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Essays in behavioral economics Chakraborty, Anujit 2017

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Essays in Behavioral EconomicsbyAnujit ChakrabortyB.E., Jadavpur University, 2009M.S., Indian Statistical Institute, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2017c© Anujit Chakraborty 2017AbstractThis thesis studies individual choice in both individualistic and interactive deci-sions, under different situations of risk, uncertainty and time delay.The first chapter of my dissertation investigates the tendency of human beingsto make choices that are biased towards alternatives in the present. I characterizethe general class of utilities which are consistent with present-biased behavior. Ishow that any present-biased preference has a subjective max-min representation,which can be cognitively interpreted as the decision maker considering the mostconservative “present equivalents” in the face of subjective uncertainty about futuretastes.The second chapter of my thesis provides desiderata of choice consistency thatexperimenters should employ while estimating time preferences from choice data.We also show how application of this desiderata can help us learn new insightsfrom previous experimental studies.The third chapter of my thesis establishes a tight relation between non-standardbehaviors in the domains of risk and time by considering a decision maker withnon-expected utility preferences who believes that only present consumption iscertain while any future consumption is uncertain. We provide the first completecharacterization of the two-way relations between i) certainty effect and presentbias, and, ii) common ratio effect and the common difference effect. A corollaryto our results is that hyperbolic discounting implies the Common Ratio Effect andthat quasi-hyperbolic discounting implies the Certainty Effect.In the fourth chapter of my thesis, I use variation in experimental design (time-discounting) and belief data from subjects to investigate the determinants of behav-ior in Finitely Repeated Prisoner’s Dilemma games.iiLay SummaryMy thesis investigates decision making under conditions of risk, uncertainty andtemporal delay. In chapters one and three, instead of studying each of these be-haviors in isolation, I provide a more comprehensive theory of human behavior bystudying the interplay of uncertainty and time as infleuncing factors in different en-vironments. Chapter two uses a meta-study over recent influential experimental pa-pers to inform the design of future experiments investigating temporal-preferences.Chapter four studies the effect of temporal delay (discounting) on human interac-tion in an environment where there is a tradeoff between individual gain and socialsurplus.iiiPrefaceA version of Chapter 2 has been published [Chakraborty, A., Calford, E.M., Fenig,G. et al. Exp Econ (2017). doi:10.1007/s10683-016-9506-z]. I was the principalcontributor on the project, and hence I am the main author on this project.Chapter 3 was jointly co-authored with Professor Yoram Halevy, and we equallycontributed to this work at every stage of the project.Chapter 4 involves experimental study and associated methods were approved bythe University of British Columbia’s Research Ethics Board [Ethics Application#H13-02107].The author confirms that all the online links provided in the bibliography werefunctional on July 7th, 2017, prior to submission.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Present Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Model and the main result . . . . . . . . . . . . . . . . . . . . . 61.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 An extension to risky prospects . . . . . . . . . . . . . . . . . . 121.4 Extension to consumption streams . . . . . . . . . . . . . . . . . 141.5 An outline of the proofs . . . . . . . . . . . . . . . . . . . . . . 161.6 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.7 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . 211.8 Properties of the representation . . . . . . . . . . . . . . . . . . 221.9 Stake dependent Present Bias . . . . . . . . . . . . . . . . . . . 231.10 Application to a timing game . . . . . . . . . . . . . . . . . . . 241.11 Choice over timed bads . . . . . . . . . . . . . . . . . . . . . . 27vTable of Contents2 External and Internal Consistency of Choices made in Convex TimeBudgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 Quantitative evaluation . . . . . . . . . . . . . . . . . . . . . . 322.3 Corner choices . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 WARP violations . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Demand and wealth monotonicity . . . . . . . . . . . . . . . . . 342.6 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 Impatience monotonicity . . . . . . . . . . . . . . . . . . . . . . 372.9 Monotonicity index . . . . . . . . . . . . . . . . . . . . . . . . . 372.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Allais meets Strotz: Remarks on the relation between Present Bias andthe Certainty Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . 424 Drivers of Cooperation in Finitely Repeated Prisoner’s Dilemma . 474.1 An overview of the literature . . . . . . . . . . . . . . . . . . . . 484.2 Experimental design . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66AppendicesA Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73A.1 Appendix to to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . 73A.2 Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . 97A.3 Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 100viList of Tables1.1 Risk vs No-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Utilities of different selves under Case 1 (Left) and Case 2 (Right) 262.1 Demand and wealth monotonicity violations as a function of num-ber of interior choices . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Joint frequency of number of interior choicesets (by subjects) andnumber of interior choicesets that do not violate (demand and wealth)monotonicity (by subject), restricted to subjects who have at leastone interior choiceset. . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Total Cooperation by treatments . . . . . . . . . . . . . . . . . . 534.2 Comparison of first and terminal periods . . . . . . . . . . . . . . 554.3 Cooperation by treatment and period . . . . . . . . . . . . . . . 564.4 Relative Frequency of Forgiving , Reciprocal, and Unfolding play 574.5 Comparison of first and terminal periods . . . . . . . . . . . . . . 594.6 Change in beliefs and Forgiving across games (1-8) . . . . . . . . 594.7 Behavior and beliefs . . . . . . . . . . . . . . . . . . . . . . . . 604.8 Logit regressions on belief variables and game dummies . . . . . 63A.1 Models of temporal behavior . . . . . . . . . . . . . . . . . . . . 73A.2 Cooperation by treatment and period, split by order of treatments 101viiList of Figures2.1 Choice list vs CTB estimates of discount factor for the 36 all-cornersubjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Evolution of cooperation . . . . . . . . . . . . . . . . . . . . . . 544.2 Evolution of cooperation (All games from Between Session) . . . 584.3 Average Beliefs (g1,g4) vs Average cooperation across Games 1-8 60A.1 The function h. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.2 Function h approximated in a sinusoidal region . . . . . . . . . . 99viiiAcknowledgementsI am indebted to my supervisor Yoram Halevy for his continuous guidance andencouragement during these years. He introduced me to Behavioral Economics,helped me through every single step of my PhD and when needed, pushed me tobelieve more in myself. I do not think it is possible to have a better supervisor. Iwould like to thank Li Hao, Mike Peters, Wei Li and Lazrak Ali. They have alwaysgiven brilliant advice and have helped me become a better researcher and a betterthinker. I acknowledge SSHRC (FAS # F11-04991) for their financial assistanceand I thank the UBC faculty and graduate community for all their support. I wishto thank Vitor Farinha Luz and Utpal Sarkar for providing valuable suggestions. Iwant to mention Sampoorna, Jacob, Oscar, Pierluca, Coral, and Timea for beingwith me through thick and thin, as true comrades. I also want to thank Bikrama-ditya for being an unwavering friend, and Dyuti, for introducing me to economicsmany years ago.ixDedicationTo my mother, who taught me how to be myself. To my father, for sharing withme his passion for academics from very early days. To Dida and Dadu, who neverstopped believing in me. To all four of them, once again, for their unconditionallove and for always supporting my dreams.xIntroductionThe discipline of Economics studies how different incentive structures and eco-nomic stimuli shape behavior. As expected of any social science, this requires a twopronged approach grounded in theory and empirics. On one hand, economists pro-vide theories of behavior progressively consistent with the human actions we ob-serve in everyday market and non-market transactions. On the other hand, economistsalso generate and utilize data obtained from surveys, census, market studies or cre-ative experiments to test and select between competing theories of human behavior.This thesis combines features of both theoretical and empirical approaches toinvestigate decision making under conditions of risk, uncertainty, and temporal de-lay. In its course, it uses existing empirical findings to motivate why particular devi-ations from classical economic assumptions are necessary for better understandingof human behavior in certain scenarios. Then, precise alternative assumptions1 areprovided to formulate new theories and the empirical implications are discussedsubsequently2. Below we provide the overarching background and motivation forthe questions and results studied in this thesis.The modal temporal preferences obtained in previous experimental studiesshow a disproportionate bias for present rewards and consumption (Loewensteinand Prelec [1992], Frederick et al. [2002]), a phenomenon known as Present Bias.Existing theories (Pollak [1968], Harvey [1986], Laibson [1997], Hayashi [2003],Ebert and Prelec [2007]) which try to account for Present Bias, require extrane-ous assumptions on behavior which are often unrealistic or too strong. In Chapter1, we ask the question if it is possible to obtain a characterization and model forPresent Biased preferences, without having to impose extraneous assumptions onbehavior. We further study the consequences of such behavior in practical settingsand its welfare implications.In situations of risk and uncertainty, the modal behavior observed is consistentwith conservative behavior under uncertainty, and subsequently a bias for certainty(Allais [1953b], Ellsberg [1961], Gilboa and Schmeidler [1989], Cerreia-Vioglioet al. [2015]). This bears a close resemblance to the temporal behavior of a biasfor present rewards (Present Bias), as introduced in the previous paragraph. The1These alternative assumptions are weaker versions of previous classical assumptions.2This is why the following essays are broadly consistent with the topic of Behavioral Economics.1Introductionsimilarity begs the following question: Are preferences under risk and time prefer-ences related, and if so, how could one formally understand this relationship? Weanswer this question in Chapter 3.On the empirical side of matters, the importance of developing new experi-mental methods for studying temporal preferences cannot be overstated. Chapter2 of this thesis provides benchmarks which should be used to evaluate recently de-veloped experimental methodologies (Andreoni and Sprenger [2012], Augenblicket al. [2015]) in isolation, as well as to compare them to older methodologies (Har-rison et al. [2002]).In light of the preceding questions, it is natural to wonder about how temporaldelay (and risk) affects human behavior in interactive environments. One suchinteractive environment is captured by the Prisoner’s Dilemma game (Roth andMurnighan [1983], Bo [2005]), where subjects face a tradeoff between individualgain and social surplus. In Chapter 4 we answer the following question: Howcould we study human behavior in interactive environments under temporal delay/discounting, and what could we extrapolate about human motivations from such astudy?2Chapter 1Present BiasExponential discounting is extensively used in economics to study the trade-offsbetween alternatives that are obtained at different points in time. Under exponen-tial discounting, the relative preference for early over later rewards depends onlyon the temporal distance between the rewards (stationarity). However, recent ex-perimental findings have called the model into question. Specifically, experimentshave shown that small rewards in the present are often preferred to larger rewardsin the future, but this preference is reversed when the rewards are equally delayed.As an example, consider the following two choices:Example 1.A. $100 today vs B. $110 in a weekC. $100 in 4 weeks vs D. $110 in 5 weeksMany decision makers choose A over B, and D over C. This specific pattern ofchoice reversal can be attributed to a bias we might have towards alternatives in thepresent, and hence is aptly called present bias or immediacy effect. This is one ofthe most well documented time preference anomalies (Thaler 1981; Loewensteinand Prelec 1992; Frederick et al. 2002). If preferences are stable across decision-times and the decision-makers are unable to ward against the behavior of theirfuture selves, the same phenomenon creates dynamic inconsistency in behavior:People consistently fail to follow up on the plans they had made earlier, especiallyif the plans entail upfront costs but future benefits. Every year many people pledgeto exercise more, eat healthier, become financially responsible or quit smokingstarting next year but fail to follow through when the occasion arrives, to their ownfrustration.There is a big literature on what kind of utility representations could rational-ize choices made by a present-biased decision maker (DM), which we succinctlysummarize in Table A.1 in Appendix I. Though all of these models capture thebehavioral phenomenon of present bias, none of them can be called the model ofpresent-biased preferences. Instead they are all models of present bias and someadditional temporal behavior that is idiosyncratic to the model.3 Moreover, these3For example, Quasi-Hyperbolic Discounting ( called β -δ discounting interchangeably) addition-3Chapter 1. Present Biasadditional behavioral features conflict across the models and are often not empir-ically well-founded. This raises the following natural question: What is the mostgeneral model of present-biased preferences? Or alternatively, what general classof utilities is consistent with present-biased behavior? Such a model would be ableto represent present-biased preferences without imposing any extraneous behav-ioral assumption on the decision maker. This paper proposes a behavioral charac-terization for such general class of utilities. We start by introspecting about whatexactly present-biased behavior implies in terms of choices over temporal objects.The following example provides the motivation for our “weak present bias” axiom.Example 2. Suppose that a DM chooses (B) $110 in a week over (A) $100 today.What can we infer about his choice between (B′) $110 in 5 weeks versus (A′) $100in 4 weeks, if we condition on the person being (weakly) present-biased?4Note that B%A, implies that a possible present-premium ($100 is available atthe present) and the early factor ($100 is available 1 week earlier) are not enoughto compensate for the size-of-the-prize factor ($110>$100). Equally delaying bothalternatives preserves the early factor and the size-of-the-prize factor, but, the al-ready inferior $100 prize further loses its potential present-premium, which shouldonly make the case for the previous preference stronger. Hence, B%A must implyB′ %A′ to be consistent with a weak notion of present bias.We use this motivation to define a Weak Present Bias axiom, which relaxes sta-tionarity by allowing for present bias but rules out any choice reversals inconsistentwith present bias5. We then show that if a decision maker satisfies Weak PresentBias and some basic postulates of rationality, then, his preferences over receivingan alternative (x, t) (that is receiving prize x at time t) can be represented in thefollowing way (henceforth called the minimum representation)V (x, t) = min u∈U u−1(δ tu(x))where δ ∈ (0,1) is the discount factor, andU is a set of continuous and increasingutility functions. The minimum representation can be interpreted as if the DM hasally assumes Quasi-stationarity: violations of constant discounting happen only in the present periodand the decision maker (DM)’s discounting between any two future periods separated by a fixed dis-tance is always constant. On the other hand, Hyperbolic Discounting (which subsumes Proportionaland Power discounting as special cases) captures the behavior of a decision maker whose discountingbetween any two periods separated by a fixed distance decreases as both periods are moved into thefuture.4The choice of monetary reward for this example is without loss of generality. The reader canreplace monetary reward with a primary reward in the example, and the main message of this examplewould still go through undeterred.5Note that we are assuming present-premium≥ 0, thus ruling out the case where it is negative,i.e, something that would be consistent with future bias. This “weak” inequality of present-premiumis conceptually equivalent to a “weak” presence of present bias.4Chapter 1. Present Biasnot one, but a set U of potential future tastes or utilities. Each potential futuretaste (captured by a utility function u ∈ U ) suggests a different present equiva-lent6 for the alternative (x, t). The DM resolves this multiplicity by considering themost conservative or minimal present equivalent. Given that the present equivalentof any prize in the present is the prize itself, the minimum representation has nocaution imposed on the present, thus treating present and future in fundamentallydifferent ways. For any prize x received at time t = 0, min u∈U (u−1(δ 0u(x))) = x7,which can be interpreted as if, immediate alternatives are not evaluated throughsimilar standards of conservativeness, as is expected of a DM with present bias.Moreover, the fact that all alternatives are procedurally reduced to present equiv-alents for evaluation and comparison, underlines the salience of the present to theDM. This is another way in which the psychology of present bias is incorporated inthe representation. Our representation nests the classical exponential discountingmodel as the special case obtained when the set U is a singleton and hence canbe considered a direct generalization of the standard model of stationary temporalpreferences.Our model of decision making nests all the popular models of present-biaseddiscounting as special cases, as those models satisfy all the axioms imposed inour analysis. However, there are several robust empirical phenomena discussed inSections 1.3 and 1.9 which temporal models like β -δ or hyperbolic discountingcannot account for, but the current model can. For example, Keren and Roelofsma[1995] show that once all prizes under consideration are made risky, they are nolonger subject to present-biased preference reversals anymore. In other words,once certainty is lost, present bias is lost too. None of the models of behaviorthat treat the time and risk components of an alternative separately (for example,any discounted expected or non-expected utility model) can accommodate suchbehavior. We extend our analysis to a richer domain of preferences over risky timedprospects and provide an extended minimum representation that can account forthis puzzling behavioral phenomenon. In Section 1.10 we show how a benevolentsocial planner can use insights from time-risk behavior to improve the welfare ofpresent-biased individuals. Another choice pattern that most temporal models failto accommodate is the stake dependence of present bias. For example, a DM mighthave a bias for the present, but he might also expend considerably more cognitiveeffort to fight off this bias when the stakes are large. His large stake choices wouldsatisfy stationarity, whereas he would appear to be present-biased in his choices6Present equivalent of an alternative (x, t) is the immediate prize that the DM would considerequivalent to (x, t). For a felicity function u defined on the space of all possible prizes x, and adiscount factor of δ , the discounted utility from (x, t) is δ tu(x). Hence the corresponding presentequivalent is u−1(δ tu(x)).7As, δ 0 = 1, u−1(δ 0u(x)) = u−1(u(x)) = x for all u ∈U .51.1. Model and the main resultover smaller stakes (see Halevy 2015 for supporting evidence). We show how ourrepresentation can accommodate such preferences in Section 1.9.The subjective max-min feature of the functional form has been used previ-ously by Cerreia-Vioglio et al. [2015] in the domain of risk preferences, thoughthey had the minimum replaced by an infimum. In their paper, Cerreia-Vioglioet al. [2015] show that if we weaken the Independence axiom to account for theCertainty Effect (Allais 1953a), we obtain a representation where a decision makerevaluates the certainty equivalent of each lottery with respect to a set of Bernoulliutility functions and then takes the infimum of those values as a measure of pru-dence. We discuss this connection in greater detail in Section 1.5 and describe howthe techniques used in our paper can be used to provide an alternative derivation oftheir main result in a reduced domain.The paper is arranged as follows: Section 1.1 defines the novel Weak PresentBias axiom and provides the main representation theorem of the paper. Section 1.2builds on the main result to provide intuition about the separation of β -δ discount-ing from Hyperbolic discounting. Section 1.3 extends the main result to a richerdomain with risk. Section 1.4 discusses extensions of the representation result toconsumption streams. We provide an intuition of the inner workings of the proofsin Section 1.5. Section 1.6 comments on the uniqueness of the results. Section1.7 surveys the literature closely related to this paper. Sections 1.8 and 1.9 discussthe testability, refutability and empirical content of our model. Sections 1.10-1.11provide applications, policy implications and extensions of the main results of thepaper. The proofs of the main theorems are included in Appendix II.1.1 Model and the main resultA decision maker has preferences% defined on all timed alternatives (x, t)∈X×Twhere the first component could be a prize (monetary or non-monetary) and thesecond component is the time at which the prize is received. Let T= {0,1,2, ...∞}or T = [0,∞) and X = [0,M] for M > 0. We impose the following conditions onbehavior.A0: % is complete and transitive.Completeness and transitivity are standard assumptions in the literature, thoughone can easily argue that they are more normative than descriptive in nature. Thefew instances of present-biased intransitive preferences studied in the economicsliterature, notably Read [2001], Rubinstein [2003] and Ok and Masatlioglu [2007]fall outside our domain of consideration due to (A0).61.1. Model and the main resultA1: CONTINUITY: % is continuous, that is the strict upper and lower contoursets of each timed alternative is open w.r.t the product topology.Continuity is a technical assumption that is generally used to derive the conti-nuity of the utility function over the relevant domain. When, T= R+, the standardβ -δ model does not satisfy continuity at t = 0.8A2: DISCOUNTING: For t,s ∈ T, if t > s then (x,s) (x, t) for x> 0 and(x,s)∼ (x, t) for x = 0. For y> x> 0, there exists t ∈ T such that, (x,0)% (y, t).The Discounting axiom has two components. The first part says that the deci-sion maker always prefers any non-zero reward at an earlier date. The second partstates that any reward converges to the zero reward (and hence, continually losesits value), as it is sufficiently delayed.A3: MONOTONICITY: For all t ∈ T (x, t) (y, t) if x> y.The Monotonicity axiom requires that at any point in time, larger rewards arestrictly preferred to smaller ones. Finally, in light of Example 1, we formally defineWeak Present Bias below.A4: WEAK PRESENT BIAS: If (y, t)% (x,0) then, (y, t+ t1)% (x, t1) for allx,y ∈ X and t, t1 ∈ T.To provide context the standard Stationarity axiom is stated below.Stationarity: (y, t1) % (x, t2) if and only if, (y, t + t1) % (x, t + t2) for all x,y ∈ Xand t, t1, t2 ∈ T.Weak present bias as defined in the fourth axiom is the most intuitive weakeningof Stationarity in light of the experimental evidence about present bias or imme-diacy effect. It allows for choice reversals that are consistent with present-bias,something that Stationarity does not allow. On the other hand, having an oppositebias for future consumption is ruled out . 9Other than all the separable discountingmodels mentioned in Appendix I, this Weak Present Bias axiom is also satisfiedby the non-separable models of present bias proposed by Benhabib et al. [2010]8Pan et al. [2015] axiomatize a model of Two Stage Exponential (TSE) discounting which incor-porates the idea of β -δ discounting while maintaining continuity.9Further, (y, t)  (x,0) and (y, t + t1) ∼ (x, t1) is also not consistent with WPB, Continuity andMonotonicity. The reason being that, by Continuity, there would exist y′ < y, (y′, t)  (x,0) and(x, t1) (y′, t+ t1). Whereas, (y, t)∼ (x,0) and (y, t+ t1) (x, t1) is allowed by the postulates A0-4.71.1. Model and the main result10 and Noor [2011]. This stands testimony to the fact that the Weak Present Biasaxiom is able to capture the general behavioral property of present bias in a verysuccinct way. Now we present our main representation result.Theorem 3. The following two statements are equivalent:i) The relation % defined on X×T satisfies axioms A0-A4.ii) For any δ ∈ (0,1), there exists a setUδ of monotonically increasing contin-uous functions such thatF(x, t) = minu∈Uδu−1(δ tu(x)) (1.1)represents the binary relation%. The setUδ has the following properties: u(0) = 0and u(M) = 1 for all u ∈Uδ . F(x, t) is continuous.Note that for any timed alternative (x, t), u−1(δ tu(x)) in (1.1) computes its“present equivalent”, the amount in the present which the individual would deemequivalent to (x, t) if u were his utility function. For all present prizes, the presentequivalents are trivially equal to the prize itself (u−1(δ 0u(x)) = x ∀u) irrespectiveof the utility function under consideration, and thus there is no scope or need forprudence. Whereas for timed alternatives in the future, wheneverU is not a single-ton, the DM chooses the most conservative present equivalent due to the minimumfunctional, thus exhibiting prudence. This is the primary intuition of how thisfunctional form treats the present differently from the future and thus incorporatespresent bias into it. A potential motivation for the minimum representation and dif-ferential treatment towards present and future, follows from Loewenstein [1996]’svisceral states argument: “..immediately experienced visceral factors have a dis-proportionate effect on behavior and tend to crowd out virtually all goals otherthan that of mitigating the action, ...but.. people under weigh, or even ignore, vis-ceral factors that they will experience in the future.” The following example showsan easy application of the theorem to represent present-biased choices.Example 4. Consider U = {u1,u2}, where,u1(x) = xa for a>0u2(x) = 1− exp(−bx) for b>010Benhabib et al. [2010] introduce the discount factor∆(y, t) ={1 t = 0(1− (1−θ)rt)(1−θ)− by t > 081.2. Special casesAlso consider, a = .99, b = .00021, δ = .91. One can easily check that aminimum representation with respect to this U would satisfy Weak Present Bias(also follows from Theorem 3). The minimum representation with respect to thisU would assign the following utilities to the timed alternatives in Example 1.V (100,0) = min(100,100) = 100V (110,1) = min(100.056,99.995) = 99.995V (100,4) = min(68.317,68.48) = 68.317V (110,5) = min(68.320,68.344) = 68.320Hence,V (100,0) > V (110,1)V (100,4) < V (110,5)Thus the minimum function with a simple U can be used to accommodatepresent biased choice reversals.1.2 Special casesThis section applies Theorem 3 to a popular model of present bias, the β -δ model(Phelps and Pollak 1968, Laibson 1997). The β − δ model evaluates each alter-native (x, t) as U(x, t) = (β + (1− β ).1t=0)δ tu(x), where u,δ ,β have standardinterpretation. 1t=0 is the indicator function that takes value of 1 if t = 0 and value0 otherwise, thus assigning a special role to the present. Given that the β−δ modelsatisfies Weak Present Bias and all the other axioms included in Theorem 1 (for thediscrete case) , any such β − δ representation must have an alternative minimumrepresentation, as shown in Theorem 3.Below, we consider the simplest possible β − δ representation with linear fe-licity function u(x) = x, T = {0,1,2, ..} and construct the corresponding WeakPresent Bias representation.Claim 5. β -δ representation with u(x) = x has an alternative minimum represen-tation.Proof. Define the functions uy : R→ R+ for all y ∈ R+:91.2. Special casesuy(x) =xβfor x≤ βδyδy+(x−βδy) 1−δ1−βδ for βδy< x≤ yx for x> yFor any y ∈R+, x≤ uy(x)≤ xβ for all x ∈R+. As uy is an increasing function,it must be that x ≥ u−1y (x) ≥ βx. Since, x ≤ uy(x), we get δ tuy(x) ≥ δ tx, whichimplies,u−1y (δtuy(x))≥ u−1y (δ tx)≥ βδ txFinally, for x = y, δ tuy(x) = δ tx< δx and, hence, uy(δ tuy(x)) = βδ tx.Therefore, V (x, t) =miny∈R+u−1y (δ tuy(x)) = (β +(1−β ).1t=0)δ tx, which fin-ishes our proof.11This shows that if we start with a rich enough set of piece-wise linear utilities,the minimum representation with respect to that set, is enough to generate behaviorconsistent with β -δ discounting. In the example above, the set values taken by theset of functions is bounded above and below at each non-zero point x of the domainby [xβ,x], and this brings us to our next result. Our next theorem characterizes thebehavioral axiom necessary and sufficient for the functions in Uδ to be similarlybounded.We start by introducing two more axioms.A5: EVENTUAL STATIONARITY: For any x> z> 0 ∈ X, there exists t1 ∈ T,such that for t ≥ 0, (z, t) (x, t+ t1) and (z,0) (xt , t1+ t) for any xt such that(x,0)∼ (xt , t).A6: NON-TRIVIALITY: For any x ∈ X, and t ∈ T, there exists z ∈ X, such that(z, t) (x,0).The last axiom basically means that the space of prizes is rich enough to haveexceedingly better outcomes, and it is only needed when X= R+, and can bedropped if X= [0,M]. (See Corollary 1)A5 is the more crucial axiom. That for any x > z > 0 ∈ X , there exists a suffi-cient delay τ1 ∈T, such that (z,0) (x,τ1) is already implied by Discounting (A2).11This is not necessarily the only possible minimum-representation of the β -δ discounting.101.2. Special casesWhat has been added is the existence of delay t1 for which we additionally have(z, t)  (x, t + t1) for all t ≥ 0: This intuitively means once the later larger prizeis “sufficiently” delayed, the relative rates at which the attractiveness of the ear-lier and later rewards fall with further delay (increasing values of t) are consistentwith stationarity. This rules out certain preference reversals that were previouslyallowed under WPB. The last and third part of the axiom, (z,0)  (xt , t1 + t) forany xt such that (x,0) ∼ (xt , t), also has the same interpretation. The A5 prop-erty provides a crucial separation between two popular classes of present-biaseddiscounting functions: β -δ discounting and Hyperbolic discounting , as only theformer satisfies it, but the latter does not. We show this more formally in Proposi-tion 25 in Appendix II.Theorem 6. Let T= {0,1,2, ...∞} and X=R+. The following two statements areequivalent:i) The relation % satisfies properties A0-A6.ii) There exists a setUδ of monotonically increasing continuous functions suchthatF(x, t) = minu∈Uδu−1(δ tu(x)) (1.2)represents the binary relation %. The set Uδ has the following properties:u(0) =0 for all u ∈ Uδ , supu u(x) is bounded above, infu u(x) > 0 ∀x > 0, infuu(z)u(x)isunbounded in z for all x> 0. F(x, t) is continuous.This theorem implies that any “minimum-representation” of hyperbolic dis-counting must require a set of functions which would take unbounded set valuesat some point of the domain. The immediate conclusion one can draw from hereis that one cannot generate any variant of Hyperbolic discounting (with any felic-ity function) with a minimum representation over a finite set U of utilities. Thistheorem also has a straightforward corollary, where we consider the prize domainX=[0,M] and drop A6.Corollary 7. Let T= {0,1,2, ...∞} and X= [0,M]. The following two statementsare equivalent:i) The relation % satisfies properties A0-A5.ii) There exists a setUδ of monotonically increasing continuous functions suchthatF(x, t) = minu∈Uδu−1(δ tu(x))represents the binary relation%. The setUδ has the following properties:u(0) = 0,u(1) = 1 for all u ∈U , infu u(x)> 0 ∀x. F(x, t) is continuous.111.3. An extension to risky prospectsProspect A Prospect B % chosing A % chosing B N1 (9,1,0) (12,.8,0) 58% 42% 1422 (9,.1,0) (12,.08,0) 22% 78% 653 (9,1,3) (12,.8,3) 43% 57% 2214 (100,1,0) (110,1,4) 82% 18% 605 (100,1,26) (110,1,30) 37% 63% 606 (100,.5,0) (110,.5,4) 39% 61% 1007 (100,.5,26) (110,.5,30) 33% 67% 100Table 1.1: Risk vs No-Risk1.3 An extension to risky prospectsIn this section, we extend the representation derived in Section 1.1 to risk. Thisextension serves the following three goals. First, it shows that the representationin Section 1 has a natural extension to simple binary lotteries, with zero beingone of the lottery outcomes. Second, through the extended representation we areable to accommodate experimental evidence that is inconsistent with most previoustemporal models of behavior. Finally, through this extension, we will be able toidentify a unique discount factor δ for any DM satisfying certain postulates ofbehavior.We start by presenting the experimental evidence from time-risk domain thatour model would be able to accommodate, but, the temporal models from Ap-pendix I would not. In the following text, we summarize each alternative by thetriplet (x, p, t) where x is a monetary prize, p is the probability with which x isattained at time t. For the first three rows (from Baucells and Heukamp [2010]), x was offered in Euros, and in the next four (taken from Keren and Roelofsma[1995].), x was offered in Dutch Guilder, t was measured in months in Columns1:3, and measured in weeks in Columns 4:7.The data can be interpreted in the following way: People have an affinity forboth certainty and immediacy. The loss in either certainty or immediacy has asimilar disproportionate effect on preferences (compare rows 5 and 6 with row 4,or rows 2-3 with row 1). Most interestingly, there is very little evidence of present-biased reversals over risky prospects (compare rows 6-7, with rows 4-5). It is thelatter finding that is at odds with most temporal models of behavior. In fact itrules out all discounted expected or non-expected utility functional forms which121.3. An extension to risky prospectsare separable in the temporal and risk components.12We will consider preferences over triplets (x, p, t) ∈X×P×T, which describethe prospect of receiving a reward x ∈ X at time t ∈ T with a probabilityp ∈ [0,1].X = [0,M] is a positive reward interval, P = [0,1] is the unit interval of probabil-ity, and T = [0,∞) is the time interval. We impose the following conditions onbehavior.B0: % is complete and transitive.B1: CONTINUITY: % is continuous, that is the strict upper and lower contoursets of each risky timed alternative are open w.r.t the product topology.B2: DISCOUNTING: For t,s ∈ T, if t > s then (x, p,s) (x, p, t) for x, p> 0and (x, p,s)∼ (x, p, t) for x = 0 or p = 0. For y> x> 0, there exists T ∈ T suchthat, (x,q,0)% (y,1,T ).B3: PRIZE AND RISK MONOTONICITY: For all t ∈ T, (x, p, t)% (y,q, t) ifx≥ y and p≥ q. The preference is strict if at least one of the two followinginequalities is strict.Note that the first four axioms are just extensions of A0-A3.B4: WEAK PRESENT BIAS: If (y,1, t)% (x,1,0) then, (y,1, t+ t1)% (x,1, t1)for all x,y ∈ X, α ∈ [0,1] and t, t1 ∈ T.B5: PROBABILITY-TIME TRADEOFF: For all x,y ∈ X, p,q,θ ∈ (0,1], andt,s,D ∈ T, (x, pθ , t)% (x, p, t+D) =⇒ (y,qθ ,s)% (y,q,s+D).The fifth axiom (used previously in Baucells and Heukamp 2012a) says thatpassage of time and introduction of risk have similar effects on behavior, and thereis a consistent way in which time and risk can be traded off across the domainof behavior. This axiom implies calibration properties as well that we will utilizein the proofs, and it will be crucial to pin down a unique discount factor δ forany DM. Additionally, (B4) when combined with (B5) captures a decision maker’sjoint bias towards certainty as well as the present, i.e, it embeds Weak Present Biasas well as Weak Certainty Bias13 in itself. This underlines the insight that once12Rows 1 and 3 also imply the same.13Weak Certainty Bias can be defined on X×P in the following fashion: If (y, p) % (x,1) then,(y, pα)% (x,α) for all x,y ∈ X and α ∈ [0,1].131.4. Extension to consumption streamsrisk and time can be traded-off, Weak Present Bias and Weak Certainty Bias arebehaviorally equivalent. Similar relations between time and risk preferences havebeen elaborated on previously by Halevy [2008], Baucells and Heukamp [2012a],Saito [2015], Fudenberg and Levine [2011], Epper and Fehr-Duda [2012] andChakraborty and Halevy [2015]. In Section 1.5, we will discuss how the WeakCertainty Bias postulate connects the current work to previous literature on riskpreferences.We are now ready for our next result.Theorem 8. The following two statements are equivalent:i) The relation % on X×P×T satisfies properties B0-B5.ii) There exists a unique δ ∈ (0,1) and a set U of monotonically increas-ing continuous functions such that F(x, p, t) = minu∈U (u−1(pδ tu(x))) representsthe relation %. For all the functions u ∈ U , u(M) = 1 and u(0) = 0. Moreover,F(x, p, t) is continuous.The next example shows a potential application of this representation in lightof Keren and Roelofsma [1995]’s experimental results.Example 9. Consider the set of functions U and parameters considered in Exam-ple 4. When applied to the representation derived in Theorem 8, they predict thefollowing choice pattern.V (100,1,0) > V (110,1,1)V (100,1,4) < V (110,1,5)V (100, .5,0) < V (110, .5,1)V (100, .5,4) < V (110, .5,5)Note that this is exactly the choice pattern obtained in the original Keren andRoelofsma [1995] experiment: time and risk affect choices in similar ways, andonce certainty is removed present bias disappears.1.4 Extension to consumption streamsIn this section, we extend the representation derived in Section 1.1 to deterministicconsumption streams. The DM’s preferences % are defined over [0,∞)T , the set ofall consumption streams of finite length T > 1. We impose the following conditionson behavior.141.4. Extension to consumption streamsD0: % is complete and transitive.D1: CONTINUITY: % is continuous, that is the strict upper and lower contoursets of each consumption stream are open w.r.t the product topology.D2: DISCOUNTING: If 0≤ s< t ≤ T −1, then(0, .. y︸︷︷︸in period s, ..,0)% (0, .. y︸︷︷︸in period t, ..,0) for y≥ 0 with the relation being strict ifand only if y> 0. Further, for y0 > x> 0, and for any sequences (y1,y2,y3, ..ym)and (n1,n2, ..,nm), where, (0, ..0, yi−1︸︷︷︸in period ni,0..,0)% (yi,0, ..,0) ∀i ∈ {1,2, ...,m} ,0< ni ≤ T −1 and ∑m1 ni = t, there exists t ∈ N such that, ym ≤ x.D3: MONOTONICITY: For any (x0,x1, ..xT−1), (y0,y1, ..yT−1) ∈ [0,∞)T ,(x0,x1, ..xT−1)% (y0,y1, ..yT−1) if xt ≥ yt for all 0≤t≤ T −1. The preference isstrict if at least one of the inequalities is strict.D4: WEAK PRESENT BIAS: If (0, .. y︸︷︷︸in period t, ..,0)% (x,0, ..,0) then,(0, .. y︸︷︷︸in period t+ t1, ..,0)% (0, . x︸︷︷︸in period t1.,0) for all x,y ∈ X and t, t1 ∈ T.Note that the first five axioms are alternative restatements of A0-A4 in the currentdomain, but the Discounting axiom warrants some independent discussion. Asbefore, the second part of the Discounting axiom states that anyperiod-consumption keeps falling arbitrarily in present-equivalent value, as oneincreases the total discounting it is subjected to. Due to the added restriction thatthe DM can only consider time delays of upto T −1 periods for T ≥ 2, we haveapproximated arbitrary delays by a sequence of delays, none greater than T −1.But, the restatement is also a stronger version of the former (under WPB) as italso imposes path independence (by stating the axiom for arbitrary sequences(ni)mi=1 of delays instead of requiring it to hold for a particular sequence of delays,that sum to t) while achieving this total discounting. This is necessary whileworking with the non-compact prize space of [0,∞).151.5. An outline of the proofsD5: STRONG ADDITIVITY: For any pair of orthogonal14 consumptionbundles (x0,x1, ..xT−1), (y0,y1, ..yT−1) ∈ [0,∞)T , if, (x0,x1, ..xT−1)∼ (z0,0, ..,0)and (y0,y1, ..yT−1)∼ (z′0,0, ..,0), then,(x0+ y0,x1+ y1, ..xT−1+ yT−1)∼ (z0+ z′0,0, ..,0).Orthogonality of consumption vectors imply that xt > 0 only if yt = 0, andyt > 0 only if xt = 0 for all t. The fifth axiom implies the standard notion ofAdditivity used in axiomatizations of additive representation of streams, and ishence named Strong Additivity.We are now ready for our next result.Theorem 10. The following two statements are equivalent:i) The relation % on [0,∞)T satisfies properties D0-D5.ii) For any δ ∈ (0,1), there exists a setUδ of monotonically increasing contin-uous functions such thatF(x0,x1, ..,xT−1) = x+T−1∑1minu∈Uδu−1(δ tu(xt))represents the binary relation%. The setUδ has the following properties: u(0) = 0and u(M) = 1 for all u ∈Uδ . F(.) is continuous.It is worth noting that this extension to streams required a strong notion ofadditivity, and hence, the resulting representation on streams is not as general asthe one derived in the previous domain. For example, the representation here doesnot nest the classical exponentially discounted additive utility representation in itsmost general form.1.5 An outline of the proofsThis section outlines the proofs of Theorems 3-8 chronologically and places themethodology used in the proofs in the context of recent literature.We will provide the outline for the case of T ∈ [0,∞), as it is less technical butconveys the main idea behind the proofs nonetheless. For any timed alternative(z,τ), there exists x ∈ X such that (z,τ)∼ (x,0). This follows from monotonicity,continuity, connectedness of the prize-domain and this guarantees that any (timed)14Two vectors are orthogonal if their dot product is zero.161.5. An outline of the proofsalternative has a well defined present equivalent with respect to %. It is easy to seethat when τ = 0, one must have z = x. Given the present equivalents with respectto % are well defined, one possible utility representation V : X×T→ R+ is thefunction that assigns to every alternative (z,τ), the present equivalent according tothe relation (z,τ) ∼ (x,0). The crux of the remaining proof lies in showing thatthere exists a set of utilities Uδ such that the previously defined V function can berewritten asV (z,τ) = x = minu∈Uδu−1(δ τu(z))The proof is constructive. For any point x∗ ∈ (0,M), we construct a function ux∗(.)in the following steps.i) We assign ux∗(0) = 0, ux∗(x∗) = 1.ii) For any x ∈ (x∗,M], we find t > 0 such that (x, t)∼ (x∗,0). Define, ux∗(x) = δ−t(for any δ ∈ (0,1) under consideration) and re-label x as xt .iii) For y ∈ (0,x∗), define ux∗(y) = min{δ τ : (xt , t + τ) ∼ (y,0)} for some t fromstep (ii).We show that the minimum is well defined in step (iii), and the constructed ux∗()is strictly increasing, continuous, and has the following crucial property: If (z, t)∼(x,0) then, δ tux∗(z) ≥ ux∗(x) and subsequently, u−1x∗ (δ tux∗(z)) ≥ x , with the weakinequality replaced by equality if x = x∗. The asymmetric construction of ux∗() onthe left and right of x∗ is crucial for this to hold.Next we define Uδ = {ux∗(.) : x∗ ∈ (0,M)}. It readily follows from the afore-mentioned property of constructed utility functions that minu∈Uδ u−1(δ tux(z)) = xwhenever (z, t)∼ (x,0).Theorem 6 builds on these methods and insights of Theorem 3. Eventual Sta-tionarity gurantees that the functions in U can be constrcuted in a way such thatfor any two points x < y there exists t1 for which u(x) > δ t1u(y) for all u ∈ U .Now when one normalizes, u(1) = 1 for all u ∈U , using the condition mentionedin the previous sentence, one additionally obtains that supu u(x) is bounded aboveand infu u(x)> 0 ∀x> 0.Theorem 8 connects time and risk in the following way: Given the Probability-Time Tradeoff axiom, the X×P×T domain is isomorphic to either of the reduceddomains of X×P or X×T. For example, there exists unique δ ∈ (0,1) such that(x, p, t) ∼ (x, pδ t ,0) and (x, p, t) ∼ (x,1, t + logδ p) for all x ∈ X and p ∈ P. Thistheorem restricts its domain to T= R+, unlike Theorem 3, which holds equally forT= N0 as well. The axioms on X×P×T domain imply completeness, transitiv-ity, continuity, risk monotonicity (Discounting respectively), Weak Certainty Bias(Weak Present Bias respectively) for a preference defined on the reduced domainof X×P (X×T respectively for T= R+). Proving Theorem 8, now reduces to171.5. An outline of the proofsproving that there is a minimum representation on X×P or X×T of the formsminu∈U (u−1(pu(x))) or minu∈U (u−1(δ tu(x))) respectively. Additionally, provingany one of the representations from the implied axioms on the relevant domain isequivalent to proving all of the representations on the respective domains. Thisflexibility is allowed by the Probability Time Tradeoff axiom. In the Appendix, weprove how the reduction from the richer domain to X×P or X×T works, and thenprove that a relation on X×P satisfies completeness, transitivity, continuity, riskmonotonicity and Weak Certainty Bias if and only if the relation on X×P can berepresented by the functional form of minu∈U (u−1(pu(x))).This result on the reduced X×P domain brings us to a very interesting con-nection that the present work has with Cerreia-Vioglio et al. [2015]. In that paper,the authors consider preferences over lotteries (L ) defined over a compact real in-terval [w,b] of outcomes. To account for violations of the Independence Axiom15based on a DM’s bias towards certainty or sure prizes16, they relax it in favor ofNegative Certainty Independence (NCI) axiom defined below.NCI: (Dillenberger 2010) For p,q ∈L , x ∈ [w,b], and λ ∈ (0,1),qD Lx =⇒ λ p+(1−λ )qD λ p+(1−λ )LxCerreia-Vioglio et al. [2015] show that if D satisfies NCI and some basic ra-tionality postulates, then there exists a set of continuous and strictly increasingfunctions W , such that the relation D can be represented by a continuous functionV (p) = infu∈W c(p,u), where c(p,u) is the certainty equivalent of the lottery p withrespect to u∈U . The proof of their theorem has the following steps: FromD, theyconstruct a partial relation D′ which is the largest sub-relation of the original pref-erence D that satisfies the Independence axiom. By Cerreia-Vioglio [2009], D′is reflexive, transitive (but possibly incomplete), continuous and satisfies Indepen-dence. Next, following Dubra et al. [2004] 17, there exists a set W of continuousfunctions on [w,b] that constitutes an Expected Multi-Utility representation of D′,that is, pD′ q if and only if Ev(p)≥ Ev(q) for all v ∈W . Now taking an infimumof the present equivalents with respect to all the functions in W yields a represen-tation that assigns to each lottery its certainty equivalent implied by the relationD.This NCI axiom when reduced to the domain of binary lotteries on X×P,conveys the same behavior as the Weak Certainty Bias axiom we have discussed15For p,q,r ∈ L, and λ ∈ (0,1), pD q if and only if λ p+(1−λ )r D λq+(1−λ )r.16We denote the lottery that gives the outcome x ∈ [w,b] for sure as Lx ∈L .17Dubra et al. [2004] define a convex cone in the linear space generated by the lotteries related byD′ and then apply an infinite-dimensional version of the separating hyperplane theorem to establishthe existence of W .181.6. Uniquenessabove and have used in the proof of our theorem. Our representation over X×Pis a minimum representation that is an exact parallel of the infimum representationobtained by Cerreia-Vioglio et al. [2015]. This is no coincidence: we provide analternative derivation of Cerreia-Vioglio et al. [2015]’s result in a reduced domainof lotteries for similar behavior and show that their infimum representation can bereplaced with a minimum representation under the implied axioms in our domain.Our proof is essentially constructive, as illustrated in Claim 5, and it does not useany intermediate results (for example, results from Dubra et al. [2004]).The similarity in functional forms naturally prompts the question: Could theproof in Cerreia-Vioglio et al. [2015] be applied directly to our representation theo-rems? The answer to the question is negative for the following two reasons. Firstly,when an NCI-like axiom (Weak Certainty Bias) is imposed on my restricted do-main of binary lotteries, the results from Cerreia-Vioglio et al. [2015] no longerfollow as corollaries of their main theorem due to the reduced strength of the im-plied axioms. This follows the usual relation between size of domain and strengthof axiom. Secondly, there is no way of starting with an appropriately defined axiomof present bias on consumption streams (instead of timed payments) and reach-ing a present-biased utility representation on streams by using the route (PresentBias)⇔(NCI)⇔(Multi EU)⇔ (Present-biased representation), under any equiva-lence of time and implicit risk necessary for the first and last steps.1.6 UniquenessThe uniqueness results discussed here are formulated keeping the main represen-tation theorem of the paper in mind, but they apply equally to the other represen-tation theorems with minor adjustments. We start with a crucial question aboutthe representation: Could we have come across an alternative representation forthe same preferences without the exponential discounting part inside the presentequivalents? For example, could we have ended up with a representation of theform:V ′(x, t) = minu∈Uu−1(∆(t)u(x)) (1.3)where ∆(t) is some time-decreasing discount function other than exponential dis-counting, for example the hyperbolic one? Note that this is an interesting question,as a positive answer would open the door to representations where the presentequivalents are taken with respect to hyperbolic or quasi-hyperbolic discounting.However, the answer is negative. If we start with any ∆(t) such that∆(t+ t1)∆(t)6=191.6. Uniqueness∆(t1) for some t, t1 , there would either 1) exist some binary relation which satis-fies all the axioms in this paper, but cannot be represented by the representation in(1.3), or 2) the representation in (1.3) with a permissible set of utilities U wouldrepresent preferences which do not satisfy at least one of the axioms in this paper,thus breaking the two-way relation between the axioms and representation.Proposition 11. Given the axioms A0-4, the representation in (1.3) is unique inthe discounting function ∆(t) = δ t inside the present equivalent function.Proof. See Appendix II.One of the limitations of representations over X×T space (the domain usedin Sections 1 and 2) is the lack of uniqueness in terms of the discount factor δ .We inherit the non-uniqueness of δ in Theorems 3-6 from Fishburn and Rubinstein[1982]. Fishburn and Rubinstein [1982] impose A0-A3 along with Stationarity onpreferences to derive a exponential discounting representation. In their represen-tation, given those conditions on preferences, and given δ ∈ (0,1) , there exists acontinuous increasing function f such that (x, t) is weakly preferred to (y,s) if andonly if δ tu(x)≥ δ su(y). They have the following result: if (u,δ ) is a representationfor a preference % then so is (v,β ) where β ∈ (0,1) and v = u logβlogδ . Same holds forour representations in Theorems 3-6: if (δ ,U ) is a representation of %, then so is(α,F ),where F is constructed by the functions v = ulogβlogδ for u ∈ U . Obviouslythis is a restriction imposed by working on the prize-time domain and we can nolonger consider δ as a measure of impatience. To put things in perspective, in aseminal paper Koopmans [1972] instead considers the richer domain of consump-tion streams, and under the additional assumptions of separability and stationarity,he derives a time-separable additive exponential discounting representation of be-havior. In Theorem 8 we provide a representation over a richer domain where thediscount factor δ ∈ (0,1) is unique.Next, we show that the set of functions in the representation in (1.1) is unique upto its convex closure. DefineF = {u : [0,M]→ R+ : u(0) = 0, u is strictly increasing and continuous}Define the topology of compact convergence on the set of all continuous func-tions from R to R. Also, let co(A) and A¯ define the convex hull and closure of theset A (with respect to the defined topology), and c¯o(A) define the convex closureof the set A.Proposition 12. IfU ,U ′ ⊂F are such that c¯o(U ) = c¯o(U ′), and the functionalform in (1.1) allows for a continuous minimum representation for both of thosesets, then, minu∈U u−1(δ tu(x)) = minu∈U ′ u−1(δ tu(x)).201.7. Related literatureProof. See Appendix II.Proposition 13. i) If there exists a concave function f ∈U , and if U1 is the subsetof convex functions inU , then minu∈U (u−1(δ tu(x))) =minu∈U \U1(u−1(δ tu(x))).ii) If u1,u2 ∈U and u1 is concave relative to u2, then, minu∈U u−1(δ tu(x)) =minu∈U \{u2} u−1(δ tu(x)).Proof. See Appendix II.1.7 Related literatureThis paper is closely linked to the literature that explores the conditions underwhich a “rational” person may have present-biased preferences. Sozou [1998],Dasgupta and Maskin [2005] and Halevy [2008] explain particular uncertaintyconditions that can give rise to present-biased behavior. While telling an uncer-tainty story sufficient to explain present bias, all these models explicitly assumethe particular structure of risk or uncertainty with relevant risk attitude, and theseassumptions are central to establishing behavior consistent with present bias inthe respective models. In this paper we deviate from this norm: we do not ex-plicitly assume any uncertainty framework or uncertainty attitude. But we stillobtain a subjective state space representation that is necessary and sufficient forpresent bias. The set of future tastes U can be considered to be the subjectivestate-space, and the decision maker considers the most conservative state depen-dent utility minu∈U u−1(δ tu(x)) to evaluate each timed alternative.Our representation looks similar to the max-min expected utility representationof Gilboa and Schmeidler [1989] used in the uncertainty or ambiguity aversionliterature, though there is no objective state space or uncertainty defined in our set-up. We have already discussed the connection of our paper with Cerreia-Vioglioet al. [2015] in terms of the similarity in representation. There are other variantsof the minimum or infimum functional in previous literature, for example, Cerreia-Vioglio [2009] and Maccheroni [2002], used in different contexts.There is also a sizable literature on the behavioral characterizations of temporalpreferences, that the current project adds to. Olea and Strzalecki [2014], Hayashi[2003] and Pan et al. [2015] characterize the behavioral conditions necessary andsufficient for β -δ discounting, Loewenstein and Prelec [1992] characterize Hyper-bolic discounting, and, Koopmans [1972], Fishburn and Rubinstein [1982] do thesame for exponential discounting. Gul and Pesendorfer [2001] study a two-periodmodel where individuals have preferences over sets of alternatives that representsecond-period choices. Their axioms provide a representation that identifies the de-cision maker’s commitment ranking, temptation ranking and cost of self-control.211.8. Properties of the representation1.8 Properties of the representationWe propose an alternative notion of “present premium” comparison below. Thepresent premium can be considered as the maximal amount of future consumptionone is willing to forego to have the residual moved to the present. For example, if(y, t)∼ (x,0), then the present premium of (y, t) is (y− x)≥ 0.Consider the following partial relation defined on the set of binary relations %over X×T.Definition 14. %1 allows a higher premium to the present than%2 if for all x,y∈Xand t ∈ T(x, t)%1 (y,0) =⇒ (x, t)%2 (y,0)The next result connects this notion of comparative present premia to our rep-resentation.Theorem 15. Let %1 and %2 be two binary relations which allow for minimumrepresentation w.r.t sets Uδ ,1 and Uδ ,2 respectively. The following two statementsare equivalent:i) %1 allows a higher premium to the present than %2.ii) Both Uδ ,1 and Uδ ,1∪Uδ ,2 provide minimum representations of %1.Proof. See Appendix II.One might wonder if there could also be a representation theorem similar toTheorem 3 for an appropriately defined Weak Future Bias axiom. Below wedefine Weak Future Bias, and provide a corresponding representation.A4*: WEAK FUTURE BIAS: If (x,0)%(y, t) then, (x, t1)% (y, t+ t1) for allx,y ∈ X and t, t1 ∈ T.This is an alternative relaxation of Stationarity that is complementary to WPB.Weak Present Bias, when combined with Weak Future Bias yields the StationarityAxiom. We now present the following result.Theorem 16. Let T = [0,∞) and X = [0,M]. The following two statements areequivalent:i) The relation % satisfies properties A0-A3 and A4*.ii) There exists a setUδ of monotonically increasing continuous functions suchthatF(x, t) = maxu∈Uu−1(δ tu(x))221.9. Stake dependent Present Biasrepresents the binary relation%. The setUδ has the following properties: u(0) = 0and u(M) = 1 for all u ∈Uδ . F(x, t) is continuous.As expected Weak Future Bias is characterized by a weakly optimistic attitude to-wards the future. The proof is similar to that of Theorem 3, and is hence omitted.Testable ImplicationsThe major testable condition in the paper comes from the Weak Present Bias ax-iom: If (y, t) % (x,0) then, (y, t + t1) % (x, t1) for all x,y ∈ X and t, t1 ∈ T. Statedin terms of the contra-positive, If (x, t1) (y, t+ t1) for some x,y ∈ X and t, t1 ∈ T,t, t1 > 0, then, (x,0)  (y, t). Intuitively speaking, this model only allows prefer-ence reversals that arise from present bias (as restricted by the Weak Present Biasaxiom). So any temporal preference that stems from any other behavioral phe-nomenon would refute the model.1.9 Stake dependent Present BiasConsider a decision maker who makes the following 2 pairs of choices.Example 17.$100 today  $110 in a week$110 in 5 weeks  $100 in 4 weeks$11 in a week ∼ $10 today$11 in 5 weeks ∼ $10 in 4 weeksBoth pairs of choices are consistent with Weak Present Bias, but there is a clas-sical choice reversal (or a local violation of Stationarity) only in the first pair.18This kind of choice is at odds with all the models of present bias that we have men-tioned other than the one in this paper, but not necessarily at odds with economicintuition. For example, if a DM’s present bias is driven by the psychological fearof future uncertainty, the higher the stake, the higher would be the manifestation ofthis fear, and the more present-biased he would appear. The opposite phenomenon,18This kind of behavior closely parallels the “magnitude effect”: in studies that vary the outcomesizes, subjects appear to exhibit greater patience toward larger rewards. For example, Thaler [1981]finds that respondents were on average indifferent between $15 now and $60 in a year, $250 now and$350 in a year, and $3000 now and $4000 in a year, suggesting a (yearly) discount factor of 0.25,0.71 and 0.75 respectively.231.10. Application to a timing gamewhen a subject appears strictly present-biased for smaller stakes but appears sta-tionary at larger stakes (for the same set of temporal values) can happen, if thesubjects get better at temporal decisions at higher stakes due to cognitive optimiza-tion. None of the models in Appendix I can account for the behavior in Example17,19 whereas, the simple minimum function mentioned in Example 4 can accountfor such choices. There is scope to run future experiments to test for such stake de-pendent behavior. The closest precedent for such an experimental design appearsin Halevy [2015] where the author finds evidence of stake dependent present bias.1.10 Application to a timing gameIn this section we are going to study dynamic decision-making games for a present-biased DM whose preferences are consistent with the time-risk relations outlinedin Keren and Roelofsma [1995]. Present-biased preferences, when extended to adynamic context20, create time inconsistent preferences, which in turn results ininefficient decision making and loss in long-term welfare. The goal of this sectionis to convince the reader about the importance of axiomatization of risk-time rela-tions, by showing that risk-time relations have important welfare implications forsuch a present-biased individual.Consider the following game adopted from O’Donoghue and Rabin [1999]. Sup-pose a DM gets a coupon to watch a free movie, over the next four Saturdays. Hehas to redeem the coupon an hour before the movie starts. His free ticket is issuedsubject to availability of tickets, and if there are no available tickets, the couponis wasted. Hence there is some risk while redeeming the coupon. The movies onoffer are of increasing quality- the theater is showing a mediocre movie this week,a good movie next week, a great movie in two weeks and Forrest Gump in threeweeks. Our DM perceives the quality of these movies as 30, 40, 60 and 90 on ascale of 0−100. In our problem, the DM can make a decision maximum 4 times,at τ = 1,2,3,4 (measured in weeks). The DM’s utility at calendar time τ fromwatching a movie of quality x with probability p at calendar time t + τ(in weeks)is given by:Uτ(x, p,τ+ t) =p100α tx for p100α t ≥ α 12(αβ) 12pβ tx for p100α t < α 1219For example, if one tries to fit a β -δ model to this data, the second pair of choices immediatelysuggest β = 1, which in turn is inconsistent with the first pair of choices.20We are imposing Time Invariance of preferences following Halevy [2015]. We will make preciseassumptions about sophitication/ naivete as we go.241.10. Application to a timing gameWhere, β = .99, α = (.99)100 ≈ .36. This utility function (which is inspired byPan et al. [2015]’s Two Stage Exponential discounting model) has the following in-terpretation: The DM has a long run weekly discount factor of .99 that sets in aftera delay of half a week for p = 1. Before reaching the cut-off, the DM is extremelyimpatient, with a smaller discount factor of α = β 100 ≈ .36, and hence is biasedtowards the present and very short-run outcomes. Similarly, the DM also propor-tionally undervalues probabilities close to 1. The utility function(s) Uτ define apreference that satisfies all the axioms in Section 1.3, and hence have a minimumrepresentation. The DM is time-inconsistent, as his preferences between futureoptions differ between any two decision periods τ1 and τ2 for τ1,τ2 ∈ {1,2,3,4}.Let us assume that the DM is aware of his future preferences, that is she is sophis-ticated, a notion pioneered by Pollak [1968]. We are going to use the followingnotion of equilibrium for this game.Definition 18. (O’Donoghue and Rabin [1999]) A Perception Perfect Strategy forsophisticates is a strategy ss = (ss1,ss2,ss3,ss4), such that such that for all t < 4, sst =Yif and only if U t(t)≥U t(τ ′) where τ ′ = minτ>t{ssτ = Y}.In any period, sophisticates correctly calculate when their future selves wouldredeem the coupon if they wait now. They then decide on redeeming the coupon ifand only if doing it now is preferred to letting their future selves do it. We considertwo cases:Case 1: Suppose, there is not much demand for movie tickets in that city, andthe DM knows that he can always book a ticket through his coupon and p = 1 forall alternatives under consideration.In this case, the unique Perception Perfect Strategy is ss = (Y,Y,Y,Y ). Theknowledge that the future selves are going to be present biased creates an unwind-ing effect: The period 2 sophisticate would choose to use the coupon towards thegood movie as he knows that the period 3 sophisticate would end up using thecoupon for the great movie instead of going for Forrest Gump due to present bias.The period 1 sophisticate in turns correctly understands that waiting now wouldonly result in watching the good movie and hence decides to see the mediocremovie right now instead.Case 2: Suppose, due to persistent demand for movie tickets in that city, andthe DM knows that redeeming a coupon results in a movie ticket in only 99% ofcases.The unique Perception Perfect Strategy is ss = (N,N,N,Y ). The unwindingfrom the previous case does not happen here due to the risk involved in redeemingthe coupon. Once the present is risky (equivalent to having a front end delay dueto Probability Time Tradeoff), the bias previously assigned to the present vanishes,251.10. Application to a timing gamet ssτ t ssτ1 2 3 4 1 2 3 4τ4 90 Yτ4 54.2 Y3 60 54.2 Y 3 36.1 53.6 N2 40 36.1 53.6 Y 2 24 35.8 53 N1 30 24 35.8 53 Y 1 18 24 35.8 52.57 NTable 1.2: Utilities of different selves under Case 1 (Left) and Case 2 (Right)stopping the unraveling. The DM waits until the final period to cash in his couponwhen the expected returns are the highest to the long run self.The Left pane of Table 1.2 is for Case 1 (p = 1), the right table is for Case 2(p= .9). The entries in the table provide Uτ(x, p, t), and explain the equilibria. Thesophisticated DM compares the quantities in red row-wise for each τ when makinga decision.It would be instructive to compare the two cases in terms of welfare implica-tions. Since present-biased preferences are often used to model self-control prob-lems rooted in the pursuit of immediate gratification, we would compare welfarefrom the long run perspective. This outcome in Case 1 is consistent with the fol-lowing general result in O’Donoghue and Rabin [1999]: When benefits are imme-diate, the sophisticates “preprorate”, i.e, they do it earlier than it might be optimal.For example, considering the long term self’s interests, given a long term weeklydiscount factor of .99 for movie quality, the equilibrium outcome of watching themediocre movie (quality of 30) in the first week, instead of Forrest Gump (qualityof 90) definitely results in sub-optimal welfare in Case 1. For example, consideringthe choices from a τ = 0 self gives U0(30,1,1) = 18, and U0(90,1,4) = 53. Onthe other hand, the introduction of a small amount of risk in Case 2, stops the un-raveling in terms of “preprorating” (preponing consumption), thus helping the DMattain the most efficient outcome in equilibrium, thus reversing the O’Donoghueand Rabin [1999] result. In fact, not only is the highest level of available welfareachieved in Case 2 after the introduction of risk, the equilibrium welfare improvesfrom Case 1 to Case 2 in the absolute sense, even though apriori Case 2 seems tobe worse than Case 1 for the DM!U0(30,1,1) = 18 < U0(90, .99,4) = 52This is an interesting application of how introducing a dominated menu ofchoices can result in absolute welfare improvement.What would happen if the DM had the same preferences Uτ(), but, instead was261.11. Choice over timed badsunaware that his preferences were dynamically inconsistent? Let us consider theextreme case (popularly called “naïveté” in the literature) where the DM thinks thathis future selves’ preferences would be identical to his current selves’. We will callsuch a DM naive, and use the following equilibrium notion to characterize theirbehavior.Definition 19. A Perception Perfect Strategy for naifs is a strategy sn =(sn1,sn2,sn3,sn4),such that such that for all t < 4, snt = Y if and only if Ut(t)≥U t(τ) for all τ > t.The naive DM, acting under his false belief of time consistency, redeems thecoupon in the current period if and only if it yields him the highest payoff amongthe remaining periods. Table 1.2 tells us that in Case 1, sn = (N,N,Y,Y ), and inCase 2, sn = (N,N,N,Y ). Thus the introduction of risk in this example also helpsa naive DM make the most efficient choice in equilibrium.1.11 Choice over timed badsMost of the discussion on Present Bias till now has been centered around timedprizes or consumption, in general objects which are desirable. The central result ofthis paper is that Present Bias (as defined in A4 in Section 1.1) over such outcomes,can be represented by a minimum representation. This section would provide us theanswers to the following two natural follow-up questions: 1) What would PresentBias look like when timed undesireable-goods or bads (for example, effort) areconcerned? 2) What would be a utility representation of such preferences?We would consider the richer domain that includes risk, without loss of gener-ality. The DM has preferences over triplets (x, p, t), which describe the prospect ofreceiving an undesirable good x ∈ X at time t ∈ T with a probabilityp ∈ [0,1]. Weimpose the following conditions on behavior.C0: % is complete and transitive.C1: CONTINUITY: % is continuous, that is the strict upper and lower contoursets of each timed alternative are open w.r.t the product topology.The first two axioms are identical to axioms B0 and B1 used in Section 1.3.C2: DISCOUNTING: For t,s ∈ T, if s> t then (x, p,s) (x, p, t) for x, p> 0and (x, p,s)∼ (x, p, t) for x = 0 or p = 0. For x> y> 0, there exists T ∈ T suchthat, (x,q,0)% (y,1,T ).27ConclusionC3: PRIZE AND RISK MONOTONICITY: For all t ∈ T, (x, p, t)% (y,q, t) ify≥ x and q≥ p. The first binary relation is strict if at least one of the 2 followingrelations are strict and if y,q> 0.Discounting and Monotonicity have been adapted in the most intuitive way.People want to delay bad outcomes and they prefer when bad outcomes are lesslikely. Also when bad outcomes are concerned, more is worse.C4: WEAK PRESENT BIAS: If (x,1,0)% (y,1, t) then, (x,1, t1)% (y,1, t+ t1)for all x,y ∈ X and t, t1 ∈ T.The Weak Present Certainty Bias requires that given the present and certaintyare special, a DM would try to avoid bad outcomes which are in the present andare certain. Moreover, loss of certainty or immediacy can only make bad outcomesbetter.C5: PROBABILITY-TIME TRADEOFF: For all x,y ∈ X, p ∈ (0,1], andt,s ∈ T, (x, pθ , t)% (x, p, t+D) =⇒ (y,qθ ,s)% (y,q,s+D).The Probability-Time tradeoff axiom is unchanged and has the same interpre-tation as before.Theorem 20. The following two statements are equivalent:i) The relation % on X×P×T satisfies properties C0-C5.ii) There exists a unique δ ∈ (0,1) and a set U of monotinically decreasingcontinuous functions such thatF(x, p, t) = maxu∈U−u−1(pδ tu(x)) =−minu∈Uu−1(pδ tu(x))represents the relation %. For all the functions u ∈U , u(M) = −1 and u(0) = 0.Moreover, F(x,p,t) is continous.ConclusionThis paper provides an intuitive behavioral definition of (Weak) Present Bias andcharacterizes a general class of utility functions consistent with such behavior. Ourutility representation can be interpreted as if a DM is unsure about future tastes andpresent bias arises as an outcome of his cautious behavior in the face of uncertainty28Conclusionabout future tastes. Given most of the previous models of present bias have ex-traneous behavioral assumptions over and above present bias which are often em-pirically unsupported, we believe that our representation theorem is an importanttheoretical development in this literature. Having a more general representationfor present bias, also helps us accommodate empirical phenomenon (for example,stake dependent present biased behavior) that previous models could not accountfor. We have extended the model to incorporate time-risk relations in behaviorand provided an example where this relation can be utilized for welfare improv-ing policy design. Given the axiomatic nature of our work, we provide simpletestable conditions necessary and sufficient for our utility representations. Theseconditions can be easily taken to the laboratory or field to be empirically tested.We hope that this paper generates further interest in theoretical and applied workdirected towards forming a better understanding of intertemporal preferences.29Chapter 2External and InternalConsistency of Choices made inConvex Time Budgets2.1 IntroductionAndreoni and Sprenger (2012, henceforth AS) introduce Convex Time Budgets(CTB) to experimentally measure intertemporal substitution. In their design thesubject faces linear experimental budgets, which allow her to choose interior al-locations between payments at two time periods (henceforth ct , ct+k). One canrationalize such interior allocations if the subject’s preferences between ct and ct+kare (weakly) convex. It thus provides a way to adjust the measurement of subjectivediscount rates for intertemporal substitution.There are some basic properties that allocations in the Andreoni and Sprengerdesign should satisfy in order to be rationalizable by a very general model of in-tertemporal choice: allocations should satisfy wealth monotonicity (normality);21ct should be weakly decreasing in interest rate (demand monotonicity);22 alloca-tions should be consistent with impatience.23,24The AS design includes nine choicesets per subject, where each choiceset is acollection of five CTB tasks between payments at t and at t + k (where t = 0,7,35and k = 35,70,98 measured in days). Eight out of the nine choicesets contain awealth shift which could be used to test for wealth monotonicity. Demand mono-tonicity is tested by the other four CTB tasks within a choiceset. Impatience is21ct and ct+k should be weakly increasing in wealth, holding interest rate constant.22ct is a weakly decreasing function of the interest rate, holding the dates t and t + k and wealthnormalized to the later date constant.23As the later (earlier) date is shifted away from the present, ct should weakly increase (decrease),holding the earlier (later) date, price ratio and wealth constant.24The various monotonicity criteria for which we evaluate the empirical demand should not beconfused with monotonicity of the utility function with respect to (ct ,ct+k) . In particular, wealth anddemand monotonicity are consequences of the very weak assumption that ct and ct+k are normalgoods.302.1. Introductiontested by comparing across choicesets belonging to the same subject.25AS included three choice lists (MPLs) that correspond to three choicesets. Eachone of these choice lists included four pairwise choices that corresponded to CTBs.In other words, on these lines of the choice list a subject was asked to make a pair-wise choice between the two points in which each CTB intersects the horizontalaxis (ct+k = 0) and the vertical axis (ct = 0). In the CTB task the menu of alloca-tions the subject was allowed to choose from included these two allocations and allinterior allocations. We use this set-up to test for violations of the Weak Axiom ofRevealed Preference (WARP), which requires that if an alternative is chosen froma menu and is available in a sub-menu then it should be chosen from the sub-menuas well. If in the pairwise choice a subject chooses one corner while in the CTB shechooses the opposite corner this contradicts WARP. The implication is that thereexist no complete and transitive preference that can rationalize these choices.In this study we document the level of adherence of choices (at the individuallevel) to the above very mild external and internal consistency requirements. Wefind a very high level of WARP violations among the many subjects who madecorner choices. Violations for all three internal measures of monotonicity are con-centrated in subjects who make interior choices and thereby take advantage of thenovel feature of Andreoni and Sprenger’s experimental design. Wealth monotonic-ity violations are more prevalent and pronounced than either demand or impatiencemonotonicity violations.We believe that the findings reported here make it very challenging for one toclaim that choices made in CTB experiments reflect on deep and stable preferences.We urge researchers to study the source of the documented problematic behavior inorder to decide if it is inherent to CTB or could be attributed to the implementationof CTB in AS (2012).25When evaluating wealth monotonicity we allow for the non-generic possibility of linear pref-erences with marginal rate of substitution between ct and ct+k equal to the gross interest rate overk days in which the wealth shift occurs, i.e. 1+ r = 1.25. In this case, the demand is a correspon-dence and wealth monotonicity as defined above need not hold (we thank Jim Andreoni and CharlieSprenger for bringing up this possibility). However, to be consistent with this knife edge case, sub-jects need to satisfy: (1) c∗t = 0 for all r > 0.25 and c∗t+k = 0 for all r < 0.25. (2) In every choiceset(t,k′) such that k′ < k: c∗t = 0 for all r≥ 0.25. (3) In every choice set (t,k′) such that k′ > k: c∗t+k′ = 0for all r ≤ 0.25. (1) follows from linearity and (2-3) follow since the daily rate changes as k varies.312.2. Quantitative evaluation2.2 Quantitative evaluation2.3 Corner choicesAlthough the CTB design allowed for interior choices, 70% of all choices weremade at the corners of the budget set. 36 of the 97 subjects made only cornerchoices. There is little within subject variation and between subject heterogene-ity among these subjects. Nineteen of these subjects had the exact same choicesequence for all tasks: they chose the later-larger reward whenever the “gross in-terest rate” was greater than 1. Four other subjects chose the later-larger reward forall 45 CTB tasks, irrespective of interest rate and time horizon.2.4 WARP violationsOut of the 36 subjects who made all corner choices in CTB, we found 43 violationsof WARP.26 This is especially impressive if one considers that 17 of them alwayschose later consumption in the CTB and switched immediately in the choice lists(always chose later consumption). Therefore WARP violations could be detectedonly among the remaining 19 subjects. The direction of WARP violations is notrandom: 34 violations are in the direction of exhibiting less impatience in CTBthan in choice list, while only 9 are in the opposite direction.Since these subjects did not exhibit any curvature in their CTB choices, wecan directly estimate their discount factor based on the 3 choice lists and the cor-responding CTBs. One should not adjust for curvature for these subjects, sincetheir intertemporal decisions did not suggest any concavity of the felicity func-tion. The results are plotted in the attached Figure 2.1.27 We find that for 11 ofthem the discount factor estimated from CTB data would be higher than the oneestimated from choice list data, while for two subjects the relation between thediscount factors would be in the opposite direction. Note that the choices madeby the 17 subjects who always chose later consumption can be rationalized with adiscount factor of 1, and one cannot form a point estimate of the discount factorsof 4 other subjects who chose always immediate consumption in at least one of thethree CTBs.2826The discussion in this subsection ignores indifferences since we believe that the evidence issystematic and cannot be accounted for by the knife-edge arguments of linear preferences.27AS’ Figure 4A is similar, but we restrict to subjects who made only corner choices and thereforethere is no need to adjust for concavity28If one estimates a quasi-hyperbolic model based on these three CTBs or CLs, the conclusions donot change. In particular, the present-bias parameter (beta) under both elicitation methods is exactly1 for 28 out of the 32 subjects.322.4. WARP violations00. 0.2 0.4 0.6 0.8 1Choice list yearly discount factor CTB yearly discount factor Choice list vs CTB estimates of discount factor Figure 2.1: Choice list vs CTB estimates of discount factor for the 36 all-cornersubjectsAmong the other 61 subjects who made at least a single interior choice in the45 CTBs tasks we find similar directional effect of WARP violations. If one ofthe three choicesets that has a comparable choice list has all corner choices, wefind 23 WARP violations in the direction of exhibiting lower impatience in CTBthan in choice list and none in the opposite direction. In choicesets with interiorCTB choices (where the potential to observe direct WARP violation is smaller) wefound 10 violations in the direction of exhibiting lower impatience in CTB than inchoice list and 5 in the opposite direction.One interpretation to WARP violation (following Ok, Ortoleva and Riella,2015) is that CTB generates some reference dependence; alternatively it is pos-sible that many subjects become really confused when presented with CTBs. Inany case, the fact that a larger menu changes optimal choice systematically can-not be reconciled with a standard model of choice rationalized by a complete and332.5. Demand and wealth monotonicitytransitive preferences, as the discounted utility model.2.5 Demand and wealth monotonicityAs the 36 subjects with all corner choices did not take advantage of the convexifi-cation offered by the CTB, we believe it would be misleading to include them inevaluating CTB for internal consistency (monotonicity). Hence, the analysis belowconcentrates on the 61 subjects with at least one interior choice.2.6 FrequencyTable 2.1 reports the frequency of choicesets that have wealth or demand mono-tonicity violations as a function of the number of interior choices made in a choic-eset.The frequency of demand monotonicity violations is below 10% for choicesetsthat contain 4 or fewer interior choices. However, more than 36% of choicesetswith all interior choices have demand monotonicity violations. The frequency ofwealth monotonicity violations is considerably higher: around half of the choice-sets with at least one interior choice have a wealth monotonicity violation.Table 2.2 reports, for the 61 subjects with at least one interior choiceset, thedistribution of subjects satisfying wealth and demand monotonicity as a functionof the number of interior choicesets. A choiceset is considered interior if at leasta single choice (out of five) is not at the corners of the budget line (ct ,ct+k > 0).29Table 2.2 reveals that more than half of the 61 subjects violate monotonicity in atleast half of their interior choicesets (the shaded entries in the table).2.7 MagnitudeThe two tables above demonstrate the high frequency of non-monotone choicesin interior choicesets, especially as a response to wealth changes. We now turnto measure the magnitude of these behaviors. We calculate the magnitude of awealth monotonicity violation by the number of tokens required to be reallocatedto eliminate the violation at the higher wealth level. Our wealth monotonicitymeasure differs substantially from that reported in footnote 25 by Andreoni andSprenger for four reasons (presented in decreasing order of importance).29Table 2.2 excludes subjects who made all corner solutions. Among the 36 subjects who madeonly corner choices, we find only one non-monotonic choiceset.342.7. Magnitude# of interiorchoices in achoiceset# ofchoice-sets# of choicesets thatexhibit demandmonotonicityviolations# of choicesets thatexhibit wealthmonotonicityviolations# of choicesets thatexhibit either wealthor demandmonotonicityviolations0 435* 1 9 101 101 10 26 342 78 5 31 343 80 6 47 484 63 6 47 475 116 42 56 76Total 873 70 216 249*324 out of the 435 choicesets with no interior choice (almost 75%) belong to the 36 subjectswith only corner solutions.Table 2.1: Demand and wealth monotonicity violations as a function of number ofinterior choices# of interior* # of monotone interior* choicesetschoicesets 0 1 2 3 4 5 6 7 8 9 Total1 0 2 22 1 0 0 13 2 0 0 2 44 0 2 0 0 1 35 1 1 0 0 2 1 56 0 2 0 0 0 1 1 47 0 0 1 1 0 0 0 2 48 0 3 2 0 0 2 1 1 0 99 1 8 5 4 2 0 2 2 0 5 29Total 5 18 8 7 5 4 4 5 0 5 61*A choiceset is considered “interior” if at least a single choice (out of 5) is not at the corners ofthe budget line.Table 2.2: Joint frequency of number of interior choicesets (by subjects) and num-ber of interior choicesets that do not violate (demand and wealth) monotonicity (bysubject), restricted to subjects who have at least one interior choiceset.352.7. MagnitudeFirst, when calculating non-monotonicity, AS mistakenly defined non-monotonicity that is expressed as an over-allocation to ct+k (and under-allocation to ct) as a negative number, while non-monotonicity that is expressed asan under-allocation to ct+k (and over-allocation to ct) as a positive number. Averag-ing these two wealth monotonicity violations cancels out at the aggregate level. Forexample, if choices are generated at random using a uniform distribution over thetoken allocated to ct , independently among the two budget lines, the AS measureof wealth monotonicity violation would equal zero in expectation. The expectedvalue of our measure would be approximately 27 tokens (out of 100 tokens).30Second, when calculating the average adjustment required to restore mono-tonicity, AS use a denominator that includes all choicesets with a wealth shift,rather than just choicesets which have a wealth monotonicity violation. We believethat the AS approach is not advisable chiefly because, by including the 36 sub-jects who made only corner choices (and had no wealth monotonicity violation),it artificially dilutes the magnitude of monotonicity violations performed by sub-jects who responded to the convexification offered by the CTB design by makinginterior choices. Third, we consider the knife edge case of linear preferences withmarginal rate of substitution between ct and ct+k equals the gross interest rate inwhich the wealth comparative statics is performed as discussed in footnote 25. Thiscorrection applies to 8 choicesets. Fourth, when calculating adjustments AS allowfor fractional token adjustments. Given that subjects were only able to allocate in-teger values of tokens we believe it is more appropriate to calculate the adjustmentvalues using whole tokens.31We find that there are 216 violations of wealth monotonicity, with an averagesize of 23.2 tokens, which is 23.2% of the experimental budget or $4.64 of ct atthe higher wealth level. That is, conditional on violating wealth monotonicity, themagnitude of the measure is almost as high as the equivalent measure calculated forrandom choice. Andreoni and Sprenger report an average adjustment of just 1.67tokens to restore wealth monotonicity. It is important to note that the canceling outoccurs mainly at the population level rather than the individual level. Allowing foronly individual-level canceling out reduces our measure to only 17.9 tokens.We calculate the magnitude of demand monotonicity violations by finding theminimal amount of ct that needs to be reallocated per choiceset to restore mono-tonicity. There are 70 choicesets with demand monotonicity violations, with anaverage size of 17.4 tokens and a value (at time t) of $3.02.3230Note that the probability that a pair of choices violate wealth monotonicity in this case is 80%.31Indeed, AS use integer number of tokens when calculating the magnitude of demand monotonic-ity violations.32AS report 8 demand monotonicity violations for the (t = 7,k = 70) choiceset with an averagemagnitude of 24.6 tokens; in comparison, we find only 7 violations with an average magnitude of362.8. Impatience monotonicityAnother measure of the degree of non-monotonicity within a choiceset is tocalculate the smallest number of choices that must be removed from a choicesetto restore monotonicity.33 For the 249 choicesets that exhibit at least one non-monotonicity, the average number of data points that must be removed is 1.2. Thisfigure includes the 179 choicesets that exhibit only wealth non-monotonicity andtherefore require the removal of only a single data point; for the 70 choicesets thatexhibit demand non-monotonicity the average number of data points that must beremoved is Impatience monotonicityTurning to impatience, there are 10 pairs of choicesets across which either t isconstant and k varies, or t + k is constant and t varies; these are the only pairs ofchoicesets in which it is possible to test for impatience. In a comparable pair ofchoicesets (in the sense described above), we test for impatience monotonicity asdescribed in footnote 23 for all pairs of choice tasks (one in each choiceset) withthe same prices.We find that 47 of the 97 subjects satisfy the impatience criterion for all 10pairs of choicesets; restricting the sample to the 61 subjects with at least one inte-rior choice, we find that only 12 subjects made choices consistent with impatiencemonotonicity, and that 17 subjects violate impatience monotonicity in at least 5 ofthe 10 choiceset comparisons.2.9 Monotonicity indexFinally, we calculate an index that measures the (approximate) minimal number ofdata points that need to be eliminated from an individual’s dataset in order to beconsistent with the three monotonicity requirements.34 Out of the 36 subjects withno interior choice, 35 subjects satisfy all monotonicity measures.35 Out of the 6123.4 tokens in this choiceset. AS appear to have erroneously included additional adjustments for ct+k,and correcting for this reduces both the number and magnitude of demand monotonicity violations.33When removing data points to restore monotonicity we also consider joint violations of demandand wealth monotonicity.34This index is close in spirit to the Houtman-Maks (1985) index which is used to calculate themaximal set of observations in a dataset that is consistent with the Generalized Axiom of RevealedPreference (GARP). Because the AS (2012) design has no power to detect violations of GARP, anychoices made in a choiceset can be rationalized by a utility function, and by Afriat’s theorem theutility function can be chosen to be increasing in (ct ,ct+k). This, however, should not be confusedwith wealth monotonicity, which is a property of the demand function.35For the other subject, one needs to remove a single choice.372.10. Conclusionsubjects with at least a single interior choice, in 22 datasets we need to remove fouror fewer choices,36 in 21 datasets we need to remove between five to nine choices(more than 10% of choices) and in an additional 18 datasets one needs to remove10 or more choices (more than 20% of the total number of choices).2.10 ConclusionAndreoni and Sprenger’s proposal to use CTB in order to measure time preferencesrepresents a potentially important methodological advance. In principle, assumingdiscounted expected utility, such a method can allow a researcher to calculate amore precise measurement of the discount function by controlling for intertemporalsubstitution. However, our examination of data gathered by Andreoni and Sprenger(2012) using this method uncovers serious problems.Subjects who made only corner choices in CTB violate WARP very frequentlyrelative to the pairwise choice benchmark. This fact suggests that corner choices inCTB cannot be interpreted as reflecting reasoned behavior or deep preferences, butare heavily influenced by confusion or some reference introduced by the CTB. As awhole, the bias of WARP violations relative to the pairwise choice benchmark is inthe direction of lower impatience (higher discount factor). This explains why ASdo not require the concavity adjustment used in other studies in order to estimatesimilar discount factors.Subjects with interior choices are broadly consistent with demand monotonic-ity (except when all choices are interior) and the evidence for impatience mono-tonicity violations is moderate. However, the high frequency and substantial mag-nitude of wealth monotonicity violations in this data suggest that interior choicesmade in CTB (responding to the convexification) may not reflect reasoned behaviorand stable preferences as well.Unfortunately, the data does not permit us to identify the source of these severeproblems. It could be systematic to CTB or a result of AS’ experimental interface.We believe that further investigation into the origin of the serious problems docu-mented in the present study is crucial for an informed interpretation of existing andnew experiments utilizing CTB.36Only 9 of the 61 subjects made choices fully consistent with monotonicity.38Chapter 3Allais meets Strotz: Remarks onthe relation between Present Biasand the Certainty Effect3.1 IntroductionAlmost all decisions involve outcomes that are uncertain, realized at different pointsin time, or both. For example, following a strict and often unpleasant diet programrequires some motivation about future gains accruing from it, which are quite of-ten uncertain. There has been persistent interest in the fields of Psychology andEconomics to understand how behaviors across risky and temporal domains mightbe related to each other. The standard approach of modeling intertemporal prefer-ences is through the use of geometric (constant, exponential) discounting in whichthe payoff of a stream of consumption is aggregated through a (delay-geometric)weighting that results in a present discounted value. This is mirrored in the riskdomain, as the canonical model for risk behavior is expected utility which aggre-gates the utility of each possible alternative by weighting it by its probability. Butthe similarities do not end here as both models contain similar inadequacies asdescriptive models. First, preferences are disproportionately sensitive to certainty(certainty effect) and to the present (present bias/immediacy effect/diminishing im-patience). Second, proportional changes in probabilities or equal changes in timedelays for timed consumption affect preferences disproportionately (common ratioeffect and common difference effect respectively).37 This two-way relation is wellaccepted in the Psychological literature [Keren and Roelofsma, 1995, Green andMyerson, 2004, Weber and Chapman, 2005, Chapman and Weber, 2006, to namea few] and there is an understanding that the existence of such mirroring behav-iors is not a mere coincidence, but points to a common fundamental property ofdecision making that manifests itself in different domains of behavior [Prelec andLoewenstein, 1991]. There are many ways in which this relation between risk and37Often times certainty effect and present bias are taken as special cases of common ratio effectand common difference effect, respectively.393.2. Backgroundtemporal behavior can be motivated. Delayed rewards or consumption can be in-herently risky, as there might be events between the current date and the promiseddate which interfere in the process of acquiring the reward/consumption. On theother hand, Rachlin et al. [1986, 2000] suggested that the certain value of proba-bilistic rewards may be expressed not directly by probabilities but by mean waitingtime, and the form of the waiting-time discount function is similar to that usedin a model of temporal behavior consistent with present bias. This relation hasalso been analyzed in more recent works in Economics [Halevy, 2008, Saito, 2011,Baucells and Heukamp, 2012b, Epper and Fehr-Duda, 2012, Saito, 2015]. Giventhis is a two way relation, none of risk or temporal behaviors have primacy over theother, so any formalization of this relation would necessarily involve the two-wayfeature discussed above. The goal of this paper is to provide a formal characteriza-tion of this relation in the most natural setting. We start by showing how previousattempts at this endeavor have failed to achieve this goal. To be more specific,we show that though the formalization in the direction from certainty effect to di-minishing impatience has been correctly posited in the literature, it is the converserelation that still lacks formal rigor. We provide a formal characterization of thetwo-way relations between i) certainty effect and present bias, and, ii) commonratio effect and the common difference effect. A corollary to our results is that hy-perbolic discounting implies the Common Ratio Effect and that quasi-hyperbolicdiscounting implies the Certainty Effect.The next section provides a brief acknowledgment to the prior unsuccessfulattempts made in this literature to establish risk-time equivalence relations. In Sec-tion 3 we suggest an intuitive extension to the existing notion of diminishing impa-tience, which when used in the analytical framework provided by Halevy [2008],re-establishes the severed connection between non-standard behavior over time andunder risk.3.2 BackgroundThe idea that diminishing impatience (hyperbolic discounting, present bias) maybe related to the certainty of the present and the risk associated with future re-wards, was formalized by Halevy [2008]. In this model, every consumption pathc = (c0,c1,c2, . . .) is subject to a constant hazard rate of termination (r). The de-cision maker chooses among consumption paths as if she calculates present dis-counted utility for every possible length of the path (periods before stopping). Thedistribution over present discounted utilities is then evaluated using Rank Depen-dent Utility (RDU) with probability weighting function g(·), which models prefer-ences that are disproportionately sensitive to certainty. The crucial behavioral ax-403.2. Backgroundiom accommodates dynamic inconsistency between optimal choices at the presentand the immediate future (t = 1) only if there is uncertainty concerning consump-tion in the immediate future, drawing an important qualitative distinction betweenthe effect of randomness in the immediate future and stochastic consumption inlater periods (t = 2,3, . . .).38 Together with other standard axioms on the DM’spreferences over stochastic consumption streams, they are then represented by theutility function:U (c,r) =∞∑t=0g((1− r)t)δ tu(ct) (3.1)where δ is a constant pure time preference parameter and u(·) is her felicity func-tion. The decision maker’s impatience at time t is then the ratio of her discountfunction at periods t and t+1. Halevy [2008] defines diminishing impatience if theimpatience is maximized at t = 0, and Theorem 1 in his paper claims equivalencebetween diminishing impatience and increasing elasticity of g(·). To prove hisclaim, Halevy [2008] proceeds in two steps. First, diminishing impatience holds ifand only if the weighting function satisfies a certain functional inequality.39 Sec-ond, he invokes an equivalence result from Segal [1987, Lemma 4.1] between theabove functional inequality and increasing elasticity of g(·). Saito [2011] correctlypoints out that Segal did not prove that increasing elasticity of the weighting func-tion is necessary for the functional inequality, and provides an example of a DMwho exhibits diminishing impatience but her weighting function’s elasticity is notincreasing (and therefore does not exhibit the common ratio effect).40 Saito [2011]attempts to establish the sought equivalence between present bias and the certaintyeffect (Claim 3 in his paper) by retaining the first part of Halevy’s argument, andnoting that the functional inequality is equivalent (by definition) to the certaintyeffect for RDU.We show that diminishing impatience as defined by Halevy [2008] and used bySaito [2011] does not imply the certainty effect. In light of this new finding, theequivalence results of Halevy [2008] and Saito [2011] reduce to a one-directionalimplication from the domain of risk to the domain of time. We provide details inAppendix.38Which is impossible to draw in a framework in which consumption occurs only in a singleperiod.39The functional inequality is a special case of Kahneman and Tversky [1979, pg. 282] sub-proportionality which characterizes common-ratio violations for RDU. Kahneman and Tversky alsostate the equivalence claimed later by Segal [1987, Lemma 4.1], which is used in the second part ofHalevy’s argument.40In particular, Saito [2011] shows that Tversky and Kahneman [1992] weighting function forgains with γ = 0.5 exhibits diminishing impatience but does not possess increasing elasticity around0 and does not satisfy the common ratio effect.413.3. Definitions and Results3.3 Definitions and Results3.3.1 The Certainty and Common Ratio EffectsLet (x, p) be a lottery that pays x with probability 0≤ p≤ 1 and 0 with probability1− p. A DM exhibits Strict Certainty Effect if for every x,y∈R+ and p,q∈ (0,1):(x,1)∼ (y,q)⇒ (x, p)≺ (y, pq). She exhibits Certainty Effect if for every x,y∈R+and p,q ∈ (0,1): (x,1) ∼ (y,q)⇒ (x, p)  (y, pq) and there exist p,q for whichthe preference is strict. If the DM’s preferences are represented by RDU thenStrict Certainty Effect is equivalent to the following restriction on the weightingfunction:41g(pq)> g(p)g(q) (3.2)A DM exhibits Strict Common Ratio Effect if for every x,y ∈ R+ and p,q ∈ (0,1),` ∈ (0,1]: (x, `) ∼ (y,q`)⇒ (x, p`) ≺ (y, pq`). She exhibits Common Ratio Effectif the implied preference is weak and there exist p,q, ` for which the preference isstrict. If the DM’s preferences are represented by RDU then Strict Common RatioEffect is equivalent to the following restriction on the weighting function:42g(`)g(p`)>g(q`)g(pq`)(3.3)3.3.2 Diminishing ImpatienceWe assume that the DM’s preferences over stochastic consumption paths satisfythe behavioral axioms in Halevy [2008] and are represented by (3.1). The discountfunction at period t is: ∆(t) = β tg((1− r)t) and her (one period) impatience at t is∆(t)/∆(t+1). In Halevy [2008] and Saito [2011], the definition of DiminishingImpatience (DI) is restricted to only one-period delay. It implies that for all naturalnumbers t: ∆(0)/∆(1)> ∆(t)/∆(t+1) which is satisfied if and only if for everyr ∈ (0,1) and t ∈ N:43g((1− r)t+1)> g(1− r)g((1− r)t) (3.4)Both Halevy [2008] and Saito [2011] state without proof that (3.2) holds if andonly if (3.4) holds. Although the direction (3.2)→(3.4) is immediate,44 we pro-vide in Appendix A.2.1 a counter-example which shows that the converse in not41Certainty Effect implies weak inequality in (3.2) for every x,y ∈ R+ and p,q ∈ (0,1) and exis-tence of p,q for which (3.2) is satisfied with strict inequality.42Common Ratio Effect implies weak inequality in (3.3) and existence of p,q, ` for which theinequality in (3.3) is strict.43Note that this is equivalent to writing g(rt+1)> g(r)g(rt) ∀r ∈ (0,1) and t ∈ N.44Define p := 1− r and q := (1− r)t423.3. Definitions and Resultstrue in general. In other words, DI as defined above does not imply the certaintyeffect for arbitrary weighting functions. Intuitively, the certainty effect implies abias towards certainty irrespective of how risky the alternative is, the dual to whichwould be a bias towards the present (t = 0) irrespective of the delay in the com-pared consumption. In evaluating the reason for the severed connection betweentime and risk preferences, we note that the definition of diminishing impatienceused in the literature focuses on a delay of a single period, thus only comparing∆(t) to ∆(t+1) as t increases from 0. This one-period definition fails to generalizeto longer delays, and thus fails to account for present bias behaviorally.45Motivated by the behavioral literature in general, and the quasi-hyperbolic dis-counting model in particular, which focus on the failure of stationarity indepen-dently of the delay under consideration,46 we suggest to compare ∆(t) to ∆(t+ k)for arbitrary k ≥ 1.Definition 21. The decision maker exhibits Delay Independent Diminishing Im-patience (DIDI) if ∆(0)∆(k)>∆(t)∆(t+k) ∀k, t ∈ N, where ∆(t) is the decision maker’s timediscounting at period t.DIDI requires impatience to diminish for all possible delays (k≥ 1), hence is astrengthening of the standard definition,47 which is satisfied by the quasi hyperbolicdiscounting model (see the Proposition below). For preferences represented by(3.1) DIDI holds if and only if for every r ∈ (0,1) and t,k ∈ N: g((1− r)t+k)>g((1− r)k)g((1− r)t).Hyperbolic discounting motivates the definition of Strongly Diminishing Im-patience as ∆(t)∆(t+1)>∆(t ′)∆(t ′+1) ∀t ′ > t ∈ N. Note that Strongly Diminishing Impatiencetoo, is restricted to only one-period delays, and hence similar to Definition 21, westrengthen this measure to be delay independent:Definition 22. The decision maker exhibits Delay Independent Strongly Diminish-ing Impatience (DISDI) if ∆(t)∆(t+k)>∆(t ′)∆(t ′+k) ∀k, t ′ > t ∈ N, where ∆(t) is the decisionmaker’s time discounting at period t.If preferences are represented by (3.1) then DISDI holds if and only if for everyr ∈ (0,1) and t < t ′, k ∈ N\{0}: g((1− r)t)g((1− r)t+k) > g((1− r)t ′)g((1− r)t ′+k) .45For further discussion and intuition see the introductory discussion in Appendix A.2.1.46Halevy [2015] provides a formal definition and recent experimental evidence for stationarity ina dynamic setting.47DI is the special case of DIDI where delay k = 1. An implication of the counter-example pro-vided in Appendix A.2.1 is that DI does not imply DIDI.433.3. Definitions and ResultsProposition. Quasi-hyperbolic discounting satisfies DIDI (but not DISDI), Hyper-bolic discounting satisfies DISDI (and hence DIDI).Proof. In case of quasi-hyperbolic discounting: U = u(c0)+ β ∑∞t=1 δ tu(ct), andfor β < 1:∆(0)∆(k)=1βδ k>βδ tβδ t+k=1δ k=∆(t)∆(t+ k)=∆(t ′)∆(t ′+ k)The last equality holds for t, t ′ > 0. Hence, quasi-hyperbolic discounting satis-fies DIDI, but not DISDI.In Hyperbolic Discounting the discount function for period t is given by ∆(t) =11+ρtfor ρ > 0. For arbitrary k, and t ′ > t,∆(t)∆(t+ k)= 1+ρk1+ρt> 1+ρk1+ρt ′=∆(t ′)∆(t ′+ k)Hence, hyperbolic discounting satisfies DISDI (and hence DIDI).3.3.3 The Relation between Risk and Time PreferencesAs noted above, the effect of risk attitude on intertemporal preferences in (3.1) isstraightforward. We summarize this relation below.Claim. Strict Certainty Effect (3.2) implies Delay Independent Diminishing Impa-tience (DIDI), and the Strict Common Ratio Effect (3.3) implies Delay IndependentStrongly Diminishing Impatience (DISDI).The following Theorem proves the converse direction (though in a weaker formthat does not substantiate an equivalence), that is - how the DM’s intertemporalpreferences determine her risk attitudes.48 The result is direct and comprehensivein the sense that it does not rely on any intermediate connections through properties(e.g, convexity, increasing elasticity) of the weighting function.Theorem. Consider a DM represented by (3.1) with continuous g(·).1. Delay Independent Strongly Diminishing Impatience implies the CommonRatio Effect (and the Certainty Effect).2. Delay Independent Diminishing Impatience implies the Certainty Effect.48Note that although the Theorem does not imply Strict Common Ratio/Certainty Effects, it isinconsistent with expected utility since even the weaker forms imply the existence of probabilitiesfor which (3.2) and (3.3) are satisfied with strict inequality.443.3. Definitions and ResultsProof. (1) Consider a sequence {mini }∞i=1 of rational numbers that converges toln plnq` ,where mi,ni are positive integers. Similarly, consider a sequence {aibi }∞i=1 of rationalnumbers that converges to ln`lnq` , where ai,bi are positive integers. Note thatln`lnq` <1, so we can choose {aibi }∞i=1 such that ai < bi. Now, given this sequences, define asequence {ri}, such that q`= rnibii , that is ri =(q`)1nibi < 1. Note that as aibi convergesto ln`lnq` , rainii = (q`)aibi converges to (q`)ln`lnq` = `. Similarly, as mini converges toln plnq` ,rmibii = (q`)mini converges to (q`)ln plnq` = p.Now using DISDI, ∀i :g(rainii )g(raini+mibii) > g(rnibii)g(rnibi+mibii)Using the continuity of g, as i→ ∞, the Common Ratio Effect follows:g(`)g(p`)≥ g(q`)g(pq`)(2) Let p,q ∈ (0,1) and assume without loss of generality that p< q. Considera sequence {mini }∞i=1 of rational numbers that converges toln plnq , where mi,ni arepositive integers. Given this sequence, define a sequence {ri}, such that q = rnii ,that is: ri = q1ni < 1. Note that as mini converges toln plnq , rmii = qmini converges toqln plnq = p.Now, ∀i :g(rmi+nii ) > g(rmii )g(rnii )g(rmii q) > g(rmii )g(q)Using the continuity of g, Certainty Effect follows: g(pq)> g(p)g(q).Corollary. Consider a DM represented by (3.1) with continuous g(·).1. Hyperbolic discounting implies the Common Ratio Effect (and the CertaintyEffect).2. Quasi-hyperbolic discounting implies Certainty Effect.453.3. Definitions and ResultsIt is important to recall that preferences represented by (3.1) are defined overconsumption streams in discrete time (following Koopmans, 1960).49 It followsthat all notions of diminishing impatience (as DI, DIDI, DISDI) are required tohold only for natural numbers, while risk preferences (Certainty Effect, CommonRatio Effect) are defined over lotteries with probabilities in the simplex. With thisinsight, it is not surprising that properties of risk preferences manifest themselves inthe time domain. The counter-example in the Appendix together with the Theoremdemonstrate that the opposite direction can be established as well, but the notionof diminishing impatience must be appropriately defined so it will not be delaydependent. We believe that these new notions (DIDI and DISDI) are very intuitiveand reflect the natural meaning of diminishing impatience. Moreover, in light ofrecent work generalizing hyperbolic discounting to continuous time [Webb, 2016]we conjecture that continuous adaptations of DIDI and DISDI will be required inorder to create the link from time to risk, though this remains for future work asthe behavioral underpinning of (3.1) are stated in discrete time.49This framework is considerably different from Fishburn and Rubinstein [1982] whose domainincludes payments of $x at time t, which is applicable to more selective environments (as bargaining).46Chapter 4Drivers of Cooperation in FinitelyRepeated Prisoner’s DilemmaPrisoner’s dilemma is probably one of the most popular games in economics as itpits two fundamental human actions against one another - Cooperation and Defec-tion. From human-beings to competing firms, from feuding countries to animalsin the eco-system- wherever there is a possible interaction of interests, one oftenfaces a choice between cooperation and defection against others, the latter com-ing at a personal gain and social cost. As a result, many economic and strategicinteractions we see around us can be represented simply as finitely repeated Prison-ers dilemma (FRPD) games. One standard result from Microeconomic Theory isthat under standard assumptions, no cooperation can be supported in any subgameperfect Nash equilibrium of a Finitely Repeated Prisoner’s dilemma(FRPD) game.Column PlayerDefect (F) Cooperate (C)Row PlayerDefect (F) b,b c,dCooperate (C) d,c a,ac>a>b>d>0This paper investigates the question of how the gains from potential future peri-ods (shadow of the future) determine behavior in FRPD games. We vary the futuregains in a 5-period FRPD game by imposing (and varying) exponential discount-ing (δ t) over periods t = 1−5 of the 5-period game. The discount factors used areδ = 1, 3/4, 3/8, 1/4. As the discounting increases (and δ decreases), for any fixedfirst-period payoff matrix, the payments diminish at a higher rate across rounds 1to 5. In a few of our treatments, we also ask subjects about their beliefs on theirpartners’ actions, to pin down the driving force behind cooperation and defectionin the games, and see how subject beliefs react to their exxperiences. The gamehorizon and payoffs are chosen in such a way that an egoist (a player with is onlyconcerned about maximizing payoffs subject to beliefs) would never cooperate5050See Appendix A.3474.1. An overview of the literatureunder the δ = 1/4 treatment. Such an egoist would also never cooperate in the lasttwo periods of the δ = 3/8 treatment, and never cooperate at all if she believesthat the her partner plays any variety of threshold strategies 51. Cooperation underreputation equilibrium Kreps et al. [1982] is possible in the δ = 1, 3/4 treatments,and it should decrease with δ . The predictions from egoistic cooperation drivenby reputation contrast sharply with that from theories that assume that subjects’actual utilities are determined by joint strategies and payoffs of them and theirpartners. For example, any theory that assumes that subjects get a fixed boost inutility from playing kind/ altruistic (cooperative) strategies would suggest that co-operation might increase disproportionately in later rounds of the low δ treatments.Similarly, Rabin [1993] would suggest that as long as reciprocity minded peopleput a high enough belief on their partner being kind in their actions, they mightreciprocate with kindness too, as long as the utility gains from their kind actionsupersede the utility loss from getting a sub-optimal payoff. So, with the mone-tary loss from sub-optimal actions diminishing in the low δ treatments (especiallyin the later rounds), subjects are more likely to be kind, especially if they expectkindness from their partners.We find the following results: 1) Cooperation in the first period of a FRPDgame decreases monotonically the more the future is discounted. First period co-operation is highly correlated with subjects’ beliefs on their partners’ cooperatingin the same period, and, their partners’ propensity to cooperate in the future inresponse to cooperation. 2) Higher the discounting (and lower the final perioddiscounted stakes), higher is the observed final-period cooperation, and this is ro-bust to subjects gaining a considerable amount of game-experience. Final periodcooperation is also highly correlated with the subject’s belief of their partners coop-erating in the final period. 3) Subjects systemcatically over-estimate their partner’spropensity of engaging in cooperative behavior. 4) Reported beliefs are consistentwith reasonable learning, and move in predictable directions in response to goodand bad outcomes. 5) Justifying aggregate subject behavior requires the use of bothegoistic and altruistic theories.In Section 4.1, we provide an overview of the related literature. Section 4.2 dis-cusses the experimental design. Section 4.3 provides the main results and analysisfrom the experimental data, and Section 4.4 concludes.4.1 An overview of the literatureThere have been many experimental studies about both finitely and infinitely re-peated Prisoner’s Dilemma games. Below we describe the relevant literature sepa-51Strategies that conditionally cooperate till some period, and revert to always defecting thereafter.484.1. An overview of the literaturerately for studies on finitely repeated PD games and infinitely repeated PD games.The effect of the scope of future cooperation on current behavior has been stud-ied in detail in the domain of infinitely repeated PD games. Infinitely repeated PDgames are implemented in the lab using a random termination protocol, i.e, aftereach period, the game ends with some predetermined probability. This probabilityof continuation is a dual of the δ used in our setting, under Expected Utility, andit determines the shadow of the future in infinitely repeated PD games. Roth andMurnighan [1983] vary the probability of continuation in their experimental settingand find that higher the probability, the greater the number of players who cooper-ated in the first round of the game. Bo [2005] replicates that higher continuationprobabilities result in higher cooperation levels, and additionally shows that it isnot just the higher number of expected periods of play, but the higher probabilityof repeated interaction that drives this behavior.52In the following, we will discuss the experimental literature that studies thedeterminants of cooperation in FRPD games. Andreoni and Miller [1993] con-trol subjects’ beliefs over the value of building a reputation (Kreps et al. [1982])by varying the probability that subjects interact with a pre-programmed opponent(a computer that plays a Tit-For-Tat strategy). In their study, cooperation fallsthrough the rounds of FRPD and higher beliefs about playing the computer aremore conducive to higher cooperation. Cooper et al. [1992] compare behavior inone-shot PDs to that in FRPDs and observe higher cooperation rates in the FRPD.The authors find evidence of both reputation building and altruism and they con-clude that neither model can explain all the features of the data on its own. Thereis some dispersed evidence about how cooperation in FRPD might be affected bythe shadow of the future. Bereby-Meyer and Roth [2006] find more cooperation inround one of FRPDs than in the one-shot games, which is equivalent of compar-ing first round cooperation rates of δ = 1 and δ = 0 in our setting. In the FRPDgames conducted by Bo [2005] first round cooperation rates are higher in gameswith a longer horizon, consistent with the hypothesis that shadow of the futuremight drive cooperation even in FRPD games. Embrey et al. [2015] identify howthe value of cooperation can be captured by the “size of the basin of attraction ofAlways Defect”, and how it is an important determinant of cooperation in FRPDgames in the previous literature. Beside their comprehensive meta-study, they alsodesign a new experiment that compares two treatments in which the horizon of therepeated game is varied, but the value of cooperation is kept constant. One canthink of our experiment as a dual to theirs, as we keep the horizon of the repeated52The paper compares first period cooperation rates from infinitely repeated games with x numberof expected periods to that of finitely repeated games having x periods, and finds that the former isbigger.494.2. Experimental designgame constant, but vary the value of cooperation. Charness et al. [2016] show thathigher monetary payoffs from cooperation are associated with substantially highercooperation rates in one shot PD games.There is also some work about how beliefs might evolve and affect PD play.For example, Kagel and McGee [2016a,b] have both individual play and team playin their FRPD games and, analysis of team dialogues show signicant discrepanciesbetween subjects’ inferred beliefs and those underlying standard models of cooper-ation in the FRPD. Cox et al. [2015] reveal second-mover histories from an earliersequential-move FRPD game to the first-mover. They unexpectedly find highercooperation rates when histories are revealed. They also provide an accompany-ing theory in which players decide on conditional cooperation based only on naiveprior beliefs about what strategy their opponent is playing.4.2 Experimental designA total of 132 subjects participated in 5 sessions between November 19, 2015 toDecember 3, 2015, where subjects played 5-period FRPD-D games, for the valuesδ = 1, 3/4, 3/8, 1/4. Each subject played under each of the four treatments in theseWithin sessions. Sessions lasted approximately one hour and a total of 132 subjectsparticipated in these sessions. Within any game, as the rounds progressed, the stagepayments diminished according to the particular δ employed in that game. Thepayoff matrix for the first period for any treatment was fixed at:Column PlayerDefect CooperateRow PlayerDefect 1200,1200 2600,200Cooperate 200,2600 2000,2000Subjects were be divided into two groups. In each match, a subject from thefirst group was matched with a new subject from the second group using turn-pikematching. Two subjects would never meet in more than a game. Within everygame or match, the subjects could see the past actions taken by their partner, butthey could not see their partner’s actions in previous games. This informationprotocol coupled with the matching procedure ensured that actions taken withina match or game should not influence their or their partner’s behavior in futurematches. The fact that no information of previous matches was provided to theirnew partners and the payoff structures in each treatment was common knowledgefor the subjects. The block of treatments were repeated twice in the same order,504.2. Experimental designso, each subject played 2 matches under each treatment, thus playing a total of(2× 4) = 8 matches/ games. At the end of the experiment, one of the 8 gameswas randomly chosen, the total points or lab currency earned by the subject in thatgame was converted into dollars at an exchange rate of 300 points = $1 and paid tothe subject. The subjects also received a $5 show-up fee.The order of treatments in the Within sessions was randomized at the sessionlevel: At the beginning of each session, a coin toss by the experimenter decided ifthe treatments in that session were arranged in the orderδ = 1, 3/4, 3/8, 1/4, 1, 3/4, 3/8, 1/4(block of treatments repeated twice) or in the opposite order. Over all, 3 sessionswere ran in the former order and 2 in the latter order. A considerable time wasspent at the beginning of each session, to make sure that the subjects understoodthe game, the payment scheme, the matching protocol and the interface thoroughly.The subject instructions and screen shots of the GUI are included in the Appendix.As a robustness checks for the results, four more sessions were run, with Be-tween design, two each for δ = 1/4 and δ = 3/4. In the four Between sessions (heldin April, 2017) the subjects played 8 games under a single δ , thus giving themmore time to learn about the particular treatment, and eliminating any possibilityof cross-treatment effects. The subjects were also asked to answer the followingfour prediction questions at the start of each game with a new partner:• How likely is your partner to play L on the first round of the this game?• How likely is your partner to play L in the very next round if you played Tin the previous round of the game?• How likely is your partner to play L in the very next round if you played Bin the previous round of the game?• How likely is your partner to play L on the very last (5th) round of the thisgame?At each question the subjects could respond on a scale of 0 to 10, and they wereadvised to enter a higher number the more likely they thought the event was. Theywere also provided the following the following reference points:• A response of 0 (lowest point of the scale) would mean "never".• 5 (midway point of the scale) would mean "as likely as getting Heads on afair coin toss/ 50-50 odds",514.3. Results• 10 (right extreme of the scale) would mean "surely".• Events more likely than “never” and less likely than heads on a fair coin toss,should be rated between 0 and 5, and so on.There was a separate paragraph in the instructions which the subjects were advisedto read only if they were more comfortable in thinking of likelihoods in termsof probabilities. This para linked how their probability assesments would map toresponses on the 0−10 scale. The prediction tasks were not incentivized, to makesure that prediction incentives could not influence FRPD-D play in any way. Atotal of 90 subjects participated in the Between sessions. The payment schemeand exchange rate in the Between treatment was identical to those in the Withintreatment, and the subjects received $6 as show-up fee.There is one more crucial aspect of the design that deserves separate men-tion. The norm in the experimental literature on repeated games is to model theshadow of the future by varying the continuation probabilities, and tie the exper-imental findings with theoretical predictions by assuming that subjects calculateexpected returns in accordance with Expected Utility Theory. Under ExpectedUtility Theory, the continuation probability is equivalent to a discount factor im-posed on future rounds of a repeated game. This norm runs almost in denial ofthe literature (Kahneman and Tversky [1979], Tversky and Kahneman [1992]) thatrejects Expected Utility Theory (EUT) based on lab evidence. Our decision to im-pose discounting instead of random termination was taken in acknowledgement ofthe rich set of findings that indicate that subjects frequently violate EUT in theirbehavior.4.3 ResultsOur analysis will be presented in three parts. We will start by analysis of the datafrom the Within sessions, then show how the main results about cooperation werereplicated in the Between session, and then finally describe the rich beliefs datafrom the Between sessions.Twenty seven of the total 132 subjects in the Within sessions always defect,whereas only one subject always cooperates. Hence, there is very little evidenceof subjects who play the game as if cooperation was a dominant strategy in thesub-game. Fifty four of the subjects cooperate only thrice among their fortydecisions. Average cooperation across the 5 periods of the Within sessions ishighest in the δ = 1 treatment, and is lower for all lower values of δ (Table 4.1,Column 1). There is very little evidence of any change in total cooperation for524.3. Resultstreatments with δ ∈ (0,1). One could interpret the data as if the discount factor ofδ = 1 is saliently different from all discount factors in terms of being a driver oftotal cooperation. This mirrors Kahneman and Tversky [1979]’s finding that thereis a discrete change in risk behavior when moving from certainty (probabilityp = 1) to risky options, but preferences are less sensitive to moving between twodifferent risky options. The average percentage cooperation in the δ = 1 treatmentis close to what was obtained (23.78%) by Bo [2005] over 10 matches in his4-period FRPD experiment.Table 4.1: Total Cooperation by treatments(1) (2)All data Block 2d1 22.20∗∗∗ 20.91∗∗∗(2.301) (2.704)d2 15.15∗∗∗ 12.88∗∗∗(1.789) (2.009)d3 15.61∗∗∗ 13.48∗∗∗(1.830) (2.231)d4 15.08∗∗∗ 13.64∗∗∗(1.720) (2.198)N 5280 2640d1_d2 0.0000570 0.000911d2_d3 0.775 0.769d3_d4 0.742 0.946Standard errors in parentheses∗ p< 0.05, ∗∗ p< 0.01, ∗∗∗ p< 0.001The same results holds when we we allow for learning and only consider thesecond block of the treatments (Column 2 in Table 4.1) in our analysis. The aver-age cooperation in the second block is lower than the average over first and secondblocks.Though the treatments of δ = 3/4, 3/8, 1/2 have very similar aggregate cooperationlevels, the strategy profiles by periods look very different across all the three treat-ments. Figure 4.1 describes the evolution of cooperation throughout the periods534.3. ResultsFigure 4.1: Evolution of cooperationfor the different peroids. The cooperation levels aggregated across all sessions areindeed ranked period by period till the first two periods. The difference in coop-eration among the treatments vanishes around the third period and thereafter flipssigns! This is why the difference between cooperation levels at δ = 3/4, 3/8, 1/2vanishes when all five periods are aggregated.The cooperation rates decline progressively and sharply across the periods inthe δ = 1, 3/4 treatments, whereas, they are stable in the lower δ treatments. Forexample, very few people start by cooperating in δ = 1/4 treatment, but the levelof cooperation remains stable throughout the later stages of a game. This is insharp contrast to the cooperation profile in the δ = 1 treatment, where the percent-age cooperation is in a steady decline throughout. This results in a cross-over ofcooperation rates between the two treatments at period 5, which explains why theδ < 1 treatments result in similar aggregate levels of cooperation. The percentagecooperation in the first period is decreasing in δ . The differences between the firstand terminal period cooperation levels across the treatments is presented in Table4.2.The first round cooperation rates are decreasing in δ , and the differences aresignificant for all pairs of discount factors, other than the smallest two, which are544.3. ResultsTable 4.2: Comparison of first and terminal periodsPeriod 1 Period 5δ1 = 1 33.71∗∗∗ 9.85∗∗∗(3.55) (1.97)δ2 = 34 26.14∗∗∗ 6.06∗∗∗(3.12) (1.70)δ3 = 38 20.08∗∗∗ 12.88∗∗∗(2.84) (2.32)δ4 = 14 17.42∗∗∗ 16.67∗∗∗(2.64) (2.45)N 1056 1056δ1 = δ2 .009 .11δ2 = δ3 .03 .006δ3 = δ4 .14Standard errors (clustered at subject level)are reported in parentheses belowLower panel reports p-values from F test for H0 : δi = δ j∗ p< 0.05, ∗∗ p< 0.01, ∗∗∗ p< 0.001554.3. Resultsunderpowered. This is consistent with the meta-analysis in Embrey et al. [2015]which finds that across experimental studies, the cooperation in the first period isincreasing in the length of the horizon, another metric for shadow of the future. Onthe other hand, the percentage cooperation in the last period of the δ = 1/4 treat-ment is significantly higher than those of the δ = 1 and δ = 3/4 treatments.Table 4.3: Cooperation by treatment and periodPeriod1 Period2 Period3 Period4 Period5 Average1 33.71 25.76 23.11 18.56 9.85 22.202 26.14 16.67 15.53 11.36 6.06 15.153 20.08 13.26 14.77 17.05 12.88 15.614 17.42 11.36 14.02 15.91 16.67 15.08Total 24.34 16.76 16.86 15.72 11.36 17.01Part of this cross-over can be attributed higher propensity of subjects cooperat-ing in the terminal period of the game in the δ = 1/4, 3/8 treatments. As consistentwith previous experiments, a big proportion of the subject pool (62 out of the 132total subjects) make such non-egoistic choices at least once, but only 17 subjectsdo it more than once. But this cannot be the whole story behind the cross-overpattern, given the cooperation rates start converging across treatments well beforethat. Below we study some of the other behavioral channels that could be drivingthe cross-over in cooperation rates.We call a choice Forgiving, if a subject cooperates while her opponent played De-fect against her cooperative move in the last period. Recooperating implies that asubject switched back to cooperation in the current period after Defecting in thelast period. Both Forgiving and Recooperating play are behaviors which wouldincrease cooperation.Similarly, we define Unfolding as responding to Cooperate-Cooperate in the previ-ous period by Defaulting in the current period. This behavior results in unfoldingthe Default-Default equilibrium one eventually expects under reputation theories,and hence the name. In the table below we report the relative frequencies and totalpossible instances of these three kinds of behavior by the Treatments.As one would expect from the previous results, subjects are significantly more for-giving in the lower δ treatments. For example, they are twice as likely to forgive inthe δ = 1/4 treatment than the δ = 3/4 treatment. 82 out of the 132 subjects indulgein Forgiving behavior at least once. There are insignificant minor differences in564.3. ResultsRecooperating and Unfolding behavior. Forgiving seems to be the major drivingforce behind the cooperation rates flipping in later periods. Here is a rough wayto understand this: Compared to δ = 1/4, there are respectively 43 and 23 moreinstances of cooperation in first period in treatments δ = 1 and δ = 3/4. Thesedifferences are almost single handedly overcome by 33 and 41 instances of addi-tional instances of Forgiveness observed in the δ = 1/4 treatment compared to thethe other two aforementioned treatments. 53In the four treatments, 82 of the total132 subjects commit Forgiving behavior at least once.The more frequent forgiving behavior at the lower δ treatments implies that smallerthe payment horizon gets, the more forgiving people are. The fact that the smallerδ treatments have smaller cooperation rates in the earlier periods and have similarrates of Reciprocal play means that it is most likely the lower costs of Forgivingthat is driving the crossing over of cooperation rates for the higher periods.Table 4.4: Relative Frequency of Forgiving , Reciprocal, and Unfolding playTreatments Forgiving Recooperating Unfolding1 0.07 0.21 0.17(789) (117) (150)2 0.05 0.23 0.16(872) (114) (70)3 0.07 0.28 0.14(884) (98) (74)4 0.10 0.22 0.14(901) (105) (50)Total 0.07 0.23 0.15(3446) (434) (344)The top entry reports the relative frequency of behaviors byTreatment. The total number of possible observations is reportedin brackets below. Treatments 1-4 are in decreasing order from1 to 1/4.Results on cooperation from Between study:The between study replicates the central finding of cross-over of cooperationtrends from the Within study, as shown in Fig 4.2. The average cooperation inthe δ = 3/4 treatment is relatively stable, whereas cooperation falls steadily in theδ = 1/4 treatment. In Table 4.5, we compare the first and last period cooperations53Fudenberg et al show that subjects are “Slow to anger and fast to forgive” in PD games whereactions are noisy. We see a similar trend here.574.3. ResultsFigure 4.2: Evolution of cooperation (All games from Between Session)of the two treatments for games while allowing for learning in the initial periods.The cross-over pattern is still highly significant.Beliefs in the Between study:In the following, we analyze the responses to the four likelihood questions. Foreach subject, let the quadruple (g1, g2, g3, g4) contain responses to the followingfour questions on the 0−10 scale:• How likely is your partner to play L in the very next round if you played Tin the previous round of the game?• How likely is your partner to play L in the very next round if you played Bin the previous round of the game?• How likely is your partner to play L on the very last (5th) round of the thisgame?Firstly, the reported likelihoods/ beliefs seem consistent with learning. For exam-ple, subjects weakly decrease their reported g1 (∆g1 ≤ 0) after a bad experience(their partner defecting in the first period of the last game) in 89% of all possibleoccasions and weakly increase g1 after a good experience (their partner cooperatingin the first period of the last game) on 86% of all occasions. These percentages are584.3. ResultsTable 4.5: Comparison of first and terminal periodsGames 2-8 Games 4-8Period 1 Period 5 Period 1 Period 5δ2 = 34 40.48∗∗∗ 11.90∗∗∗ 40∗∗∗ 9.52∗∗∗(6.12) (2.78) (6.25) (2.73)δ4 = 14 23.80∗∗∗ 21.73∗∗∗ 18.75∗∗∗ 18.33∗∗∗(4.72) (4.06) (4.80) (4.19)δ2 = δ4 .03 .049 .008 .08Standard errors (clustered at subject level) are reported in parentheses belowLower panel reports p-values from F test for H0 : δi = δ j∗ p< 0.05, ∗∗ p< 0.01, ∗∗∗ p< 0.001Table 4.6: Change in beliefs and Forgiving across games (1-8)Quarter Three Quartersg1 g2 g3 g4 Forgiving g1 g2 g3 g4 Forgiving(game-1) -.41 -.26 -.10 -.36 -.007 -.19 -.26 -.13 -.23 -.013(.07) (.07) (.06) (.07) (.005) (.07) (.07) (.06) (.06) (.004)constant 5.3 6.5 2.4 5.2 .14 4.9 5.8 2.8 3.8 .12(.39) (.39) (.26) (.38) (.03) (.37) (.36) (.29) (.37) (.02)N 384 384 384 384 1194 336 336 336 336 96990% and 91% respectively in case of g4. Note that ∆gi = 0 is consistent with learn-ing, as we only observe subject responses on a discrete grid, and for small changesin gi we might not see any changes in their reported beliefs. We can summarize theevolution of beliefs across the games by running a regression of the beliefs againstthe variable (game-1). The coefficient of regression can be read as the averagechange in beliefs after every passing game, whereas the coefficient on “constant”provides us average beliefs at the start of the session. As seen in Table 4.6, on av-erage, subjects get more pessimistic about their partners as the session goes on andmore games are played. Our finding are in contrast to Cox et al (2015) who findthat subjects might have unsophisticated priors. Further, the learning of thresholdstrategies (that defection should be followed by defection) is slower in the Quartertreatment. There only seems to be significant learning away from Forgiving in theδ = 3/4 treatment only.In Figure 4.3, we plot the evolution of Period 1 belief g1 (and and Period 5belief g4) against the average cooperation in that period, across the two treatments,594.3. ResultsFigure 4.3: Average Beliefs (g1,g4) vs Average cooperation across Games 1-8Table 4.7: Behavior and beliefsReported belief N % Coopg10-4 358 16%5 198 48%6-10 164 53%g40-4 429 13%5 155 21%6-10 136 30%throughout games 1−8. Under the assumption that the subjects report their prob-abilistic assessments about the population in their response, we can meaningfullycompare it to the actual average response of the population. Other than in the caseof first period cooperation in the Three quarters treatment, subjects systematicallyoverestimate the how often cooperation takes place.Optimistic beliefs about partner’s actions are also highly associated with coop-erative behavior by the players themselves, as we show in Table 4.7. Given thatsubjects were provided reference points for 0, 5, 10 on their response scale, weuse the most natural way to tabulate the belief data, and look at subject responses.Fisher’s exact test and the chi-squared test result in a rejection of the null of equalrelative proportions of cooperation with p-values of zero for the tabulations, andsuggest that more optimistic a subject was about her partner’s responses, the morelikely they were to cooperate.To take the analysis a step further, in Table 4.3 we run a logit regression of604.3. Resultsfirst and last period cooperation on the self-reported beliefs, game and treatmentdummies. The standard errors are clustered at the subject level, as in the rest of thisstudy. Even after controlling for the treatments and the games, beliefs over partnercooperating in the first period and partner’s propensity to reciprocate to coopera-tion are significant drivers of first period cooperation. Also, as expected higher theodds of partner cooperating after facing a defection, lower is the cooperation inthe first period. On the other hand, it seems that among the belief variables, beliefabout partner cooperating in the final period is the sole determinant of last periodcooperation. This result is highly intuitive. We have focussed on forgiving behav-ior from only the first 4 periods to make sure we do not double account Period 5behavior. Forgiving behavior (in the first four periods) decreases with increasingbeliefs about partner cooperating in the first period, and increases with higher be-liefs about the partner cooperating in the last period. The former is consistent withpossible disenchantment driven aversion to forgiving, and the latter is consistentwith forgiving being driven by optimism about partner’s future play.614.4. Conclusion4.4 ConclusionThis paper povides a systematic analysis of how the the shadow of the future mightaffect cooperation in FRPD games. The discount factor, which is a measure ofthe shadow of the future is a significant determinant of first-round cooperation inFRPD games. First period cooperation decreases as the discount factor decreases,consistent with reputation play. But, latter period cooperation is instead drivenby behavior consistent with theories of altruism, fairness and reciprocal kindness(Rabin [1993]), and cooperation increases with decreasing discount factor. We findthat cooperation in both the first and the last period is driven by beliefs about thepartner reciprocating. We also find that subjects are generally over-optimistic intheir beliefs about their partners, but their beliefs move in reasonable directionswhen confronted with good or bad news. Finally, the subject population containsboth subjects who look like they are playing egoistic reputation equilibrium, andplayers whose utility respond to altruistic and reciprocity motives.624.4. ConclusionTable 4.8: Logit regressions on belief variables and game dummies(1) (2) (3)Coop in Period 1 Coop in Period 5 Forgiving in Period<5main1.game 0 0 0(.) (.) (.)2.game -0.0615 0.206∗ -0.261∗(0.254) (0.309) (0.150)3.game -0.299∗ 0.103 -0.337(0.282) (0.322) (0.195)4.game -0.252∗ -0.508∗ -0.660∗∗∗∗(0.291) (0.402) (0.179)5.game -0.0443∗ -0.566∗ -0.533∗∗(0.286) (0.376) (0.214)6.game -0.408∗ -0.343 -0.297∗(0.316) (0.384) (0.187)7.game -0.368∗ -0.613∗ -0.489∗∗∗(0.317) (0.387) (0.188)8.game -0.389∗ -0.271∗ -0.601∗∗∗(0.314) (0.346) (0.200)T 0.902∗∗ -0.661∗∗ -0.139(0.386) (0.319) (0.178)g1 0.216∗∗∗∗ -0.0437 -0.0555∗(0.0610) (0.0479) (0.0324)g2 0.159∗∗ 0.0241 0.0166(0.0636) (0.0533) (0.0303)g3 -0.0892∗ -0.0210 0.0255(0.0807) (0.0676) (0.0411)g4 0.0956 0.128∗∗ 0.0658∗(0.0705) (0.0572) (0.0360)_cons -2.915∗∗∗∗ -1.431∗∗∗ -1.081∗∗∗∗(0.593) (0.461) (0.260)N 720 720 1594Standard errors in parentheses∗ p< 0.1, ∗∗ p< 0.05, ∗∗∗ p< 0.01, ∗∗∗∗ p< 0.00163ConclusionThe first chapter provides a novel, testable weakening of classical assumptions toderive a new theory that would best fit Present Biased behavior. We show that anypresent-biased preference has a max-min representation, which can be cognitivelyinterpreted as if the decision maker considers the most conservative present equiv-alents in the face of uncertainty about future tastes. We compare our theory withexisting theories of temporal behavior (Koopmans [1972], Fishburn and Rubin-stein [1982], Harvey [1986], Laibson [1997], Ebert and Prelec [2007]) to discussthe dimensions along which it is an improvement over the latter. We also discusshow previously unsupported behavioral anomalies from the domain of time andrisk can now be addressed through this new theory, and how knowledge of suchanomalous behavior can in turn be used for better policy design. The third chapterof the thesis shows how time-delay and risk are behavioral duals of one another,and a formal study of any one also conveys fundamental insights about the other.Though this duality had been hypothesized previously in the literature (Green andMyerson [2004], Baucells and Heukamp [2012b], Halevy [2008], Saito [2015]), weprovide the first formal derivation of the duality relationship. We show how underthe assumptions of Non-Expected Utility and constant hazard rate, one can derivea bias from the present or a bias for certainty from one another, thus completingthe characterization result in Halevy [2008]. In chapters one and three, instead ofstudying behaviors under risk or time-delay in isolation, we provide a more holistictheory of human behavior by studying the joint interplay of uncertainty and timeas influencing factors.Chapters two and four contribute to the literature on the empirical study of pref-erences. Chapter two uses a meta-study over recent influential experimental papersto inform the design of future experiments investigating temporal-preferences. Weprovide desiderata of choice consistency that experimenters should employ whileestimating time preferences from choice data. We also show how the applicationof our desiderata can help us learn new insights from recent experimental studies.Chapter four introduces a novel experimental design to study the effect of tempo-ral delay (discounting) on human interaction in an environment where there is atradeoff between individual gain and social surplus. We find that subject behavioris driven by a combination of altruistic and selfish motives, and selfish motives64Conclusionsometimes drive cooperative play, especially when the future gains from coopera-tion is large enough. Altruistic play is common only when returns from selfish playare very low. 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Thefollowing two statements are equivalent:54We take the idea of tabular presentation from Abdellaoui et al (2010).Model Author(s) ∆(t)0 Exponential discounting Samuelson (1937) (1+g)−t ,g> 01 Quasi-hyperbolic discounting Phelps and Pollak (1968) (β +(1−β )t=0)(1+g)−t ,β < 1,g> 02 Proportional discounting Herrnstein (1981) (1+gt)−1,g> 03 Power discounting Harvey (1986) (1+ t)−α ,α > 04 Hyperbolic discounting Loewenstein and Prelec (1992) (1+gt)−α/γ ,α > 0,g> 05 Constant sensitivity Ebert and Prelec (2007) exp[−(at)b],a> 0,1> b> 0Table A.1: Models of temporal behavior73A.1. Appendix to to Chapter 1i) The relation % defined on X×T satisfies properties A0-A4.ii) For any δ ∈ (0,1) there exists a set Uδ of monotinically increasing contin-uous functions such thatF(x, t) = minu∈Uδ(u−1(δ tu(x)))represents the binary relation%. Moreover, u(0) = 0 and u(M) = 1 for all u ∈Uδ .F(x, t) is continuous.Proof: We start by showing (ii) implies (i). To show Weak Present Bias, wefollow the following steps(y, t)% (x,0)=⇒ minu∈Uδ (u−1(δ tu(y)))≥minu∈U (u−1(u(x)))=⇒ minu∈Uδ (u−1(δ tu(y)))≥ x=⇒ u−1(δ tu(y))≥ x ∀u ∈Uδ=⇒ δ tu(y)≥ u(x) ∀u ∈Uδ=⇒ δ t+t1u(y)≥ δ t1u(x) ∀u ∈Uδ=⇒ u−1(δ t+t1u(y))≥ u−1(δ t1u(x)) ∀u ∈Uδ=⇒ minu∈Uδ (u−1(δ t+t1u(y)))≥minu∈Uδ (u−1(δ t1u(x)))=⇒ (y, t+ t1)% (x, t1)To show Monotonicity and Discounting, let us show (x, t)  (y,s), when, eitherx > y and t = s, or, x = y and t < s. As all the functions u ∈ Uδ are strictlyincreasing, and δ ∈ (0,1),δ tu(x) > δ su(y) ∀u ∈Uδ⇐⇒ u−1(δ tu(x)) > u−1(δ su(y)) ∀u ∈Uδ⇐⇒ minu∈Uδu−1(δ tu(x)) > minu∈Uδu−1(δ su(y))⇐⇒ (x, t)  (y,s)For proving the second statement under Discounting, start with any u1 ∈ Uδ . Forz> x> 0, and δ ∈ (0,1) there must exist t big enough such that74A.1. Appendix to to Chapter 1u1(x) > δ tu1(z)⇐⇒ u−11 (u1(x)) > u−11 (δ tu1(z))⇐⇒ x > minu∈Uδu−1((δ tu(z))Hence, there exists t big enough such that (x,0) (z, t).That % satisfies continuity follows directly from the definition of continuity on theutility function.Now, we will prove the other direction of the representation theorem. We will firstdeal with the case of T=[0,∞). A similar proof technique would be used in theproof of Theorem 8.Proof for the case when T=[0,∞).Proof. For every x∗ ∈ (0,M), we are going to provide an increasing utility functionux∗ on [0,M] which would have δ τux∗(x)≥ ux∗(y) if (x,τ)% (y,0). Additionally itwould also have δ tux∗(xt) = ux∗(x∗) for all (x∗,0)∼ (xt , t).Fix ux∗(x∗) = 1, ux∗(0) = 0.For any x ∈ (x∗,M], by Discounting there exists a delay T large enough, such that(x∗,0) (x,T ). Hence, it must be true that (x,0) (x∗,0) (x,T ). By Continuitythere must exist t(x) ∈ T such that, (x, t(x))∼ (x∗,0). Define the utility at x asux∗(x) = δ−t(x) (A.1)It would be helpful to introduce the following additional notation to move seam-lessly between prizes and time in terms of indifference of time-prize pairs w.r.t(x∗,0). For t > 0, define xt as the value in (x∗,M] such that (xt , t)∼ (x∗,0). Usingcontinuity, we can say that all points in the interval (x∗,M] can be written as xt forsome t > 0. This notation essentially implements the inverse of the t(x) functiondefined in the previous paragraph.Now, for x ∈ (0,x∗), defineux∗(x) = inf{δ τ : There exists t such that (xt , t+ τ)∼ (x,0)} (A.2)Firstly, we will show that the infimum in (A.2) can be replaced by minimum. Letthe infimum be obtained at a value I = δ τ∗ . Consider a sequence of delays {τn} thatconverge above to τ∗, and (xtn , tn+ τn)∼ (x,0). Clearly, {tn} is the correspondingsequence of t’s in (A.2). Note that tn ∈ [0, tmax] where (x∗,0) ∼ (M, tmax). Hence,{tn} must lie in this compact interval, and must have a convergent subsequence75A.1. Appendix to to Chapter 1{tnk}. If t∗ is the corresponding limit of {tnk}, we know that t∗ ∈ [0, tmax]. Similarly,xt can be considered a continuous function in t (this also follows from the continu-ity of%). Therefore, xtnk → xt∗ when tnk→ t∗. Thus, we have (xtnk , tnk +τnk)∼ (x,1)for all elements of {nk}. As, nk → ∞, xtnk → xt∗ , tnk + τnk → t∗+ τ∗. Then, usingthe continuity of %, (xt∗ , t∗+ τ∗)∼ (x,1). Hence, the infimum can be replaced bya minimum.Now we will show that the utility defined in (A.1) and (A.2) has the followingproperties : 1) It is increasing. 2) δ tux∗(xt) = ux∗(x∗) for all (x∗,0) ∼ (xt , t). 3)(x,τ) % (y,0) implies δ τux∗(x) ≥ ux∗(y), 4) u is continuous. The first two proper-ties are true by definition of u. We will show the third and fourth in some detail.Consider (x,τ)% (y,0). In the case of interest, τ > 0 and hence, x> y.Now let x > y > x∗. Let, u(y) = δ−t1 , which means, (y, t1) ∼ (x∗,0) . Given(x,τ)% (y,0), we must have(x,τ+ t1)% (y, t1)∼ (x∗,0)Hence, if (x, t2)∼ (x∗,0), then,t2 ≥ τ+ t1⇐⇒ ux∗(x) = δ−t2 ≥ δ−(τ+t1)⇐⇒ δ τux∗(x) ≥ δ−t1 = ux∗(y)If, x> x∗ > y, the proof follows from the way the utility has been defined.Let y < x < x∗. Let, ux∗(x) = δ t1 , which means, (xt , t + t1) ∼ (x,0) for some xt ∈[x∗,M]. Given (x,τ)% (y,0), we must have(xt , t+ t1+ τ)% (x,τ)% (y,0)Hence, ux∗(y)≤ δ τ+t1 = δ τux∗(x).Now we turn to proving the continuity of ux∗ . The continuity at x∗ from the right,or on (x∗,M] is easy to see.Next, for any r = δ s ∈ (0,1), definef (r) = sup{y : (xt , t+ s)∼ (y,0)}= yˆ (A.3)The supremum can be replaced by a maximum, and the proof is similar to the onebefore. Suppose there is a sequence of {yn} that converges up to a value yˆ, and,(xtn , tn+ s)∼ (yn,1). Note that tn lies in a compact interval [0, tmax], and hence hasa convergent subsequence tnk that converges to a point in that interval tˆ ∈ [0, tmax].Now, xt is continuous in t (in the usual sense), and hence, xtn also converges to xtˆ .Further, ynk → yˆ as nk→ ∞. Therefore, using, (xtnk , tnk + s) ∼ (ynk ,0), as, nk→ ∞,76A.1. Appendix to to Chapter 1it must be that (xtˆ , tˆ + s) ∼ (yˆ,0). Hence, the supremum in (A.3) must have beenattained from xtˆ , and hence the supremum can be replaced by a maximum. Furthergiven this is a maximum, we can say that yˆ∈ (0,x∗). The f function is well defined,strictly increasing and is the inverse function of ux∗ over r ∈ (0,1) to (0,x∗), in thesense that, u( f (r)) = r. This function can be used to show the continuity of u atthe point x∗.Finally, the function u can be easily normalized to have ux∗(M) = 1. (By dividingthe function from before by ux∗(M).)Now, consider Uδ = {ux∗() : x∗ ∈ (0,M]}. By construction of the functions, itmust be that(x, t)% (y,0) ⇐⇒ δ tu(x)≥ u(y) ∀u ∈Uδ(x, t)∼ (y,0) ⇐⇒ δ tu(x)≥ u(y) ∀u ∈Uδand δ tuy(x) = uy(y) for some uy ∈UδFor any (z,τ), consider the sets {(y,0)∈X×T : (y,0)% (z,τ)} and {(y,0)∈X×T :(z,τ) % (y,0)}. Both are non-empty, as (z,0) belongs to the first one and (0,0) inthe second one. Both sets are closed in the product topology. Their union is con-nected, and hence there exists an element in their intersection, i.e, there exists ay1 ∈ X such that (y1,0) ∼ (x, t). By monotonicity this y1 must be unique. There-fore there must exist a continuous present equivalent utility representation for %.We show this formally in the next two paragraphs.Given % is complete, transitive and satisfies continuity, there exists a continuousfunction F¯ :X×T→ R such that F¯(a)≥ F¯(b) if and only if a% b for a,b∈X×T.(Following Theorem 1, Fishburn and Rubinstein [1982]).We define G : X→ R as G(x) = F¯(x,0). The function G would be strictly mono-tonic and continuous. Also define F : X×T→ R as F(x, t) = G−1(F¯(x, t)). Asany alternative has a unique present equivalent, F is well defined, is a monotoniccontinuous transformation of F¯ (hence represents %) and F(x,0) = x for all x ∈X.By definition the function F assigns to every alternative its present equivalent asthe corresponding utility. Therefore, the present equivalent utility representation iscontinuous.We will show that the function W defined below also assigns to every alternative(z,τ) an utility exactly equal to its present equivalent.W (x, t) = minu∈Uδu−1(δ tu(x)) = F(x, t)Consider any (z,τ) ∼ (y1,0). By definition of Uδ and by construction of its con-stituent functions, it must be that for all u ∈Uδ , δ τu(z)≥ u(y1) and there exists a77A.1. Appendix to to Chapter 1function uy1 such that δ τuy1(z) = u(y1). This is equivalent to the following state-ment: For all u ∈ Uδ , u−1(δ τu(z)) ≥ y1 and there exists a function uy1 such thatu−1y1 (δτuy1(z)) = y1.Therefore, W (z,τ) = minu∈Uδ u−1(δ τu(z)) is continuous utility representation ofthe relation %.Proof for the case of T={0,1,2, ...}.This proof would be more technical and we will break down the proof of thiscase into the following Lemmas.Lemma 23. Under Axioms A0-A4, for a fixed x0, and any xt and t such that (xt , t)∼(x0,0), there exists a continuous strictly increasing function u such that δ tu(xt) =u(x0) and δ tu(z1)≥ u(z0) for all (z1, t)% (z0,0). Further, u(0) = 0, u(M) = 1.Proof. By the Discounting axiom, we know that there exists a smallest integern≥ 1 such that (x0,0)% (M,n). Choose x∗0 = x0. For 0 < t < n , find x∗t such that(x∗0,0)∼ (x∗t , t) . If (x0,0) (M,n), choose xn = M.We define x∗−1 in the following wayx∗−1 = min{x ∈ X : (x,0)% (x∗j , j+1), j = 0,1,2, ...n}The idea is to look at the present equivalents of (x∗j , j+1) and take the maximumof those present equivalents. The alternative way to express the same is to look atthe intersection of the weak upper counter sets of (x∗j , j+1) on X×{0}, and thentake the minimal value from that set.Next we will use this to define x∗−2, then use x∗−1 and x∗−2 to define x∗−3. In general,for i ∈ {−1,−2,−3...} define x∗i recursively as the minimum of the set{x ∈ X : (x,0)% (x∗j , j− i), j = i+1, i+2, ...n}The definition uses the same idea as before. We consider the intersection of theweak upper counter sets of (x∗j , j− i) on X×{0} and take its minimum. The set isnon-empty (x∗0 belongs to it, for example), closed and the minimum exists due tothe continuity, monotonicity and discounting properties.Next we show that for every x∗i with i≤−1, there exists j ∈ {0,1, ..n} such that(x∗j , j− i) % (x∗i ,0). The proof is by induction. For i = −1, it is immediate fromthe definition. Suppose, it holds for all i ≥ −m. Consider x∗−i−1. By construc-tion, there must exist k ∈ {−m,−m+ 1, ..n} such that (x∗−i−1,0) ∼ (x∗k ,k+ i+ 1).78A.1. Appendix to to Chapter 1If k ∈ {0,1, ..n} we are done already. If not, by the induction hypothesis, thereexists j ∈ {0,1, ..n} such that (x∗j , j− k) % (x∗k ,0), which gives, (x∗j , j+ i+ 1) %(x∗k ,k+ i+1), and hence, (x∗j , j+ i+1)% (x∗−i−1,0), completing the proof.Next we will show that the sequence {...x∗−2,x∗−1,x∗0,x∗1,x∗2, ..} is converges belowto 0. Suppose not (we are going for a proof by contradiction), that is there existsw> 0 such that xi ≥ w for all i ∈ Z. As, M > z> 0, there must exist t1 big enoughsuch that (z,0) (M, t1). Consider the element x∗−t1 from the sequence in consider-ation. Using the result from the previous paragraph, it must be true that there existsj ∈ {0,1, ..,n}, such that (x∗j , j+ t1) % (x∗−t1 ,0). Now, as M ≥ x∗j , we must have,(M, t1)% (x∗−t1 ,0) (z,0), which provides a contradiction.Consider any y0 ∈ (x∗0,x∗1).We are going to find a y1,y2, ..yn−1 recursively.Finding y1: For each point y ∈ (x1,x2], take reflections of length 1, i.e, find xysuch that (y,1)∼ (xy,0). Note that, (x∗1,0) (y,1) (x∗0,0). Hence, xy ∈ (x∗0,x∗1).Let, xx2 be the reflection for the point x2. For any y ∈ (x∗1,x∗2], f (y) = x∗0 +(xy−x∗0)(x∗1− x∗0)(xx2− x∗0). Now, for y0 ∈ (x∗0,x∗1), define y1 as f−1(y0).We can check that this method satisfies the 2 following conditions:1) Consider two such sequences, one starting from y10, and another from y20,with y10 > y20. We will have y11 > y21.2) All points in intervals (x∗1,x∗2) are included by some y1 from the sequence.This follows from monotonicity and discounting too.Now, the recursive step:For each point y ∈ (x∗i ,x∗i+1], take reflections of length j ∈ {i, i−1..,1} conditionalon those reflections being in the corresponding (x∗i− j,x∗i+1− j] intervals. For any y,at least one of these reflections must exist, and in particular the one with length ialways exists, as (x∗1,0) (x∗i+1, i)% (y, i) and (y, i) (x∗i , i)∼ (x∗0,0).Now, for each such reflection, find the corresponding sequence of {y0,y1, ..yi−1}it belongs to, and denote the smallest y0 from that collection of sequences as xy ∈[x∗0,x∗1]. Note that xxi+1 ≤ x∗1. Define the 1 : 1 strictly increasing function f from(xn−1,xn] to (x∗i ,x∗i+1] in the following way: For any y ∈ (x∗i ,x∗i+1] , f (y) = x∗0 +(xy− x∗0)(x∗1− x∗0)(xxi+1− x∗0). Now, define yi as f−1(y0). The conditions mentioned aboveare still satisfied for the extended sequence.For i ≤ −1, define yi recursively in the following way. Start by finding y′i as theminimum of the set{y ∈ X : (y,0)% (y j, j− i), j = i+1, i+2, ...n}Define x′−i as the minimum of the set79A.1. Appendix to to Chapter 1{y ∈ X : (y,0)% (y j, j− i), j = i+1, i+2, ...n−1}Finally, defineyi = x∗i+1− (x∗i+1− y′i)(x∗i+1− x∗i )(x∗i+1− x′i)(A.4)Given y10 > y20 determines the order of y1t > y2t , for t ∈ {1,2, ..n−2}, our inductiveprocedure make sure this holds true for all t ≤−1 too.One can check for covering properties of the sequences by induction. Suppose allpoints in the intervals (x∗i ,x∗i+1) are covered by yi for some sequence, for i ≥ j forsome integer j. We are going to show that all points in (x∗j−1,x∗j) are also cov-ered by y j−1 for some sequence. Take any point y ∈ (x∗j−1,x∗j), and consider itscorresponding y′ as defined in Equation A.4. Consider the reflections from pointy′ of sizes 1, ..n− j+ 1, i.e, the points at those temporal distances which are in-different to it, conditional on being in the corresponding intervals. By the induc-tion hypothesis, each of those reflection end points must be coming from somey0 ∈ (x∗0,x∗1). Take the sequence with smallest y0, and that sequence would resultin having y ∈ (x∗j−1,x∗j) as its next element.Now, define u onX as follows: Set u(x∗n)= u(x∗n)= 1. For the sequence ...,x∗−2,x∗−1,x∗0,x∗1, .., let u(x∗i ) = δ i−n for all positive and negative integers i. Next, let us define u on(x∗n−1,x∗n) as any continuous and increasing function with inf(x∗n−1,x∗n)u(x) = δ =u(x∗n−1) and sup(x∗n−1,x∗n)u(x) = 1 = u(M). We can extend each dual sequence withsome as u(yi) = δ i−nu(y0). This finishes the construction of a u that satisfies theconditions mentioned in the Lemma.Lemma 24. Under Axioms A0-A4, there exists a continuous present equivalentutility function F :X×T→ R that represents%. Moreover, for δ ∈ (0,1) , F(z,τ)=minu∈Uδ u−1(δ τu(z)) for some set Uδ of strictly monotonic, continuous functions,u(0) = 0 and u(M) = 1 for all u ∈Uδ .Proof. Consider the set Uδ of all strictly monotonic, continuous functions u suchthat δ tu(z1)≥ u(z0) for all (z1, t)% (z0,0), u(0) = 0 and u(M) = 1. By the previousLemma, this set is non-empty, and for any (z1, t) ∼ (z0,0) includes a function u,such that δ tu(z1) = u(z0). By construction of the functions, it must be that(x, t)% (y,0) ⇐⇒ δ tu(x)≥ u(y) ∀u ∈Uδ(x, t)∼ (y,0) ⇐⇒ δ tu(x)≥ u(y) ∀u ∈Uδand δ tuy(x) = uy(y) for some uy ∈Uδ80A.1. Appendix to to Chapter 1For any (z,τ), consider the sets {(y,0)∈X×T : (y,0)% (z,τ)} and {(y,0)∈X×T :(z,τ) % (y,0)}. Both are non-empty, as (z,0) belongs to the first one and (0,0) inthe second one. Both sets are closed in the product topology. Their union is con-nected, and hence there exists an element in their intersection, i.e, there exists ay1 ∈ X such that (y1,0) ∼ (x, t). By monotonicity this y1 must be unique. There-fore there must exist a continuous present equivalent utility representation for %.We show this formally in the next two paragraphs.Given % is complete, transitive and satisfies continuity, there exists a continuousfunction F¯ :X×T→ R such that F¯(a)≥ F¯(b) if and only if a% b for a,b∈X×T.(Following Theorem 1, Fishburn and Rubinstein [1982]).We define G : X→ R as G(x) = F¯(x,0). The function G would be strictly mono-tonic and continuous. Also define F : X×T→ R as F(x, t) = G−1(F¯(x, t)). Asany alternative has a unique present equivalent, F is well defined, is a monotoniccontinuous transformation of F¯ (hence represents %) and F(x,0) = x for all x ∈X.By definition the function F assigns to every alternative its present equivalent asthe corresponding utility. Therefore, the present equivalent utility representation iscontinuous.We will show that the function W defined below also assigns to every alternative(z,τ) an utility exactly equal to its present equivalent.W (x, t) = minu∈Uδu−1(δ tu(x)) = F(x, t)Consider any (z,τ) ∼ (y1,0). By definition of Uδ and by construction of its con-stituent functions, it must be that for all u ∈Uδ , δ τu(z)≥ u(y1) and there exists afunction uy1 such that δ τuy1(z) = u(y1). This is equivalent to the following state-ment: For all u ∈ Uδ , u−1(δ τu(z)) ≥ y1 and there exists a function uy1 such thatu−1y1 (δτuy1(z)) = y1.Therefore, W (z,τ) = minu∈Uδ u−1(δ τu(z)) = F(z,τ) is a continuous utility repre-sentation of the relation %.Proposition 1: Given the axioms A0-4, the representation form in (1.3) is uniquein the discounting function ∆(t) = δ t inside the present equivalent function.Proof. We start with the case where ∆(t) is such that∆(t+ t1)∆(t)< ∆(t1) for some81A.1. Appendix to to Chapter 1t, t1. Consider any singleton U = {u}.(y, t) ∼ (x,0)=⇒ u−1(∆(t)u(y)) = x=⇒ ∆(t)u(y) = u(x)=⇒ ∆(t+ t1)u(y) = ∆(t+ t1)∆(t) u(x)< ∆(t1)u(x)=⇒ u−1(∆(t+ t1)u(y)) < u−1(∆(t1)u(x))=⇒ (x, t1)  (y, t+ t1)Hence, the relation implied by the representation contradicts Weak Present Bias.Now assume the opposite, let there exists some t, t1 > 0 such that∆(t+ t1)∆(t)>∆(t1) . Now suppose we started with a relation % which has (y, t) ∼ (x,0) as wellas (y, t+ t1)∼ (x, t1) for all t, t1 and some x,y. (This does not necessarily mean thatthe person’s preferences satisfy stationarity in the broader sense as we do not askthis from all x,y.) We will show below that such preferences cannot be representedby the functional form we started with for any set of functions U .(y, t) ∼ (x,0)=⇒ minu∈U(u−1(∆(t)u(y))) ≥ minu∈U(u−1(u(x))) = x=⇒ ∆(t)u(y) ≥ u(x) ∀u ∈U=⇒ ∆(t+ t1)u(y) ≥ ∆(t+ t1)∆(t) u(x)> ∆(t1)u(x) ∀u ∈U=⇒ u−1(∆(t+ t1)u(y)) > u−1(∆(t1)u(x)) ∀u ∈U=⇒ minu∈U(u−1(∆(t+ t1)u(y))) > minu∈U(u−1(∆(t1)u(x)))=⇒ (y, t+ t1)  (x, t1)This completes our proof.Proposition 2: If U ,U ′ ⊂F are such that c¯o(U ) = c¯o(U ′), and the func-tional form in (1.1) allows for a continuous minimum representation for both ofthose sets, then, minu∈U u−1(δ tu(x)) = minu∈U ′ u−1(δ tu(x)).Proof. We will prove this in 2 steps.First we will show that for any set A, minu∈A u−1(δ tu(x)) = minu∈A¯ u−1(δ tu(x)),where A¯ is the closure of the set A.It is easy to see the direction that minu∈A u−1(δ tu(x))≥minu∈A¯ u−1(δ tu(x)).We will prove the other direction by contradiction. Suppose, minu∈A u−1(δ tu(x))>82A.1. Appendix to to Chapter 1minu∈A¯ u−1(δ tu(x)). This would imply that there exists v ∈ A¯\A and some ε >0, such that v−1(δ tv(x)) + ε < u−1(δ tu(x)) for all u ∈ A. By definition of thetopology of compact convergence and given that v belongs to the set of limitpoints of A, there must exist a sequence of functions {vn} ⊂ A which convergesto v in the topology of compact convergence , i.e, for any compact set K ⊂ R+,limn→∞ supx∈K |vn(x)−v(x)|= 0. It can be shown that under this condition, v−1n (δ tvn(x))would also converge to v−1(δ tv(x))where vn ∈U .55 This constitutes a violation ofv−1(δ tv(x))+ε < u−1(δ tu(x)) for all u∈A. Hence, it must beminu∈A u−1(δ tu(x))=minu∈A¯ u−1(δ tu(x)).As a second part of this proof, we will show that for any set A, minu∈A(u−1(δ tu(x)))=minu∈co(A)(u−1(δ tu(x))).It is easy to see that minu∈A(u−1(δ tu(x)))≥minu∈co(A) u−1(δ tu(x)), as A⊂ co(A).We will again use proof by contradiction to show the opposite direction. We as-sume that there exists a u¯ ∈ c¯o(A) and (x, t) ∈ X×T, such that u¯ = ∑ni=1λiui,∑ni=1λi = 1 and u¯−1(δ su¯(y))<mini u−1i (δsui(y)). This would imply that ui(u¯−1(δ su¯(y)))<δ sui(y) for all i.Now,δ su¯(y) = δ s∑iλiui(y)= ∑iλiδ sui(y)> ∑iλiui(u¯−1(δ su¯(y)))= u¯(u¯−1(δ su¯(y)))= δ su¯(y)This gives us a contradiction. Note that the equality right after the inequality comesfrom the definition of u¯.Hence, we have, minu∈A u−1(δ tu(x)) = minu∈co(A) u−1(δ tu(x)).Proposition 3: i) If there exists a concave function f ∈U , and ifU1 is the sub-set of convex functions inU , then minu∈U u−1(δ tu(x)) =minu∈U \U1 u−1(δ tu(x)).ii) If u1,u2 ∈U and u1 is concave relative to u2, then, minu∈U u−1(δ tu(x)) =minu∈U \{u2} u−1(δ tu(x)).55As, vn→ v in the topology of compact convergence, vn→ v point wise, hence, δ tvn(x)→ δ tv(x).Now, as v−1n → v−1 compact convergence (proof later in the appendix), v−1n (δ tvn(x))→ v−1(δ tv(x)).83A.1. Appendix to to Chapter 1Proof. If a function u is convex,u−1(δ tu(x)) = u−1(δ tu(x)+(1−δ t)u(0))≥ u−1(u(δ tx+(1−δ t)0))= δ txSimilarly for concave f , we would have, f−1(δ t f (x))≤ δ tx which completes theproof of part (i). Note that this result is expected given concave functions give riseto more conservative present equivalents.For part (ii), note thatu−11 (δtu1(x)) = u−11 (δtu1(u−12 (u2(x))))≤ u−11 (u1(u−12 (δ tu2(x))))= u−12 (δtu2(x))Where the inequality arises from the fact that u1 is concave relative to u2.Proposition 25. Eventual stationarity is satisfied by β -δ discounting, but not hy-perbolic discounting.Now for any x> z> 0 ∈ X , choose t1 > log 1δ(u(x)u(z)).t1 > log 1δ(u(x)u(z))⇐⇒(1δ)t1 >u(x)u(z)=⇒ u(z) > δ t1u(x)> βδ t1u(x)=⇒ βδ tu(z) > βδ t+t1u(x)(z, t)  (x, t+ t1)Also, (x,0)∼ (xt , t) implies, u(x) = βδ tu(xt), which implies,u(z) > δ t1u(x) = βδ t+t1u(xt)(z,0)  (xt , t+ t1)This shows that β −δ does indeed satisfy A5.Now consider the simple variant of Hyperbolic discounting model when α = γ = 1.Fix any felicity function u and x > z > 0 ∈ X . We will show that there does notexist t1,such that (z, t) (x, t+ t1) for all t ≥ 0.84A.1. Appendix to to Chapter 1(z, t)  (x, t+ t1) for all t ≥ 0⇐⇒ u(z)1+ t>u(x)1+ t+ t1for all t ≥ 0⇐⇒ 1+ t+ t11+ t>u(x)u(z)for all t ≥ 0⇐⇒ 1+ t11+ t>u(x)u(z)for all t ≥ 0Note that the last statement is not possible, as for fixed t1 the LHS↓ 1 as t ↑ ∞,whereas, the RHS is always a fixed number, that is strictly greater than one. Hence,hyperbolic discounting does not satisfy A5.Theorem 6: The following two statements are equivalent:i) The relation % satisfies properties A0-A6.ii) There exists a set Uδ of monotinically increasing continuous functions suchthatF(x, t) = minu∈Uu−1(δ tu(x))represents the binary relation%. The setU has the following properties: u(0) = 0for all u∈U , supu u(x) is bounded above, infu u(x)> 0 ∀x, infuu(z)u(x)is unboundedin z for all x> 0.Proof : Going from (ii) to (i) :That (ii) implies Monotonicity, Discounting, Weak Present Bias and Continuity hasalready been shown in the proof of Theorem 3.Showing Eventual Stationarity: Given supu u(x) is bounded above and infu u(x)> 0, for any choice of x,z > 0 and δ ∈ (0,1) there exists t1 > 0 big enough such thatinfu u(z)> δ t1 infu u(x). This would imply that, for all u ∈U ,u(z) > δ t1u(x)and, hence, (z,0) (x, t1).Now, for t > 0 consider xt such that (xt , t) ∼ (x,0). By the representation, thisimplies that there exists u1 ∈U such thatδ tu1(xt) = u1(x)=⇒ δ t+t1u1(xt) = δ t1u1(x)< u1(z)=⇒ minuu−1(δ t+t1u1(xt)) ≤ u−11 (δ t+t1u1(xt))< u−11 (u1(z)) = minu u−1(u(z))85A.1. Appendix to to Chapter 1Hence, (z,0) (xt , t+ t1).Similarly, for all u ∈U ,δ tu(z) > δ t+t1u(x)=⇒ minuu−1(δ tu(z)) > minuu−1(δ t+t1u(x))Hence, (z, t) (x, t+ t1).Showing Non-triviality: We have that infuu(z)u(x)is unbounded in z for all x> 0.Therefore, for any x, and t ∈ T, there exists z, such thatinfuu(z)u(x)> δ−t=⇒ u(z)u(x)> δ−t ∀u ∈U=⇒ δ tu(z) > u(x) ∀u ∈U=⇒ u−1(δ tu(z)) > u−1(u(x)) ∀u ∈U=⇒ minuu−1(δ tu(z)) > minuu−1(u(x))(z, t)  (x,0)To go from the direction (i) to (ii) of Theorem 6, one needs to follow Lemma 26-28.Lemma 26. Under Axioms A1-A6, for any (x0, t),(xt ,0) such that (x0, t) ∼ (xt ,0)in the original relation, there exists u∈U such that δ tu(xt) = u(x0) and δ tu(z1)≥u(z0) for all (z1, t) ≥ (z0,0). Moreover, u is strictly monotonic, continuous, andu(0) = 0, u(1) = 1.Proof. We will prove it for t = 1, x0,xt > 0 and then show the general guidelinefor a general t.We define the following procedure: Choose x∗0 = 1. Find x∗1 such that (x∗0,0) ∼(x∗1,1) . We can do it because of the Non-Triviality assumption. Clearly, x∗1 = x1.Next find x∗−1 = max{x−1,x′−1} where (x∗0,1) ∼v (x−1,0) and (x∗1,2) ∼v (x′−1,0).The value x−1 > 0 exists because, (x∗0,0)  (x∗0,1)  (0,1), coupled with the factthat % is continuous. Same with x′−1.Note that x∗0 > x∗−1 by discounting. Next going in the opposite direction, we findx∗2 = min{x2,x′2,x′′2}, where, (x∗1,0) ∼v (x2,1), (x∗0,0) ∼v (x′2,2) and (x∗−1,0) ∼v(x′′2 ,3). Next we find x∗−2,x∗3,x∗−3,x∗4, ... sequentially. Thus one can find a sequence...x∗−3 < x∗−2 < x∗−1 < x∗0 < x∗1 < x∗2...86A.1. Appendix to to Chapter 1We will show that this sequence is unbounded above and converges below to 0 .Consider any z< x∗0. By A5, there must exist t1 such that (z,0) (x∗0, t1). and givenfor any t > 0, (x0,0)% (x∗t , t), by monotonicity, it must hold that (z,0) (x∗t , t+t1).By definition of x∗−1, either (x∗−1,0)∼ (x∗0,1) or (x∗−1,0)∼ (x∗1,2), if not both. So,by WPB, either (x∗0, t1) % (x∗−1, t1− 1) or (x∗1, t1 + 1) % (x∗−1, t1− 1), and hence,either (z,0)  (x∗−1, t1− 1) . One can use the construction of the sequence, andinduction, here on, to show that, for any general 0 < i < t1, (z,0)  (x∗−i, t1− i).Hence, it must be that x∗−t1 ≤ z, which proves that the sequence converges belowto zero. To show that the sequence is unbounded above, one uses a similar trick.Consider z > x∗0. There must exist t2 such that (x∗0, t) (z, t + t2) for all t ≥ 0, andgiven for any t > 0, (x∗−t ,0)% (x∗0, t), by monotonicity, it must hold that (x∗−t ,0)%(x∗0, t)  (z, t + t2). By definition of x∗1, (x∗1,1) ∼ (x∗0,0)  (z, t2). So, by WPB, itmust be that (x∗1,0)  (z, t2− 1). (z < x∗1 is trivial and hence neglected). One canuse the construction of the sequence, and induction, here on, to show that, for anygeneral 0< i< t2, (x∗i ,0) (z, t2− i). Hence, it must be that x∗t2 ≥ z, which provesthat the sequence diverges to infinity.Consider any y0 ∈ (x∗0,x∗1). We find y′−1 such that (y′−1,0)∼ (y∗0,1). Finally,y∗−1 = x∗0− (x∗0− y′−1)(x∗0− x∗−1)(x∗0− x−1)∈ (x∗−1,x∗0).The upper bound on y∗−1 comes from the fact that (x∗0 > y′−1) and the lower boundcomes from the fact that y′−1 is bounded below by x−1. Note that for y∗0, yˆ0 ∈(x∗0,x∗1), y∗0 > yˆ0 if and only if y∗−1 > yˆ−1. And finally, for any y∗−1 ∈ (x∗−1,x∗0) thereexists a y∗0 ∈ (x∗0,x∗1) corresponding to it.Next we will define an inductive procedure to find the other points in such se-quences. Let S be the set of all such sequences. The induction hypothesis is thatfor every y∗0 ∈ (x∗0,x∗1) we have already defined a corresponding chain56 Si = y∗−i <...y∗−3 < y∗−2 < y∗−1 < y∗0 < y∗1 < y∗2.. < y∗i−1, i≥ 2 such that i) y∗n ∈ (x∗n,x∗n+1) for allthe elements of all the chains. ii) If we compare the nth elements of 2 chains theyare always similarly ranked, regardless of the value of n. iii) If the last elementconstructed is y∗i for i ∈ N then, any point in (xn,xn+1) for n ∈ {−i, ..i−1} is partof exactly one chain inSi.Finding y∗i where i≥ 1: Note that we can write x∗i = min{x1i ,x2i ,x3i ...x2ii }57, where(x1i ,1)∼ (x∗i−1,0),(x2i ,2)∼ (x∗i−2,0)..,(x2i−1i ,2i−1)∼ (x∗−i+1,0). Similarly, x∗i+1 =min{x1i+1,x2i+1,x3i+1...x2i+1i+1 }.Define, x′i+1 =min{x1i+1,x2i+1,x3i+1...x2ii+1}≥ x∗i+1.De-56A set paired with a total order.57We are using one extra comparison than that existed in the original construction of the sequence,and this is to make sure that x∗i has 2i comparisons in its construction, just like y∗i . Given the struc-ture of the sequence we can always add more comparisons than the original, but never have fewercomparisons.87A.1. Appendix to to Chapter 1fine y′i = max{y1i ,y2i ,y3i , ...y2ii } where (y1i ,1) ∼ (y∗i−1,0),..,(y2ii ,2i) ∼ (y∗−i,0). Fi-nally, y∗i = x∗i +(y′i− x∗i )(x∗i+1− x∗i )(x′i+1− x∗i )∈ (x∗i ,x∗i+1). By monotonicity, yni ∈ (xni ,xni+1)for all n ∈ {1,2, ..,2i}. Therefore, y′i ∈ (x∗i ,x′i+1). Therefore, y∗i ∈ (x∗i ,x∗i+1), theupper bound comes from the fact that x′i+1 > y′i and the lower bound comes fromthe fact that y′i is bounded below by x∗i . Note that for y∗0, yˆ∗0 ∈ (x∗0,x∗1), y∗0 > yˆ∗0 ifand only if y∗i > yˆ∗i . And finally, for any yˆ∗i ∈ (x∗i ,x∗i+1) there exists a yˆ∗0 ∈ (x∗0,x∗1)corresponding to it. The last part can be shown constructively.Finding y∗−i−1 where i ≥ 1: Note that x∗−i = max{x1−i,x2−i,x3−i...x2i+1−i }58, where(x1−i,0)∼v (x∗−i+1,1),(x2−i,0)∼v (x∗−i+2,2)..,(x2i−i,0)∼v (x∗i ,2i). Similarly, x∗−i−1 =max{x1−i−1,x2−i−1, .. x2i+1−i−1,x2i+2−i−1}.Define, x′−i−1 = max{x1−i+1,x2−i+1,x3−i+1, ...x2i+1−i+1}≤ x∗−i−1.Define y′−i−1 =max{y1−i+1,y2−i+1,y3−i+1, ...y2i+1−i+1} where (y1−i+1,0)∼v (y∗−i+2,1), ....,(y−i+1,0) ∼v (y∗i ,2i+ 1). Finally, y∗−i−1 = x∗−i− (x∗−i− y′−i−1)(x∗−i− x∗−i−1)(x∗−i− x′−i−1)∈(x∗−i−1,x∗−i).By monotonicity, yn−i−1 ∈ (xn−i−1,xn−i) for all n∈{1,2, ..,2i+1}. There-fore, y′−i−1 ∈ (x′−i−1,x∗−i). Therefore, y∗−i−1 ∈ (x∗−i−1,x∗−i), the upper bound comesfrom the fact that x∗−i > y′−i−1 and the lower bound comes from the fact that y′−i−1is bounded below by x′−i−1. Note that for y∗0, yˆ0 ∈ (x∗0,x∗1), y∗0 > yˆ0 if and only ify∗−i−1 > yˆ−i−1. And finally, for any yˆ−i−1 ∈ (x∗−i−1,x∗−i) there exists a yˆ0 ∈ (x∗0,x∗1)corresponding to it. The last part can be shown inductively. Fix yˆ′−i−1. Find thepoints (whenever possible) zn ∈ (x∗n,x∗n+1) for n ∈ {−i,−i+1,−i+2, ..i} such that(yˆ′−i−1,0) ∼v (zn,n+ i+ 1). Note that we can always find atleast one such zn.59Next, using the induction hypothesis we can map all the zn’s to a y∗0 ∈ (x∗0,x∗1). Wetake the maximum of all such y∗0s and define it as yˆ∗0. One can check that start-ing from this (yˆ−i+1, ..yˆ0, yˆ1..yˆi) would indeed result in ending with the yˆ′−i−1 westarted with. 60Now, define u onX as follows: Set u1(x∗0) = 1. For the sequence ..,x∗−1,x∗0,x∗1, .., letu(x∗i ) = δ i for all positive and negative integers i. Next, let us define u! on (x∗−1,1)as any continuous and increasing function with inf(x∗−1,1)u1(x) = δ = u(x∗−1) and58As before, we are using one extra comparison than that existed in the original construction ofthe sequence.59There exists k such that (x∗−i,0) ∼v (x∗−i+k,k). In general, (x∗−i,0) %v (x∗−i+k,k).This implies(yˆ′−i,0) v (x∗−i+k,k) and (x∗−i+1+k,k) v (yˆ′−i,0). Hence, there exists z−i+k ∈ (x∗−i+k,x∗−i+k+1)such that (yˆ′−i,0)∼v (zn,n+ i).60Suppose not. Given our definition of yˆ0, starting from this (yˆ−i+1, ..yˆ0, yˆ1..yˆi) would give usyˆ′′−i ≥ yˆ′−i. Let, yˆ′′−i > yˆ′ and (yˆ′′−i,0)∼ (yˆ−i+k,k) for yˆ−i+k ∈ (x−i+k,x−i+k+1), this being the relationthat binds while defining yˆ′′−i. Given, (yˆi+k,k)v (yˆ′−i,0) and (yˆ′−i,0) (x′−i,0)%v (x∗−i+k,k), therewould exist (yˆ′′−i,0)∼ (yˆ∗−i+k,k) for yˆ−i+k ∈ (x∗−i+k,x∗−i+k+1) and yˆ−i+k < yˆ′−i+k which would be acontradiction.88A.1. Appendix to to Chapter 1sup(x∗−1,1)u1(x) = 1 = u(1). We can extend each dual sequence with some y0 ∈(x∗−1,1) as u(yi) = δ iu(y0). Finally, define U(x, t) = δ tu1(x)u1(1)to ensure u1(1) = 1(note that u1(1)> 0).It is important to note here that the utility defined retains all the monotonicity, dis-counting and present bias properties. Consider any (y, t) % (x,0) in the originalrelation. The element x must belong to one of the sequences defined above. If xtis the corresponding element to the right in that sequence separated by a distanceof t, then, by construction we must have u(x) = δ tu(xt) and (x,0) % (xt , t). Bymonotonicity, it would be true that y> xt and hence, u(x)< δ tu(y).Now we will extend the logic above to a more general case of (x0, t),(xt ,0) suchthat (x0, t)∼ (xt ,0) for t > 1.For i ∈ {1, ..t}, let xi be such that (x0,0) ∼ (xi, i). We define the following pro-cedure: Choose x∗0 = x0, the same x0 that was provided in the statement of thisLemma. Find x∗1 such that (x∗0,0) ∼ (x∗1,1) . Of course, x∗1 = x1. Next use theiterative format used in Lemma 2 to find x∗2,x∗3, ..x∗t .At each of these steps, by WPB, one would get, x∗i = xi, ending with x∗t = xt . Weprovide a brief outline for this, the proof requires induction.Let, x∗2 =min{x2,x′2}, where, (x2,2)∼v (x∗0,0) and (x′2,1)∼v (x∗1,0). By WPB, thelatter implies, (x′2,2)%v (x∗1,1). By definition of x∗1, (x∗0,0)∼v (x∗1,1). Putting it alltogether,(x′2,2)%v (x∗1,1)∼ v(x∗0,0)∼v (x2,2)Hence, x′2 ≥ x2, and x∗2 = x2.Similarly, let x∗3 = min{x3,x′3,x′′3}, where, (x3,3) ∼v (x∗0,0), (x′3,2) ∼v (x∗1,0) and(x′′3 ,1)∼v (x∗2,0).(x′3,3)%v (x∗1,1)∼ v(x∗0,0)∼v (x3,3)Also,(x′′3 ,3)%v (x∗2,2)∼ v(x∗0,0)∼v (x3,3)And so on. Note that the sequence in which the elements are being found till nowhas been different that that in Lemma 2. Here on, find the sequence elements in thefollowing order x∗−1,x∗t+1,x∗−2,x∗t+2, .... using the iterative procedure as Lemma 2.For any y0 ∈ (x∗0,x∗1), find similar sequences in the same order as we derived thesequence x∗.Now, define u on X as before to finish the proof. Note that any u such constructedis strictly monotonic, continuous, and u(0) = 0, u(1) = 1.89A.1. Appendix to to Chapter 1Lemma 27. Under Axioms A1-A6, there exists a set of functions U such that,for all u ∈ U , u is strictly monotonic, continuous, and u(0) = 0, u(1) = 1, andδ tu(z1) ≥ u(z0) for all (z1, t) ≥ (z0,0). Moreover, i) for any (x, t) ∼ (y,0), thereexistsu∈U such that δ tu(xt)= u(x0). ii) For x> 0, infu∈U u(x)> 0, supu∈U u(x)<∞Proof. Consider the set U consisting of all functions u constructed from all theindifference relations ∼ in (26). It would suffice to show that infu∈U u(x) > 0,supu∈U u(x)< ∞.First we will show that infu∈U u(x) > 0. This is trivial for points above x = 1.Consider 0< x< 1. Suppose we are constructing a function that would respect therelation (x0,0)∼ (xt , t).By A5, there exists t1 such that (x, t)  (1, t + t1) for all t ≥ 0 and for any yi suchthat such that (1,0) ∼ (yi, i) for i ≥ 0, (x,0)  (yi, t1 + i). Consider the followingcases:CASE 1: Consider x0 < x < 1. By A5, there exists t ≥ 1, such that in the se-quence constructed, xt−1 < x≤ xt . Note that given the construction of the sequencefor (x,0)∼ (xi, i), it must be that for any (xp,xq), p< q, (xp,0)% (xq,q− p)). Bymonotonicity, using xt−1 < x ≤ xt , for any point xi in the sequence , |i| ≤ t, onehas (xi,0)% (xt , t− i)% (x, t− i) . Hence, for any element xi of the sequence withi≤ 0, (xi,0)% (x, t− i) (1, t1+ t− i), with the last inequality coming from A5.61Hence, the x(t+t1)th element of the sequence must be weakly to the right of 1. Thus,u(x)≥ 1δ t1+1.CASE 2: Consider x< x0< 1. By construction of the dual sequence {..x−1,x0,x1, ..},it must be that x−t1 ≤ x and xt1 ≥ 1. Thus, u(x)≥1δ 2t1. 62Hence, u(x)≥ 1δ 2t1for all u ∈U .Now, showing that supu∈U u(x) < ∞. This is trivial for points x ≤ 1. Considerx > 1. By A4, there exists t1 such that (1,0)  (x, t1) and for any y such that(x,0) ∼ (y, i), i ≥ 0 ,(1,0)  (y, t1 + i). Suppose we are constructing a functionthat would respect the relation (x0,0) ∼ (xt , t), and in the process construction adual sequence {..x−1,x0,x1, ..}. There are two cases as before.CASE 1: Consider x0 > x > 1. By A4, there exists t ≥ 1, such that in thesequence constructed, one has x−t ≤ x < x−t+1. As before, given the constructionof the sequence for (x,0)∼ (xi, i), it must be that for any (xp,xq), p< q, (xp,0)%61The property we are using implicitly without proving is the following: In our constructed se-quences, xi always is a direct reflection from {x0,x−1,x−2, ..} when i is positive, and a direct reflec-tion of {x0,x1,x2..}when i is negative. This follows from WPB.62One can make the bound tighter.90A.1. Appendix to to Chapter 1(xq,q− p).) By monotonicity, using x−t ≤ x, for any point xi in the sequence ,|i| ≤ t, (x,0) % (x−t ,0) % (xi, i+ t) . Hence, for any element xi of the sequencewith i≥ 0, (1,0) (xi, t1+ i+ t). Thus, u(x)≤ 1δ t1+1 .CASE 2: Consider x> x0> 1. By construction of the dual sequence {..x−1,x0,x1, ..},it must be that x−t1 ≤ 1 and xt1 ≥ x. Thus, u(x)≤1δ 2t1.Hence, u(x)≤ 1δ 2t1for all u ∈U .Lemma 28. Under Axioms A1-A6, there exists a continuous present equivalentutility function F : X×T→ R that represents %. F is monotonically increasing inx and monotonically decreasing in t.Proof. The first part of this proof is very similar to Lemma 24, and we will omit ithere. By construction of the set U , V (x, t) = minv∈U v−1(δ tv(x)). Moreover, forall u ∈U , u(0) = 0, u(1) = 1, infu∈U u(x)> 0, supu∈U u(x)< ∞ for x> 0.Finally, from A6, for any x> 0, and t ∈ T, there exists z such that (z, t) (x,0).δ tu(z) > u(x) ∀u ∈U=⇒ u(z)u(x)> δ−t ∀u ∈U=⇒ infuu(z)u(x)≥ δ−t ∀u ∈UBut we had started with arbitrary t. Hence, infuu(z)u(x)is unbounded above for anyx> 0.Theorem 8: The following two statements are equivalent:i) The relation % satisfies properties B0-B5.ii) There exists a continuous function F : X×P×T→ R such that (x, p, t) %(y,q,s) if and only if F(x, p, t) ≥ F(y,q,s). The function F is continuous, in-creasing in x, p and decreasing in t ∈ T. There exists a unique δ ∈ (0,1) anda set U of monotinically increasing continuous functions such that F(x, p, t) =minu∈U u−1(pδ tu(x)) and u(0) = 0 for all u ∈U .Proof. Showing that (ii) implies (i) :Continuity and monotonicity of % follow from the continuity and monotonicity of91A.1. Appendix to to Chapter 1F . Weak Present Bias follows as before.B5 can be shown in the following way:(x, pθ , t) % (x, p, t+D)=⇒ minuu−1(pθδ tu(x)) ≥ minuu−1(pδ t+Du(x))=⇒ θ ≥ δD=⇒ minuu−1(qθδ su(y)) ≥ minuu−1(qδ s+Du(y))=⇒ (y,qθ ,s) % (y,q,s+D)We will prove the direction (i) to (ii) in the following three steps.Step 1: Recall the Probability Time Tradeoff axiom: for all x,y ∈ X, p,q ∈(0,1],and t,s ∈ T, (x, pθ , t)% (x, p, t+∆) =⇒ (y,qθ ,s)% (y,q,s+∆).This axiom has calibration properties that we will use. Given Monotinicity, (x,1,0)(x,1,1)  (x,0,0) for any x ∈ X. By continuity, there must exist δ ∈ (0,1) suchthat (x,δ ,0)∼ (x,1,1). Note that Probability-Time Tradeoff Axiom helps us write(x,δ ,τ + 1) ∼ (x,1,τ) for all x ∈ X and τ ∈ T, and further extend it to (x,q, t) ∼(x,qδ t ,0). For integer t’s this follows by induction.For any integer b > 0, there exists ∆(1b) = δ1 ∈ P such that (x,δ1,0) ∼ (x,1,1b).Now applying Probability Time Tradeoff (PTT) repeatedly b times, (x,1,1) ∼(x,δ b1 ,0), which implies, δ1 = δ1b . For any ratio of positive integers (rational num-ber) t =ab, ∆(ab) = δab . This argument can be extended to all real t > 0. Thiscrucially helps us pin down δ as the discount factor.Henceforth, we are going to concentrate on finding a representation of the reduceddomain of X× [0,1]. Note that this reduced domain can also be conceptually seenas the set of all binary lotteries that have zero as one of the outcomes.Step 2: The rest of the proof will have a similar flavor to the ones the reader hasalready encountered. For every x∗ ∈X, we are going to provide an increasing utilityfunction u on [0,M]which would respect all the relations of the form (x, p)% (y,1),i.e, have pu(x)≥ u(y) and also have pu(y) = u(x∗) for all (x∗,1)∼ (y, p).Fix x∗, u(0) = 0 and u(x∗) = 1. For x ∈ (x∗,M], defineu(x) = {1p: (x, p)∼ (x∗,1)} (A.5)and,xq = {x : (x,q)∼ (x∗,1)} (A.6)The expressions in (A.5) and (A.6) exist due to the continuity of %.Now, for x ∈ (0,x∗), define92A.1. Appendix to to Chapter 1u(x) = inf{p(q) : (xq,qp(q))∼ (x,1),q≤ 1} (A.7)First, we will show that the infimum in (A.7) can be replaced by minimum. Con-sider a sequence of probabilities {pn} that converge below to p∗, and (xqn , pnqn)∼(x,1). Note that qn ∈ [qmax,1] where (x∗,1) ∼ (M,qmax). Hence, {qn} must bebounded by this closed interval, and must have a convergent subsequence {qnk}.Let q∗ be the corresponding limit, and we know that q∗ ≥ qmax. Similarly, xq can beconsidered continuous in q (this also follows from the continuity of %). Therefore,xqnk → xq∗ as qnk → q∗. Also, it must be that pnk → p∗ as qnk → q∗. Thus, we have(xqnk , pnk qnk) ∼ (x,1) for all elements of {nk}. Then, using the continuity of %,(xq∗ , p∗q∗)∼ (x,1).u(x) = inf{p : (xq, pq)∼ (x,1)}= min{p : (xq, pq)∼ (x,1)}= p∗Now we will show that the utility defined in (A.5) and (A.7) has the followingproperties : 1) It is increasing. 2) p1u(x1)= u(y1) 3) (x, p)% (y,1), implies pu(x)≥u(y) 4) u is continuous. The first two properties are true by definition of u. We willshow the third in some detail. Consider (x, p) % (y,1). In the case of interest,p < 1 and hence, x > y. Now let x > y > x∗. Let, u(y) = 1/p1, which means,(y, p1)∼ (x∗,1) . Given (x, p)% (y,1), we must have(x, pp1)% (y, p1)∼ (x∗,1)Hence,u(x) ≥ 1pp1⇐⇒ pu(x) ≥ 1p1= u(y)If, x> x∗ > y, the proof follows from the way the utility has been defined.Let y< x< x∗. Let, u(x) = p1, which means, (xq, p1q)∼ (x,1) for some xq. Given(x, p)% (y,1), we must have(xq, p1qp)% (x, p)% (y,1)Hence, u(y)≤ pp1.Now we turn to proving the continuity of u. The continuity at x∗ from the right iseasy to see.Next, for any r ∈ (0,1), definef (r) = sup{x ∈ [0,x∗) : (xq,qr)∼ (x,1)} (A.8)93A.1. Appendix to to Chapter 1The supremum can be replaced by a maximum, and the proof is similar to the onebefore. Suppose there is a sequence of {xn} that converges up to a value xˆ, and,(xq(n),q(n)r) ∼ (xn,1). Note that q(n) lies in a closed interval, and hence has aconvergent subsequence that converges to a point in that interval. Let us call thispoint qˆ. Now, x is continuous in q (in the usual sense), and hence, xn and xq(n) alsohas a convergent subsequence. The convergent subsequence {xq(n)} and {xn} musthave the same limit point, let us call it xqˆ, a point in [x∗,M]. Hence, the supremumin (A.8) must have been attained from xqˆ, and hence the supremum can be replacedby a maximum. The f function is well defined, strictly increasing and is the inversefunction of u over r ∈ (0,1). This function can be used to show the continuity of uat the point x∗.Finally, the function u can be easily normalized to have u(M) = 1.Step 3: In this step, we construct theU set as in Theorem Theorem 3, to completethe proof.Theorem 10: The following two statements are equivalent:i) The relation % on [0,∞)T satisfies properties D0-D5.ii) For any δ ∈ (0,1), there exists a set Uδ of monotinically increasing contin-uous functions such thatF(x0,x1, ..,xT−1) = x+T−1∑1minu∈Uδu−1(δ tu(xt))represents the binary relation%. The setUδ has the following properties: u(0) = 0and u(M) = 1 for all u ∈Uδ . F(.) is continuous.Proof: Going from (ii) to (i), we will show how the representation satisfies D5and the second property in D2.Suppose, (x0,x1, ..,xT−1) and (y0,y1, ..,yT−1) are orthogonal. Therefore,F(x0+ y0,x1+ y1, ..,xT−1+ yT−1) = F(x0,x1, ..,xT−1)+F(y0,y1, ..,yT−1)= F(z0,0, ..,0)+F(z′0,0, ..,0)= z0+ z′0= F(z0+ z′0,0, ..,0)To see how Discounting can be derived, start by assuming y0 > x > 0, and choosea function u1 ∈U . As δ ∈ (0,1), there must exist t such that δ tu1(y0)< u1(x) andhence, u−11 (δtu1(y0)) < x. For any sequences (y1,y2,y3, ..ym) and (n1,n2, ..,nm),where, (0, ..0, yi−1︸︷︷︸in period ni,0..,0)% (yi,0, ..,0) ∀i∈{1,2, ...,m}, one must have δ niu1(yi)≥94A.1. Appendix to to Chapter 1u1(yi−1) ∀i ∈ {1,2, ...,m}.Multiplying all these inequalities gives us,δ∑niu1(y0) ≥ u1(ym)⇐⇒ ym ≤ u−11 (δ∑niu1(y0))= u−11 (δtu1(y0))< u−11 (u1(x))= x=⇒ ym ≤ xNow to show the proof for the direction (i) to (ii), we start by following thesame steps we used in the proof of Theorem 3 to derive the set Uδ . There aretwo points to be noted during the construction of functions in Uδ . First, onlycomparisons upto lengths of T −1 periods need to be considered. Secondly, in theconstruction of each function u∈Uδ , the fact that the interative construction spansover R≥0 is guaranteed by the second part of the Discounting axiom. The additiverepresentation across periods follows from using induction and the D5 axiom.Theorem 5: Let %1 and %2 be two binary relations which allow for minimumrepresentation w.r.t sets Uδ ,1 and Uδ ,2 respectively. The following two statementsare equivalent:i) %1 allows a higher premium to the present than %2.ii) Both Uδ ,1 and Uδ ,1∪Uδ ,2 provide minimum representations for %1.Proof. The direction from (i) to (ii): Consider any (x, t)∈X×T such that (x, t)∼1(y,0). Using (i), we must have, (x, t)%2 (y,0).Hence,minu∈Uδ ,2u−1(δ tu(x)) ≥ y=⇒ minu∈Uδ ,1∪Uδ ,2u−1(δ tu(x)) = yHence,minu∈Uδ ,1∪Uδ ,2u−1(δ tu(x)) = minu∈Uδ ,1u−1(δ tu(x)) (A.9)To go in the opposite direction, let us assume, (x, t)%1 (y,0).95A.1. Appendix to to Chapter 1Given, (A.9), it must be thatminu∈Uδ ,1∪Uδ ,2u−1(δ tu(x)) = minu∈Uδ ,1u−1(δ tu(x))≥ y=⇒ u−1(δ tu(x)) ≥ y ∀u ∈Uδ ,1∪Uδ ,2=⇒ u−1(δ tu(x)) ≥ y ∀u ∈Uδ ,2=⇒ minu∈Uδ ,2u−1(δ tu(x)) ≥ yHence, (x, t)%2 (y,0), which completes the proof.Proposition 29. Let fn be a set of bijective, increasing, continuous functions. Letfn → f “locally uniformly”/ compactly (equivalent notions in Rn.), where f isbijective, increasing, continuous. Then, f−1n → f−1 compactly.Proof. Consider the composite function gn = fno f−1. Note that gn is also bijec-tive, increasing, continuous. As fn converges locally uniformly to f , gn convergeslocally uniformly to the identity function g(x).To see this, note that for any ε1 > 0supx∈[c,d]|gn(x)−g(x)| = supx∈[c,d]| fn( f−1(x))− f ( f−1(x))|= supy∈[ f−1(c), f−1(d)]| fn(y)− f (y)|≤ ε1for n≥ N0 for some N0.Choose an interval [a,b]. Now, there would exist n1,n2 such that gn(a−1)≤ a andgn(b+ 1) > b for n ≥ n1 and n ≥ n2 respectively. Similarly, there exists n3 suchthat supx∈[a−1,b+1] |gn(x)−g(x)|< ε for n≥ n3.Finally, for N ≥ max{n1,n2,n3}supx∈[a,b]|g−1n (x)−g(x)| ≤ supx∈[gn(a−1),gn(b+1)]|g−1n (x)− x|= supt∈[a−1,b+1]|g−1n (gn(t))−g(t)|= supt∈[a−1,b+1]|t−g(t)|< εTherefore, g−1n = f o f−1n converges locally uniformly to the identity function. There-fore, f−1n converges locally uniformly to f−1.96A.2. Appendix to Chapter 3A.2 Appendix to Chapter 3A.2.1 Diminishing Impatience does not imply the Certainty EffectWe start by providing a basic intuition of why DI as characterized by (3.4) failsto imply the certainty effect. To complete a proof in the direction from DI tocertainty effect one is required to approximate arbitrary probabilities used in lot-teries by the total hazard rate of termination over one or multiple periods. Morespecifically, one needs to approximate the ratio of probabilitiesg(p)g(pq),g(1)g(q)in therelation (3.2) by the relative hazard rates between two consecutive time-periodsin (3.4),g((1− r)t)g((1− r)t+1) ,g(1)g(1− r) respectively, for some hazard rate r. Given (3.4),we are restricted to establishing the certainty effect relation for p,q combinationswhich can be approximated as integer exponents of each other, hence the resultdoes not generalize to the certainty effect. Under DISDI we are approximatingg(p)g(pq),g(1)g(q)by the relative hazard rates between arbitrarily spaced time-periodsin (3.4),g((1− r)t)g((1− r)t+k) ,g(1)g((1− r)k) . Hence, we are allowed to approximate p,q bydifferent integer exponents of the hazard rate and hence rational exponents of eachother (for example, when, p = rk,q = rt , then p = qkt ). Given the rationals aredense in reals (and the integers are not!), a sequence ofkt’s can approximate lnq pand this allows the relation from time to risk be established for general p,q andcontinuous g. The following counter-example provides the vital step that DI doesnot imply DISDI.If (3.4) implied (3.2), then (3.4) would also imply that ∀r ∈ (0,1) and ∀m,n∈Ng(rm+n)> g(rm)g(rn) (A.10)We rewrite this problem in an additive form by defining f (x) =−log(g(e−x))⇐⇒g(x) = e− f (−logx). Then f : (0,∞)→ (0,∞) is differentiable and increasing, justlike the function g. The inequalities under consideration are now:∀t ∈ N and∀r ∈ (0,1), g(rt+1) > g(r)g(rt)⇐⇒ e− f(−log(rt+1)) > e− f (−log(rt))e− f (−log(r))⇐⇒ f (−(t+1)log(r)) < f (−tlog(r))+ f (−log(r))Now, defining x :=−log(r) ∈ (0,∞) for r ∈ (0,1).f ((t+1)x)< f (tx)+ f (x) (A.11)97A.2. Appendix to Chapter 3Further, the boundary conditions g(0) = 0 and g(1) = 1 translate to f (0) = 0 andf (∞) = ∞.63Similarly, (A.10) converts tof (mx+nx)< f (mx)+ f (nx) ∀x ∈ (0,∞)and ∀m,n ∈ N (A.12)Summing it up, (3.4) implies (A.10), if and only if (A.11) implies (A.12). Thenext step is to propose a function f which would satisfy (A.11) on all points of itsdomain, but violate (A.12) for some x ∈ R and some m,n ∈ N.Instead of providing the function f , we propose it’s derivative h, so f can becalculated as f (x) =∫ x0 h(x)dx.64 Let, k = 201+sin(pi/2−.0001) and δ = 50kpi cos(pi/2−.0001)≈ .157.Let,h(x) =11+(1− x)δ For x< 11+ k2 +k2 sin100pi(1+pi/2−.0001100pi − x) For 1≤ x≤ 1.005+ pi/2−.0001100pi1 For 1.005+ pi/2−.0001100pi < x< 2− .0054+3sin100pi(x−2) For 2− .005≤ x≤ 2+ .0057 For 2+ .005< x< 2.5− .0054+3sin100pi(2.5− x) For 2.5− .005≤ x≤ 2.5+ .0051 For 2.5+ .005< x< 3− .0054+3sin100pi(x−3) For 3− .005≤ x≤ 3+ .0057 For 3+ .005< x< 5− .0054+3sin100pi(5− x) For 5− .005≤ x≤ 5+ .0051 For x> 5+ .005f is increasing, twice differentiable and f (∞) = ∞. h(x) is plotted in Figure A.1.We next show that (A.11) holds.Lemma 30. ∀t∈ N, ∀x ∈ R,∫ x0 h(x)dx> ∫ (t+1)xtx h(x)dx.Proof. The most intuitive way to check the claim would be to notice that the sinu-soids introduced hardly perturb the area under the curve. Figure A.2 illustrates thepoint in a more clear fashion by considering the function h for a small part of thereal line. For all practical purposes, one could go about checking the inequalitiesby replacing the sinusoid (in black) in Figure 1 by a corresponding discontinuous63Using the extended real line (R∪∞)64Recall that f (0) = 0.98A.2. Appendix to Chapter 3Figure A.1: The function h.function(h¯(x) = 7 for x ≤ 2.5, h¯(x) = 1 for x > 2.5 as drawn in red). The areabetween the two curves in [2.495,2.5] is only (.005∗3− 3100pi )≈ .005. Therefore,as long as the inequalities hold with a large enough margin, this intuitive methodof approximating sinusoids with flat lines works fine. The area between the twocurves in [2.5,2.505] is also (.005 ∗ 3− 3100pi ). Thus, the two approximations in[2.495,2.505] are equal and opposite in direction, and the areas under the red andblack curves in this region are equal. During our analysis, in some cases therewill be multiple approximations in opposite directions which would partially orcompletely cancel each other out.Figure A.2: Function h approximated in a sinusoidal regionUtilizing this intuition more rigorously, one can create upper bounds and lowerbounds on∫ (t+1)xtx h(x)dx and∫ x0 h(x)dx respectively to complete the proof.For 0 < x ≤ 1, ∫ x0 h(x)dx > ∫ (t+1)xtx h(x)dx is obvious, as [0,x] contains the highestvalues obtained by h(x) on the real line.For, 1 < x ≤ 53 ,∫ x0 h(x)dx =∫ 10 h(x)dx+∫ x1 h(x)dx >12(11+ 11+ δ )+ (x− 1) =99A.3. Appendix to Chapter 410+ δ2 + x.65 The inequality holds because h(x) ≥ 1 with strict inequality for1≤ x< 1.005+ pi/2−.0001100pi , and hence∫ x1 h(x)dx> x−1. In the interval [tx,(t+1)x],h(x) ≤ 7 and after mutual canceling out there are no more than 3 sinusoidal per-turbations which could increase the area under the curve. Hence,∫ (t+1)xtx h(x)dx <7x+3(.015− 3100pi ) = 6x+x+3(.015− 3100pi )≤ 6(53)+x+3(.015− 3100pi ) = 10+x+3(.015− 3100pi ).For 53 ≤ x≤ 2,∫ x0 h(x)dx> 10+δ2 +x as before. On the other hand, using the sameline of logic as before,∫ 2xx g(x)dx< 1.x+6[(4−3)+(2.5−2)]+3.(.015− 3100pi ) =9+x+3.(.015− 3100pi ). Similarly,∫ 3x2x h(x)dx≤ 1.x+6[5−2.53 ]+3.(.015− 3100pi )=10+ x+3.(.015− 3100pi ).Similarly for larger values of x, it can be shown that∫ x0 h(x)dx >∫ (t+1)xtx h(x)dx.(follows trivially for x≥ 5.)Now complete the counter-example:∫ 20h(x)dx< 12+δ2+{.01∗10+(.015− 3100pi)}< 14−2(.015− 3100pi)=∫ 53h(x)dxThe first inequality follows from setting an upper bound on the sinusoidal pertur-bation introduced around 1.66 Therefore, f (5) > f (2)+ f (3), which provides uswith the counter-example to equation (A.12) and hence, to equation (A.10). Inother words, as (A.11) does not imply (A.12), (3.4) does not imply (A.10), andhence, (3.4) does not imply (3.2).That is, even if for all t ∈N and for all r ∈ (0,1) : g((1−r)t+1)> g((1−r)t)g((1−r)) it does not imply that ∀p,q ∈ (0,1): g(pq)> g(p)g(q) .A.3 Appendix to Chapter 4Cooperation in low δ treatmentsThe high and low discount factors would have different predictions under reputa-tion equilibrium. First consider, δ ≤ 3/8. Any egoistic player who believes that theother player conditionally cooperates till first defection with probability ρ0, con-templates the following before making a choice in the first period of the game:Suppose the subject is in Period 1. The lowest possible payoff from Defecting rightaway in this game is65δ = 50kpi cos(pi/2− .0001) = .157 (approximately)66This particular sinusoid dies down after 1.005+ pi/2−.0001100pi < 1.01 and never rises above theh(x) = 1 line by more than 6 units.100A.3. Appendix to Chapter 4Table A.2: Cooperation by treatment and period, split by order of treatmentsPeriod1 Period2 Period3 Period4 Period5 Average1 39.19 29.05 27.03 23.65 12.16 26.222 34.46 18.24 15.54 10.14 4.73 16.623 22.30 13.51 15.54 15.54 10.81 15.544 14.86 7.43 12.16 14.86 14.19 12.701 26.72 21.55 18.10 12.07 6.90 17.072 15.52 14.66 15.52 12.93 7.76 13.283 17.24 12.93 13.79 18.97 15.52 15.694 20.69 16.38 16.38 17.24 19.83 18.10The first 4 rows are from sessions run in decreasing order of δ .The last 4 rows are from sessions run in increasing order of δΠD = ρ0.2600+(1−ρ0)1200+1200(δ +δ 2+δ 3+δ 4)The highest possible payoff from Cooperating is bounded above by 67ΠC = ρ0.2000+(1−ρ0)200+ 2000(δ +δ 2+δ 3)+2600δ 4︸ ︷︷ ︸payoff from cooperating all the way and defecting on Period 5Now,ΠD > ΠC⇐⇒ ρ0.2600+(1−ρ0)1200+1200(δ +δ 2+δ 3+δ 4) > ρ0.2000+(1−ρ0)200+2000(δ +δ 2+δ 3)+2600δ 4⇐⇒ ρ0.600+(1−ρ0)1000 > 800(δ +δ 2+δ 3)+1400δ 4The LHS is atleast 600. The RHS is increasing in δ and the maximum possiblevalue of RHS is 482 for δ ≤ 3/8. Hence, any rational player should Defect rightaway under δ ≤ 3/8. The same argument would hold if the subject was in any otherperiod of the game.Now, instead assume δ ≥ 3/4. If ΠC is the highest possible return from Cooperting67Under the considered δ range, this does strictly better that defecting earlier.101A.3. Appendix to Chapter 4in the present, then,ΠC ≥ ρ0.2000+(1−ρ0)200+ 2600δρ0+1200δ (1−ρ0)︸ ︷︷ ︸minimum second round payoff given beliefs+ 1200(δ 2+δ 3+δ 4)︸ ︷︷ ︸minimum continuation payoff from last 3 periodsFor reasonably large ρ0, Πc > ΠD. A similar analysis holds for later periods also.In general, when beliefs about the other player being a cooperative behavioral typeis high, Cooperation is always justifiable for δ ≥ 3/7.A similar analysis would show that there exist no possible beliefs on the partnerthat should induce an egoist to cooperate in any period of the δ = 1/4 treatment. Theeasiest wat to see it is to note that the lowest possible one time gain from defectionvs cooperation in any period, overpowers any possible future gains, irrespective ofthe beleifs held over the other player’s action.102


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