an f i : Θi → Y for each i such that  ui (f i (θi ), (θi , θ−i )) ≥ ui (f i (m), (θi , θ−i )) ∀m ∈ Θi , ∀θ−i ∈ Θ−i , ∀θi ∈ Θi then whenever F is level-k implementable it is also level-k implementable in the uniform level 0 equilibrium. Proof By the revelation principle, level-k implementation means that F can be implemented with a direct mechanism.  ¯ Therefore, we have a mechanism �c, M = ×i {Θi × {0, 1, 2, . . . k}}� where each player  truthfully reveals her type (θi , k)  Now consider a new mechanism h, where h = c everywhere except for h((θi , 1), (θ−i , 0)) =  f (θi ) ∀θ−i ∈ Θ−i . i  Consider the following function  g(θi ) = ui (f i (θi ), θ) − ui (f i (m), θ) � + (ui (h((θi , 1), (θ−i , k)), θ) − ui (h(m, (θ−i , k)), θ)) ¯ θ−i ×k∈Θ−i ×{0,...k}  We know the first part of the sum ui (f i (θi ), θ) − ui (f i (m), θ) > 0 ∀m ∈ Θi , θ ∈ Θ.  Thus, choose a di (θi ) for each payoff type, such that the following inequality holds for ¯ all θ� ∈ Θi and m ∈ θ� × {0, . . . , k}:  52  � � g(θi ) = di (θi ) ui (f i (θi ), θ) − ui (f i (θ� ), θ) � + (ui (h((θi , 1), (θ−i , v)), θ) − ui (h(m, (θ−i , v)), θ)) ≥ 0  ¯ θ−i ×v∈Θ−i ×{0,...k}  Then add D = max{di (θi )} duplicate level 0 strategies, L0i , to the message space for θi ,i  each individual i to make a new mechanism with the new augmented message space Mi�  and outcome function c� , where c� = h for all m ∈ M and c� ((θi , k), L0−i ) = f (θi ) for all ¯ and for all i ∈ I. θi × k ∈ Θi × {0, . . . , k} Proof follows by induction on k:  1 � | |M−i  ˆ  2  4D · ui (c� ((θi , 1), L0−i ), θ) + 1 = |M � |  ˆ  2  1 = |M � |  ˆ  2  ˆ  2  3  ui (c� ((θi , 1), (θ−i , v)), θ)5 dp(θ−i |θi )  ¯ θ−i ×v∈Θ−i ×{0,...k}  i  4D · ui (f (θi ), θ) +  1 ≥ |M � |  X  X  3  ui (h((θi , 1), (θ−i , v)), θ)5 dp(θ−i |θi )  ¯ θ−i ×v∈Θ−i ×{0,...k}  4D · ui (f i (θ� ), θ) +  X  ui (h(m, (θ−i , v)), θ)5 dp(θ−i |θi )  ¯ θ−i ×v∈Θ−i ×{0,...k}  �  4D · ui (c (m, L0−i ), θ) +  3  X  �  ∀θ� ∈ Θi  3  ui (c (m, (θ−i , v)), θ)5 dp(θ−i |θi )  ¯ θ−i ×v∈Θ−i ×{0,...k}  ¯ ∀θi ∈ Θi and for all i ∈ I. The inequality follows since g(θi ) ≥ 0. ∀ m ∈ Θi × {0, . . . , k}, Therefore, all level 1 types have an incentive to tell the truth in the uniform level 0 equilibrium.  Now, suppose level k − 1 types truth-tell and consider a level k type 1 < k ≤ k¯ with  payoff type θi ˆ  ui (h((θi , k), (θ−i , k − 1)), θ)dp(θ−i |θi )  =  ˆ  ui (c((θi , k), (θ−i , k − 1)), θ)dp(θ−i |θi )  ≥  ˆ  ui (c(m, (θ−i , k − 1)), θ)dp(θ−i |θi )  =  ˆ  ui (h(m, (θ−i , k − 1)), θ)dp(θ−i |θi )  ¯ ∀ m ∈ Θi × {0, . . . , k}. �  Whenever the assumptions for Proposition 3.6 are satisfied, the assumptions for Proposition 3.7 are also satisfied. This means that we can also find a mechanism that implements F in the uniform level 0 equilibrium that will be robust to any beliefs about payoff types. 53  Chapter 4  Rationality and consistent beliefs: theory and experimental evidence 4.1  Introduction  Over the past 20 years there has been an accumulation of experimental evidence that challenges the empirical validity of Nash equilibrium predictions when subjects interact in novel environments. This evidence not only prompts the need for alternative solution concepts to explain behavior in one-shot games, but an understanding of why Nash equilibrium fails in the laboratory can provide valuable insight into the plausibility of alternative solution concepts. Nash equilibrium does not simply provide a description of behavioral predictions but embodies assumptions about players’ rationality and beliefs that underpin these predictions. There are three epistemic conditions that govern strategic behavior in a Nash equilibrium. The first is rationality: whether a player plays a best response to her beliefs. The second is a belief about the rationality of others: whether a player believes others are rational. And, the third is consistent beliefs about strategies: whether a player believes others hold correct beliefs about the action she is playing.1 Theorists and experimentalists have developed and explored alternative models that can account for failures of these types of epistemic conditions.2 However, tests of these 1 I consider only complete information games which implicitly assume common knowledge of payoffs. I discuss the relaxation of this assumption in Section 7. 2 The two most prominent models are quantal response equilibrium (QRE) and level-k models. QRE relaxes rationality assumptions by allowing players to make mistakes when they best respond but maintains the Nash assumption of consistent beliefs (McKelvey and Palfrey 1995). Level-k models maintain the assumption that players are rational, but allow players to underestimate the rationality of others, thus also implicitly relaxing the assumption of consistent beliefs (Nagel 1995; Stahl and Wilson 1995; and CostaGomes and Crawford 2006).  54  alternative models only jointly test relaxations of Nash assumptions in combination with additional structural assumptions that these models impose. The approach in this paper permits separate identification of these epistemic conditions without imposing additional structural assumptions. Untangling the role that rationality, beliefs about others’ rationality, and consistent beliefs play in strategic reasoning permits direct comparisons of solution concepts and may even suggest new ways to model strategic reasoning. Should we model players as rational agents or as making systematic mistakes when they best respond? Should we maintain the Nash assumption that players have consistent beliefs about strategies or should we relax this constraint in our modeling? This paper uses theory and an experiment to identify which of the three epistemic conditions hold at the individual level. The analysis focuses on a special class of games, ring games, that allow us to separately identify these conditions. The theoretical analysis uses the tools of epistemic game theory to isolate behavioral predictions in ring games under different combinations and formal statements of the above three epistemic conditions. The experimental analysis uses strategic choice data from a carefully chosen set of ring games that permit us to identify whether players are rational, up to which order they believe others are rational, and whether they hold consistent beliefs. To the best of my knowledge, this is !"#$%&'()*+,-).%/') the first paper to use ring games (or network games in general) to study strategic reasoning.3  Q10,$'!A!  Q10,$'!L!  Q10,$'!M!  ! ! "#$!%&'()!&*$+)&%,&+-!'$()'&.)&/+!011/2(!3(!)/!&(/10)$!)#$!4$#05&/'01!&671&.0)&/+(!/%! Figure 4.1: 3-player ring game )#$!'0)&/+01&),!0((367)&/+(!%'/6!)#$!0((367)&/+!/%!./+(&()$+)!4$1&$%(8!9+!/'*$'!)/! 3+*$'()0+*!2#,!2$!+$$*!0+!&*$+)&%,&+-!'$()'&.)&/+:!./+(&*$'!)#$!%/11/2&+-!$7&()$6&.! The end goal of this paper is to use choice data to make inferences about players’ .#0'0.)$'&;0)&/+!%/'!<0(#!$=3&1&4'&368!>360++!0+*!?'0+*$+43'-$'!@ABBCD!0+*! epistemic conditions (i.e. their belief structures). This is an inherently difficult problem (#/2!)#0)!&+!./671$)$!&+%/'60)&/+!-06$(:!&%!0!(34E$.)F!&(!'0)&/+01:!4$1&$5$(!)#$!/)#$'! C!)#$+!)#0)!(34E$.)!63()!710,!0+*! 710,$'!&(!'0)&/+01:!0+*!(0)&(%&$(!./+(&()$+)!4$1&$%( as there are generally many different beliefs that lead to the same behavior. This paper !! 4$1&$5$!)#0)!#$'!/77/+$+)!710,(!0!70&'!/%!()'0)$-&$(!)#0)!%/'6!0!<0(#!$=3&1&4'&368 focuses on ring games as they allow us to isolate the behavioral implications of different ! belief structures. The advantage of ring games lies in their unique opponent structure. A "#$!./64&+$*!0((367)&/+(!/%!'0)&/+01&),!0+*!./+(&()$+)!4$1&$%(!*$)$'6&+$!<0(#! ring$=3&1&4'&36!710,8!"#$'$%/'$:!&%!0!710,$'!710,(!0!<0(#!$=3&1&4'&36!&+!0+!@+G'/3+*D! game is essentially a series of two-player normal form games. However, the opponent */6&+0+.$!(/15041$!-06$!2$!.0++/)!*$)$'6&+$!&%!@&D!'0)&/+01&),!0+*!@+GAD)#G/'*$'! structure is relaxed relative to standard game forms. In the 3-player ring game in Figure 4$1&$%!&+!'0)&/+01&),!#/1*(H!/'!&%!@&&D!'0)&/+01&),:!4$1&$%!&+!)#$!'0)&/+01&),!/%!/)#$'(!0+*! 4.1, Player 1’s payoff depends on the action of Player 2. Player 2’s payoff depends on the ./+(&()$+)!4$1&$%(!#/1*8!"#$'$%/'$:!)#$!4$#05&/'01!&671&.0)&/+(!/%!1/2$'G/'*$'! 3 For a review of the literature on network games, or the analogous concept in computer science (graphical '0)&/+01&),!0+*!./+(&()$+)!4$1&$%(!.0++/)!4$!($70'0)$*!%'/6!)#/($!/%!#&-#$'G/'*$'! games) see Jackson (2005) and Kearns (2007) '0)&/+01&),8! ! >360++!0+*!?'0+*$+4$'-$'I(!.#0'0.)$'&;0)&/+!#/1*(!4$.03($!)#$!./+(&()$+)!4$1&$%! 55 0((367)&/+!&67/($(!'$()'&.)&/+(!/+!#&-#$'G/'*$'!4$1&$%(8!J/'!$K0671$:!&+!0+!+G 710,$'!-06$!)#&(!6$0+(!)#0)!710,$'!AI(!4$1&$%(!04/3)!710,$'!L!*$)$'6&+$!2#0)!710,$'! M!4$1&$5$(!04/3)!710,$'!L8!"#3(:!&)!&(!+/)!+$.$((0',!)/!&67/($!)#$!0((367)&/+(!/+! #&-#$'G/'*$'!4$1&$%(!@&8$8!&)!&(!+/)!+$.$((0',!)/!0((36$!)#0)!710,$'!A!4$1&$5$(!)#0)! 710,$'!M!4$1&$5$(!710,$'!L!&(!'0)&/+01D8!N/2$5$':!&+!60+,!$+5&'/+6$+)(!)#&(! '$()'&.)&/+!&(!+/)!($+(&41$8!"#$!'&+-!-06$!&(!0+!$K0671$!/%!(3.#!0+!$+5&'/+6$+)8!9+! )#$!'&+-!-06$:!710,$'!A!&+)$'0.)(!/+1,!!"#!$%&'()!2&)#!710,$'!L8!9)!&(!4$#05&/'011,!  action of Player 3. And, Player 3’s payoff depends on the action of Player 1. Ring games are particularly effective at isolating the behavioral implications of different orders of rationality because the structure of ring games allows us to isolate beliefs at different orders. In general, higher orders of rationality may pin down behavior, but do not rule out the event that behavior arises under lower-orders of rationality. This is necessarily true in all games, as lower-order rationalizable sets necessarily contain higher-order rationalizable sets.4 However, the implications of different orders of rationality can be separated under the assumption that a lower-order rational player does not respond to changes in higher-order beliefs. For example, we assume a rational player responds to changes in her own payoffs but not to changes in her opponent’s payoffs (her 1st-order beliefs). This is reasonable as a rational player does not account for the relationship between her opponent’s payoffs and actions and hence, likely ignores her opponent’s payoffs entirely. The behavioral implications of different orders of rationality can then be separated under this assumption by considering behavior in a carefully chosen set of ring games that differ only in higher-order payoffs. The structure of ring games allows us to isolate higher-order beliefs from lower-order beliefs allowing us to consider different games (with different rationalizable actions) that have the same lower-order payoffs.5 Ring games and their role in separately identifying the three epistemic conditions will be discussed in detail in the next section. This paper uses strategic choice data to obtain individual-level estimates of the three epistemic conditions. We find that 94 percent of subjects behave consistently with rationality: they do not choose strictly dominated actions. 72 percent of subjects behave consistently with rationality and the belief that others are rational. 44 percent of subjects behave consistently with rationality and the 2nd-order belief that others are rational. And, 20 percent of subjects behave consistently with rationality and the 3rd-order belief that others are rational. The epistemic conditions sufficient for Nash equilibrium in two-player games are: rationality, belief in others’ rationality and consistent beliefs about strategies. This means that approximately 28 percent of subjects do not satisfy the sufficient rationality conditions for Nash equilibrium and approximately 72 percent satisfy the sufficient rationality conditions. However, of those 72 percent of subjects that satisfy the sufficient rationality conditions for Nash equilibrium, none of them satisfy consistent beliefs. Not a single subject satisfies all three sufficient epistemic requirements for Nash equilibrium. The results of this experiment provide support for the key features of the level-k and cognitive hierarchy models; heterogeneous levels of rationality and inconsistent beliefs are important features of strategic reasoning at the individual level. This paper proceeds as follows. The next subsection discusses related literature. Section 4 Rationalizable sets include all actions supported by rationality assumptions. This sets will be defined and discussed further in the next section. 5 This identification strategy cannot be implemented in standard game forms (bimatrix games) because there is a tight link between higher-order and lower-order beliefs. Thus, there dos not exist different games (with different rationalizable actions) that differ only in higher-order beliefs.  56  4.2 introduces the ring game and motivates the experimental design through an example. Section 4.3 formalizes the epistemic concepts of rationality and consistent beliefs and the identifying assumptions used to separately identify the three epistemic conditions. Section 4.4 discusses the experimental design. Section 4.5 gives the experimental results. And, Section 4.6 provides a discussion of the alternative solution concepts of rationalizability, quantal response equilibrium, and level-k and cognitive hierarchy models and of the alternative approach of relaxing common knowledge of payoff assumptions. Omitted proofs can be found in Appendix B.1.  4.1.1  Related literature  The methodology here differs substantially from previous experimental work that tests between alternative solution concepts. Existing game theoretic work compares different solution concepts by jointly testing a package of assumptions. The two most prominent of these alternative solution concepts are quantal response equilibrium (QRE) and the class of level-k and cognitive hierarchy models. QRE relaxes rationality assumptions by allowing players to make mistakes when they best respond but maintains the Nash assumption of consistent beliefs (McKelvey and Palfrey 1995). Level-k and cognitive hierarchy models maintain the assumption that players are rational in that they perfectly best respond to their beliefs, but relax the Nash assumption of consistent beliefs by allowing players to hold different beliefs about the rationality of others (E.g. Nagel 1995; Stahl and Wilson 1995; Costa-Gomes and Crawford 2006; and Camerer et al. 2004). Both of these alternative solution concepts require the analyst to impose untestable structural assumptions that are generally difficult to separate from epistemic failures themselves.6 Wright and Leyton-Brown (2010) provide the most exhaustive test to date between a set level-k and QRE models by performing a meta-analysis of existing normal-form game experimental data using standard structural methodology. Level-k, cognitive hierarchy and QRE perform equally well. The approach in this paper differentiates between alternative solution concepts by testing epistemic conditions directly. Untangling the role that rationality and consistent beliefs play in strategic reasoning will allow direct comparison of alternative solution concepts without relying on untestable structural assumptions that risk introducing bias. The experimental results provide support for level-k and cognitive hierarchy models. Subjects are rational; yet underestimate the rationality of others.7 Subjects do not satisfy consistent beliefs. Further, 6 Level-k  and cognitive hierarchy models always impose a specification for the behavior of level-0 types. This specification then anchors the beliefs of all other levels by having players play an action that is a finitely iterated best response to the level-0 behavior. Similarly, QRE imposes the structural assumption that players’ mistakes are determined according to a logistic error structure. Players then noisily best respond to other players’ noisy best responses. 7 These results are similar to Weizsäcker (2003) that finds that at the aggregate level subjects tend to attribute less rationality to others than to themselves. Weizsäcker uses a variant of the QRE model that has two parameters, one that captures a player’s own likelihood of making a mistake when best responding and one she attributes to the likelihood her opponent makes mistakes.  57  the orders of rationality that are estimated in this paper can be thought of as ‘levels’ in the level-k literature. Level-1 types are rational, level-2 types are rational and believe others are rational, level-3 types are rational and satisfy 2nd-order belief in rationality, and so on. The estimated order of rationality distribution thus gives an estimate of the level-k level distribution that is independent of the level-0 specification.8 The estimated distribution in this paper puts more weight on higher levels than in typical level-k experiments, which find most weight on level 1 and level 2 types. This suggests that specifying level-0 as uniformly random may bias estimates of the level distribution. This paper is also closely related to Healy (2011) and Costa-Gomes and Weizsäcker (2008). Both Healy’s work and this paper test epistemic conditions of Nash equilibrium in the lab. The main differences between these papers is the methodology. Healy does an aggregate level analysis and estimates epistemic conditions by eliciting subjects’ beliefs about actions, payoffs, and rationality and eliciting subjects’ beliefs about their opponents’ beliefs about actions and payoffs. He finds that Nash equilibrium fails because players are often wrong in their beliefs about what other players are doing and that at times subjects are not playing the games we think they are.9 Costa-Gomes and Weizsäcker elicit beliefs and choices in normal form games. They find that players tend not to best respond to their own stated beliefs, however they also find that players behave differently when asked to state beliefs. Whether or not belief elicitation affects strategic behavior remains an open question. See Healy (2011) for a more complete discussion of these issues. Thus, this paper takes a different approach, obtains individual level estimates, and uses a design specifically chosen to allow us to identify epistemic conditions from choice data. This approach is motivated by the intended application - modeling strategic choice. In addition, we are able to get an estimate of a player’s order of rationality, which would be difficult under belief elicitation as we would need to elicit beliefs up to the 3rd order. This approach allows us to find the correlation between higher-orders of rationality and consistent beliefs - even sophisticated players (satisfying high-orders of rationality) do not satisfy consistent beliefs. This paper is related to the literature on iterated dominance (Beard and Beil 1994; Andrew et al. 1994; Huyck et al. 2002; Ho et al. 1998; and Costa-Gomes et al. 2001), as we estimate a player’s order of rationality based on her choices in dominance solvable games. The existing literature tends to focus on violations of iterated dominance, with the exception of Ho et al. (1998) which measures a subject’s capability to perform levels of iterated dominance as her level-k level. The current paper measures subjects order of 8 Burchardi and Penczynski also provide an estimate of the level-k distribution that is independent of the level-0 specification. They incentivize communication between group members and analyze the communication in order to determine subject’s depth of reasoning. 9 This means that the assumption of common knowledge of payoffs does not hold. However, Healy finds this is true only for particular games, like the Prisoner’s Dilemma. He finds support for the payoff assumptions imposed in this paper in games where there seems to be no obvious role for other-regarding preferences, particularly in dominance solvable games.  58  rationality by classifying subjects into levels of iterated dominance using a more general identifying assumption than the level-k structure. The results in this paper are consistent with the results in this literature that finds that people do not tend to violate dominance (do not play dominated strategies) but may violate higher orders of dominance.  4.2  Example  The main goal of this paper is to separate to which extent the three assumptions: (1) rationality, (2) beliefs about rationality, and (3) consistent beliefs contribute to strategic reasoning. Are people rational? To what order do people believe others are rational? And, do people have consistent beliefs? In this section, I illustrate the challenge in separately estimating these three features of reasoning and motivate the ring game as a solution to this problem. Consider the problem of estimating a subject’s order of rationality. Bernheim (1984), Pearce (1984), and Tan and Werlang (1988) establish the strategies that can be rationally played when a subject has (k-1)th-order belief in rationality10 : in any complete information game, a subject who is rational and satisfies (k-1)th-order belief in rationality must play a kth-order rationalizable action. In many games (and in all the games considered in this paper) a strategy is kth-order rationalizable if and only if it survives k rounds of iterated deletion of strictly dominated strategies. For succinctness, we often refer to a subject who is rational and satisfies (k-1)th-order belief in rationality as satisfying kth-order rationality.  !  "#$%&'!(! ! $! 1!  "#$%&'!)*+!$,-./0+  ! ! !  ! ! !  ! ! !  "#$%&'!)! ! $! 1!  "#$%&'!(*+!$,-./0+  $! (2!  2!  !  1!  2!  (3!  !  !  !  !  !  !  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  !  ! ! !  !  ! ! !  $! (3!  2!  1!  2!  3!  !  !  !  !  ! Figure 4.2: B1: rationalizable bimatrix game ! Consider the game !B1 in 4.2. This for player ! Figure "#$%&'!(! ! game ! is ! 2nd-order "#$%&'!)!rationalizable ! 1 and 1st-order rationalizable player !2. If! player rational, she ! ! for ! 2! is "#$%&'!(*+!$,-./0+ "#$%&'!)*+!$,-./0+ ! must play action a. If player 1 is rational she ! can ! play $! either 1! a !or b.! If she ! satisfies $! 1!2nd-order rationality (is !  $! (2!  2!  !  "#$%&'!)*+!$,-./0+  10 kth-order  "#$%&'!(*+!$,-./0+  !  rational and believes player 2 is rational) then she must play action a.  !  !  !  $! (3!  2!  !  !  !  ! ! !  ! ! !  belief in rationality corresponds to holding finite-orders of beliefs about the rationality of others. For example, 1st-order belief in rationality means you believe your opponent is rational. 2nd-order belief in rationality means you1! believe your opponent believes her opponent is rational, and, so on. 2!that(3! ! 1! 2! 3! This is formalized in Section 4.3.  !  !  !  59  ,-./0+  "#$%&'!)! "#$%&'!4*+!$,-./0+! $! 1!  $! (2!  2!  !  $! (2!  2!  ! ! ! !  ! ! !  ! ! !  "#$%&'!4! "#$%&'!(*+!$,-./0+! $! 1!  ! ! !  $!  (3!  !  !  ! ! !  ,-./0+  "#$%&'!(! "#$%&'!)*+!$,-./0+! $! 1!  !  ! ! !  ,-./0+  ! ! !  !  ! !  2!  ! ! ! ! ! ! !a action  ! "#$%&'!)! ! "#$%&'!(*+!$,-./0+! ! player $! 1.1! She may have played a as  !  ! "#$%&'!(! ! "#$%&'!)*+!$,-./0+! ! $! play1!the subject  !  ! ! Suppose we observe! a  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  because she is 2nd-order rational. rule out $! (2! However, 2! ! we cannot $! (3! 2! the possibility that the  subject only satisfies lower orders of rationality. This is because the sets of rationalizable actions necessarily contain1! one another. 2! (3!The! 1st-order 1! rationalizable 2! 3! set contains the 2nd-  order rationalizable set which contains the 3rd-order rationalizable set and so on. This leads  ! ! to an identification problem.  "#$%&'!(! ! $! 1!  "#$%&'!)*+!$,-./0+  !  !  !  ! ! !  ! ! !  ! ! !  "#$%&'!(*+!$,-./0+  !  !  "#$%&'!)! ! $! 1!  !  "#$%&'!(*+!$,-./0+  !  ! ! !  !  $! (2!  2!  !  1!  (3!  !  2!  "#$%&'!)*+!$,-./0+  ! ! !  !  !  !  $!  2!  3!  1! (3!  2!  structured way: player 1 has the same payoffs in both B1 and B2. In other words, player 1 has equivalent 1st-order payoffs in games B1 and B2 (but has different higher-order payoffs). !  1!  2!  (3!  !  1!  2!  (3!  1!  2!  This structure along with an identifying assumption allows us to solve our identification  3!  separate the behavioral implications of 2nd-order rationality from 1st-order rationality.  ! can be separated Following this logic, the1! behavioral implications of 1! kth-order 2! (3! ! 2! rationality (3! 1! (3!  identification assumption seems empirically valid. In our experimental data, subjects follow the ! 83 ! percent ! of !the time. ! In! addition, ! ! the assumption ! ! is supported ! ! by !the restrictions of this assumption limited depth of reasoning! literature. If a subject has a finite depth of reasoning k, then she is rational and satisfies (k-1)th-order belief in rationality and will ignore any information that has to be processed at k+1 depths of reasoning. !For example, if a subject is rational (and not higher-order rational) because she has a depth of reasoning of ! 1 then she will not base her decision on any payoffs besides her own. There is empirical support for this type of behavior from experiments that analyze the information search patterns of subjects. Costa-Gomes et !al. (2001), Costa-Gomes and Crawford (2006), Wang et al. (2009), Brocas et al. (2009), Camerer et al. (2002), ! and Johnson et al. (1993) all analyze strategic behavior by investigating the information search pattern of subjects. They find a correlation between the play of kth-order rationalizable strategies and patterns of !search that are associated with k depths of reasoning.  !  60  ! ! ! ! ! !  ! "#$%&'!4*+!$,-./0+  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  !  ! ! order payoffs. For example, if a rational subject does not respond to changes in 2nd-order 11 ! play ! the "#$%&'!(! ! in !both! games. "#$%&'!)! ! ! "#$%&'!4! payoffs, then she should same action Under !this assumption, a ! ! ! ! ! ! ! ! "#$%&'!)*+!$,-./0+ "#$%&'!(*+!$,-./0+! rational subject would play either (a, a) or! (b, b) as player 1"#$%&'!4*+!$,-./0+ in games !B1 and B2 respectively, $! 1! ! ! ! ! $! 1! ! ! ! ! however a subject who is 2nd-order rational (or higher) would play action profile (a,$!b). 1! ! then allow us to Observing the action profile of a subject in both games B1 and B2 would $! (2! 2! ! $! (2! 2! $! 2! 3! !  ! lower-order ! ! ! ! subjects ! ! do not ! respond ! to! changes ! in! higher! problem. We assume that rational  11 This  ! ! !  ! "#$%&'!4*+!$,-./0+  ! "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  !  ! ! ! ! ! ! ! ! ! ! Figure 4.3: B2: rationalizable bimatrix game ! This problem can be! resolved by observing related sets of ! "#$%&'!(! ! behavior ! ! in"#$%&'!)! ! games. ! Consider ! "#$%&'!4! ! ! ! rationalizable ! 1 !and"#$%&'!(*+!$,-./0+ the game B2 in Figure! 4.3.! This game ! is 2nd-order 1st"#$%&'!)*+!$,-./0+ "#$%&'!4*+!$,-./0+! for! player ! $! must 1! play !action! b. !If player order rationalizable for! player ! 2.$! If player 1! 2 ! is rational, ! ! she $! 1! 1 is 2nd-order rational then she must play action b. In addition, B2 is !related to B1 in a $! (2! 2! ! $! (2! 2! $! (3! 2!  !  2!  !  !  !  ! "#$%&'!(! "#$%&'!)*+!$,-./0+! $! 1!  ! ! !  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  ! ! !  ! ! !  "#$%&'!)! "#$%&'!(*+!$,-./0+! $! 1!  !  ! ! !  !  ! ! !  $! (2! 3! ! by looking $! (2! 3! that differ only in kth-order payoffs from lower-orders of rationality at games (and higher) but have different kth-order rationalizable implications. However, no such  1!  3!  (3!  !  1!  3!  2!  bimatrix games exist. This is because there is a tight link between higher- and lower-order payoffs in! bimatrix her 1st-, 3rd-, 5th-order payoffs, ! ! games. ! A !player’s ! !payoffs ! determine ! and so on. ! While her opponent’s payoffs determine her 2nd-, 4th-, 6th-order payoffs, and  ! so on. Higher-order payoffs cannot change independently of lower order payoffs. Therefore, ! "#$%&'!(! ! payoffs "#$%&'!)! if any two! bimatrix games have! the !same up to at! least the 2nd-order they must be ! (and ! the! same ! rationalizable "#$%&'!)*+!$,-./0+ "#$%&'!(*+!$,-./0+! the same! game hence! have implications).  ! ! solves $! this 1!problem ! by ! making ! $! use of 1! a novel class of games: ring games. A This paper !  $! (2!  3!  !  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  !  ring game is essentially a series of two-player games that have a unique opponent structure.  $! (2!  3!  In a standard 2-player game, player 1 and player 2 are each other’s mutual opponent. But, in a ring game, player 1’s opponent is player 2 but player 2 has an entirely different opponent,  1!  3!  (3!  !  1!  3!  2!  player 3. The opponent structure of ring games allows us to solve the identification problem  ! allows ! us ! to induce ! ! ! in ! higher-order ! ! payoffs independently of lower-order because it changes ! ! !  ! ! !  ! ! !  "#$%&'!)! ! $! 1!  "#$%&'!4*+!$,-./0+  "#$%&'!(*+!$,-./0+  $! (2!  3!  !  1!  3!  (3!  !  !  !  !  !  !  !  $! (2!  3!  1!  3!  (3!  !  !  !  !  ! ! ! ! ! !  ! ! !  ! ! !  "#$%&'!(*+!$,-./0+  "#$%&'!4! ! $! 1!  ! ! !  $!  (2!  3!  !  1!  3!  2!  !  !  !  !  !  !  !  "#$%&'!(! ! $! 1!  "#$%&'!)*+!$,-./0+  !  ! ! !  "#$%&'!)*+!$,-./0+  ! ! !  !  !  "#$%&'!4*+!$,-./0+  payoffs. !  Figure 4.4: R1: rationalizable ring game  ! "#$%&'!4*+!$,-./0+  ! "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  !  ! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!4! ! Consider game R1 in 1’s payoff own action ! the ! "#$%&'!)*+!$,-./0+ ! !4.4.! Player ! ! upon ! her ! Figure "#$%&'!4*+!$,-./0+ ! depends "#$%&'!(*+!$,-./0+ ! !and the action of player 2. Player 2’s payoff depends upon her own action and the action $! 1! ! ! ! ! $! 1! ! ! ! ! $! 1! ! of player 3. And, player 3’s payoff depends upon her action and ! the action of player 1. $! (2! 3! ! $! (2! 3! $! 3! 2! ! R1 is 1st-order rationalizable for player 3, 2nd-order rationalizable for player 2, and 3rd-order rationalizable for player 1. Thus, player 3 must !play a if she is rational, player 2  1!  3!  (3!  !  1!  3!  (3!  1!  (2!  3!  !  must play a if she is 2nd-order rational and player 1 must play a if she is 3rd-order rational (rational,! believes ! !her opponent ! ! is rational ! ! and ! believes ! her! opponent ! ! believes ! her! opponent !  ! is rational). Game R2, in Figure 4.5, is 1st-order rationalizable for player 3, 2nd-order ! rationalizable for player 2, and 3rd-order rationalizable for player 1. Player 3 must play b if  ! she is rational, player 2 must play b if she is 2nd-order rational and player 1 must play b is ! she is 3rd-order rational.  ! In addition, R1 and R2 are related to each other in a structured way. Player 1 has ! 1st- and 2nd-order payoffs in games R1 and R2 but the games have different the same 61  3!  !  1!  3!  (3!  !  !  !  !  !  !  ! ! !  ! ! !  ! ! !  ! !  !  "#$%&'!4*+!$,-./0+  $!  1!  "#$%&'!)*+!$,-./0+  !  $!  ! !  $! (2!  3!  1!  3!  (3!  !  !  !  !  ! ! !  ! ! !  ! ! !  ! !  ! !  $!  1!  ! !  $!  (2!  3!  !  3!  2!  !  !  !  !  1!  !  !  !  ! ! ! !  ! ! !  ! ! !  !  !  "#$%&'!(*+!$,-./0+  !  $! (2!  !  "#$%&'!)*+!$,-./0+  !  ! !  "#$%&'!4*+!$,-./0+  1!  ! !  "#$%&'!(*+!$,-./0+  ! !  !  3!  !  1!  3!  (3!  !  !  !  !  !  !  $! (2!  3!  1!  3!  (3!  !  !  !  !  ! !  ! ! !  !  $! (2!  "#$%&'!4! "#$%&'!(*+!$,-./0+! $! 1!  "#$%&'!4*+!$,-./0+  !  "#$%&'!)! "#$%&'!4*+!$,-./0+! $! 1!  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+  !  "#$%&'!(! "#$%&'!)*+!$,-./0+! $! 1!  $!  3!  2!  !  1!  (2!  3!  !  !  !  !  !  !  ! Figure 4.5: R2: rationalizable ring game ! ! rationalizable implications. This is possible because the relaxed opponent structure of the ! ring game allows us to break the tight link between higher- and lower-order payoffs that ! bimatrix games impose. R1 and R2 differ only in higher-order payoffs. Thus, these games ! separate the behavioral implications of 3rd-order rationality and 2nd-order rationality under  the identification assumption that a 2nd-order rational subject will not respond to changes in 3rd-order payoffs. Under our identifying assumption a subject that only satisfies 2ndorder rationality will play either (a, a) or (b, b) as player 1 in games R1 and R2 respectively, however a subject who is 3rd-order rational (or higher) would play action profile (a, b). Observing the behavior of player 1 in R1 and R2 will allow us to separate the implications of 3rd-order rationality from lower-orders of rationality. Each additional player in the ring game allows an additional degree of independence between beliefs. For example, a 4-player ring game could be used to separate 4th-order rationality from lower-orders of rationality. Once a subject’s order of rationality is identified, a separate set of games can then be used to estimate whether or not a player has consistent beliefs. This is done using a characterization of Nash equilibrium due to Aumann and Brandenburger (1995) and Perea (2007).12 In any 2-player complete information game if a player satisfies 2nd-order rationality and consistent beliefs13 , then she has to play an action consistent with a Nash equilibrium. Thus to estimate whether consistent beliefs hold, we will consider a bimatrix game that has a unique Nash equilibrium but where all actions are rationalizable. For all subjects that satisfy 2nd-order rationality, if they play the Nash equilibrium we will assign them as having consistent beliefs. For any subject that satisfies 2nd-order rationality, failure to play a Nash equilibrium is an indication of a failure of consistent beliefs. 12 This  characterization is a slight variation on both Aumann and Brandenburger (1995) and Perea (2007). Essentially, we rewrite Aumann and Brandenburger’s characterization using the epistemic type space framework in order to test a characterization that tests restrictions on one player’s belief hierarchy. This effectively provides epistemic conditions for a ‘Nash equilibrium in beliefs’. It does not rely on players having a way to coordinate their beliefs, and instead looks at the internal consistency requirements on one player’s belief hierarchy which would lead her to believe that she and her opponents play actions that are consistent with a Nash equilibrium. This is similar to Perea’s characterization, however we strengthen the notion of consistent beliefs that Perea applies. The proof can be found in Appendix A. 13 If a player plays strategy s, then she has consistent beliefs if she believes her opponent believes she is playing s.  62  In the following section, we define what we mean by rationality, higher-orders of rationality and consistent beliefs. We formalize our identification assumptions and show that orders of rationality can be separately identified independent of consistent beliefs in ring games.  4.3  The model  In this section, we apply the tools of epistemic game theory in order to model a player’s beliefs about the strategies and payoffs of others. This structure then allows us to formally define rationality and player’s beliefs about each others’ rationality, actions, payoffs, and beliefs. Type spaces are regularly used to model incomplete information games. In such a type space, a player’s type represents her beliefs about the payoff types of others. An epistemic type space is analogous, except a player’s types represent her beliefs about the payoffs and strategies of others. First, we formally define a ring game. Definition. An n-player ring game Γ is a tuple Γ = �I = {1, . . . , n}; S1 , . . . , Sn ; π1 , . . . , πn ; Θ1 , . . . , Θn ; p� where I is a finite set of players, Si is a finite set of actions for player i, Θi is  a set of payoff types for each player, πi : Si × Sp(i) × Θi → R represents the payoff function for each player and p : I → I is function representing the opponent of player i with the restriction that p(i) = 1 + imodn.  Given a game, we can define an epistemic type space which describes the beliefs of each player about the strategies and payoffs of the other players. Definition. Let Γ = �I = {1, . . . , n}; S1 , . . . , Sn ; π1 , . . . , πn ; Θ1 , . . . , Θn ; p� be an n-player game. A finite Γ-based epistemic type space is a set �T1 , . . . , Tn ; b1 , . . . , bn ; sˆ1 , . . . , sˆn ; θˆ1 , . . . , θˆn � where Ti is a finite set, bi : Ti → �(T−i ) is an injective function, sˆi : Ti → �(Si ), and θˆi : Ti → Θi .  Every player has a set of types Ti . The function sˆi defines a strategy for each type. The function θˆi defines a payoff type for each type. The function bi represents each type’s beliefs about the types of her opponents, T−i . Therefore bi (ti ) together with the function sˆ defines type ti ’s beliefs about the strategies of her opponents and bi (ti ) together with the function θˆ defines type ti ’s beliefs about the payoff types of her opponents.  4.3.1  Identifying assumptions  Formally we make three identifying assumptions. The first two assumptions place restrictions on the type space and the third is the behavioral assumption discussed in Section 4.2. 63  The first assumption restricts the game to a game of complete information - each player has a single payoff type determined by the payoffs of the game. This assumption imposes that differences in behavior will come from differences in beliefs about rationality and consistency rather than differences in payoff types. Further, it allows us to assume that exogenous changes in payoffs cause exogenous changes in players beliefs. It is a standard assumption in experimental economics - preferences are induced by the game payoffs. A1: Θi = {θi } The second assumption places a restriction on first-order beliefs. Players form first-order beliefs only about their direct opponents. This assumption only makes sense in ring games (which we focus our attention on), this is because in a ring-game, a player’s payoff depends only on the action of her direct opponent. For example, in a 3-player ring game, player 1 cares what player 3 does only to the extent that she cares what player 2 believes player 3 does. In other words, player 1’s beliefs about what player 3 does has no influence on her expected payoffs. A2: bi : Ti → �(Tp(i) ) for all i ∈ I The behavioral assumption discussed in the previous chapter assumes that a player with a finite order of rationality will not respond to changes in higher order beliefs. Under assumption A2, changing the payoffs of a player that is n links away from player i will only affect player i’s nth-order beliefs and not her lower order beliefs. Thus A1 and A2, build the epistemic foundation to apply our behavioral assumption A3 (that requires that higher-order beliefs can be changed independent of lower-order beliefs in ring games). The last assumption is a behavioral assumption that allows us to separate the behavioral implications of different orders of rationality and consistent beliefs. Essentially, this is the assumption that players with finite-orders of rationality do not respond to changes in higherorder beliefs. In order to define this assumption we must define higher-order beliefs about payoffs. Every game can be characterized by its payoff hierarchy for each player. Under as¯ i (Γ) of player i for any complete sumption A1 and A2, we can define the payoff hierarchy h information game Γ: ¯ i (Γ) = {θpk (i) }∞ . h k=0 ¯ i (Γ), her 1st-order beliefs about payoffs are The payoffs of player i are determined by h 0 i ¯ (Γ), her 2nd-order beliefs about payoffs are determined by h ¯ i (Γ) and so determined by h 1  2  on. Payoff hierarchies of complete information games are relatively redundant. For a 2¯ 1 (Γ) = {θ1 , θ2 , θ1 , θ2 , . . .}. player complete information ring game (i.e. a bimatrix game), h  However, increasing the number of player’s in the ring game permits us to provide a richer set 64  ¯ 1 (Γ) = {θ1 , θ2 , θ3 , θ1 , θ2 , θ3 , . . .}. of payoff hierarchies. For example, in a 3-player ring game, h Thus, each additional player in the ring game adds an additional degree of freedom to define higher-order beliefs independently of lower-order beliefs. The third assumption imposes a behavioral restriction that requires a player that satisfies only kth-order rationality to not respond to changes in her mth-order beliefs about payoffs whenever m is greater than k. A3: A player i who satisfies rationality and (k − 1)th-order belief in rationality (but not kth-order belief in others rationality) will not respond to changes in her mth-order beliefs about payoffs for all m > k (i.e. play the same action in any two games, Γ and ¯ i (Γ) = h ¯ i (Γ� ) for all j ≤k) Γ� , such that h j  j  In the next subsection, we define precisely what is meant by rationality, higher-order beliefs about rationality and consistent beliefs.  4.3.2  Epistemic conditions  4.3.2.1  Rationality  We can define the expected utility of a type ti by14 ui (si , ti ) ≡  �  bi (ti )(tp(i) )π(si , sˆ(tp(i) )).  tp(i) ∈Tp(i)  Given this specification for utility, a type is rational if the action sˆi (ti ) is a best response for player ti given her beliefs. Definition. A type ti is rational if sˆi (ti ) maximizes player i’s expected payoff under the measure bi (ti ). That is, if ui (ˆ si (ti ), ti ) ≥ ui (s� , ti ) for all s� ∈ Si . We are interested in whether types are rational, but also if they believe others are rational and so on. First, we say a player believes an event if she places probability 1 on that event happening. Definition. A type ti believes an event E ⊆ Tp(i) if bi (ti )(E) = 1. Let the set B i (E) = {ti ∈ Ti |bi (ti )(E) = 1} be the set of types for player i that believe event E.  Higher-order beliefs about the rationality of others can then be defined in recursively in the following way. Define 14 Let  π(si , sˆ(tp(i) )) be redefined in the standard way whenever sˆi is a mixed-strategy.  65  Ri1 Rim+1  = {ti ∈ Ti |ti is rational} m = Rim ∩ B i (Rp(i) )  )"# )$# )"# )$# Definition. If ti ∈ Rim+1 then we say that ti satisfies rationality and mth-order belief  !(#  in rationality.  !%# 4.6!for& a 3-player ring game. If t1 is Consider the following type structure in Figure (# !"# belief!$#in rationality, that means t'#'#' rational and satisfies 1st-order and t2 is rational. 1 is rational #  !(&&#that means t1 is rational, t2 is If t1 is rational and satisfies 2nd-order belief !in%&rationality, # rational, and t3 is rational. If t1 is rational and satisfies kth-order belief in rationality she !(&&&rational can only put positive weight on types that are themselves in any of her kth-order # beliefs. )"#  )$#  )%#  )"#  !"#  !$#  !%#  !(# '#'#'#  Figure 4.6: A type structure for a 3-player ring game )"# )$# )%# )"# 4.3.2.2  !"#  Consistent beliefs  !%#  !$#  !"# '#'#'#  The assumption of consistent beliefs places restrictions on player’s beliefs about the strategies of others. Consistent beliefs ensure that player i believes her opponent believes she is playing the action she is actually playing. In order to capture this restriction, consistent beliefs are defined by two conditions. First, players must be correct about their opponents’ beliefs in some manner. And, second, players must believe their opponents are correct about her beliefs. A player has correct beliefs if she believes she is correct about her opponent’s beliefs. In other words, she has the same beliefs about her opponent’s beliefs as her opponent. Definition. A type ti has correct beliefs if whenever she places positive probability on type tj and bj (tj )(E) = p for event E ⊆ Tp(j) then ti believes that his opponent believes that event E occurs with probability p.  Like with rationality, we want to be able to discuss whether a player believes other players have correct beliefs and so on. Define  66  Ci1 Cim+1  = {ti ∈ Ti |ti has correct beliefs} m = Cim ∩ B i (Cp(i) )  Definition. If ti ∈ Cim then we say that ti satisfies correct beliefs to the mth-order. In addition, a player must believe that her opponents are correct about her own beliefs. If this is true, we will say that beliefs are self-referential. Self-referential beliefs are the condition that whenever a type ti believes an event, then she believes her opponent believes she believes it. In other words, she believes her opponent is correct about her beliefs. However, in the case of an n-player ring game, a player’s second-order beliefs do not refer back to her own type. For example, in the 3-player ring game, player 1 forms firstorder beliefs about player 2, second-order beliefs about player 3 and third-order beliefs about player 1. Therefore, player 1’s 1st-order beliefs are trivially self-referential in this case because second-order beliefs are about player 3 and not player 1. Thus, no type space will violate self-referential beliefs for player 1 (when n > 2). However, the relevant case is what player 1 believes that player 2 believes that player 3 believes Because, of the particular partner structure assumed (i.e. pn (i) = i) we need to only check whether a player’s nth-order beliefs are self-referential in an n-player ring game. We formally define a player’s higher-order belief function gi in the following way. Let player i’s first-order beliefs be given by gi1 : Ti → �(Tp(i) ), defined by gi1 (ti ) = bi (ti ). Recursively define player i’s (m+1)th-order beliefs gim+1 : Ti → �(Tpm+1 (i) ) by gim+1 (ti )(E) =  �  tpm (i) ∈Tpm (i)  � � m gim (ti ) tpm (i) bp (i) (tpm (i) )(E)  for all E ⊂ Tpm+1 (i) . Definition. A type ti has self-referential beliefs (to the nth-order) if whenever bi (ti )(E) = p for some E ⊆ Tp(i) then g n (ti )(E) = p. The term consistent beliefs refers to the joint assumption that beliefs are both correct and self-referential. Consider the type structure in Figure 4.7 for a 3-player ring game. Type t1 has consistent beliefs - she believes that player 2 believes that player 3 believes she is of type t1 . This captures the Nash assumption of consistent beliefs. Type t1 plays strategy sˆ1 (t1 ) and she believes that player 2 believes that player 3 believes that she plays strategy sˆ1 (t1 ). Definition. A type ti has consistent beliefs (to the nth-order) Γ if she has correct belief to the nth-order and self-referential beliefs  67  !(&&&# )"#  )$#  )%#  )"#  !"#  !$#  !%#  !(# '#'#'#  )"#  )$#  )%#  !"#  !$#  !%#  )"#  !"# '#'#'#  Figure 4.7: A consistent type structure for a 3-player ring game  4.3.3  Separating rationality and consistent beliefs  This section establishes that the behavioral implications of different orders of rationality and consistent beliefs can be formally separated in ring games under the three identifying assumptions. Assumption A1 assumes that preferences are determined by the game payoffs. No predictions would be possible without some restrictions on the relationship between the game payoffs and player’s preferences. We assume the simplest restriction in this paper.15 A1 allows us to assume that exogenous changes to payoffs induce exogenous changes in a player’s beliefs (as described by the game payoffs). Assumption A2 allows us to assume that changing the payoffs for a player pk (i) changes only player i’s kth-order (or higher) beliefs about payoffs. Assumption A1 and A2 allow us to apply assumption A3. Assumption A3 acts as an exclusion restriction that separates the behavioral implications of different orders of rationality. Without an exclusion restriction, the behavioral predictions of different orders of rationality necessarily nest one another. A player that satisfies only lower-order beliefs about the rationality of others could always play an action that a player that satisfies higher orders of rationality could play. Under A3, a player with a lower order of rationality would never play the same set of actions as a player with a higher order of rationality in sets of games that differ only in higher order beliefs. Thus, by observing strategic choices in specifically chosen sets of games, it is possible to separately identify different orders of rationality. This was demonstrated with examples in Section 4.2. Further, under A2, we are ensured that a player’s order of rationality can be identified independently of consistent beliefs in ring games (in n-player ring games that are rationalizable in n rounds or less). Without the assumption A2, existing characterizations of Nash equilibrium apply to the ring game (Aumann and Brandenburger 1995; Perea 2007). When the following restrictions hold for a player: (i) rationality, (ii) 1st-order belief in rationality, and (iii) consistent beliefs then she has to play a Nash equilibrium. Consider the 3-player ring games from Section 4.2. If A2 does not hold, then rationality, 1st-order belief in rationality and consistent beliefs will ensure a player will play the Nash equilibrium strategy. However, rationality, and 2nd-order belief in rationality will also ensure a player will play the Nash equilibrium strategy. Thus, the behavioral implications of higher-order rationality are not separated from those of lower-order rationality and consistent beliefs. This is not 15 This  assumption is discussed further in Section 4.6.1.  68  true once A2 is imposed. Under A2, rationality, 2nd-order belief in rationality and consistent beliefs will not ensure a player will play an action consistent with a Nash equilibrium. To see this consider a player with the following type structure as in Figure 4.8. Type t1 satisfies 1st-order belief in rationality and has consistent beliefs. However, it is rational for t1 to play action b, an action that is not consistent with a Nash equilibrium. '"#  '$#  '%#  '"#  !"#  !$#  !%#  !"# &#&#&#  ( ###  ( ###  ( ###  ( ###  Figure 4.8: A consistent type structure for a 3-player ring game Proposition 4.1 formally states that in ring games, under A2, we need to combine consistent beliefs with higher orders of rationality to ensure that a Nash equilibrium is played. In a 3-player ring game, rationality, 2nd-order belief in rationality and consistent beliefs are sufficient conditions for Nash equilibrium.16 Proposition 4.1. Consider an n-player ring game Γ =< I = {1, . . . , n}; S1 , . . . , Sn ; π1 , . . . , πn ; Θ1 , . . . , Θn ; p > and a Γ-based epistemic type space < T1 , . . . , Tn ; b1 , . . . , bn ; sˆ1 , . . . , sˆn >.  Suppose type ti satisfies the following conditions: (i) rationality, (ii) (n − 1)th-order belief in rationality and (iii) self-referential beliefs and nth-order correct beliefs. Then it must be � that σ defined by σi = sˆi (ti ) and σpk (i) (s) = sˆpm (i) (tpm (i) )(s) · gim (ti )(tpm (i) ) for tpm (i) ∈Tpm (i)  all s ∈ Spm (i) , k ∈ {1, . . . , I − 1} constitutes a Nash equilibrium.  These conditions are all required. Rationality and (n-1)th-order belief in rationality are needed in order to guarantee a Nash equilibrium action will be played in an n-player ring game (that is dominance solvable in n rounds or less). No combination of lower-order rationality and consistent beliefs will be sufficient. In other words, the set of actions that are supported by rationality, kth-order beliefs in rationality and consistent beliefs is the same as the set of actions supported only by rationality and kth-order belief in rationality as long as k<n. Consistent beliefs have no bite in ring games unless the player satisfies rationality and at least (n-1)th-order belief in rationality. 16 When  n = 2, A2 holds trivially and these conditions collapse to those of Aumann and Brandenburger.  69  4.4  Experimental design  This section discusses the details of the experimental design that allows us to separately identify the three key epistemic conditions: rationality, beliefs about rationality and consistent beliefs. The design is motivated by the results of the previous sections.  4.4.1  Rationality  Each subject’s order of rationality is estimated from strategic choice data in 2 4-round rationalizable 4-player ring games. This separates subjects into five rationality categories: irrational (R0), rational (R1: 1st-order rational), rational and 1st-order belief in rationality (R2: 2nd-order rational), rational and 2nd-order belief in rationality (R3: 3rd-order  ! "#$%&'!+,-!$./012-  ! "#$%&'!+,-!$./012-  ! "#$%&'!+,-!$./012-  !  ! ! 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The games are similar to ! ! ! ! ! ! ! $! 3! .! ! ! $! 3! .! ! $! 3! .! ! $! 3! .! the ! ! ! .! )! 6! 4! .! )! ()! (6! .! )! 6! 4! .! )! ()! games. (6! ! Section ! ! games G1 $! R1 6! and (5! R2 7! from $! +! 4.25!except 4! that $! and 6! G2 (5! are 7! 4-player, $!3 action +! 5! 4! ! Adding an additional! player permits identification of a player’s order of rationality up to 3! )! (5! (4! ! 3! )! ()! )! ! 3! )! (5! (4! ! 3! )! ()! )! the 4th-order. Each subject’s order of rationality is estimated by her choice pattern in the 8 ! .! )! 6! 4! ! .! )! ()! (6! ! .! )! 6! 4! ! .! )! ()! (6! games games × 4 player Only the payoffs of Player 4 change across these two ! ! (2 "#$%&'!(! ! ! positions). ! "#$%&'!)! !! ! "#$%&'!),-!$./012-! "#$%&'!*,-!$./012-! ! the ! same games. Thus, player 1! has 3rd-order beliefs about payoffs, player 2 has the same ! ! ! $! 3! .! ! ! ! $! 3! .! ! (5! 7! !! ! $!! +!"#$%&'!)! 5! 4! ! $!! 6!"#$%&'!(! 70 "#$%&'!),-!$./012-! "#$%&'!*,-!$./012-! ! ! ! ! ! 3! )! ()! )! 3! )! (5! (4! ! ! $! 3! .! ! ! ! $! 3! .! .! )! 6! 4! ! .! )! ()! (6! $! 6! (5! 7! ! $! +! 5! 4! ! ! 3! )! (5! (4! ! 3! )! ()! )! ! .! )! 6! 4! ! .! )! ()! (6! !  2nd-order Rational (R2)  a  a  !  !  b  b  !  !  3rd-order Rational (R3) 4th-order Rational (R4)  a a  a a  a a  "  b b  b b  b b  "  a  b  Action G1 G2 Type P4 P3 P2 P1 P4 P3 P2 P1 2nd-order beliefs about payoffs, and player 3 has the same 1st-order beliefs about payoffs. " " player a ! rational c (satisfying ! " "the assumptions A1-A3) The Rational predicted(R1) actions for a kth-order are given in Table 4.1. A(R2) R1 subject and G2 when she plays as 2nd-order Rational a aplays! the#samec action b in!G1 # player 1, 2, or Rational 3. A R2 (R3) plays the ⌘ she plays as player 2 or 3rd-order a same a action b ⌘in G1 c and b G2awhen player 1. And, a R3 subject plays the same action in G1 and G2 when she plays as player 1.17  4th-order Rational (R4)  a  a  b  a  c  P1 Type  G1  Rational (R1)  b  a  c  P2 G2  G1  P3 G2  G1  P4 G2  G1 G2  (a,a)(b,b)(c,c) (a,a)(b,b)(c,c) (a,a)(b,b)(c,c)  (a,c)  2nd-order Rational (R2) (a,a)(b,b)(c,c) (a,a)(b,b)(c,c)  (a,b)  (a,c)  3rd-order Rational (R3)  (a,a)(b,b)(c,c)  (b,a)  (a,b)  (a,c)  4th-order Rational (R4)  (a,c)  (b,a)  (a,b)  (a,c)  Table 4.1: Predicted actions under rationality and assumptions A1-A3 A player’s order of rationality is estimated by matching her action profile in the 8 games  Rational + PE + Consistent to one of the predictions in Table 4.1. Consider the 5 types: R0, R1, R2, R3 and R4. The  action of each type is determined A1-A3 and Type R1 by R2assumptions R1 R2 R1 her order R2 of rationality over the set of 8 games as given in Table 4.1. Let ai ∈ {a, b, c}8 be an 8-tuple representing  < 3rd-order Rational a,b,cof a,b,c (a,a)(b,b)(c,c) (a,a)(b,b)(c,c) the action of subject i in each the 8 games. Each of the types, R1-R4, has a predicted action 3rd-order (or set ofRational actions) for ak ∈ Sk ⊂ {a, b, c}8 . For example, type a theseb 8 games,(a,b) (a,b)  R4 has a unique prediction. She must play the 4th-order rationalizable strategy in each of the 8 games, S4 = {(a, b, a, a, c, a, b, c)}. Type R3 has a unique prediction for 6 of the 8 games, but can play a number of different strategies for the other 2 games, S3 =  {(a, b, a, a, a, a, b, c), (c, b, a, a, c, a, b, c), (b, b, a, a, b, a, b, c)}. Notice that A1-A3 generate an exclusion restriction and ensure that S4 ∩ S3 = ∅. We can define similar sets for types R1 and R2 such that Sn ∩ Sm = ∅ for any n �= m ∈ {1, 2, 3, 4}.  If a subject’s action profile matches one of the predicted action profiles of type Rk, then  we would assign them as that type. However, a subject’s action profile does not always coincide precisely with a predicted action profile so we would like to allow for the possibility that a player might make an error. We allow subjects to make at most 1 error. If they are within 1 error to an exact type match of type Rk (i.e. the action played in one of the 8 17 We estimate a player’s order of rationality from a set of finite games. In these games, the behavioral implications of a belief with probability 1 will be the same for a belief with probability p, for high enough p. Relaxations of a 1-belief in rationality to a p-belief in rationality is discussed in Appendix B.2. We show that allowing for a relaxation to p-belief does not change the identification of orders of rationality or consistent beliefs.  71  *Uniform L0 14 choices 13 choices 12 choices 11 choices  L1 0 2 10 14  L2 0 0 2 6  L3 0 0 0 5  L4 0 3 10 17  n = 63 0 5 22 42  games can be changed so that the action profile matches a predicted action profile of type *G3/G4 Stability Rk) then we assign that subject as that type. If a subject makes 2 or more errors, then we assign them as an irrational (R0) Player type. 1If a subject’s profileBoth is within 1 error of 2 n=63 Playeraction 2 types, we assign3 them to the type where assumption A3 of 3 actions 28 the error is21due to not following 8 2 of 3 actions 61 62 60 rather than rationality (in practice this means that if an action profile was 1 error away 5 of 6 actions 41 from two different types, they were assigned the lowest type).  Assignment # of subjects assigned # of action profiles Exact type match 51 40 Type match with 1 error 24 290 R0 (2+ errors) 5 6231 Total  80  6561  Table 4.2: Subjects assigned by category The number of subjects assigned by exact type match and by error is given in Table 4.2. Each subject could play 3 possible actions in each of the 8 games for a total of 6561 possible action profiles. The rate of exact type match is extremely high, 51 of the 80 subjects were assigned to types because their action profiles matched one of the 40 action profiles of types R1-R4 exactly. An additional 24 subjects were assigned to types R1-R4 because their action profiles matched one of R1-R4’s action profiles by 1 error (290 possible action profiles match type R1-R4 with 1 error). And, the remaining 5 subjects were assigned as R0 types if they deviated from the predicted profiles of R1-R4 types by more than 1 error. The assignment mechanism can be generated by a finite mixture model where each subject has some probability of being each of the five types and the probability is chosen to maximize the likelihood that the subject’s action profile is generated by the given type. This approach closely follows Costa-Gomes and Crawford (2006). Details are discussed in Appendix B.3.  4.4.2  Consistent beliefs  Next, we test whether a player has consistent beliefs about strategies based on strategic choices in game Game G3 in Figure 4.11. G3 is a 2-player ring game. There is a unique Nash equilibrium (a, a) in game G3. We apply the characterization of Nash equilibrium to estimate consistent beliefs. If a subject satisfies rationality, 1st-order belief in rationality, and consistent beliefs then she must play the Nash strategy a in game G3 (as both player’s 1 and 2). If a subject satisfies rationality and 1st-order belief in rationality as determined from games G1 and G2 (all 72  5! (6!  !  "#$%&'!)! ! $! 1! ,! "#$%&'!(*+!$,-./0+  $! 5!  3!  6!  1! )! (3! (6! ,! )! ()! (5!  3! 6!beliefs. (5! ! immediate unless you have 1! consistent  1! (5! )! (6!  !  Strategic choices from games G1, G2 and G3! permit us to categorize subjects into groups ,! (6! ()! 5! ,! 3! (5! (6! ! ,! 3! (5! (5! that are: boundedly rational (R0 and R1 types), sufficiently rational with inconsistent belief  !  (R2, R3, and R4 types who do not satisfy consistent beliefs) and sufficiently rational with  ! "$! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!:! consistent belief (R2, R3, types that beliefs). This "#$%&'!)*+!$,-./0+ ! satisfy "#$%&'!:*+!$,-./0+ ! "#$%&'!2*+!$,-./0+! ! and ! R4 ! ! consistent ! ! ! classification ! categorizes players into! 3 epistemic satisfy ! ! ! rational ! do ! not ! $! 1!types: ,! boundedly $! 1! types ,! that ! $! 1! ,! the sufficient rationality conditions required to play Nash equilibrium in 2-player games, $! 6! )3! ()! ! $! (2! (6! 2! ! $! )3! (2! 6!  ! ! ! !  ! ! 1! games 3! 6!but(5! 1! consistent )3! 6! (2! (5!that )! Nash equilibrium in 2-player do not satisfy beliefs, and 1! those  (6!  !  3! (5! (5!  !  "#$%&'!:*+!$,-./0+  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+!  !  ! ! ! !  !  ! ! Figure 4.11: G3: 2-player ring game ! ! R2, R3 and R4 subjects), and fails to play to play a in G3, then it must be because the ! assumption of consistent ! beliefs fails. The action a is also7$-./0$#.8$1#&!9&-+! a rationalizable action. A player who satisfies any level of rationality ! could play a. Because of this, our estimate of consistent beliefs will and not consistent beliefs "#! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!:! necessarily be an upper bound on the proportion of players who satisfy consistent beliefs. "#$%&'!)*+!$,-./0+ ! "#$%&'!:*+!$,-./0+ ! "#$%&'!2*+!$,-./0+ ! ! ! ! ! ! ! ! ! To alleviate overestimation, (a, a) was chosen to be somewhat undesirable. Playing a in G3 ! ! $! 1! ,! ! ! ! $! 1! ,! ! ! ! $! 1! ,! only ever gives moderate payoffs and as player! 2, there is the possibility! of getting a zero $! 6! )3! ()! $! (2! (6! 2! $! )3! (2! 6! payoff. In other words, believing your opponent believes you are playing a should not be  "#$%&'!:*+!$,-./0+  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+!  !  !  !  1! )3! 6! (2!  sufficiently rational types that satisfy the sufficient rationality conditions required to play  ! requirements. satisfy both the rationality ,! and(6! consistent ()! 5!belief ,! 3! (5! (6!  ,!  ! ! !  ! ! !  "#$%  "#$%&'!(  $!  1  $! ()! (  1! 6! ( ,! ! ! !  5! (  "#$%  "#$%&'!(  $!  1  $! 6! (  1! 5! (  ,! ()! (  !  Laboratory ! implementation !  ! ! 1! design 3! 6! in (5! 1! )3! 1! session. (5! )! We used a within-subjects which subjects played6!all(2! games in each  (6!  !  ! Each subject played the games G1, ()! G2 and player ,! (6! 5! G3! in each ,! of3!the(5! (6!positions, ,!for a3!total (5! (5!  !  ! 73  "#$%&'!:*+!$,-./0+  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+!  informed of one another’s decisions.  !  6!  jects made all decisions !through ! $!an subjects, subjects did not interact $! 6!  !  ! ! In! order ! ! ! $!across online 1! ,!interface. $! to1!ensure ,! independence 1! with one another during the experiment and were not ! ! )3! ()! $! (2! (6! 2! $! )3! (2!  ! ! ! !  ,!  ! ! ! !  Sessions were conducted ISIT computer undergraduate the "#!in !Arts"#$%&'!(! ! labs ! !with "#$%&'!)! ! students ! ! at"#$%&'!:! University of British Columbia. No subject more than Sub"#$%&'!)*+!$,-./0+ ! participated ! one! treatment. "#$%&'!2*+!$,-./0+ ! ! ! ! ! ! in "#$%&'!:*+!$,-./0+ ! !  "#$%&'!2*+!$,-./0+  4.4.3  !  ! ! ! !  ,! )!  "!  ! ! !  "#$%&'!2*+!$,-./0+  1! )! ()! )!  !!  ! ! !  !  $! 2! (3! 4!  ! ! !  "#$%&'!2*+!$,-./0+  "#$%&'!(! ! $! 1! ,! "#$%&'!)*+!$,-./0+  !  ! ! !  "#$%&'!)*+!$,-./0+  "#$%&'!(*+!$,-./0+!  ! ! !  ! ! !  "#$%  "#$%&'!(  $!  1  $! ()! (  1! 6! ( ,!  5! (  of 10 games.18 The within subject design allows us to estimate the level of rationality and consistent beliefs for each subject. Subjects played the games in a random order without feedback. Subjects were required to spend at least 90 seconds on each of the games. Once subjects made choices in all games they were given the opportunity to revise their choices (without any feedback). If subjects chose to revise their choices they could review each of the games (in the same original order) and make changes to their choices. Subjects were paid a $5 showup fee and received payment from their choices in one of the games. One of the games was randomly selected for payment at the end of the experiment. Subjects were randomly and anonymously assigned to 2 or 4 player groups (depending on whether the game to be paid was a 2- or 4-player game). Subjects were paid based on their choice and the choices of their group members in the selected game and received the dollar value of their payoff in the game. Subjects were paid in cash at the end of the experiment. The average session lasted 45 minutes and the average subject earned approximately $17 dollars (maximum payment was $25 and minimum payment was $7, including showup fee). Payments were in Canadian dollars. Instructions were read aloud by the experimenter at the beginning of the session. The instructions to all subjects were the same. Subjects then completed a short understanding quiz to make sure they understood the instructions. Complete instructions and quiz can be found in Appendix B.4. The dataset includes observations from 6 sessions, with between 12 and 16 subjects in each. In total, 80 subjects are represented in the dataset.  4.5  Experimental results  Figure 4.12 gives the proportion of subjects who satisfy each order of rationality. 94 percent of subjects are rational. That is they play a dominant strategy whenever the game has one. Thus, 6 percent of the population is irrational. The 1st-order rational types are those subjects who are rational but not 2nd-order rational. They account for 22 percent of the subject pool. The 2nd-order rational types satisfy rationality and 1st-order belief in rationality but not 2nd-order belief in rationality. 27 percent of subjects are 2nd-order rational. 25 percent of subjects are 3rd-order rational: satisfying rationality and 2nd-order belief in rationality but not 3rd-order belief in rationality. And, lastly, 20 percent of the subject pool are at least 4th-order rationality: satisfying rationality and at least 3rd-order belief rationality. The estimated levels of rationality suggest that approximately 28 percent of subjects fail to play Nash equilibrium because of a failure of bounded rationality. This includes subjects 18 4  additional, related games were played by each subject. The data from those games is not analyzed in this paper  74  )445( 2+3,%"-."( /#$%&#'(  :45(  1"-,%"-."( /#$%&#'(  845( 745( 645(  0&-,%"-."( /#$%&#'(  245( 145(  )*+,%"-."( /#$%&#'(  !"#$#"%#&'#(')*+,-./0'  945(  045( )45(  !""#$%&#'( 1"2-"0'#('34%#&456/7'  45(  Figure 4.12: Subjects classified by order of rationality who are irrational and those who are rational but fail to account for the rationality of others. The sufficient epistemic conditions for Nash equilibrium (in 2-player games) require a subject to be rational and to have at least 1st-order beliefs that others are rational. 72 percent of the population satisfies the sufficient rationality conditions for Nash equilibrium. Next, we estimate whether each subject satisfies consistent beliefs based on their choices in game G3. Subjects are categorized into 3 epistemic types: (1) boundedly rational types: types that irrational or are rational but fail to account for the rationality of others to the 1st-order, (2) sufficiently rational/inconsistent types: types that satisfy the sufficient rationality conditions of Nash equilibrium but do not satisfy consistent beliefs, and (3) sufficiently rational/consistent types: types that satisfy all three sufficient conditions for Nash equilibrium. Not a single subject in our sample satisfies all 3 sufficient conditions for Nash equilibrium. It is also important to notice that while failure of Nash equilibrium is due in part to bounded rationality, failure to play Nash equilibrium rarely occurs because subjects are not rational. In addition, the failure to play Nash equilibrium is largely due to a failure of having consistent beliefs. 28 percent of the subject pool is boundedly rational and 72 percent of the subject pool is sufficiently rational with inconsistent beliefs. The proportion of subjects who played each of the three actions {a, b, c} for each of the  10 games are given in Figure 4.14. The bold numbers highlight the Nash action in each game. While a small proportion of subjects did play the Nash equilibrium strategy a as player 1 in game G3, only 1 subject played a as player 2. None of the subjects played a as both player 1 and player 3 in game G3.19 19 The  Nash equilibrium action is played rarely in G3. Only 1 player played the Nash equilibrium action  75  R4  81%  a  b  c  G1P1  .825  .025  .150  G1P2  .075  .900  .025  G1P3  .912  .088  0  .950 !"#$#"%#&'#(')*+,-./0'  G1P4  G2P1  .562  G2P2 G2P3  +!"#  .037  .013  *!"# )!"# (!"#  .025  .413  .475  &!"#.475  .050  .163  %!"#.787  .050  G2P4  '!"#  $!"#  0  .013  .987  !"#  ,-./01023#456-/52#  G3P1  .300  .150  .550  G3P2  .012  .338  .650  7.89:1/;23#456-/52#<# 7.89:1/;23#456-/52#<# :/9-/=:=;1/;#>12:1?=# 9-/=:=;1/;#>12:1?=# 1$20/-32.'45$-0'  Figure 4.13: Subjects classified by epistemic type  G1  G2  G3  P1  P2  P3  P4  P1  P2  P3  P4  P1  P2  a  .825  .075  .912  .950  .562  .475  .163  0  .300  .012  b  .025  .900  .088  .037  .025  .475  .787  .013  .150  .338  c  .150  .025  0  .013  .413  .050  .050  .987  .550  .650  Figure 4.14: Proportion of subjects playing each action by game  4.5.1  Secondary results  During the experiment, each subject answered a short 5-question quiz after the instructions were read aloud. The quiz was designed to test the subject’s understanding of the game structure and to make sure they understood the instructions. The quiz can be found in Appendix B.4. 16 subjects failed the quiz. Table 4.3 breaks down the number of subjects who failed the quiz by order of rationality. 60 percent of the irrational subjects (R0) failed the quiz. 35 percent of the 1st-order rational (R1) subjects failed the quiz. Approximately, 16 percent of both 2nd-order (R2) and 3rd-order rational (R3) types failed. None of the 4th-order rational subjects (R4) failed. This suggests an explanation for the boundedly as player 2. This of course does not mean that no one will play actions consistent with a Nash equilibrium in other games. We can easily come up with games where we would expect over 90 percent of subjects to play the Nash equilibrium (i.e. games that are dominance solvable in 1 round). We however likely do not want to think of players satisfying consistent beliefs in some games and not others. That is because, our estimation of consistent beliefs based on observing Nash equilibrium behavior only determines an upper bound on the proportion of players who have consistent beliefs. Thus, it really only matters what the lowest proportion of Nash equilibrium compliance is across games.  76  *G1 and G2: Only subject who pass quiz rational subjects - they did not have a clear understanding of the games they were asked to play.20  Rationality + PE + errors Rationality + PE Rationality  R0 of Order Rationality  R1  2  15  R0 R1 5 R2 R3 5 R4  R2  R3  Fail 14  18  3 6 13 4 3 25 0  9 18  R4  n  Percent 16  65  60% 35% 13 17% 16% 13 0%  16 19  56 80  Table 4.3: Subjects who failed quiz, classified by order of rationality Subjects were also givenwho the play opportunity to in revise their Players same action G1 and G2,choices by typeafter completing all of *Uniform L0 P2 P3 each of the P4 games (in total the games for the firstP1time. The subjects could review the original order) and make changes if they desired. 39 subjects review theirL4 choices. Table L20 chose toL3 n = 63 L0 1 1L1 0 1 4.4 givesL1the estimated orders of rationality based off of initial choices versus final choices. 14 choices 0 0 0 0 0 11 9 7 0 11 13 choices 2 0 0 3 5 The rows give the estimated orders of rationality based off initial choices and the columns L2 12 17 0 0 20 L3 8 0 0 0 11 12 choices 2 choices. The 0 estimated10distribution 22is give the estimated orders of rationality10based off final L4 0 014 0 0 11 11 choices 6 5 17 42 largely unaffected by the opportunity to review. The estimated order of rationality changes for only 11 subjects from initial to final choices. 7 subjects increase their order of rationality and 4 subjects decrease order of rationality. *G3/G4their Stability  Inital Choices  n=63 R0 3 of 3 actions R0 4 2 of 3 actions R1 1 5 of 6 actions R2 0 R3 R4  0 0  PlayerFinal 1 Choices Player 2 R1 R2 R3 28 21 0 0 1 61 62 16 2 0 0  0 22 1 0  0 2 16 0  R4 0 0 4 0 12  Both #Review 8 2 60 7 41 17 6 7  Assignment # of subjects assigned of action profiles Table 4.4: Order of rationality determined by initial versus# final choices Exact type match  51  40  Subjects played thematch gameswith in a 1random Type error order. But,24it could be that the order 290 systematically affected players strategic R0 (2+ errors)behavior. The most likely 5 scenario is that experience 6231 playing as player 4 in Games G1 and G2 will enable a player to sort out the dominance solvable nature of the games and lead them to have higher orders Total 80 of rationality. This does 6561not appear to be the case. Table 4.5 lists the estimated distributions of boundedly rational (R0 and R1 types) and sufficiently rational (R2, R3, and R4) types under different order assumptions. 20 Arguably, we may want to limit our analysis only to subjects who pass the quiz if we are interested in investigating the strategic reasoning of subjects who understand the game structure. This changes the results in the predictable way: 20 percent of subjects are characterized as boundedly rational and 80 percent are sufficiently rational but have inconsistent beliefs.  77  Boundedly Rational Sufficiently Rational N  games  games  games  games  games  .2750  .2444  .3333  .3438  .3529  .2857  .7250  .7556  .6667  .6562  .6471  .7143  80  45  21  32  17  28  We cannot reject the hypothesis that the sample distribution (2)-(6) are different from (1) at any standard levelsisusing the unaffected Pearson's chi-squared test statistic. The estimated ordersignificant distribution largely by the order in which subjects play Chi-squared statistic 5% significance: 3.841 the games. Therestatistic is no systematic effect6.635 of experiencing the player 4 (P4) game as the first Chi-squared 1% significance: few games (columns (2)-(4)) or as the last few games (columns (5)-(6)). Also, we cannot if wehypothesis use Yate's correction smallrestricted sample sizes --> still cannot reject(columns the null at (2)-(6)) reject Even the null that any for of the sample distributions all standard significance levels. are equivalent to the full sample distribution (column (1)) under any standard significance  levels using the Pearson chi-squared test statistic. (1)  (2) 1 P4 game in first 4 games  (3) Both P4 in first 6 games  (4) (5) (6) Both P4 in Both P4 in Both P4 in first 8 last 6 last 8 games games games  Type  All orders  Boundedly Rational  .28  .24  .33  .34  .35  .29  Sufficiently Rational  .72  .76  .67  .66  .65  .71  N  80  45  21  32  17  28  Table 4.5: Sample distributions under different game orders The estimated distribution of orders depends upon the assumption A3. If A3 holds then subjects are not misidentified. However, if A3 does not hold, a subject’s order of rationality may be mispecified. However, it is possible to bound our observed rationality distribution under different assumptions on subjects’ behavior. For example, suppose a player satisfied kth-order of rationality, did not satisfy A3, but played all actions in the kthorder rationalizable set with equal probability. Then the probability that a kth-order rational player is really a (k-1)th-order rational player is at most 1/9. Additionally, the behavior of irrational types (R0) have not been specified. If an irrational type plays randomly, then there is a 1/81 chance that an irrational type would get assigned as a R1 type, a 1/243 chance an irrational type would get assigned as an R2 type, and so on. We might expect then that 1 of our 80 subjects was misidentified as a R1 type rather than a R0 type.21  4.6  Discussion  There are a number of alternative solution concepts to Nash equilibrium. Popular alternatives are rationalizability, quantal response equilibrium (QRE), and level-k and cognitive 21 It is also possible that an irrational type does not play randomly and rather responds to incentives in some manner, but is not rational. Wright (2013) finds that defining L0 as taking into account salient features like highest payoff improves the fit of level-k models in normal-form games. For example, it is possible to imagine an irrational type as one that chooses the action that gives him the chance at the highest payoff in each game. In our design, this type of irrationality is not distinguished from rationality. At the extreme, this could shift our type estimation down by one, thus 45 percent of subjects will be identified as 2nd-order rational and higher while 55 percent identified as rational or irrational.  78  hierarchy models. Each of these solution concepts relaxes at least one of the assumptions underlying Nash equilibrium. Rationalizability relaxes the consistent belief assumption of Nash equilibrium but requires players to satisfy common knowledge of rationality: satisfy rationality and kth-order belief in rationality as k tends to infinity (Bernheim, 1984). QRE relaxes the rationality requirements of Nash equilibrium but maintains the consistent belief component. Level-k and cognitive hierarchy models maintain the assumption that subjects are rational but relax the consistent belief assumption of Nash equilibrium by allowing players to hold heterogenous beliefs about the rationality of others. In addition to imposing different assumptions on players’ rationality and belief structures, these alternative solution concepts impose additional structural assumptions. Rationalizability, in the most general terms, imposes no additional assumptions, but only makes precise predictions in rationalizable games. When testing the model it is often supposed that players play all possible rationalizable actions with equal probability. The QRE model imposes a logistic error structure to describe the probability of error in best responding. Cognitive hierarchy and level-k solution concepts impose a particular behavior for level-0 types and particular assumptions about the beliefs of each level about the levels of other players. The success of these alternative models is typically judged by the likelihood of a given model to explain a given data set evaluated at likelihood-maximizing parameter estimates. Thus, each solution concept is judged based on the complete package of assumptions about rationality, beliefs and additional structural assumptions. The standard methodology makes it difficult to separately identify the role that each of these assumptions plays in explaining a given data set. It is particularly difficult to separate between the level-k, cognitive hierarchy and QRE models because the structural assumptions these solution concepts impose tend to lead to the same behavior in many environments. For example, because subjects always make mistakes under the QRE model, players end up best responding to a mixed strategy that puts positive weight on all actions. But, in the level-k model, if level-0 types play all actions with equal probability, as is the typical specification for level-0 behavior, level-1 subjects end up best responding to a mixed strategy with positive weight on all actions. Further, both models allow for players to believe others are making mistakes (and to make mistakes themselves). This is explicitly modeled in the QRE framework as systematic mistakes when best responding. However, because level-0 types can be irrational, as long as higher levels always put some weight on level-0 types (this is true in cognitive hierarchy models), every level-k type believes his opponents make errors with some probability. And, each type in the level-k and cognitive hierarchy model is allowed to make an error when best responding when the models are fitted to the data under maximum likelihood. The approach in this paper allows us to abstract from the additional structural assumptions and directly assess the assumptions of rationality and beliefs that underlie these  79  solution concepts. Individual level rationality estimates rule out the ability of QRE and rationalizability to explain the experimental data based simply on the assumptions these solution concepts embody. We estimate that 94 percent of subjects are rational.22 This rules out the assumption that subjects make systematic mistakes. Likewise, rationalizability requires homogeneity in players’ beliefs about the rationality of others. This again is ruled out by our rationality estimates. The level-k model is the most promising as it incorporates the assumption that players are rational, relaxes the assumption of consistent beliefs, and allows for heterogeneity in subjects’ beliefs about the rationality of their opponents. We take a closer look at these solution concepts below. Definition. Consider an n-player ring game. The strategy σ ∈ �(S1 ) × · · · × �(SI ) is ∞ � ¯ k , where R ¯ k is the rationalizable if for all i ∈ I and for all a ∈ supp{σi }, then a ∈ R k=0  kth-order rationalizable set defined earlier.  i  i  QRE is typically modeled by assuming players make mistakes according to a logistic error function. Definition. Consider an n-player ring game. The strategy σ ∈ �(S1 ) × · · · × �(SI ) is a  quantal response equilibrium (QRE) of Γ if for all i ∈ I and for all a ∈ supp{σi }, then exp(λ · ui (a, σp(i) )) σi (a) = � exp(λ · ui (si , σp(i) )) si ∈SI  The level-k model is defined below. This paper does not differentiate between level-k and cognitive hierarchy models (hence we consider only the former for simplicity). Definition. Consider an n-player ring game Γ =< I = {1, . . . , I}; S1 , . . . , SI ; π1 , . . . , πi ; p >. The strategy σ = σ1 × · · · × σI , where σi : {0, . . . , k} → �(Si ) is a level-k equilibrium in Γ if for all i ∈ I, for all n ∈ {1, 2, . . . , k} and for all a ∈ supp{σi (l)}  ui (a, σp(i) (n − 1)) ≥ ui (a� , σp(i) (n − 1)) ∀a� ∈ Si The majority of our subject pool satisfies rationality, which does not fit QRE assumptions unless the precision parameter, λ, is high (people do not make mistakes in best response often). But, even if a high λ could explain the high proportion of rational subjects, QRE is not able to predict the pattern of behavior across the 8 games described by G1 and G2 for any value of the precision parameter. Figure 4.15 gives the proportion of subjects who play the rationalizable actions in each of the player positions in games G1 and G2. The empirical frequency of playing the rationalizable strategy increases across the player 22 In addition, 51 subjects are assigned to R1-R4 types because they match a predicted type profile exactly (no mistakes in best-responding in 8 games). The remaining matched subjects make 1 error, though this error is predominately (21 of 24 subjects) in not matching the A3 assumption (thus not necessarily a mistake in best-responding).  80  !"#$#"%#&'#(')*+,-./0'!1234&5' 62%#&21472+1-')/"2/-53'  !++,"  !"#$%&'""  ("  )"  1+,"  !"#$%&'" *"  ("  0+," /+," .+,"  )"  -+,"  *"  *+," )+," (+," !+," +," 2!"  2(" 829-'  Figure 4.15: Proportion of subjects who play rationalizable action in G1 and G2 positions (position 1 to 4) in both games G1 and G2. The QRE predictions are not consistent with aggregate data for any λ. For any value of λ, the QRE equilibrium does not predict that the rationalizable action will be played most often by player 4, then player 3, player 2 and player 1 in descending order. The QRE predictions are given in Figure 4.16.23 The  +$  !"#$%$&'&()*#+*!'%)&,-*.%/#,%'&0%$'1* 2("%(1-)*  !"#$%$&'&()*#+*!'%)&,-*.%/#,%'&0%$'1* 2("%(1-)*  behavior is clearly inconsistent with the rationalizable solution concept as well.  !"*$ !")$ !"($  -./012$+$  !"'$  -./012$,$  !"&$  -./012$#$ -./012$%$  !"%$  +$ !"*$ !")$ !"($  ,-./01$+$  !"'$  ,-./01$2$  !"&$  ,-./01$#$ ,-./01$%$  !"%$ !"#$  !"#$ !$ !"+&$!"#$!"%&$!"'$!"(&$!"*$+"!&$+",$+"#&$+"&$  !$  3%4$5%*  !"+&$  !"#$  !"%&$  !"'$  !"(&$  3%4$5%*  Figure 4.16: QRE predictions for games G1 and G2 Level-k models predict the aggregate behavior in Figure 4.15. Consider any distribution over levels L1, L2, L3 and L4. L4 types play the rationalizable strategy as player 1. L3, and L4 types play the rationalizable strategy as player 2. L2, L3 and L4 types play the rationalizable strategy as player 3. L1, L2, L3 and L4 types play the rationalizable strategy as player 4. No matter what the distribution over levels, or the specification for L0, the proportion of subjects playing the rationalizable strategy will weakly increase over the player positions 1-4. Further, depending on the specification of L0, the level-k model can capture the differences in rationalizable play across games G1 and G2. If we specify L0 behavior 23 In  addition, a high precision would not account for the failure of consistent beliefs.  81  as uniformly random, all levels will play the rationalizable action in G1. However, in G2 only L4 subjects play the rationalizable action as player 1, only L4 and L3 subjects play the rationalizable action as player 2, only L2, L3, and L4 subjects play the rationalizable action as player 3, and all levels play the rationalizable action as player 4. In addition, kth-order rationality and assumption A3 captures the main assumptions defining the behavior of an Lk type. L1 types are rational and do not respond to changes in 1st-order beliefs as they best respond to a fixed level-0 behavior. L2 types are 2nd-order rational and and do not respond to changes in 2nd-order beliefs. L3 types are 3rd-order rational and do not respond to changes in 3rd-order beliefs, and so on. Thus, the distribution of orders of rationality estimated in Figure 4.12 gives us a ‘L0’ independent estimate of the level-k distribution. The distribution of levels in Figure 4.12 puts more weight on higher levels than is generally estimated in level-k experiments. This could suggest that the usual assumption of uniform L0 biases the estimation of the level distribution. The experimental design in this paper also allows for a direct test of the limited depth of reasoning assumption that underlies the level-k and cognitive hierarchy models. These models assume that players do not base optimal behavior on higher-order beliefs. This assumption has not previously been tested directly.24 Our unique experimental design allows us to asses this assumption because it isolates changes in higher-order beliefs while keeping lower-order beliefs constant. Approximately 80 percent of our subject pool does not respond to changes in 3rd-order beliefs.  4.6.1  Beliefs about payoffs  Throughout this paper, we have restricted our attention to complete information games which implicitly assume that common knowledge of payoffs holds (assumption A1). In this section, I relax assumption A1 and discuss in which ways the results of the previous sections are affected by relaxing this assumption. As discussed earlier, each complete information game can be represented by a payoff hierarchy defined by ¯ i (Γ) = {θpk (i) }∞ . h k=0 ¯ i represents player i’s hierarchy of beliefs about payoffs for the specified The sequence h 24 Analysis of subjects’ search patterns (over payoffs) suggests that reasoning conforms to differential and limited depths of reasonings (Costa-Gomes et al. (2001); Costa-Gomes and Crawford 2006; Brocas et al. 2009). But in standard 2-player games, a subject with any depth of reasoning greater than two must look up both her own payoffs and her partners payoffs (tight link between higher- and lower-order beliefs about payoffs in standard games). Thus, structural assumptions over the order in which players search the same payoff information are necessary to differentiate between different depths of reasoning. The ring game simplifies this inquiry because a player’s lower- and higher-order beliefs are no longer linked and players with different depths of reasoning base behavior on different payoffs (assumptions about the order of search is no required) and we get insight into reasoning from choice data alone.  82  game payoffs. However, under relaxations of A1 this belief hierarchy may not be consistent with a type ti ’s beliefs that are defined from a given type space. We can formally define each types beliefs about payoff types. For any event F ⊂ Θpm (i) ,  m ≥ 1, we can define player i’s mth-order beliefs about the payoff types of others from the ˆ Define hm : Ti → �(Θpm (i) ) higher-order belief function g m and the payoff function θ. i  hm i (ti )(E) =  i  �  gim (ti )(tpm (i) )  tpm (i) ∈Tpm (i) :θˆpm (i) (tpm (i) )∈E  Thus hi (ti ) = {hki }∞ k=0 represents type ti ’s hierarchy of beliefs about the payoff types  of others. Type ti ’s payoff type is h0i (ti ), she believes that her opponent p(i) has payoffs  h1i (ti ) ∈ �(Θp(i) ), she believes that her opponent p(i) believes that her opponent p(p(i)) has payoffs determined by h2i (ti ) ∈ �(Θp2 (i) ), and so on.  In theory, a type could hold any hierarchy of beliefs about payoff types Θ as specified  by the type space. However, in this paper an important restriction is to types whose beliefs ¯ i. coincide with the actual payoff hierarchy h Definition. A type ti satisfies correct beliefs about payoffs to the mth-order if ¯ k }m−1 . {hk }m−1 = {h i  k=0  i  k=0  It is possible to show that for any n-player ring game, we can relax the assumption about common knowledge of payoffs to correct beliefs about payoffs to the nth-order. This result is given in Proposition 4.2.25 Proposition 4.2. Consider an n-player ring game Γ =< I = {1, . . . , n}; S1 , . . . , Sn ; π1 , . . . , πn ; θ1 , . . . , θn ; p > and a Γ-based epistemic type space < T1 , . . . , Tn ; Θ1 , . . . , Θn ; b1 , . . . , bn ; θˆ1 , . . . , θˆn ; sˆ1 , . . . , sˆn >. Suppose type ti satisfies the following conditions: (i) rationality, (ii) (n − 1)th-order belief in rationality, (iii) consistent beliefs, and (iv) correct beliefs about  payoffs to the nth-order. Then it must be that σ defined by σi = sˆi (ti ) and σpk (i) (s) = � sˆpm (i) (tpm (i) ) · gim (ti )(tpm (i) ) for all s ∈ Tpm (i) , k ∈ {1, . . . , I − 1} constitutes a tpm (i) ∈Tpm (i)  Nash equilibrium.  Thus, if the assumption of common knowledge of payoffs is relaxed to correct beliefs about payoffs to the 4th-order, none of the results in this paper will change. This means that we can allow for any higher-order uncertainty about payoffs and this will not affect the interpretation of the results. As Yildiz and Weinstein (2007) show, allowing for higher-order uncertainty while maintaining correct beliefs about payoffs to some finite-order ensures that the set of Bayesian Nash equilibria is equivalent to the rationalizable set. Thus, higher-order uncertainty (plus 4th-order rationality) will require all subjects to play the rationalizable action in G1 and G2. This means, higher-order uncertainty about payoffs alone cannot 25 The  proof is contained in the proof of Proposition 4.1.  83  rationalize the pattern of data we observe in Figure 4.15. We would still need to relax rationality assumptions on top of allowing for higher-order uncertainty in payoffs in order to explain the experimental data in games G1 and G2. Alternatively, instead of allowing only for higher-order uncertainty we could allow for payoff uncertainty at lower-orders. However, the set of payoff types would have to be nonstandard in order to rationalize the data in this experiment. To see this, suppose the set of payoff types Θi = Θ for each player includes all payoff types that maintains that preferences over outcomes are monotone. Under this restriction, Θ could include all expected utility types with different risk attitudes. What this may not include is types with other-regarding preferences. Still, allowing for payoff heterogeneity as specified in this form is not capable of explaining this experimental data unless we additionally relax rationality assumptions as well. To see this, consider the rationalizable sets for each payoff type defined as follows. ¯ i1 (θi ) = R  �  si ∈ Si |∃µ ∈ �(Sp(i) ) such that si ∈ argmax{ui (s, µ, θi )} s∈Si  �  ¯ 1 (θi ) is the 1st-order rationalizable set for payoff types θi and is set of all actions The set R i that are rational for player i with payoff type θi . Define the the kth-order rationalizable set as follows:       � ¯ ik (θi ) = si ∈ Si |∃µ ∈ �  ¯ k−1 (θ) such that si ∈ argmax{ui (s, µ, θi )} R R p(i)   s∈Si θ∈Θp(i)  Therefore the set  ∞ � ¯ k (θi ) represents all rationalizable actions for payoff type θi and R i  k=1  hence represents all the possible action that could be played in some Bayesian Nash equilibrium determined by the payoff types Θ.  Since player 4 has a dominant strategy in G1 and G2, this means that the 1st-order ra¯ 1 (θ)(G1) = R ¯ 4 (θ)(G1) = {a} and R ¯ 1 (θ)(G2) = tionalizable sets for player 4 are singletons, R 4  4  ¯ 4 (θ)(G2) = {c} ∀θ ∈ Θ4 . But, then this implies that the 2nd-order rationalizable sets for R ¯ 2 (θ)(G1) = {a} and R ¯ 2 (θ)(G2) = {b} ∀θ ∈ Θ3 , and so on with player 3 are singletons, R 3  3  ¯ 3 (θ)(G1) = {b} and R ¯ 3 (θ)(G2) = {a} ∀θ ∈ Θ3 , and R ¯ 4 (θ)(G1) = {a} and R ¯ 4 (θ)(G2) = {c} R 2 2 1 1 ∀θ ∈ Θ1 . This means that the set of rationalizable outcomes when we allow richness in the payoff types is still the unique set of rationalizable outcomes generated under the assumption  of common knowledge of payoffs. Rationality assumptions still need to be relaxed on top of allowing for richness in payoff types in order to explain the experimental data in games G1 and G2. Allowing for payoff heterogeneity, however, may change our interpretation of the data  84  from G3. If subjects do not play the Nash equilibrium in G3, then we assumed that they had inconsistent beliefs. This, however, is based on the assumption about correct beliefs about payoffs to the 2nd-order. If players have different payoff types than it might be that they have consistent beliefs but different beliefs about payoffs. For example, if player 1 has the payoff type determined by the monetary payoffs in the experiment but believes that player 2 has some other payoff type. Then, if player 1 plays c, believes player 2 is playing c and believes player 2 believes she is playing c, it is possible for player 1 to be rational, have consistent beliefs and believe player 2 is rational as long as player 2’s payoff type is such that c is a best response to c for player 2.  85  Chapter 5  Conclusion I have investigated implications of limited depths of reasoning in games throughout this dissertation. I have investigated theoretical implications of imposing limited depths of reasoning via two applications: the role of information in coordination games and optimal mechanism design. And, I have empirically investigated features of strategic reasoning in the lab and found that behavior can best be described by finite orders of rationality and inconsistent beliefs; features consistent with limited depth of reasoning models. Limited depth of reasoning models have two mechanisms which affect behavior compared to standard models like Nash and Bayesian equilibrium. First, these models impose a nonresponsiveness to higher-order beliefs and second, they allow for inconsistent beliefs. Both of these mechanisms have implications for behavior. In Chapter 2, I show that the effects of public and private information may be significantly different in models where players have bounded depths of reasoning because of the non-responsiveness to higher-order beliefs. Because, of this private information may decrease coordination under limited depths of reasoning compared to an equilibrium analysis. This is because, under equilibrium, the coordinating role of private information only operates through the effects of higher-order uncertainty. In Chapter 3, I show that studying optimal mechanism design under limited depths of reasoning significantly alters the set of social choice correspondences that can be implemented because of the relaxation of consistent beliefs. Level-k implementation weakens the implementation requirements relative to Bayesian implementation because it relaxes the cross-player consistency requirement imposed by Bayesian implementation. And, in Chapter 4, I show that limited depths of reasoning and inconsistent beliefs are both empirical features of strategic reasoning. Importantly, even though the precise predictions of standard solution concepts generally stem from the assumptions of infinite depths of reasoning and consistent beliefs, relaxing these assumptions does not mean any behavior can occur. The main element pinning down  86  behavior in limited depth of reasoning models is the specification of L0 behavior. The freedom to specify L0 behavior outside of the model is one of the main criticisms and drawbacks of the limited depth of reasoning literature. 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Econometrica, 75:365–400.  92  Appendix A  Appendix to Chapter 2 A.1  Omitted proofs  Proof of Proposition 2.1 The expected payoff gain an L1 type who receives signal x can be written as ˆ ∞for � � � L1 π (x) = D φ α + β(θ − µx − (1 − µ)y) [1 − Q(θ|x, y)] dθ − t −∞  This expression is continuous and strictly decreasing in x by A1 and A2. To see this notice that there are two effects to consider. First, 1−Q(θ|x, y) is weakly decreasing in x for a given θ. Second, the mean of the distribution of θ shifts as x increases which puts more weight on higher values of θ, which means more weight is put on smaller values of 1 − Q(θ|x, y). The two effects work in the same direction. Also, π L1 (−∞) > 0 and π L1 (∞) < 0 by properties of the normal distribution. As a result, L1 types play a threshold strategy with cutoff κ1 determined implicitly by equation (2.2). Claim: Lk types play threshold strategies with cutoff κk ∀ k ≥ 1. This claim is proved by induction on k.  It is true for L1 types by the above argument. Suppose it is true for Lk-1 types. √ Lk types expect a proportion Φ( β(κk−1 − θ)) of players to attack. Since Φ is continuous and strictly decreasing in θ, we can find a unique θ¯k such that an Lk type thinks there is a successful attack if and only if θ ≤ θ¯k where θ¯k is determined implicitly by equation (2.4). The expected payoff gain for an�Lk type who receives signal x is then�given by � π Lk (x, θ¯k ) = D 1 − Φ( α + β(µx + (1 − µ)y − θ¯k )) − t.  π Lk is continuous, strictly decreasing in x, π Lk (−∞) > 0 and π Lk (∞) < 0. As a result, Lk types play according to a threshold strategy with cutoff κk determined by implicitly by 93  equation (2.3). The result follows by induction. Proof of Proposition 2.2 Consider the case for level-k types. Let the distribution of types be specified by λ1 , λ2 , . . . where λk is the proportion of Lk types. Let LK(θ) be the proportion of all Lk types that attack, LK(θ) =  ∞ �  λk Φ  k=1  �� � β(κk − θ)  LK(θ) is continuous and strictly decreasing in θ. Thus, there exists a unique θ¯ such that ¯ = θ¯ and there is a currency crisis iff θ ≤ θ. ¯ LK(θ) We want to know how θ¯ changes when some parameter z varies. If ∂ θ¯ ∂z  LK(θ) is shifted to the right (left) and hence Claim:  ∂κk ∂D  > 0,  < 0 and  ∂κk ∂t  ∂κk ∂y  Show by induction on k.  ∂κk ∂z  > 0(< 0).  > 0(< 0) ∀k, then  < 0 ∀k ≥ 1  Consider k = 1. Let F =D  ˆ  0  1  �  1−Φ  �� �� α + β(µκ1 + (1 − µ)y − l) q(l|κ1 , y)dl − t  The cutoff is implicitly determined by F = 0.  ∂F By the proof of Proposition 2.1 ∂κ < 0. 1 � � �� ´ √ 1 ∂F α + β(µκ1 + (1 − µ)y − l) q(l|κ1 , y)dl > 0 , ∂D = 0 1 − Φ ∂F ∂t ∂F ∂y  = −1 < 0 ,  < 0. This is analogous to  ∂F ∂x .  Therefore, by the implicit function theorem,  ∂κ1 ∂D  > 0,  ∂κ1 ∂t  < 0 and  ∂κ1 ∂y  < 0.  Now suppose the claim is true for k − 1.  Equations (2.3) and (2.4) jointly determine κk . From the equation (2.4) we have: √ dθ¯Lk βφ(·) dκk−1 √ = dz 1 + βφ(·) dz Totally differentiate equation (2.3)� taking into account the�effect on the cutoff θ¯Lk : �� � � t dθ¯Lk ∂κk α + βµφ(·) dκk − + α + βφ(·) dD = 0 ⇒ >0 2 D dD ∂D � � �� � � 1 dθ¯Lk ∂κk dt = 0 ⇒ α + βµφ(·) dκk + − α + βφ(·) <0 D dt ∂t 94  � � �� � � dθ¯Lk ∂κk α + βµφ(·) dκk + α + βφ(·) (1 − µ) − dy = 0 ⇒ <0 dy ∂y  The claim follows by induction. Thus, by the argument above:  ∂ θ¯ ∂D  > 0,  ∂ θ¯ ∂t  < 0 and  ∂ θ¯ ∂y  < 0.  Proof of Proposition 2.3 Consider the optimal strategy for a sophisticated type who believes all other sophisticated types are playing according to a threshold strategy with cutoff κs . Let the distribution of types be specified by λ1 , λ2 , . . . where λk is the proportion of Lk types. A type Lk plays according to a threshold strategy with cutoff κk . The proportion of players attacking is given by L(θ) = (1 − γ)  ∞ �  λk Φ  k=1  ��  � � β(κk − θ) + γΦ( β(κs − θ))  There is a unique value θ¯s such that L(θ¯s ) = θ¯s . There is a currency crisis if and only if ∞ �√ � � θ ≤ θ¯s . Rearranging the expression, L(θ¯s ) = θ¯s , and letting S(θ) = λk Φ β(κk − θ) , we can get the following relationship between κs and θ¯s , 1 κs = θ¯s + √ Φ−1 β  �  k=1  � 1 ¯s 1 − γ ¯s θ − S(θ ) . γ γ  An equilibrium cutoff κs is the solution to the following equation � � � D 1 − Φ( α + β(µκs + (1 − µ)y − θ¯s ) = t.  The aggregate size of attack θ¯s is then determined implicitly by equation α √ (y − θ¯s ) + Φ−1 β  �  � √ � � 1 ¯s 1 − γ ¯s α + β −1 D − t √ θ − S(θ ) = Φ . γ γ D β  The left hand side of this expression is continuous in θ¯s , positive for low θ¯s and negative for high θ¯s . As a result, there always exists a solution. In other words, there exists at least one threshold equilibrium. There will be a unique symmetric threshold equilibrium whenever there is a unique solution θ¯s to this � equation. � 1−γ α 1 −1 Let G(θ) = √β (y − θ) + Φ γ θ − γ S(θ) . Then,  dG dθ  =  1 γ  � + 1−γ γ (−S (θ))  φ(Φ−1 ( γ1 θ−( 1−γ γ )L(θ))) √ �  −  √α . β  Since φ ≤ 1/ 2π and −S (θ) > 0 hold, G is strictly increasing whenever  √  2π ≥  √α γ β  .  Uniqueness follows from an iterated deletion of strictly dominated strategies argument. Let  95  π s (x, κ) be the expected gain for a sophisticated player attacking, given that she observes signal x and all other sophisticated players are playing threshold strategies with a cutoff κ. � � � π s (x, κ) = D 1 − Φ( α + β(µx + (1 − µ)y − θ¯s (κ) − t Define the function b : R → R to be the value b(κ) that solves the equation π s (b(κ), κ) = 0. b is well-defined, continuous and strictly increasing by arguments above. Therefore, b(k) has a unique fixed point, κs . Define the function bk : R → R recursively by bk (κ) = b(bk−1 (κ)) for k ≥ 2.  Consider strategies that survive 1 round of deletion of strictly dominated strategies. First, consider strategies that are dominated by playing attack at some signals. The best payoff that a player could achieve if not attacking at those signals is if all sophisticated types played not-attack for all signals. That is, as if all sophisticated types played the threshold strategy with cutoff −∞. The best response to sophisticated types playing with a cutoff −∞ is to play a threshold strategy with a cutoff b(−∞). This means that any strategy for which  not-attack is played for any x < b(−∞) is a dominated strategy. Following an analogous argument we can show that any strategy for which attack is played for any x > b(∞) is a dominated strategy. Thus, the set of strategies that survive 1 round of iterated deletion of strictly dominated strategies look like: � attack play not−attack  if x<b(−∞) if x>b(∞)  We can now repeat the argument on this smaller set of strategies and show that the set of strategies that survive k-rounds of iterated deletion of strictly dominated strategies look like: play  �  attack not−attack  if x<bk (−∞) if x>bk (∞)  By the argument in the proof of Proposition 2.4 for a function b that is strictly increasing, continuous, bounded and has one fixed point: bn (x0 ) → κs as n → ∞ for any x0 ∈ R. This  means that there is a unique strategy that survives iterated deletion of strictly dominated strategies - the threshold strategy with cutoff κs . Proof of Corollary 2.1 Let z be the signal received (i.e.. either z = x or z = y). The expected payoff gain of an L1 is given by ˆ type ∞ � L1 πz (z) = D φ( β(θ − z))[1 − Q(θ|z)]dθ − t. −∞  By similar arguments as in Proposition 2.1, π L1 is strictly decreasing and continuous in z. Therefore, L1 types play threshold strategies with cutoff κz1 determined by πzL1 (κz1 ) = 0. Players attack if signal z ≤ κz1 and do not attack otherwise.  96  Claim1: Lk types play threshold strategies with cutoff κX k ∀ k ≥ 1 in the private information game  This claim is proved by induction on k. It is true for L1 types by the above argument. Suppose it is true for Lk-1 types. √ Lk types expect a proportion Φ( β(κX k−1 − θ)) of players to attack. Since Φ is continuous and strictly decreasing in θ, we can find a unique θ¯k such that an Lk type thinks there is a √ successful attack if and only if θ ≤ θ¯k where θ¯k is determined implicitly by Φ( β(κX k−1 − k k ¯ ¯ θ )) = θ . The expected payoff gain for an Lk type�who receives signal �x is then given by � π Lk (x, θ¯k ) = D 1 − Φ( β(z − θ¯k )) − t.  π Lk is continuous, strictly decreasing in x, π Lk (−∞) > 0 and π Lk (∞) < 0. As a result, Lk types play according to a threshold strategy with cutoff κX k uniquely determined by π Lk (κX , θ¯k ) = 0 k  The result follows by induction. Claim2: Lk types play threshold strategies with cutoff κY ∀ k ≥ 1 in the public information game  This claim is proved by induction on k. It is true for L1 types by the above argument. Suppose it is true for Lk-1 types. Lk types expect everyone to attack if y ≤ κY .  If y > κY , payoff to attacking is D · P r(θ ≤ 0) − t (< 0 since L1 payoff is negative)  If y ≤ κY then payoff to attacking is D · P r(θ ≤ 1) − t (> 0 since L1 payoff is positive)  The result follows by induction. Proof of Proposition 2.4  Define the function f : RxR →� R by  � � ¯ f (x, κ) = D 1 − Φ( α + β(µx + (1 − µ)y − θ(κ))) −t √ ¯ ¯ ¯ where θ(κ) is implicitly determined by θ(κ) = Φ( β(κ − θ(κ))).  Define the function b : R → R by letting b(κ) be the value that solves f (b(κ), κ) = 0. By  argument in proof of Proposition 2.1, b is a well-defined function. By the implicit function ∂ θ¯ ∂b theorem ∂κ > 0 and ∂κ > 0. So, b is a strictly increasing, continuous function. Notice that f (¯ x, x ¯) = 0 defines the equilibrium cutoff x ¯. This means that b has a unique fixed point. Define the function bk : R → R recursively by bk (κ) = b(bk−1 (κ)) for k ≥ 2. Now let κ1 be such that κ1 < b(κ1 ).  97  Consider the following real sequence {bn (κ1 )}∞ n=1 . This sequence is increasing, since b is  an increasing function and κ1 < b(κ1 ). The sequence is also bounded since b is a bounded function (bounded above by x ¯) Since {bn (κ1 )} is bounded, continuous, and increasing,�it has some �limit, call it c. c = lim bn+1 (κ1 ) = lim b (bn (κ1 )) = b lim bn (κ1 ) = b(c) n→∞  n→∞  n→∞  where the third equality follows from continuity of b. Since, b has a unique fixed point c = x ¯. We can find the analogous result for any κ1 such that b(κ1 ) < κ1 . The proof follows by noticing that κk = bk−1 (κ1 ). Proof of Proposition 2.5 L1’s expected payoff is determined by D  ˆ �  1−Φ  ��  �� β(x − l) dQ − t  The expected payoff of the equilibrium threshold player can be represented by D  ˆ �  1−Φ  ��  �� β(x − l) dP − t  Expected payoffs are an increasing function of l and a decreasing function of x. Therefore, κ1 (Q) > κ1 (P ) since Q F.S.D P . We can show that  δκ δα  √ > 0 whenever φ ( α(κ − ρ)) (κ − ρ) < 0. This is true whenever  ρ > κ. Therefore, the result holds whenever κ < 0.5.  The threshold for a level-1 player is determined by Φ  ��  � D−t β(κ − ρ) = D  The L.H.S is an increasing function of κ and is equal to 0.5 when κ = ρ. As long as D−t t  < .5 ⇒ κ < .5  Proof of Proposition 2.6 See Bannier (2002) for proof of the equilibrium result. From arguments in proof of Proposition 2.2, if  ∂κk ∂α  < 0 ∀k, then  ∂ θ¯ ∂α  < 0. So we only really  have to consider the effect of changing α on the cutoff points of each type. Claim: ∃ y¯k s.t  ∂κk ∂α  < 0 if y > y¯k ∀k ≥ 1  Show by induction on k. Start with k=1:  The cutoff is implicitly determined by F = 0.  98  F =D  ˆ  1  0  So  � ´ 1 �√ φ α + β(µκ1 + (1 − µ)y − l) (−µκ1 + (1 + µ)y − l) q(l)dl 0 = − � ´ 1 �√ 2β 0 φ α + β(µκ1 + (1 − µ)y − l) q(l)dl  ∂κ1 ∂α ∴  δκ1 δα  “ “p ”” 1−Φ α + β(µκ1 + (1 − µ)y − l) q(l|x, y)dl − t  µ 1+µ κ1  < 0 if y >  +  1 1+µ  = y¯1  Now suppose the claim is true for k − 1.  Equations (2.3) and (2.4) jointly determine κk . From (2.3) we have θ¯k  =  µκk + (1 − µ)y − √  1 Φ−1 α+β  „  D−t D  «  And, from (2.4) we have: Φ  „p „ „ ««« 1 D−t −1 β κk−1 − µκk − (1 − µ)y + √ Φ D α+β  Totally differentiate this expression in order to find ∂κk ∂α  =  √ α+β βφ(·) ∂κk−1 ´· ·` + √ β ∂α 1 + βφ(·)  We know that  ∂κk−1 ∂α  µκk + (1 − µ)y − √  1 α+β  Φ  −1  „  D−t D  «  ∂κk ∂α  1 1 1 −1 κk − y− 1 Φ (α + β) (α + β 2β(α + β) 2  < 0 if y > y¯k−1 . √ � � α+β −1 D−t . Since, 2β Φ D  k And, ∂κ < 0 if y > κk − ∂α √ � � −1 D−t y > κk − α+β 2β Φ D Let y¯k = max{¯ yk−1 , y¯k� }. ∂κk Then ∂α < 0 if y > y¯.  =  ∂κk ∂y  „  D−t D  «!  < 0 there is a unique y¯k� such that  if y > y¯k� .  Therefore, the claim follows by induction. Now, let y¯ = max{¯ y1 , y¯2 , . . . , y¯K }, where K is the largest level-k type in the model. Thus, by the argument above: The case for  A.2  ∂ θ¯ ∂α  ∂ θ¯ ∂α  < 0, if y > y¯.  > 0 follows analogously.  Limited depth of reasoning model  The extension from level-k thinking to a general limited depth of reasoning model follows almost immediately. L1 types best respond only to L0. As a result, the behavior of an L1 type is unchanged under any limited depth of reasoning model. Suppose L1 types play a threshold strategy with cutoff κ1 . Let λki be the proportion of Li types that an Lk type believes she is playing against. Claim: Lk types play threshold strategies with cutoff κk ∀ k ≥ 1. This claim is proved by induction on k.  99  It is true for L1 types trivially, analysis doesn’t change. Suppose it is true for Lk-1 types. First consider the case where λk0 �= 0. Lk types believe the proportion of players attacking √ √ is given by λk0 l0 + λk1 Φ( β(κ1 − θ)) + . . . + λkk−1 Φ( β(κk−1 − θ)). Thus, Lk types believe √ √ they get a payoff D if λk0 l0 + λk1 Φ( β(κ1 − θ)) + . . . + λkk−1 Φ( β(κk−1 − θ)) ≥ θ. Using the distribution Q, we can then write the expected payoff of an Lk type as π  Lk  (x, κ1 , . . . , κk−1 ) =  ˆ  ∞  φ(  −∞  p  0  2  α + β(µx + (1 − µ)y − θ)) @1 − Q 4  0 1 31 k−1 X k p 1 @θ − A 5A dθ λ Φ( β(κ − θ)) |x, y i i λk 0 i=1  π Lk is continuous, strictly decreasing in x by A1 and A2 and π Lk (−∞) > 0 and π Lk (∞) < 0. To see that π Lk is strictly decreasing notice that 1 − Q is weakly decreasing in x for a given  θ. As x increases, the mean of the normal distribution shifts, placing more weight on higher values of θ. So more weight is placed on lower values of 1 − Q. The two effects work in the same directions. As a result, Lk types play a threshold strategy with a cutoff κk where κk is uniquely determined by πσLk (κk , κ1 , . . . , κk−1 ) = 0. Now consider the other case where λk0 = 0. Lk types believe the proportion of players √ √ attacking is given by λk1 Φ( β(κ1 − θ)) + . . . + λkk−1 Φ( β(κk−1 − θ)). Thus, Lk types believe √ √ an attack is successful if λk1 Φ( β(κ1 − θ)) + . . . + λkk−1 Φ( β(κk−1 − θ)) ≥ θ. The LHS is continuous and strictly decreasing in θ. Therefore ∃ a unique θ¯Lk such that there is a successful attack iff θ ≤ θ¯Lk . The payoff for an Lk type is then � expected � given by � Lk Lk Lk ¯ ¯ π (x, θ ) = D 1 − Φ( α + β(µx + (1 − µ)y − θ )) − t.  π Lk is continuous, strictly decreasing in x, π Lk (−∞) > 0 and π Lk (∞) < 0. As a result, Lk types play according to a threshold strategy with cutoff κk uniquely determined by π Lk (κk , θ¯Lk ) = 0. The result follows by induction.  A.3  General model  Consider a binary action game with a continuum of players. Players can either play action 1 or action 0. Payoffs depend upon the proportion of players playing action 1, denoted by l, and some parameter θ. Payoffs are given by the function u(a, l, θ), where a ∈ {0, 1}, l ∈ [0, 1] and θ ∈ R. Let the function π : [0, 1]xR → R be the payoff gain from playing action 1, defined as π(l, θ) = u(1, l, θ) − u(0, l, θ).  In the incomplete information version of this game, players do not directly observe θ,  instead each player receives a private signal xi where xi = θ + σ�i �i is distributed with cdf F (·) and density f (·)  100  θ is distributed with cdf P (·) and density p(·) F (·) and P (·) determine the conditional distribution of θ, denoted by g(θ|x). The conditional density g can be derived from F and P in the following way g(θ|x) =  p(θ)f ´∞  � x−θ �  p(θ)f −∞  σ � x−θ � σ  dθ  .  The following assumptions are imposed on the payoff function π and the information structure. Assumption A1 defines the game with common knowledge of θ as one of strategic complementarities - the incentive to choose action 1 is increasing in the proportion of other players who play action 1. A2 ensures that the incentive to choose action 1 is increasing in the state. A1: Action Monotonicity: π(l, θ) is nondecreasing in l A2: State Monotonicity: π(l, θ) is nondecreasing in θ Assumption A3 ensures that there is some range of fundamentals over which each action is a strictly dominant strategy in the incomplete information game. ´∞ A3: Expected Dominance: There exists x ∈ R and x ¯ ∈ R such that −∞ g(θ|x)π(l, θ)dθ < 0 ´∞ ∀l and x ≤ x and −∞ g(θ|x)π(l, θ)dθ > 0 ∀ l ∈ [0, 1] and x ≥ x ¯  Assumptions A4 and A5 are technical assumptions. A4 guarantees that our expected payoffs are continuous. Assumption A5 is essentially a restriction on the noise distribution F . It requires that g(θ|x) monotonically put more weight on higher values of θ as x increases. This is a restriction satisfied by most signal technologies. ´∞ A4: Continuity: F is continuous. −∞ g(θ|x)π(l, θ)dθ is continuous with respect to x, l and density g.  A5 : Strict Monotone Likelihood Ratio Property: For any x1 > x0 and θ1 > θ0 , the following holds  g(θ1 |x1 ) g(θ1 |x0 )  >  g(θ0 |x1 ) g(θ0 |x0 ) .  Proposition. Let assumptions A1,A2, A3, A4, and A5 hold. Then given some behavior for L0, say q(l) , there exists a unique cutoff, κk , for each k ≥ 1, such that the behavior of type Lk is � given by 1 if x ≥ κk play 0 o.w Proof By induction on k. Consider k=1. The expected payoff gain from playing action 1 for an L1 type who receives a private signal x, given the behavior of L0 is 101  ˆ  π L1 (x) =  ∞  −∞  ˆ  1  g(θ|x)q(l)π(l, θ)dldθ.  0  Notice that π L1 is strictly increasing and continuous in x by assumptions A2, A4, and A5. A3 ensures π L1 (−∞) < 0 and π L1 (∞) > 0. Therefore, L1 types play according to a threshold strategy with cutoff κ1 determined by π L1 (κ1 ) = 0. Assume true for k-1. Consider k. The expected payoff gain of an Lk type given that he receives signal x and that he expects all other players to play threshold strategies with the cutoff κk−1 is π(x, κk−1 ) =  ˆ  ∞  −∞  g(θ|x)π(1 − F (  kk−1 − θ ), θ)dθ. σ  The function π is strictly increasing and continuous in x, which follows from A1,A2,A4 and A5. A3 ensures π L1 (−∞) < 0 and π L1 (∞) > 0. Therefore, there is a unique cutoff κk where π(κk , κk−1 ) = 0. Best responses for Lk types are to play threshold strategies with cutoff κk . The result follows by induction.  A.4  Discrete model  Proposition. Suppose the behavior of L0 types is given by the continuous cdf Q(l) on [0, 1]. (i) Suppose we are in information treatment PI. Then there exists a unique κk , for each k ≥ 1, such � that the behavior of type Lk is given by 1 if x ≥ κk play 0 o.w (ii) Suppose we are in information treatment CI. Then there exists a unique κ such that the behavior � of type Lk, for any k ≥ 1, is given by 1 if y ≥ κ play 0 o.w Proof (i) By induction on k. Consider k=1. The expected payoff gain from playing action 1 for an L1 type who receives a private signal x, given the behavior of L0 is  102  π L1 (x) = =  1 2�  ˆ  x+10  1 2�  ˆ  x+10  x−10  x−10  ˆ  1 a(θ)−1 15  θq(l)dldθ − t  � � �� a(θ) − 1 θ 1−Q dθ − t 15  Notice that π L1 is strictly increasing and continuous in x. Also, for θ low enough 1 − Q = 0 and for θ high enough, 1 − Q = 1. Thus, there exists xl , xh such that π L1 (xl ) < 0 and π L1 (xh ) > 0. Therefore, L1 types play according to a threshold strategy with cutoff κ1 determined by π L1 (κ1 ) = 0. Assume true for k-1. Consider k. The expected payoff gain of an Lk type given that he receives signal x and that he expects all other players to play threshold strategies with the cutoff „ «– κk−1 is ˆ x+10 » π Lk (x, κk−1 ) =  1 2�  x−10  θ 1 − Bin a ˆ(θ) − 2, n − 1,  θ + � − κk−1 2�  dθ − t  The function π Lk is strictly increasing and continuous in x and there exists xl , xh such that π Lk (xl ) < 0 and π Lk (xh ) > 0. Therefore, there is a unique cutoff κk where π(κk , κk−1 ) = 0. Best responses for Lk types are to play threshold strategies with cutoff κk . The result follows by induction. (i) By induction on k. Consider k=1. The expected payoff gain from playing action 1 for an L1 type who receives a precise public signal y is  π  L1  (y) =  ˆ  1 a(y)−1 15  yq(l)dl − t  � � �� a(y) − 1 = y· 1−Q −t 15 Notice that π L1 is strictly increasing and continuous in y and there exists yl , yh such that π L1 (yl ) < 0 and π L1 (yh ) > 0. Therefore, L1 types play according to a threshold strategy with cutoff κ determined by π L1 (κ) = 0. Assume true for k-1. Consider k. Lk types expect everyone to attack if y ≥ κ. If y < κ, payoff to attacking is y · 0 − t < 0  If y ≥ κ then payoff to attacking is y · 1 − t > 0 The result follows by induction.  103  A.5  Round 1 estimates  MODEL:  All Types  Level-k  Equilibrium  Log-Likelihood  -1179.3  -1187.2  -1536.4  L1 L2 L3 E  .6377  .6687  (.0391)  (.0452)  .1276  .1616  (.0358)  (.0254)  .1035  .1698  (.0363)  (.0459)  .1312  1  (.0362)  λ1 λ2 λ3 λE �1 �2 �3 �E  n  .1130  .0981  (.0112)  (.0175)  .0000  .0000  (.0812)  (.0384)  0.0310  .1348  (.0586)  (.0746)  .0878  .0270  (.1333)  (.0021)  0.9099  .9797  (.0439)  (.0377)  1.0000  1.0000  (.1472)  (.0358)  1.0000  1.0000  (.1022)  (.0868)  .4565  .7415  (.2640)  (.0403)  3150  3150  3150  *Notes: bracketed numbers are bootstrapped standard errors clustered at the session level.  Table A.1: Aggregate type classification (round 1)  104  Appendix B  Appendix to Chapter 4 B.1  Omitted proofs  Proof of Proposition 4.1 We formally prove this result with a series of lemmas. Lemma B.1. For any event E ⊂ Tpk (i) �  gik (ti )(E) =  t∈Tp(i)  k−1 bi (ti )(t) · gp(i) (t)(E)  Proof. First consider k=2: �  gi2 (ti )(E) =  t∈Tp(i)  =  �  t∈Tp(i)    bi (ti )(t)   �  t� ∈Tpk−2 (p(i))  1 bi (ti )(t) · gp(i) (t)(E)    bp(i) (t� )(E)  Now suppose true for k-1 and show true for k: gik (ti )(E) =  �  t� ∈Tpk−1 (i)  =  �  t∈Tp(i)  =  �  t∈Tp(i)     �  t∈Tp(i)    bi (ti )(t)     k−1  k−2 � bi (ti )(t) · gp(i) (t )(t� ) · bp  �  t� ∈Tpk−2 (p(i))  k−1 bi (ti )(t) · gp(i) (t)(E)  105  k−2  k−2 gp(i) (t)(t� ) · bp  (i)  (t� )(E)  (p(i))    (t� )(E)  Proof follows by induction.� Lemma B.2. Consider a type ti who has correct beliefs. Then there must exist a type tp(i) ∈ Tp(i) such that bi (ti )(tp(i) ) = 1. Proof. Let ti be a type with correct beliefs. Then, there exists some type tp(i) such that bi (ti )(tp(i) ) > 0. Consider an arbitrary event E ⊂ Tp(i) and let bp(i) (tp(i) )(E) = p. Type ti believes that  player p(i) believes E with probability p. Since this holds for every event E and since bp(i) : Tp(i) → �(Tp(i) ) is injective, it must be that bi (ti )(tp(i) ) = 1.� Lemma 1 tells us that if a type has correct beliefs, then there exists some type tp(i) such that ti believes his opponent is type tp(i) . Lemma B.3. Consider a type ti who has correct beliefs to the mth-order. Then, there j  must exist a type ti+j ∈ Tpj (i) ∀j ∈ {1, . . . , m} such that bp  (i)  (ti+j )(tj+i+1 ) = 1 for all  j ∈ {1, . . . , m − 1}. Proof.  Show by induction on j: if ti ∈ Cik then ti+j ∈ Cpk−j j (i) .  Consider j = 1. Let ti be a type with correct beliefs. Then, there exists some type ti+1 such that bi (ti )(ti+1 ) = 1 from Lemma B.2. k−1 Since ti ∈ Cik and bi (ti )(ti+1 ) = 1 this implies that ti+1 ∈ Cp(i) .  Assume true for j − 1  Since ti+j−1 ∈ Cik−j+1 , let ti+j be such that bi (ti )(ti+1 ) > 0, this implies that ti+j ∈ Cpk−j j (i) . Since j ≤ k − 1, we have ti+j ∈ Cp1j (i) .  j  Therefore, there exists some type ti+j+1 ∈ Tpj+1 (i) such that bp  (i)  (ti+j )(ti+j+1 ) = 1�  Lemma B.4. Consider a type ti who has self-referential beliefs and correct beliefs to the nth-order. Then, gin (ti )(ti ) = 1. Proof. Since ti has correct beliefs to the nth-order, by Lemma B.3 there exists a type tpj (i) ∈ j  Tpj (i) such that bp n  such that bp  (i)  (i)  (tpj (i) )(tpj (i) ) = 1 ∀j ∈ {1, . . . , n − 1} and there exists a type t�i ∈ Ti  (tpn (i) )(t�i ) = 1.  Choose an arbitrary event E ⊂ Tp(i) and suppose type ti believes E with probability p. Since ti has self-reflective beliefs it must be that gin+1 (ti )(E) = p since pn (i) = i.  Since this holds for all events E and the function bi are injective for all i ∈ I, it must be that gik (ti )(Si × ti ) = 1.  106  From Lemma B.1 and Lemma B.3 we know that n gin+1 (ti )(E) = bi (ti )(tp(i) ) · gp(i) (t)(E) n−1  = bi (ti )(tp(i) ) · · · bp  (i)  = bi (t�i )(E)  (tpn−1 (i) )(t�i ) · bi (t�i )(E)  Therefore, since bi is injective, it must be that gin+1 is an injective function and ti = t�i ⇒ gin (ti )(ti ) = 1. �  Now, we are ready to prove our main proposition. Proof of Proposition 4.1. Let ti be a type that satisfies conditions (i) and (ii). From Lemma B.4 we know that there exists tj ∈ Tj for all j ∈ I such that bj (tj )(tp(i) ) = 1.  Now define the probability distribution σj ∈ �(Sj ) by σj = sˆj (tj ) for all j ∈ I. � It is also true that σpk (i) (s) = gik (ti )(tpk (i) ) since gik (ti )(tpk (i) ) = b (ti )(tp(i) ) · b i  Lemma B.4.  tpk (i) ∈Tpk (i) :ˆ sk(i) (tpk (i) )=s  p(i)  k−1  (tp(i) )(tp2 (i) ) · · · bp  (i)  (tpk−1 (i) )(tpk (i) ) which follows from Lemma B.1 and  Further, we know that type ti has correct beliefs about payoffs to the nth-order, i.e. ˆ θj (tj ) = θj for all j ∈ I. Since ti is rational it must be true that for any s ∈ supp{σi } ui (s, µp(i) , θi ) ≥ ui (a� , µp(i) , θi )  ∀a� ∈ Si .  Further, since ti ∈ Rin ⇒ tj ∈ Rj1 for all j �= i  Therefore, for any j �= i and for any action s ∈ supp{σj }  uj (s, σp(j) , θj ) ≥ ui (s� , σp(j) , θj )  ∀s� ∈ Sj .  Therefore, σ is a Nash Equilibrium.�  107  B.2  Relaxing belief to p-belief  In the experiment ran in this paper, we technically do not estimate whether a subject’s beliefs hold with probability 1. This is because for finite games, the behavioral implications of a belief with probability 1 will be the same for a belief with probability p, for high enough p. In this appendix, we show that allowing for a relaxation of belief to that of p-belief (belief with probability 1 to belief with probability p) is consistent with our experimental results. For example, throughout this paper we have described 1st-order belief in other’s rationality as believing you opponent is rational with probability 1. However, because the behavioral implications of belief that your opponent is rational with probability 1 are the same as the epistemic condition of believing that your opponent is rational with at least probability p for some high enough p, we want to relax the notion of belief to that of p-belief. Rationality and consistent beliefs are likewise relaxed to allow for the notion of p-belief. We can say that a type p-believes an event if he believes that event with probability at least p. Definition. A type ti p-believes an event E ⊆ Tp(i) if bi (ti )(E) ≥ p. Let the set Bpi (E) = {ti ∈ T i |bi (ti )(E) ≥ p} be the set of types for player i that p-believe event E.  We define the following sets in order to define higher-order p-beliefs about the rationality of others. A type satisfies p-belief in rationality if he puts weight at least p on types of his opponent that are rational.  1 Ri,p m+1 Ri,p  = {ti ∈ T i |ti is rational} m m = Ri,p ∩ Bpi (Rp(i),p )  m+1 Definition. If ti ∈ Ri,p then we say that ti satisfies rationality and mth-order p-belief  in rationality.  The requirement that a player has correct beliefs about her opponent can also be relaxed. Correct beliefs for type ti requires there to exist a type tp(i) ∈ Tp)i) such that ti believes tp(i)  with probability 1. Correct p-beliefs only requires type ti to believe tp(i) with probability at least p. Definition. A type ti has correct p-beliefs if there exists a type tp(i) ∈ Tp(i) such that bi (ti )(tp(i) ) ≥ p. Define  108  1 Ci,p m+1 Ci,p  = {ti ∈ T i |ti has correct beliefs to probability p} m m = Ci,p ∩ Bpi (Cp(i),p )  m Definition. If ti ∈ Ci,p then we say that ti satisfies correct p-beliefs to the mth-order.  Similar to correct beliefs, we can define a relaxed version of self-referential beliefs. For example, in a 3-player game, a type ti has self-referential beliefs if her third-order beliefs put probability 1 on type ti . A type ti has self-referential p-beliefs if her third-order beliefs put at least probability p on type ti . Definition. A type ti has self-referential p-beliefs if gin+1 (ti )(ti ) ≥ p. In an n-player game, we say a subject has consistent p-beliefs if she has correct p-beliefs to the nth-order and self-referential p-beliefs. m It can be shown that Ri,p = Rim for each of the players in Games G1 and G2 as long as p ≥ 78 (p ≥ 34 for player 1). In other words, the behavioral predictions under rationality and  kth-order p−belief in rationality are the same as the behavioral predictions under rationality  and kth-order 1-belief in rationality (when p is sufficiently high). Thus, we do not identify 1-belief in rationality but rather a lower-bound p¯ such that a player satisfies a p¯-belief in rationality. Given this we observe that a R4 subject satisfies rationality and 3rd-order belief in rationality, a R3 subject satisfies rationality and 2nd-order R2 subject satisfies rationality and 1st-order  7 8 -belief  7 8 -belief  3 4-  in rationality, a  in rationality, and a R1 subject satisfies  rationality. Thus, we do not actually estimate whether a subject believes others are rational with probability 1, but rather that they believe others are rational with at least probability  7 8.  Our identification of consistent beliefs, relies along the assumption that subject’s satisfied the sufficient rationality conditions for Nash Equilibria in G3. Thus, it is necessary to relax the sufficient epistemic characterization of Nash equilibrium to allow for a p-belief in rationality. This is established in the two results below. It is show that that for any p ≥ 67 , as long as a subject is rational and satisfies 1st-order p-belief in rationality and consistent  p-beliefs then she must play the Nash strategy a in game G3 (as player 1 and 2). Thus, taking subjects who are classified as R2 or higher for games G1 and G2, ensure that they satisfy rationality and 1st-order  7 8 -belief  in rationality. Thus, if they fail to play the Nash  equilibrium in G3 it must be because they fail to satisfy consistent 67 -belief (in other words, consistent p-beliefs requires: if player 1 plays strategy s then she must believe player 2 believes she is playing s with at least probability p). Proposition. Consider an n-player ring game Γ =< I = {1, . . . , I}; S1 , . . . , SI ; π1 , . . . , πI ; p > and a Γ-based epistemic type space < T1 , . . . , Tn ; b1 , . . . , bn ; sˆ1 , . . . , sˆn >. There 109  exists a p¯ such that if ti satisfies (i) rationality, (ii) (n − 1)th-order p-belief in rationality, and (iii) consistent p-beliefs, for some p ≥ p¯, then there exists some Nash equilibrium σ such that sˆi (ti ) = σi . Proof. Type ti has self-reflective p-beliefs and correct p-beliefs to the nth-order. Then there exists tj ∈ Tj for all j �= i such that bj (tj )(tp(j) ) ≥ p for all j ∈ I. � Define the probability distribution µj ∈ �(Sj ) by µj (s) = gim (ti )(tj ) for all tj ∈Tj :ˆ sj (tj )=s  s ∈ Sj j �= i.  Define the probability distribution σi = sˆi (ti ) and σj = sˆj (tj ) for all j �= I. Since ti is rational, it must be true that for all s ∈ supp{σi } ui (s, µp(i) ) ≥ ui (a� , µp(i) )  ∀a� ∈ Si .  Further, since uj = (1 − p)  ui (s, σp(i) ) as p → 1.  �  t∈Tj /tj :ˆ sj (t)=s  gim (ti )(t) + pσj it must be that ui (s, µp(i) ) →  Therefore, there exists a p¯1 such that ui (s, σp(i) ) ≥ ui (a� , σp(i) ) ∀a� ∈ Si and ∀p ≥ p¯1 .  Further, since ti ∈ Rin ⇒ tj ∈ Rj1 p¯ ≥ 12 . Therefore, for any j �= i and for any action s ∈ supp{σj }  uj (s, µp(j) ) ≥ ui (s� , µp(j) )  ∀s� ∈ Sj .  Thus, by the same continuity argument, there exists a p¯j such that uj (s, σp(j) ) ≥ ui (s� , σp(j) )  ∀s� ∈ Sj and ∀p ≥ p¯j .  Let p¯ = max{ 12 , p¯1 , . . . , p¯I }. Then, σ is a Nash Equilibrium. � Proof for G3: This proof establishes that p-belief with p >  6 7  is sufficient to guarantee that (a, a) is the  unique action profile in G3. We will regularly refer to the notion of q-dominance. Definition. In a 2-player game, an action profile a ∈ S1 × S2 is q-dominant for player i if for any µ ∈ �(S−i ) with µ(a−i ) ≥ q, then  ai ∈ argmax{ui (a, µ, θi )} a∈Si  110  $*  ./0*  $*  1/23* 3/4*  !"#$%$&'&()*!'%)+"*,*!'%)-*%* ,/0*  5*  %*  2/,*  2/0*  5* $*  3/4*  !"#$%$&'&()*!'%)+"*2*!'%)-*$*  %*  5*  !"#$%$&'&()*!'%)+"*2*!'%)-*$*  ,/0*  !"#$%$&'&()*!'%)+"*,*!'%)-*$*  !"#$%$&'&()*!'%)+"*,*!'%)-*$*  ./0* 1/23*  ./0*  !"#$%$&'&()*!'%)+"*,*!'%)-*%*  %* 2/,*  5*  !"#$%$&'&()*!'%)+"*2*!'%)-*%* 2/0*  $*  Figure B.1: Best response correspondences for game G3  %* ./0*  !"#$%$&'&()*!'%)+"*2*!'%)-*%*  First, from Figure B.2 notice that for player 1: (i) (a, a) is 34 -dominant (ii) (b, b) is 57 -dominant (iii) (c, c) is  4 13 -dominant  And, for player 2: (i) (a, a) is 57 -dominant (ii) (b, c) is 12 -dominant (iii) (c, b) is 12 -dominant Player 1: Now, consider the behavior of a type t1 that satisfies a (i),(ii) and (iii) with p-belief, p > 67 . There exists a rational type t2 such that b1 (t1 )(t2 ) ≥ p and b2 (t2 )(t1 ) ≥ p. Now, consider 2 cases:  (i) Suppose t1 plays b i.e. (ˆ s1 (t1 )(b) = µ1 (b) > 0) Since t1 is rational and believes opponent playing according to µ2 = sˆ2 (t2 ): µ2 (b) ≥  5 7  µ2 (b) ≥  2 7 5 7 2 7  − 35 µ2 (a)  Therefore: µ2 (a) ≤ µ2 (c) ≤  This also means that µ1 (a) + µ1 (b) ≥ p ⇒ µ1 (c) ≤ 1 − p.  Now, consider the behavior of the rational type t2 . We know that with µ1 (c) ≤ 1 − p, as  long as p > 67 , then any best response will be to mix between only a and c. ⇒ µ2 (a)+µ2 (c) ≥ p ⇒ µ2 (b) ≤ 1 − p. Contradiction.  (i) Suppose t1 plays c i.e. (ˆ s1 (t1 )(c) = µ1 (c) > 0) Since t1 is rational and believes opponent playing according to µ2 = sˆ2 (t2 ): u2 (b) ≤  9 13  Therefore:  −  11 13 p1  111  µ2 (b) ≤  µ2 (a) ≤ µ2 (c) ≥  9 13 3 4 1 4  This also means that µ1 (a) + µ1 (c) ≥ p ⇒ µ1 (b) ≤ 1 − p.  Now, consider the behavior of the rational type t2 . We know that with µ1 (b) ≤ 1 − p, as long as p > 76 , then any best response will be to mix between only a and b. ⇒ µ2 (a)+µ2 (b) ≥ p ⇒ µ2 (c) ≤ 1 − p. Contradiction.  Player 2: Consider a type t2 that satisfies a (i),(ii) and (iii) with p-belief. p.  So, we know that there exists a rational type t2 such that b2 (t2 )(t1 ) ≥ p and b1 (t1 )(t2 ) ≥ Now, consider 2 cases: (i) Suppose t2 plays b i.e. sˆ2 (t2 )(b) = µ2 (b) > 0 Since t2 is rational and believes opponent playing according to µ1 = sˆ1 (t1 ): Type t2 can mix between all three, so no conditions on µ2 . Now, consider the behavior of the rational type t1 . (i) Rational type t1 is mixing between a and b Therefore: µ2 (b) ≥  µ2 (a) ≤ µ2 (c) ≤  2 7 5 7 2 7  And, µ1 (a) + µ1 (b) ≥ p ⇒ µ1 (c) ≤ p. This means that rational type t2 can only mix  between a and c. Therefore µ2 (a) + µ2 (c) ≥ p ⇒ µ2 (b) ≤ p . This is a contradiction. (ii) Rational type t1 is mixing between a and c Therefore µ2 (b) ≤  µ2 (a) ≤ µ2 (c) ≥  9 13 3 4 1 4  And, µ1 (a) + µ1 (c) ≥ p ⇒ µ1 (b) ≤ p. This means that rational type t2 can only mix  between a and b. Therefore µ2 (a) + µ2 (b) ≥ p ⇒ µ2 (c) ≤ p . This is a contradiction. (i) Suppose t2 plays c i.e. sˆ2 (t2 )(c) = µ2 (c) > 0 Since t2 is rational and believes opponent playing according to µ1 = sˆ1 (t1 ): Type t2 can mix between all three, so no conditions on µ2 . Now, consider the behavior of the rational type t1 . (i) Rational type t1 is mixing between a and b Therefore: µ2 (b) ≥  2 7  112  µ2 (a) ≤ µ2 (c) ≤  5 7 2 7  And, µ1 (a) + µ1 (b) ≥ p ⇒ µ1 (c) ≤ p. This means that rational type t2 can only mix  between a and c. Therefore µ2 (a) + µ2 (c) ≥ p ⇒ µ2 (b) ≤ p . This is a contradiction. (ii) Rational type t1 is mixing between a and c Therefore 9 µ2 (b) ≤ 13 µ2 (a) ≤ 34  µ2 (c) ≥  1 4  And, µ1 (a) + µ1 (c) ≥ p ⇒ µ1 (b) ≤ p. This means that rational type t2 can only mix  between a and b. Therefore µ2 (a) + µ2 (b) ≥ p ⇒ µ2 (c) ≤ p . This is a contradiction. �  113  B.3  Assignment Mechanism  Consider 5 types: R0, R1, R2, R3 and R4. The action of each type is determined by assumptions A1-A3 and her order of rationality over the set of 8 games as given in Table 4.1. Let ai ∈ {a, b, c}8 be an 8-tuple representing the action of subject i in each of the 8 games. Each of the types, R1-R4, has a predicted action (or set of actions) for these 8 games,  ak ∈ Sk ⊂ {a, b, c}8 . For example, type R4 has a unique prediction. She must play the 4th-order rationalizable strategy in each of the 8 games, S4 = {(a, b, a, a, c, a, b, c)}. Type R3 has a unique prediction for 6 of the 8 games, but can play a number of different strategies for  the other 2 games, S3 = {(a, b, a, a, a, a, b, c), (c, b, a, a, c, a, b, c), (b, b, a, a, b, a, b, c)}. Notice  that A1-A3 generate an exclusion restriction and ensure that S4 ∩ S3 = ∅. We can define similar sets for types R1 and R2. The following property holds: Property1: For any n �= m ∈ {1, 2, 3, 4}, Sn ∩ Sm = ∅. Based on this exclusion restriction, a finite mixture model can be used to estimate a player’s order of rationality. This approach assumes that all subjects have some probability, πk , of being each type k. Given the large number of subjects that follow their type’s prediction exactly, we assume a subject follows her type’s prediction and make errors according to a spike error structure. This specification closely follows Costa-Gomes and Crawford (2006). If a subject is of type k, she follows one of type k’s predicted action profiles with probability 1 − �k and makes an error in any of the 8 games with probability �k , where �k ∈ [0, 12 ). If she makes error in one of the 8 actions, she plays either of the other strategies with equal probability (so each other strategy with probability 12 ). We can define the error density of type k ∈ {1, 2, 3, 4} for a given action profile ai by dk (ai , �k ) = (1 − �k )nik (  �k 8−nik ) 2  where nik = max {I T (ai , ak )·I(ai , ak )} and I : {a, b, c}8 ×{a, b, c}8 → {0, 1}8 is an indicator ak ∈Ak  function that equals 1 at index j whenever aij = akj .  Based on this error density, we can define subject i’s log-likelihood by lnLi (π, �, ai ) = ln  �  4 �  k=1  �  πk dk (ai , �k ) .  (B.1)  Whenever ai = ak for some ak ∈ Ak and some k ∈ {1, . . . , 4}, then �ˆk = 0 and π ˆk = 1  maximizes (B.1). Thus, if a subject plays an action profile that coincides exactly with a predictions of type k, she would be assigned type k. In addition, maximum likelihood can be used to assign other action profiles that do not match a type’s predictions exactly, as there exist some a ∈ {a, b, c}8 such that a ∈ / A1 ∪ A2 ∪ A3 ∪ A4 . However, we will not be  able to fully separate the implications of all action profiles. Depending on the action profile, 114  the maximum likelihood parameters may not be unique. In this circumstance, subjects are assigned to the lowest type. This imposes that the rationality requirements take precedence over the A3 assumption for each type. We assign a subject as R0 whenever her action profile deviates from all of the predicted action profiles of types R1-R4 by more than 1 error.1 The following is true of the assignment mechanism. Assignment Mechanism: Consider a subject i who plays action profile ai ∈ {a, b, c}8 . The following is true about the parameter π ˆ: (i) π ˆk = 1 if nik = 8 (ii) π ˆk = 1 if nik ∈ max{ni1 , ni2 , ni3 , ni4 }, nik ≥ 7 and k < j for any nij �= nik ∈ max{ni1 , ni2 , ni3 , ni4 }  (iii) π ˆ0 = 1 if max{ni1 , ni2 , ni3 , ni4 } < 7 1 Under our model, a player has a higher likelihood of being type R1-R4 than R0 whenever she makes 2 errors or less. However, we assign a subject to R0 based on more than one error to bias the mechanism in the direction of identifying R0 types.  115  B.4  Instructions and quiz **Do not use the BACK or REFRESH Buttons**  Instructions You 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 earn a considerable amount of money, which will be paid to you in cash at the end of the experiment. To insure best results for yourself please DO NOT COMMUNICATE with the other participants at any point during the experiment. If you have any questions, or need assistance of any kind, raise your hand and one of the experimenters will approach you.  The Basic Idea  Player 2's Earnings Player 3's actions  d  e  f  g  h  i  a  10  4  16  d  12  16  4  b  20  8  0  e  0  12  8  c  4  18  12  f  4  4  20  Player 3's Earnings Your actions  Player 3's actions  Your Earnings Player 2's actions  Player 2's actions  Your actions  You will play 14 games. In each of these games, you will be randomly matched with other participants currently in this room. For each game you will choose one of three actions. Each other participant in your game will also choose one of three actions.  a  b  c  g  20  12  8  h  6  8  18  i  0  16  4  Your earnings will depend on the combination of your action and player 2's action. These earnings possibilities will be represented in a table like the one above. Your action will determine the row of the table and player 2's action will determine the column of the table. You may choose action a, b, or c and player 2 will choose action d, e, or f. The cell corresponding to this combination of actions will determine your earnings. For example, in the above 3-player game, if you chose a and player 2 chooses d, you would earn 10 dollars. If instead player 2 chose e, you would earn 4 dollars. Player 2 and Player 3's earnings are listed in the other two tables. Player 2 may choose action d, e or f and Player 3 may choose action g, h, or i. Player 2's earning depends upon the action he chooses and the action player 3 chooses. Player 3's earnings depend upon the action he chooses and the action you choose. For example, if you choose c, player 2 chooses e, and player 3 chooses h then you would earn 18 dollars, player 2 would earn 12 dollars and player 3 would earn 18 dollars. When you start each new game, you will be randomly matched with different participants. We do our best to ensure that you and your counterparts remain anonymous. The earnings tables in a new game are not always the same as in the previous game, so you should always look at the earnings carefully at the beginning of each game. You will be required to spend at least 90 seconds on each game. You may spend more time on each game if you wish.  Earnings You will earn a show-up payment of $5 for arriving to the experiment on time and participating. In addition to the show-up payment, one game will be randomly selected for payment at the end of the experiment. Every participant in this room, will be paid based on their actions and the actions of their randomly chosen group members in the selected game. Any of the games could be the one selected. So you should treat each game like it will be the one determining your payment. You will be informed of your payment, the game chosen for payment, what action you chose in that game and the action of your randomly matched counterpart only at the end of the experiment. You will not learn any other information about the actions of other player's in the experiment. The identity of your randomly chosen counterparts will never be revealed.  Frequently Asked Questions Q1. Is this some kind of psychology experiment with an agenda you haven't told us? Answer. No. It is an economics experiment. If we do anything deceptive or don't pay you cash as described then you can complain to the campus Human Subjects Committee and we will be in serious trouble. These instructions are meant to clarify how you earn money, and our interest is in seeing how people make decisions.  116  **Do not use the BACK or REFRESH Buttons**  Player 2's Earnings Player 3's actions  d  e  f  g  h  i  a  10  4  16  d  12  16  4  b  20  8  0  e  0  12  8  c  4  18  12  f  4  4  20  Player 3's Earnings Your actions  Player 3's actions  Your Earnings Player 2's actions  Player 2's actions  Your actions  Quiz  a  b  c  g  20  12  8  h  6  8  18  i  0  16  4  Consider the above game. You are Player 1. Your earnings are given by the blue numbers. You may choose a or b or c.  1. Your earnings depend on your action and the action of which other player? (a)  Player 1  (b)  Player 2  (c)  Player 3  2. Suppose you choose a, Player 2 chooses f, and Player 3 chooses i. What will your earnings be? (a)  10  (b)  0  (c)  16  (d)  6  3. Suppose Player 2 chooses d and Player 3 chooses h. Which action will give you the highest earning? (a)  a  (b)  b  (c)  c  4. Suppose you choose c. What is your highest possible earning? (a)  20  (b)  18  (c)  4  5. Suppose you choose b. What is your lowest possible earning? (a)  0  (b)  4  (c)  8  Continue  117  B.5  Raw data  Subject  Rationality  ExactMatch  G1P1  G1P2  G1P3  G1P4  G2P1  G2P2  G2P3  G2P4  G3P1  G3P2  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55  0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3  0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1  1 2 2 3 3 1 1 1 1 1 1 3 3 1 1 1 1 1 1 1 3 3 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 3 1 3 3 1 1 1 1 1 1 1  3 3 1 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2  2 2 1 2 2 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  3 2 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 2 1 1 3 3 1 1 1 1 1 1 3 1 1 1 1 1 1 1 3 3 3 3 3 3 1 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 1 3 2 1 1 1 1 1 1 1 1 1  1 3 2 2 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1  2 3 2 2 1 2 3 3 3 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  1 3 3 3 1 1 1 1 3 3 3 2 3 1 2 3 1 1 1 1 3 1 3 3 3 3 3 3 3 3 1 2 3 3 3 2 3 3 3 3 1 2 3 1 1 2 3 3 3 3 2 1 3 1 1  2 3 2 3 3 3 3 3 3 2 3 3 2 3 3 3 2 2 2 3 2 1 3 3 3 2 3 2 3 3 2 3 2 3 2 2 3 2 3 2 3 3 3 2 2 3 3 3 2 3 3 3 3 2 3  118  Subject  Rationality  ExactMatch  G1P1  G1P2  G1P3  G1P4  G2P1  G2P2  G2P3  G2P4  G3P1  G3P2  56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80  3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4  1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  1 1 1 1 1 1 1 1 1 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1  2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2  3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3  3 2 3 3 1 1 3 1 3 2 3 2 3 1 2 3 3 2 3 3 3 1 3 1 3  2 3 3 3 3 2 3 3 3 3 3 2 2 3 2 3 3 3 2 3 2 3 3 3 3  1  1  2  1  1  3  1  2  3  1  1  Nash Equilibrium  Notes: ExactMatch=1 means that the action profile matches one of type R1-R4's predicted profiles exactly in games G1 and G2. Actions 1='a' , 2='b', 3='c'  !  Table B.1: Raw Data  119  

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