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Essays on decision-making under risk Freeman, David 2013

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Essays on Decision-Making UnderRiskbyDavid FreemanB.A., The University of Guelph, 2006M.A., The University of British Columbia, 2007A 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)October 2013c? David Freeman 2013AbstractChapter 2 of this thesis studies the testable content of models of expectations-basedreference-dependence. The main results of this chapter characterize a model based onK?szegi and Rabin?s (2006) preferred personal equilibrium. This model is shown to bebehaviourally equivalent to a version of the shortlisting model of Manzini and Mari-otti (2007) in environments without risk. Environments with risk motivate novel ax-ioms that are conceptually consistent with expectations-based reference-dependence.These axioms are shown to behaviourally characterize a restricted version of the pre-ferred personal equilibrium model of decision-making. Additional results characterizethe choice behaviour of K?szegi and Rabin?s (2006, 2007) personal equilibrium andchoice-acclimating personal equilibrium concepts.In the presence of background risk and under the Reduction of Compound Lotter-ies axiom, non-expected utility preferences cannot capture descriptively reasonablerisk aversion over small stakes without producing implausible risk aversion over largestakes (Safra and Segal 2008). Motivated by experimental evidence, Chapter 3 as-sumes that compound lotteries are evaluated recursively. The main results of thischapter show that non-expected utility theories generate ?as-if? narrow bracketingover small-stakes gambles despite defining utility over final wealth, and can be con-sistent with empirically reasonable risk aversion over both small and large stakes.Chapter 4 uses list elicitation with varying probabilities to experimentally studychoices between lotteries for a population of online workers. We document that listelicitation significantly diminishes risk aversion compared to binary choice elicita-tion. We show that this observation is consistent with a decision maker who hasnon-expected utility preferences, but when list elicitation is employed, reduces thecompound lottery induced by her choices and the external randomization device usedto determine payment. As a result, list elicitation distorts the inferences that can bedrawn about non-expected utility preferences.iiPrefaceAll chapters of this thesis are original and unpublished work. I am the sole author ofChapters 2 and 3 of this thesis.Chapter 4 is coauthored with Yoram Halevy and Terri Kneeland, and is a part of alarger collaborative project. The experiment was jointly designed by all three authors,and the interpretation of our results and editing the paper were also collaborativeefforts. The online interface was designed and implemented by Terri Kneeland. Theexperiments were run by myself and Terri Kneeland. I was responsible for dataanalysis and the details of the theoretical explanation of our results, and was primarilyresponsible for writing the paper. Grant applications, funding, and ethics approvalwere the sole responsibility of Yoram Halevy. Ethics approval under the project title?Mechanical Turk - Decision Theory 1? approved by the Behavioural Research EthicsBoard of the University of British Columbia (certificate number H11-01719).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Revealed Preference Foundations of Expectations-Based Reference-Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Background: expectations-based reference-dependence . . . . 82.1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Two examples and a motivating result . . . . . . . . . . . . . . . . . 112.2.1 Formal setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Mugs, pens, and expectations-based reference-dependence . . 122.2.3 IIA violations under K?szegi-Rabin under PPE . . . . . . . . 132.2.4 The testable implications of K?szegi-Rabin under PE: a nega-tive result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Revealed preference analysis of PPE . . . . . . . . . . . . . . . . . . 152.3.1 Technical prelude . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Revealed preference analysis without risk . . . . . . . . . . . 162.3.3 Revealed preference analysis with risk . . . . . . . . . . . . . 192.3.4 Sketch of proof and an intermediate result . . . . . . . . . . . 232.3.5 A definition of expectations-dependence and its implications . 24ivTable of Contents2.3.6 Limited cycle property of a PPE representation . . . . . . . . 252.4 Special cases of PPE representations . . . . . . . . . . . . . . . . . . 262.4.1 K?szegi-Rabin reference-dependent preferences . . . . . . . . 262.4.2 Reference lottery bias and dynamically consistent non-expectedutility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Alternative models of expectations-based reference-dependence: anal-ysis of PE and CPE representations . . . . . . . . . . . . . . . . . . 302.5.1 Characterization of PE . . . . . . . . . . . . . . . . . . . . . 302.5.2 Characterization of CPE . . . . . . . . . . . . . . . . . . . . 313 Calibration without Reduction for Non-Expected Utility . . . . . 333.1 Theory: RNEU risk preferences with background wealth risk . . . . 363.1.1 Non-expected utility over lotteries . . . . . . . . . . . . . . . 363.1.2 Recursive non-expected utility . . . . . . . . . . . . . . . . . 373.1.3 Wealth risk as a compound lottery . . . . . . . . . . . . . . . 383.1.4 Nonreduction and narrow bracketing . . . . . . . . . . . . . . 393.1.5 Non-reduction and small-stakes risk aversion . . . . . . . . . 403.1.6 The Rabin critique? . . . . . . . . . . . . . . . . . . . . . . . 413.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 List Elicitation of Risk Preferences . . . . . . . . . . . . . . . . . . . 464.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.1 List elicitation versus binary choice . . . . . . . . . . . . . . . 514.2.2 The independence axiom . . . . . . . . . . . . . . . . . . . . 524.2.3 Treatment effects . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.4 Comparison to a student subject pool . . . . . . . . . . . . . 544.3 Theory: binary choice versus list elicitation . . . . . . . . . . . . . . 544.3.1 Karni and Safra (1987): Reduction of Compound Lotteries . . 554.3.2 Segal (1988): non-standard view of the compound lottery formedby the list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57vTable of Contents4.4.1 Related literature on elicitation mechanisms and incentive-compatibility574.4.2 Discussion of our results . . . . . . . . . . . . . . . . . . . . . 584.4.3 Experiments on Mechanical Turk . . . . . . . . . . . . . . . . 59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61AppendicesA Proofs for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70B Proofs for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C Examples that rationalize our results . . . . . . . . . . . . . . . . . . 96C.0.4 Rationalizing our data using ROCL . . . . . . . . . . . . . . 96C.0.5 Non-standard application of Compound Independence . . . . 99viList of Tables2.1 Example of reference-dependent preferences . . . . . . . . . . . . . . 132.2 Testing IIA Independence . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Testing Transitive Limit . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Two choice correspondences . . . . . . . . . . . . . . . . . . . . . . . 243.1 Non-expected utility theories . . . . . . . . . . . . . . . . . . . . . . . 373.2 Calibration results - small and large stakes risk aversion . . . . . . . . 444.1 Binary choice versus list elicitation . . . . . . . . . . . . . . . . . . . 474.2 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Subjects by treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5 Answers to line 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Fanning in vs out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.7 Tests for payment mechanism and order effects . . . . . . . . . . . . . 534.8 mTurk vs. student subjects . . . . . . . . . . . . . . . . . . . . . . . 54viiAcknowledgementsNumerous faculty members at UBC have been extremely helpful at different timesduring my years as both an MA and PhD student. I?m extremely grateful to myprofessors and classmates for their support on my journey through graduate school.I would like to especially thank my committee members Mike Peters and Li Hao fortheir suggestions and feedback.I would like to thank my advisor, Yoram Halevy. Yoram dedicated a substantialamount of time to helping me develop as an economist, and struck a fine balancebetween providing encouragement and pushing me to raise the bar for myself.My mother and father, Cathy and Michael Freeman, have always encouraged andsupported my education from day one all the way through to my PhD. I?m incrediblylucky to have had such support.To Luba: we made it. I am thankful for your love and support, for your havingfaith in me and in us.I graciously acknowledge financial support from SSHRC in the form of a CanadaGraduate Scholarship.viiiChapter 1IntroductionMany important economic decisions involve elements of risk. Examples include insur-ance, portfolio, and occupational choices. Thought experiments and laboratory ex-periments in economics and psychology have identified robust evidence of behaviourthat is not consistent with economists? workhorse model - expected utility. Threeparticularly robust findings are that people make decisions that: (i) depend on ref-erence points (Kahneman and Tversky, 1979), (ii) violate the Independence Axiom(Allais, 1953), and (iii) exhibit risk aversion over small stakes (Rabin, 2000). Yet thechallenge of incorporating this evidence into economic models has raised issues thatremain unsolved.Reference-dependenceExperiments have identified that choice behaviour can depend on how alternativescan compare to a reference point both in risky and riskless choice tasks (Kahnemanand Tversky, 1979; Kahneman, Knetsch, and Thaler, 1990; Tversky and Kahneman,1991). Kahneman and Tversky?s (1979) prospect theory, its extension to risklessenvironments (Tversky and Kahneman, 1991), and subsequent work (Masatliogluand Ok, 2005; K?szegi and Rabin, 2006; Sagi, 2006) show that reference-dependencecan be incorporated in economic models of decision-making. These models dropthe standard assumption that a decision-maker has a single complete and transitivepreference relation and instead allow the decision-maker?s preferences to depend onher reference point. However, reference points are not a directly observable economicvariable. In lab experiments, the researcher is able to control the framing of decisionsin a way that would induce subjects? reference points. However, researchers cannot dothis in natural economic environments. This poses a challenge to the testing and useof these models in real economic environments, or in lab settings where the reference1Chapter 1. Introductionpoint is not explicitly primed.The traditional assumption in applying reference-dependent models is that adecision-maker?s status quo determines her reference point, as in prospect-theoreticmodels that were designed to explain behaviour in lab experiments. But Kahnemanand Tversky (1979, p. 286) acknowledge the limitations of this approach in realeconomic settings:?So far in this paper, gains and losses were defined by the amounts ofmoney that are obtained or paid when a prospect is played, and the ref-erence point was taken to be the status quo, or one?s current assets. Al-though this is probably true for most choice problems, there are situationsin which gains and losses are coded relative to an expectation or aspira-tion level that differs from the status quo. For example, an unexpectedtax withdrawal from a monthly pay check is experienced as a loss, not asa reduced gain.?Recently, K?szegi and Rabin (2006) proposed a model of expectations-based reference-dependent decision-making, consistent with Kahneman and Tversky?s suggestion thatexpectations may determine reference points in environments where a decision-makerexpects her status-quo to change. However, expectations are not observed in standardeconomic data, and so the critique remains: models of reference-dependence in gen-eral, and expectations-based reference-dependence in particular, allow the decision-maker?s utility function to depend on unobserved reference points. Do these modelshave testable implications, or can they explain almost anything?The Independence AxiomThe normatively-appealing Independence Axiom is the substantive axiom that givescomplete, transitive, and continuous preferences an expected utility representation.But a set of thought experiments by Allais (1953) suggests that many people would failto satisfy the Independence Axiom. The central feature of Allais? thought experimentsis a distinction between choice between a pair of risks, and choice when a certainreward is available.2Chapter 1. IntroductionThe decision theory literature in economics responded to the ?Allais paradox? byrelaxing the Independence Axiom to accommodate the choice patterns suggested byAllais. Such non-expected utility models include rank-dependent utility (Quiggin,1982; Yaari, 1987), weighted utility (Chew, 1983), and disappointment aversion (Gul,1991).1In models that do not satisfy the Independence Axiom, there are multiple waysto model how a decision-maker treats multiple sources of risk. When applying non-expected utility models, it is natural to view multiple risks as forming a multi-stagecompound lottery in the sense of Segal (1990). Suppose we take a non-expected utilitypreference over single-stage lotteries as a starting point. Segal suggests two waysto extend single-stage lottery preferences to the domain of compound lotteries, theReduction of Compound Lotteries and Compound Independence axioms. Under theReduction of Compound Lotteries axiom, a decision-maker behaves as-if she reducesa compound lottery to its probabilistically-equivalent single-stage lottery, and thenapplies her lottery preferences to evaluate the resulting single-stage lottery. Underthe Compound Independence Axiom, a decision-maker behaves as-if she evaluatesa compound lottery recursively, that is, she first applies her lottery preferences toevaluate the certainty equivalent of each second-stage lottery, and then applies herlottery preferences again to the resulting single-stage lottery.There is evidence for and against both axioms. The Reduction of CompoundLotteries axiom is used in Karni and Safra?s (1987) explanation of preference rever-sals in the Becker, DeGroot, and Marschak (1964) mechanism for eliciting certaintyequivalents, and a failure of Compound Independence is central to their argument.Segal (1990) shows that the Compound Independence is consistent with Kahnemanand Tversky?s isolation effect. Halevy (2007) finds additional evidence against theReduction of Compound Lotteries axiom. Segal (1990) shows that preferences onsingle-stage lotteries satisfy both assumptions simultaneously if and only if they areexpected utility. This discussion suggests that in some environments, the CompoundIndependence axiom is the more descriptively appropriate assumption, while in other1I use the term ?non-expected utility? to refer to these axiomatic models, as distinct frompsychologically-derived models like Kahneman and Tversky?s (1979) prospect theory which makethe more radical departure of defining utility over gains and losses, rather than over final outcomes.3Chapter 1. Introductionenvironments the Reduction of Compound Lotteries axiom may be more appropriate.Small-stakes risk aversion and the Rabin critiqueThought experiments (Samuelson, 1963; Rabin, 2000), lab experiments, and insurancechoices suggests that people demonstrate quantitatively significant risk aversion overstakes of tens and hundreds of dollars. For example, most people would turn downa gamble with a 50% chance of winning $11, and a 50% chance of losing $10. Thischoice pattern seems reasonable, and on its own does not contradict any axioms ofexpected utility.However, suppose that a decision-maker would make the same choice even if shewere much richer or much poorer. If she would turn down this gamble at any possi-ble wealth level, Rabin shows that if she has expected utility preferences she wouldalso turn down any gamble with a 50% chance of losing $100, even if it involved a50% chance of winning an infinite amount of money. This level of risk aversion isdescriptively implausible.In expected utility, risk aversion is captured in a concave utility-for-wealth func-tion. Rabin suggests that either reference-dependent models or non-expected utilitymodels could address his critique of expected utility. In reference-dependent mod-els that incorporate loss aversion and narrow bracketing, the decision-maker weighslosses more than equal-sized gains in evaluating a gamble, and ignores all other risksin her economic environment. Loss aversion can generate substantial small-stakes riskaversion over gambles that involve both gain and loss outcomes; probability weight-ing (as in prospect theory) can generate small-stakes risk aversion over gambles thatinvolve only gains. Popular non-expected utility models, including rank-dependentutility and disappointment aversion, can also display small-stakes risk aversion with-out being susceptible to Rabin?s argument, since in these models risk aversion oversmall-stakes comes primarily from non-linear probability weighting.Because most people face substantial pre-existing risks, and the decision to take orturn down an offered small-stakes gamble involves multiple sources of risk. As previ-ously discussed, reference points are not directly observable, and reference-dependentmodels require assumptions about choice bracketing. Non-expected utility models4Chapter 1. Introductionavoid the former issue, and allow for two disciplined approaches to the latter issue.Multiple sources of risk are naturally modelled as a compound lottery, and eitherthe Reduction of Compound Lotteries axiom or the Compound Independence axiomcan tractably extend the domain of non-expected utility models to compound lotter-ies. As reviewed previously, it is not a-priori obvious which assumption is the morenatural descriptive assumption in real economic decisions.Safra and Segal (2008) show that Rabin?s argument extends to smooth versionsof non-expected utility; but most commonly applied non-expected utility models,including rank-dependent utility and disappointment aversion, do not satisfy theirsmoothness assumption. However, Safra and Segal offer an additional result thatshows that that in the presence of background risk, a major class of non-expectedutility preferences are subject to a Rabin critique. While their result applies to rank-dependent utility and disappointment aversion, their argument implicitly relies onthe assumption that a decision-maker satisfies the Reduction of Compound Lotteriesaxiom when evaluating the compound lottery formed by a gamble and her pre-existingrisks.Thesis outlineChapter 2 of this thesis provides behavioural foundations for models of expectations-based reference-dependence (? la K?szegi and Rabin (2006)). When restricted to ax-ioms and model restrictions apply equally in environments with and without risk, eachcommonly-used model of preferred personal equilibrium decision-making of K?szegiand Rabin (2006) cannot be distinguished from a versions of the shortlisting model ofManzini and Mariotti (2007). The analysis shows that environments with risk providethe natural testing ground for models of expectations-based reference-dependence.The chapter offers three new axioms in environments with risk that are conceptuallymotivated by expectations-based reference-dependence. Theorem 2.1 uses these ax-ioms to provide a tight characterization of the preferred personal equilibrium modelof expectations-based reference-dependence with functional form restrictions that arenatural to environments with risk. Related results characterize alternative models ofexpectations-based reference-dependence in environments with and without risk. An5Chapter 1. Introductionadditional contribution of the analysis is a choice-based definition of expectations-dependence based on a type of violation of the Independence Axiom.Chapter 3 of this thesis revisits the problem of risk-taking in the presence of back-ground risk under non-expected utility preferences over single-stage lotteries. Theanalysis assumes that a decision-maker recursively evaluates the compound lotteryformed by a gamble offered on top of pre-existing risks. The results of this chaptershow that this assumption allows non-expected utility models like rank-dependentutility and disappointment aversion to simultaneously produce descriptively reason-able levels of risk aversion over small and large stakes for a decision-maker who facesbackground risk. Indeed, a decision-maker with recursive non-expected utility prefer-ences over compound lotteries behaves as-if she brackets narrowly over small-stakes,a result that is made precise in Theorems 3.1 and 3.2. These positive results standin stark contrast to Safra and Segal?s (2008) negative results assuming reduction ofcompound lotteries.List elicitation has become a popular experimental design for eliciting preciseinformation about risk preferences. In list elicitation, a subject makes a list of binarychoices between lotteries, and one of those binary choices is randomly selected andthe choice played for real to determine the subject?s payment from the list. Thisexperimental design makes payment the result of a compound lottery. If subjectsevaluate this compound lottery recursively, then they answer each question in the listthe same way they would answer if they made a single binary choice that determinedtheir payment for sure. However, if subjects have non-expected utility preferencesand do not evaluate compound lotteries recursively, perhaps because they reducecompound lotteries, then the design of list elicitation will lead subjects to makedifferent binary choices than if they faced a single choice. Chapter 4 visits thisproblem of experimental design for eliciting risk preferences. The main result of thischapter is that list elicitation sigificantly affects subjects? choices. This result suggeststhat it is difficult to draw unambiguous inferences about risk preferences from dataelicited using list elicitation.6Chapter 2Revealed Preference Foundations ofExpectations-BasedReference-Dependence2.1 IntroductionSeminal work by Kahneman and Tversky introduced psychologically and experimen-tally motivated models of reference-dependence to economics. A limitation preventingthe adoption of reference-dependent models is that reference points are not a directlyobservable economic variable. Kahneman and Tversky (1979) acknowledge that whileit may be natural to assume that a decision-maker?s status quo determines her ref-erence point in their experiments, it is not appropriate in many interesting economicenvironments. The lack of a generally applicable model of reference point forma-tion in economic environments has hindered applications of reference-dependence toeconomic settings.K?szegi and Rabin (2006) propose a model in which a decision-maker?s recently-held expectations determine her reference point. Their solution concept for endoge-nously determined reference points has made their model convenient in numerouseconomic applications, including risk-taking and insurance decisions, consumptionplanning and informational preferences, firm pricing, short-run labour supply, labourmarket search, contracting under both moral hazard and adverse selection, and do-mestic violence.2 In many of these applications, observed behaviour that appears im-2K?szegi and Rabin (2007); Sydnor (2010); K?szegi and Rabin (2009); Heidhues and K?szegi(2008, Forthcoming); Karle and Peitz (2012); Crawford and Meng (2011); Abeler, Falk, G?tte,and Huffman (2011); Pope and Schweitzer (2011); Eliaz and Spiegler (2013); Herweg, Muller, andWeinschenk (2010); Carbajal and Ely (2012); Card and Dahl (2011).72.1. Introductionpossible to explain using standard models naturally fits the intuition of expectations-based reference-dependence.Little is known about the testable implications of expectations-based reference-dependence in more general settings in spite of the large number of applications.It has been suggested that models of expectations-based reference-dependence mayhave no meaningful revealed preference implications, and that their success comesfrom adding in an unobservable variable, the reference point, used at the modeller?sdiscretion (Gul and Pesendorfer, 2008). The results here confront this claim: mod-els of expectations-based reference-dependence do have economically meaningful andtestable implications for standard economic data. The revealed preference axioms ofthis paper completely summarize the implications of a widely-applied version of themodel.The main contribution of this paper is to provide a set of revealed preferenceaxioms that constitute necessary and sufficient conditions for a model of expectations-based reference-dependence. Commonly-used cases of K?szegi and Rabin?s model arespecial cases of the model studied here. The revealed preference axioms clarify how themodel can be tested against both the standard rational model and against alternativebehavioural theories.As in existing models of reference-dependence, behaviour is consistent with max-imizing preferences conditional on the decision-maker?s reference point. The mainchallenge of the analysis is that expectations are not observed in standard economicdata. Under expectations-based reference-dependence, the interaction between opti-mality given a reference point and the determination of the reference point as rationalexpectations can generate behaviour that appears unusual since expectations are notobserved. Axioms justified by the logic of expectation-dependent decisions are shownto summarize the testable content of this unusual behaviour.2.1.1 Background: expectations-based reference-dependenceThe logic of reference-dependence suggests that rather than using a single utilityfunction, a reference-dependent decision-maker has a set of reference-dependent utilityfunctions. The utility function v(?|r) defines the decision-maker?s utility function82.1. Introductiongiven reference lottery r. When the reference lottery r is observable, as in the casewhere a decision-maker?s status quo is her referent, standard techniques can be appliedto study v(?|r). But when the reference lottery is determined endogenously and isunobserved, as in the case where the reference lottery is determined by the decision-maker?s recent expectations, an additional modelling assumption is needed. To thatend, K?szegi and Rabin (2006) introduce two solution concepts - personal equilibriumand preferred personal equilibrium - that capture the endogenous determination ofthe reference lottery for models with expectations as the reference lottery.In an environment in which a decision-maker faces a fully-anticipated choice set D,rational expectations require that the decision-maker?s reference lottery correspondswith her actual choice from D. In such an environment, the set of personal equilibriaof D provides a natural set of predictions of a decision-maker?s choice from a set D:PEv(D) = {p ? D : v(p|p) ? v(q|p) ?q ? D} (2.1)The personal equilibrium concept has the following interpretation. When choosingfrom choice set D, a decision-maker uses her reference-dependent preferences v(?|r)given her reference lottery (r) and chooses argmaxp?Dv(p|r). When forming expecta-tions, the decision-maker recognizes that her expected choice p will determine thereference lottery that applies when she chooses from D. Thus, she would only expecta p ? D if it would be chosen by the reference-dependent utility function v(?|p), thatis, if p ? argmaxq?Dv(q|p). The set of personal equilibria of D in (2.1) is the set of allsuch p.There may be a multiplicity of personal equilibria for a given choice set. Indeed,if reference-dependence tends to bias a decision-maker towards her reference lottery,multiplicity is natural. At the time of forming her expectations, a decision-maker eval-uates the lottery p according to v(p|p), which reflects that she will evaluate outcomesof lottery as gains and losses relative to outcomes of p itself. The preferred personalequilibrium concept is a natural refinement of the set of personal equilibria based on adecision-maker picking her best personal equilibrium expectation according to v(p|p):PPEv(D) = argmaxp?PE(D)v(p|p) (2.2)92.1. IntroductionK?szegi and Rabin (2006) adopt a particular functional form for v. They assumethat given probabilistic expectations summarized by the lottery r, a decision-makerranks a lottery p according to:vKR(p|r) =?k?ipimk(xki ) +?k?i?jpirj??mk(xki )?mk(xkj )?(2.3)In (2.3), mk is a consumption utility function in ?hedonic dimension? k; differenthedonic dimensions are akin to different goods in a consumption bundle, but specifiedbased on ?psychological principles?. The function ? is a gain-loss utility function whichcaptures reference-dependent outcome evaluations.The K?szegi-Rabin model with the preferred personal equilibrium concept hasbeen particularly amenable to applications, since the model?s predictions are pinneddown by (2.3) and (2.2). However, little is known about how the K?szegi-Rabin modelbehaves except in very specific applications.This paper focuses on expectations-based reference-dependent preferences with thepreferred personal equilibrium concept as in (2.2). Theorem 2.1 provides a completerevealed preference characterization of the choice correspondence c that equals theset of all preferred personal equilibria of a choice set, c(D) = PPEv(D). The modelof decision-making equivalent to the axioms does not restrict v to the form in (2.3)but does require that v be jointly continuous in its arguments, v(?|r) satisfy expectedutility, and v satisfy a property related to disliking mixtures of lotteries.The tight characterization of the PPE model of expectations-based reference-dependence in Theorem 2.1 may come as a surprise relative to previous work (e.g.Gul and Pesendorfer 2008; K?szegi 2010).3 The analysis here also provides additionalsurprising connections. First, the PPE representation is related to the shortlisting rep-resentation of Manzini and Mariotti (2007), a connection clarified in Proposition 2.2.3Gul and Pesendorfer (2008) show that with the personal equilibrium concept and without usingany lottery structure, the reference-dependent preferences of K?szegi and Rabin (2006) have notestable implications beyond an equivalence with a choice correspondence generated by a binaryrelation. K?szegi (2010) initially proposed the personal equilibrium concept studied here but providesonly a limited set of testable implications, and suggested that a complete revealed preference maynot be possible: ?I do not offer a revealed-preference foundation for the enriched preferences?it isnot clear to what extent the decisionmaker?s utility function can be extracted from her behavior.?102.2. Two examples and a motivating resultSecond, there is a tight connection between expectations-based reference-dependenceand failures of the Mixture Independence Axiom; violations of Independence of Irrel-evant Alternatives (IIA) are sufficient but not necessary for expectations-dependentbehaviour in the model (Proposition 2.3).2.1.2 OutlineSection 2.2 provides two examples that motivate expectations-based reference-dependence,and a result that illustrates the limits to the model?s testable implications in envi-ronments without risk. Section 2.3 provides axioms and a representation theoremfor PPE decision-making, and suggests a way of defining expectations-dependence interms of observable behaviour. Section 2.4 explores special cases of the model, includ-ing K?szegi-Rabin and a new axiomatic model of expectations-based reference lotterybias. Section 2.5 shows how the analysis can be adapted to study PE decision-makingand also to decision-making under K?szegi and Rabin?s (2007) choice-acclimating per-sonal equilibrium (CPE).2.2 Two examples and a motivating result2.2.1 Formal setupLet ? denote the set of all lotteries with support on a given finite set X, with typicalelements p, q, r ? ?. Let D denote the set of all finite subsets of ?, a typical D ? Dis called a choice set. The starting point for analysis is a choice correspondence,c : D ? D, which is taken as the set of elements we might observe a decision-makerchoose from a set D. Assume ? ?= c(D) ? D, that is, a decision-maker always choosessomething from her choice set.Define the mixture operation (1? ?)D+ ?D? := {(1? ?)p+ ?q : p ? D, q ? D?}.112.2. Two examples and a motivating result2.2.2 Mugs, pens, and expectations-basedreference-dependenceThe classic experimental motivation for loss-aversion in riskless choice comes from theendowment effect. An example of an endowment effect comes from the experimentalfinding that randomly-selected subjects given a mug have a median willingness-to-accept for a mug that is double the median willingness-to-pay of subjects who werenot given a mug (Kahneman, Knetsch, and Thaler, 1990). This classic experimentprovides no separation between status-quo-based and expectations-based theories ofreference-dependence since subjects given a mug could expect to be able to keep itat the end of the experiment.To separate expectations-based theories of reference-dependence from status-quobased theories, Ericson and Fuster (2011) design an experiment in which all subjectsare endowed with a mug, and subjects are told that there is a fixed probability(either 10% or 90%) they will receive their choice between a retaining the mug orinstead obtaining a pen, and with the remaining probability they will retain the mug;the conditional choice must be made before uncertainty is resolved.4 Subjects ina treatment with a 10% chance of receiving their choice must expect to receive amug with at least a 90% chance, and consistent with expectations-based reference-dependence, 77% of these subjects? conditionally choose the mug. In contrast, only43% of subjects conditionally choose the mug in the treatment in which subjectsreceived their chosen item with a 90% chance.The Mixture Independence axiom below adapts of von-Neuman and Morgenstern?saxiom to a choice correspondence.Mixture Independence. (1? ?)c(D) + ?c(D?) = c((1? ?)D + ?D?) ?? ? (0, 1)The median choice pattern in Ericson and Fuster?s experiment has {?mug, 1?} =c(.9{?mug, 1?}+ .1{?mug, 1? , ?pen, 1?}) but {?mug, .1; pen, .9?} = c(.1{?mug, 1?}+4This paper interprets the subjects? choice as being between two lotteries, each of which involvesthe prize of the mug with a fixed probability (10% or 90%) and the prize chosen by the subject withthe remaining probability. An alternative interpretation of the experimental setup is that subjectsface a lottery over choice sets, one of which is a singleton, and must choose from the non-singletonchoice set before the lottery is resolved. For a result on the formal relationship between these choicespaces, see Ortoleva (2013).122.2. Two examples and a motivating resultTable 2.1: Example of reference-dependent preferencesv(p|?) v(q|?) v(r|?)v(?|p) 1000 900 1050v(?|q) -1350 0 -75v(?|r) -1575 -450 -262.50.9{?mug, 1? , ?pen, 1?}). This choice pattern suggests an intuitive and empiricallysupported violation of Mixture Independence that is consistent with expectation-bias.2.2.3 IIA violations under K?szegi-Rabin under PPEConsider a decision-maker with a K?szegi-Rabin v as in (2.3), with linear utility andlinear loss aversion:56m(x) = x, ?(x) =???x if x ? 03x if x < 0When faced with a set of lotteries, suppose that our decision-maker chooses hispreferred personal equilibrium lottery as in (2.2).Consider the three lotteries p = ?$1000, 1?, q = ?$0, .5; $2900, .5?, and r =?$0, .5; $2000, .25; $4100, .25?. As broken down in Table 2.1, the decision-maker?schoice correspondence, c, is given by {p} = c({p, q}), {q} = c({q, r}), {r} =c({p, r}), and {q} = c({p, q, r}).Choice from binary sets reveals an intransitive cycle. Because of this, there is nopossible choice from {p, q, r} is consistent with preference-maximization! Considerthe Independence of Irrelevant Alternatives (IIA) axiom below, which Arrow (1959)shows is equivalent to maximization of a complete and transitive preference relation.IIA. D? ? D and c(D) ?D? ?= ? =? c(D?) = c(D) ?D?.5I would like to specially thank Matthew Rabin for suggesting this example.6Linear loss aversion is used in most applications of K?szegi-Rabin, and the chosen parameteri-zation is broadly within the range implied by experimental studies.132.2. Two examples and a motivating resultIn the K?szegi-Rabin PPE example, adding the lottery r to the set {p, q} generates aviolation of IIA, since r is not chosen yet affects choice from the larger set. Given fixedexpectations r, our decision-maker?s behaviour would be consistent with the standardmodel: she would maximize v(?|r). The decision-maker exhibits novel behaviour be-cause her expectations, and hence preferences, are determined endogenously in achoice set. However, rational expectations combined with preferred personal equi-librium put quite a bit of structure on the decision-maker?s novel behaviour. Theaxiomatic analysis that follows will clarify the nature of such structure.2.2.4 The testable implications of K?szegi-Rabin under PE:a negative resultThe preceding example demonstrates that the K?szegi-Rabin model with PPE gen-erates choice behaviour that cannot be rationalized by a complete and transitivepreference relation. Gul and Pesendorfer (2008) suggest that compared to the stan-dard rational model, this may be the only revealed preference implication of theK?szegi-Rabin model when paired with the personal equilibrium solution criteria in(2.1). Gul and Pesendorfer take as a starting point a finite set X of riskless elements,a reference-dependent utility v : X ?X ? ?, and offer the following result:Proposition 2.1. (Gul and Pesendorfer 2008). The following are equivalent: (i)c is induced by a complete binary relation, (ii) there is a v such that c(D) = PEv(D)for any choice set D, (iii) there is a v that satisfies (2.3) such that c(D) = PEv(D)for any choice set D.Proof. (partial sketch)If c(D) = {x ? D : xRy ?y ? D} then define v by: v(x|x) ? v(y|x) if xRy, andv(y|x) > v(x|x) otherwise. Then, {xRy ?y ? D} ?? {v(x|x) ? v(y|x) ?y ? D}.By reversing the process, we could construct R from v. Thus (i) holds if and only if(ii) holds.Gul and Pesendorfer cite K?szegi and Rabin?s (2006) argument that the set ofhedonic dimensions in a given problem should be specified based on ?psychologicalprinciples?. Since X has no assumed structure, Gul and Pesendorfer infer hedonic142.3. Revealed preference analysis of PPEdimensions from c and the structure imposed by (2.3). Their construction shows anyv has a representation in terms of the functional form in (2.3).The analysis that follows uses two assumptions that allow for a rich set of testableimplications of expectations-based reference-dependence. First, c is defined on a sub-sets of lotteries over a finite set. The structure of lotteries in choice sets placesadditional observable restrictions on expectations in a choice set and additional in-formation on behaviour relative to expectations. New axioms make particular useof this lottery structure to trace the observable implications of expectations-basedreference-dependence.Second, the main analysis looks for the revealed preference implications of pre-ferred personal equilibrium. The sharper predictions of preferred personal equilibriumlead to different testable implications of the PPE based model expectations-basedreference-dependence in the absence of risk.This choice space does not allow the analysis to say anything insightful about theset of hedonic dimensions of the problem. In light of Gul and Pesendorfer?s result,the representation here does not seek any particular structure on the v that repre-sents reference-dependent preferences. The analysis considers the particular structureimposed by the functional form (2.3) as a secondary issue for future work.2.3 Revealed preference analysis of PPE2.3.1 Technical preludeDefine distance on lotteries using the Euclidean distance metric, dE(p, q) :=??i(pi ? qi)2,and the distance between choice sets using the Hausdorff metric,dH(D,D?) := max?maxp?D?minq?D?dE(p, q)?, maxq?D??minp?DdE(p, q)??.It will be useful to offer a few definitions in advance of the analysis. For anyset T with typical element t, let {t?} denote a convergent net indexed by a set (0, ??]and with limit point t; t? will be used to denote the ? term in the net.7 Define7A net in a set T is a function t : S ? T for some directed set S (Aliprantis and Border, 1999).152.3. Revealed preference analysis of PPEcU(D) as the upper hemicontinuous extension of c; that is, cU(D) := {p ? D :?{p?, D?} such that p? ? c(D?), p? ? p, D? ? D}. For p ? ? and ? > 0, letN ?p := {p? ? ? : dE(p, p?) < ?} denote a ?-neighbourhood of p. For any binaryrelation R, let clR denote its closure, defined by: p(clR)q if ?{p?} ? p, {q?} ? qsuch that for each ? > 0, p?Rq?. For any finite set D and binary relation R, definem(D, R) := {p ? D : ?q ? D such that qRp but not pRq} as the set of undominatedelements in D according to binary relation R.2.3.2 Revealed preference analysis without riskIgnoring restrictions specific to risks, the classic IIA axiom provides the point ofdeparture from standard models. The two axioms below allow for failures of IIAthat can arise from the endogenous determination of expectations and preferences ineach choice set. For this section, restrict attention to axioms and restrictions on therepresentation in (2.2) that do not make use of the particular economic structure oflotteries, except for the continuity of ?.The following Expansion axiom is due to Sen (1971).Expansion. p ? c(D) ? c(D?) =? p ? c(D ?D?)Expansion says that if a lottery p is chosen in both D and D? then it is chosenin D ? D?. This seems weak as both a normative and a descriptive property, andis an implication of variations on the Weak Axiom of Revealed Preference (see Sen(1971)). Expansion rules out the attraction and compromise effects, in which an agentchooses p over both q and r in pairwise choices, but chooses q from {p, q, r}.8 In theattraction effect, r is similar to, but dominated by q and attracts the decision-makerto p in {p, q, r}; in the compromise effect, q is a compromise between more extremeoptions p and r in the choice set {p, q, r}.The Weak RARP (RARP for Richter?s (1966) Axiom of Revealed Preference9)is in the spirit of the classic axioms of revealed preference (like WARP, SARP, and8See Simonson (1989) for evidence on attraction and compromise effects. Ok, Ortoleva, andRiella (2012) provide a model of the attraction effect that captures this phenomenon.9Richter refers to his axiom as ?Congruence?. I use RARP to emphasize the close connectionwith WARP, SARP, GARP, etc. For more on the connection between these axioms, see Sen (1971).162.3. Revealed preference analysis of PPEGARP) albeit with an embedded continuity requirement. In particular, the axiomweakens (a suitably continuous version of) RARP.Define p ??Rq if p ? c(D) and q ? cU(D?) for some D, D? with {p, q} ? D ? D?.The relation ??R is defined whenever sometimes p is chosen when q is available, andsometimes q is choosable (in the sense that q ? cU(D?)) when p is available. Thestatement p ??Rq holds when p is weakly chosen over q in a smaller set, but q is weaklychoosable over p in a set that is larger in the sense of set inclusion. Define p ??Wq ifthere exist p0 = p, p1, ..., pn?1, pn = q such that (pi?1, pi) ? cl ??R for i = 1, ..., n. Thatis, ??W is the continuous and transitive extension of ??R.Weak RARP. p ? c(D), q ? cU(D?), q ? D ? D?, and q ??Wp =? q ? c(D)The crucial implication of Weak RARP is captured by its main economic implication,Weak WARP : if p = c({p, q}) and p ? c(D) then q /? c(D?) whenever p ? D? ?D.10 Manzini and Mariotti (2007) offer an interpretation in terms of constrainingreasons : an agent might choose p over q in a smaller set, like {p, q}, yet might havea constraining reason against choosing p in a larger set D. However, if we observe pchosen from a large set D, then any D? that is a subset of D contains no constrainingreason against choosing p. Thus, her choice in D? should be minimally consistentwith her choice in {p, q} and she should not choose q.Weak RARP strengthens the logic of Weak WARP in two ways. Weak WARPallows only WARP violations consistent with the existence of constraining reasons,and takes choices from smaller sets - which can fewer constraining reasons - as thedeterminant of choice in the absence of constraining reasons. The main way WeakRARP strengthens Weak WARP is by imposing that choice among unconstrainedoptions is determined by a transitive procedure.11Weak RARP as stated also strengthens a transitive version of Weak WARP byimposing continuity in two ways. Taking the topological closure of ??R and then takingthe transitive closure imposes that choice among unconstrained options is determined10The following proof that Weak RARP implies Weak WARP may help clarify the connection.Suppose p ? c(D), p ? D? ? D, and q ? c(D?). Then q ??Wp, and so if p ? c({p, q}), Weak RARPimplies that q ? c({p, q}) as well. Thus Weak RARP implies that if p = c({p, q}) and p ? c(D) hold,q ? c(D?) could not hold.11In this regard, Weak RARP is closely related to the ?No Binary Cycle Chains? axiom ofCherepanov, Feddersen, and Sandroni (Forthcoming).172.3. Revealed preference analysis of PPEby a rationale that is both transitive and continuous. This imposes a restriction thatis economically natural relative to the topological structure of lotteries. The secondcontinuity aspect of Weak RARP is that if p ? cU(D), p is seen as chooseable fromD. That is, if it is revealed that there is no reason to reject p? from D? when p? andD? are ?arbitrarily close? to p and D respectively, then Weak RARP assumes thatthere is no reason revaled to reject p from D (even if p is not chosen at D). Thesetwo strengthenings in Weak RARP are natural given the topological structure of thespace of lotteries (and many other choice spaces).Formally, say that a PPE representation in (2.2) is continuous if v is jointlycontinuous. Proposition 2.2 (i) ?? (ii), clarifies the link between the Expansion andWeak RARP axioms on one hand, and the PPE decision-making on the other hand.Manzini and Mariotti (2007) characterize a shortlisting representation, c(D) =m(m(D,P1), P2) for two asymmetric binary relations P1, P2, in terms of two axioms,Expansion and Weak WARP.12 If P2 is transitive and both P1 and P2 are continuous,say that P1, P2 is a continuous and transitive shortlisting representation.13 Proposition2.2 (ii) ?? (iii), provides a link between a version of the shortlisting model of Manziniand Mariotti and the PPE representation in (2.2).Proposition 2.2. (i)-(iii) are equivalent: (i) c satisfies Expansion and Weak RARP,(ii) c has a continuous PPE representation, (iii) c has a continuous and transitiveshortlisting representation.Proof. (ii) ?? (iii)Consider the following mapping between a continuous PPE representation v anda continuous and transitive shortlisting representation:v(q|p) > v(p|p) ?? qP1pv(p|p) > v(q|q) ?? pP2qFor v and P1, P2 that satisfy this mapping, m(D, P1) = PEv(D), andm(m(D, P1), P2) =PPEv(D).12Manzini and Mariotti (2007) and follow-up papers assume that c is a single-valued choice func-tion, which simplifies their analysis.13This terminology is different from Au and Kawai (2011) and Horan (2012) who discuss short-listing representations in which both P1 and P2 are transitive.182.3. Revealed preference analysis of PPEIt remains to verify that joint continuity in v is equivalent to continuity of P1 andP2 - the full argument is in the appendix.The v in a PPE representation characterized by Proposition 2.2 is highly non-unique: any v? that satisfies v?(q|p) > v?(p|p) ?? v(q|p) > v(p|p) and has v?(p|p) =u(p) for some u that represents P2 in the shortlisting representation also represents thesame c. Put another way, v includes information about how a decision-maker wouldchoose between any two lotteries p and q given any reference lottery r. However, if thedecision-maker?s rational expectations determine her reference lottery, as in a PPErepresentation, choices give us no direct information about a decision-maker wouldchoose between p and q given any reference lottery r /? {p, q}.2.3.3 Revealed preference analysis with riskThe result in Proposition 2.2 did not consider the possibility of adopting stronger ax-ioms or restrictions on v that are suitable when working with choice among lotteriesbut may not be economically sensible in other domains. But the evidence supportingexpectations-based reference-dependence in Ericson and Fuster (2011) suggests thatenvironments with risk provide a natural environment for studying expectations-basedreference-dependence. This section explores the possibility of a stronger characteri-zation in environments with risk.Environments with risk enable a partial separation between expectations andchoice. Suppose we view the mixture (1 ? ?)q + ?D as arising from a lottery overchoice sets that gives the singleton choice set {q} with probability 1 ? ? and giveschoice set D with probability ?. Under this interpretation, fraction 1 ? ? of expec-tations are fixed at expecting q and we also observe the decision-maker?s conditionalchoice from D. The three axioms below make use of variations on this interpretation.The Induced Reference Lottery Bias Axiom uses this partial separation betweenexpectations and choice. The axiom requires that if p is chosen in a choice set D,then p would also be conditionally chosen from D when some of the expectations arefixed at p, as in any mixture of the form (1 ? ?)p + ?D. This is a natural axiomto adopt under expectations-based reference-dependence: fixing expectations at p atleast partially fixes the reference-lottery weakly towards p; if the decision-maker is192.3. Revealed preference analysis of PPEbiased towards her reference-lottery, this should bias her towards choosing p.Induced Reference Lottery Bias. p ? c(D) implies p ? c((1 ? ?)p + ?D) ?? ?(0, 1).Notice that Induced Reference Lottery Bias allows for the violation of Mixture In-dependence observed by Ericson and Fuster (2011), but rules out a violation in theopposite direction.IIA Independence weakens the Mixture Independence Axiom to a variation thatonly implies a restriction on behaviour in the presence of IIA violations, with anembedded continuity requirement.IIA Independence. If p ? c(D) and ?? ? (0, 1] such that p /? c(D?((1??)p+?q)) ?r and p ??Wr, then ?? > 0 such that ??? ? (0, 1], ?p? ? N ?p, ?q? ? N?q , and?D? ? (1? ??)p? + ??q?, p? /? c(D?).The spirit of Weak RARP is the requirement that in the absence of constraining rea-sons, c is consistent with maximizing ??W , derived from choice from smaller choice sets.The choice pattern p ? c(D), p /? c(D?q) ? r, and p ??Wr then reveals that q blocks p.14This revealed blocking behaviour only appears when the model violates IIA. The IIAIndependence axiom requires that in this case, any mixture between q and p also pre-vents p from being chosen from any choice set. The logic of expectations-dependencethen requires that the agent would not choose p when it involves a conditional choiceof p over q.Remark 2.1. A simple test of IIA Independence that could detect behaviour inconsis-tent with expectations-dependence would be to find p, q,?, D with p ? c(D), {p, q}?c(D?q) = ? but p ? c(D?((1??)p+?q)). Table 2.2 shows two possible choice corre-spondences that describe a decision-maker who finds candy too tempting to turn downfor an apple whenever she had been expecting to eat but who can avoid temptationby planning in advance to abstain from snacking. Choice correspondence c capturesa decision-maker who can exert limited self-control against the expectations-inducedtemptation to go for candy, and is inconsistent with the IIA Independence axiom.Choice correspondence c? cannot exert this limited self-control, and is consistent withthe axiom.14In the appendix, it is shown that this choice pattern is ruled out by Weak RARP and Expansion.202.3. Revealed preference analysis of PPETable 2.2: Testing IIA IndependenceD c(D) c?(D){?apple, 1? , ?don?t eat, 1?} {?apple, 1?} {?apple, 1?}{?candy, 1? , ?apple, 1? , ?don?t eat, 1?} {?don?t eat, 1?} {?don?t eat, 1?}{?apple, .9; candy, .1? , ?apple, 1?} {?apple, 1?} {?apple, .9; candy, .1?}The continuity requirement embedded in IIA Independence slightly strengthensrestriction on c when adding q to the choice set prevents p from being conditionallychosen. The IIA Independence axiom requires that in this case, lotteries close to pprevent lotteries close to q from being conditionally chosen as well.Say that q is a weak conditional choice over r given p, qR?pr, if there exists a net{p?, q?, r?} ? p, q, r such that (1 ? ?)p? + ?q? ? c((1 ? ?)p? + ?{q?, r?}) for each ?. Aconditional choice involves a choice between q and r for when expectations are closeto p.Transitive Limit. qR?pr and rR?ps =? qR?ps.If IIA violations are only driven by the behavioural influence of expectations and theirendogenous determination, then the agent?s behaviour should be consistent with thestandard model when her expectations are fixed. The Transitive Limit axiom says thatconditional choice behaviour should look like the standard model when expectationsare almost fixed, although the axiom only imposes this restriction on weak conditionalchoices.Remark 2.2. As with continuity axioms, the Transitive Limit axiom is not exactlytestable. However, the axiom is approximately testable. The choice sets in Table2.3 provide an approximate test of Transitive Limit; c? is consistent with what wewould expect if the choice correspondence satisfies Transitive Limit. However, thechoice pattern displayed by c is approximately inconsistent with Transitive Limit,and suggests that c would violate this axiom.Formally, say that a PPE representation is an EU-PPE representation if v(?|p)takes an expected utility form for any p ? ?. Say that v dislikes mixtures if v(p|p) ?v(q|p) and v(q|q) ? max [v(p|p), v(p|q)] imply that ?? ? (0, 1), v((1? ?)p + ?q|(1??)p + ?q) ? max [v(p|p), v(p|(1? ?)p + ?q)].212.3. Revealed preference analysis of PPETable 2.3: Testing Transitive LimitD = .9{?mug, 1?} + .1 c(D) c?(D){?pen, 1? , ?mug, 1?} {?mug, 1?} {?mug, 1?}{?candy, 1? , ?mug, 1?} .9{?mug, 1?} + .1{?candy, 1?} {?mug, 1?}{?candy, 1? , ?pen, 1?} .9{?mug, 1?} + .1{?pen, 1?} .9{?mug, 1?} + .1{?pen, 1?}Theorem 2.1. c satisfies Weak RARP, Expansion, IIA Independence, Induced Ref-erence Lottery Bias, and Transitive Limit if and only if it has a continuous EU-PPErepresentation in which v dislikes mixtures.The full proof is in the appendix, and is discussed in the next subsection.Corollary 2.1. Given a continuous EU-PPE representation v for c, any other con-tinuous EU-PPE representation v? for c satisfies v?(q|p) ? v?(r|p) ?? v(q|p) ? v(r|p)and v?(p|p) ? v?(q|q) whenever p ??Wq.Corollary 2.1 clarifies that a continuous EU-PPE is unique in the sense thatany v, v? that represent the same c must represent the same reference-dependentpreferences.15 This definition of uniqueness captures that the underlying reference-dependent preferences are uniquely identified, but says nothing about the cardinalproperties of reference-dependent utility functions. In an EBRD, v plays roles in bothdetermining the set of of personal equilibria, and selecting from personal equilibria.The second part of Corollary 2.1 clarifies that this second role places a restrictionthat any v representing c must represent the same ranking of personal equilibria, atleast when that ranking is revealed from choices.Remark 2.3. In the representation in Theorem 2.1, any p chosen in D is (i) an elementof D, and (ii) is in argmaxq?Dv(?|p). A more general model might allow a decision-makerto randomize among elements of her choice set. An alternative representation mighthave the decision-maker?s reference lottery involve a randomization among elementsin D, or perhaps only elements in c(D). However, Theorem 2.1 proves that if csatisfies the five axioms it has a representation in which it is as-if the decision-makernever views herself as randomizing among elements of D.15A stronger uniqueness result is possible, since (i) each v(?|p) satisfies expected utility and thushas an affinely unique representation, (ii) joint continuity of v in the representation restricts theallowable class of transformations of v.222.3. Revealed preference analysis of PPE2.3.4 Sketch of proof and an intermediate resultThe first part of the proof takes R?p and characterizes a v such that v(?|p) representsR?p. By Transitive Limit and because R?p is continuous by construction, such a v(?|p)exists. A sequence of lemmas show that the definition of R?p and Transitive Limitaxiom imply the existence of a jointly continuous v such that v(?|p) represents R?p andsatisfies expected utility.Crucial to proof is providing a link between behaviour captured by v and behaviourin arbitrary choice sets. Consider an alternative axiom, Limit Consistency, which wasnot assumed in Theorem 2.1 but which would have been a reasonable axiom to adopt.First, define Rp as the asymmetric part of R?p.Limit Consistency. qRpp implies p /? c(D) whenever q ? D.The statement qRpp says that q is always conditionally chosen over p when expec-tations are almost fixed at p. Limit Consistency requires that a decision-maker whoalways conditionally chooses q over p when her expectations are almost fixed at pwould also never choose p when q is available. This is consistent with the logic ofexpectations-dependence. If instead qRpp but p were chosen over q in some set D,then the decision-maker would choose p over q when her expectations are p eventhough she always conditionally chooses q over p when her expectations are almostfixed at p; such behaviour would be inconsistent with expectations-dependence andis ruled out.The lemma below establishes that the axioms in Theorem 2.1 imply Limit Con-sistency.Lemma. Expansion, Weak RARP, and Induced Reference Lottery Bias imply LimitConsistency.The sufficiency part of the proof of Theorem 2.1 proceeds by using Expansion,Weak RARP, Limit Consistency, and v constructed from R?p to show that c(D) =PPEv(D). This gives the following intemediate result, a characterization of an EU-PPE representation in terms of Weak RARP, Expansion, IIA Independence, LimitConsistency, and Transitive Limit.232.3. Revealed preference analysis of PPETable 2.4: Two choice correspondencesc c?.9{?pen, 1?} + .1{?pen, 1? , ?mug, 1?} {?pen, 1?} {?pen, 1?}.9{?mug, 1?} + .1{?pen, 1? , ?mug, 1?} {?mug, 1?} {?mug, .9; pen, .1?}Theorem 2.2. c satisfies Weak RARP, Expansion, IIA Independence, Limit Consis-tency, and Transitive Limit if and only if it has a continuous EU-PPE representation.Notice than in any EU-PPE representation, expected utility of v(?|p) and jointcontinuity of v will imply that v(q|p) > v(r|p) =? qRpr. With this observation inhand, the necessity of Limit Consistency follows obviously from the representation.The remainder of the proof of the above Theorem follows from the proof of Theorem2. A definition of expectations-dependence and itsimplicationsSay that c exhibits expectations-dependence at D,?, p, q, r for ? ? (0, 1) and p, q, r ?? if (1 ? ?)p + ?r ? c((1 ? ?)p + ?D) but (1 ? ?)q + ?r /? c((1 ? ?)q + ?D).Interpret (1 ? ?)p + ?r ? c((1 ? ?)p + ?D) as involving a conditional choice of rfrom D, conditional on fraction 1 ? ? of expectations being fixed by p. Say thatc exhibits strict expectations-dependence at D,?, p, q, r for D ? D, ? ? (0, 1), andp, q, r ? ? if there is a ?? > 0 such that for all r?, D? pairs such that r? ? D? andmax?dE(r?, r), dH(D?, D)?< ?, (1 ? ?)p + ?r? ? c((1 ? ?)p + ?D?) for all ? < ??but (1 ? ?)q + ?r? /? c((1 ? ?)q + ?D?) for all ? < ??. This behavioural definitionof expectations-dependence provides a tool for identifying and eliciting expectations-dependence, as illustrated by the example below.Example (mugs and pens). Fix ? = .1, let p = ?pen, 1?; q = ?mug, 1?, r = p,and D = {p, q}.Table 2.4 shows the values that two choice correspondences, c and c?, take onthe menus (1 ? ?)p + ?D = {?mug, 1? , ?mug, .9 ; pen, .1?} and (1 ? ?)q + ?D =242.3. Revealed preference analysis of PPE{?mug, .1; pen, .9? , ?pen, 1?}. Of these two choice correspondences, c exhibits expectations-dependence given D,?, p, q, r, while c? does not.?The definition of exhibiting expectations-dependence bears striking relation tothe Mixture Independence axiom. Indeed, expectations-dependence as defined is atype of violation of Mixture Independence. Proposition 2.3 below clarifies the linkbetween a exhibiting expectations-dependence, properties of a continuous EU-PPErepresentation, and violations of the IIA axiom.Proposition 2.3. c with a continuous EU-PPE representation strictly exhibits expectations-dependence if and only if v(?|p) is not ordinally equivalent to v(?|q) for some p, q ? ?.In addition, c with a continuous EU-PPE representation that violates IIA exhibitsstrict expectations-dependence.The first part of Proposition 2.3 highlights how expectations-dependence in cis captured in a PPE representation. There is a tight tie between expectations-dependence and failures of Mixture Independence in a PPE representation, and thesecond part of Proposition 2.3 shows that a failure of IIA implies, but is not necessaryfor, expectations-dependence.The mugs and pens example shows how one might study expectations-dependencebased on the definition. Ericson and Fuster?s (2011) data violate Mixture Indepen-dence in a way consistent with expectations-based reference-dependence, and Propo-sition 2.3 shows that any PPE representation representing their median subject?sbehaviour must exhibit expectations-dependence.2.3.6 Limited cycle property of a PPE representationThe characterization in Theorem 2.1 is tight. However, it is possible that some struc-ture already imposed on the problem implies additional structure on v. Proposition2.4 shows that this is indeed the case.Say that a PPE representation satisfies the limited cycle inequalities if for anyp0, p1, ..., pn ? ?, v(pi|pi?1) > v(pi?1|pi?1) for i = 1, ..., n, then v(pn|pn) ? v(p0|pn).252.4. Special cases of PPE representationsProposition 2.4. Any PPE representation satisfies the limited cycle inequalities.Moreover, if v is jointly continuous, satisfies the limited-cycle inequalities, dislikesmixtures, and v(?|p) is EU for each p ? ?, then v defines an EU-PPE representationby (2.2).Proof. Take any p0, p1, ..., pn ? ?, with v(pi|pi?1) > v(pi?1|pi?1). The ith term in thissequence implies by the representation that pi?1 /? c({p0, ..., pn}); since c({p0, ..., pn}) ?=? by assumption it follows that pn = c({p0, ..., pn}). This implies, by the represen-tation, that v(pn|pn) ? v(pi|pn) for all i = 0, 1, ..., n ? 1, which implies the desiredresult.Conversely, for any v that satisfies the three given restrictions, the limited cycleinequalities imply that PE(D) is non-empty for any D ? D. Thus by Theorem 2.2,v defines a EU-PPE representation.Munro and Sugden (2003) mention the limited cycle inequalities (their AxiomC7), and defend the limited cycle inequalities based on a money-pump argument. Incontrast, the limited cycle inequalities emerge here as a consequence of the assumptionthat c(D) is always non-empty combined with the reference-dependent preferencerepresentation. If one considers a class of choice problems in which the agent alwaysmakes a choice, the limited cycle inequalities are a basic consequence of this and theagent?s endogenous determination of her reference lottery, regardless of the normativeinterpretation of the inequalities.2.4 Special cases of PPE representations2.4.1 K?szegi-Rabin reference-dependent preferencesIt may not be apparent at first glance whether K?szegi-Rabin preferences in (2.3)satisfy the limited-cycle inequalities that a PPE representation must satisfy to gener-ate a non-empty choice correspondence. K?szegi and Rabin (2006) cite a result dueto K?szegi (2010, Theorem 1) that a personal equilibrium exists whenever D is con-vex, or equivalently, an agent is free to randomize among elements of any non-convex262.4. Special cases of PPE representationschoice set. It is unclear whether or when this restriction is necessary to guarantee theexistence of a non-empty choice correspondence.K?szegi and Rabin suggest restrictions on (2.3). In particular, applications ofK?szegi-Rabin have typically assumed linear loss aversion, which holds when thereare ? and ? such that:?(x) =????x if x ? 0??x if x < 0(2.4)where ? > 1 captures loss aversion and ? ? 0 determines the relative weighton gain/loss utility. Proposition 2.5 shows that under linear loss aversion, K?szegi-Rabin preferences with the PPE solution concept are a special case of the more generalcontinuous EU-PPE representation.Proposition 2.5. K?szegi-Rabin preferences that satisfy linear loss aversion satisfythe limited cycle inequalities and dislike mixtures.Proposition 2.5 is alternative result to K?szegi and Rabin?s (2006) Proposition1.3, and to my knowledge provides the first general proof that a personal equilibriumthat does not involve randomization always exists in finite sets for this subclass ofK?szegi-Rabin preferences.While commonly used versions of K?szegi-Rabin preferences can provide the v ina PPE representation, there are (pathological?) cases of K?szegi-Rabin preferencesthat cannot.Proposition 2.6. Not all K?szegi-Rabin preferences consistent with (2.3) satisfy thelimited cycle inequalities.2.4.2 Reference lottery bias and dynamically consistentnon-expected utilityExpectations-based reference-dependence is the central motivation to considering thePPE representation. Now equipped with some understanding of the revealed prefer-ence implications of a PPE representation, we might take the preference relations ?L272.4. Special cases of PPE representationsand {?p}p?? as primitives, where ?p is the preference relation corresponding to v(?|p),and p ?L q corresponds to the ranking v(p|p) ? v(q|q). With these primitives, wecan study axioms that capture reference lottery bias. This is similar to the standardexercise in the axiomatic literature on reference-dependent behaviour (e.g. Tverskyand Kahneman (1991; 1992); Masatlioglu and Ok (2005; 2012); Sagi (2006)). In thatvein, consider the Reference Lottery Bias axiom below, which is closely related to the?Weak Axiom of Status Quo Bias? in Masatlioglu and Ok (2012).Reference Lottery Bias. p ?L q =? p ?p qI offer three interpretations of Reference Lottery Bias. The first interprets ?L asrepresenting the preferences that take into account that expecting to choose and thenchoosing lottery p leads to p being evaluated against itself as the reference lottery.Under this interpretation, if an agent would want to choose p over q, knowing that thischoice would also determine the reference-lottery against which they would evaluateoutcomes, then the agent would also choose p over q when p is the reference lottery.The second interpretation (along the lines of Masatlioglu and Ok (2012)) is that?L captures reference-independent preferences; in this second interpretation, if p ispreferred to q in a reference-independent comparison, then when p is the referencelottery, p is also preferred to q. According to either interpretation, Reference LotteryBias imposes that ?p biases an agent towards p relative to ?L. This seems likea natural generalization of the endowment effect for expectations-based reference-dependence.A third interpretation emphasizes ?L as the ranking of lotteries induced by theagent?s ex-ante ranking of choice sets when restricted to singleton choice sets. Underthis interpretation, an agent who wants to choose a lottery from a choice set accordingto her ex-ante ranking would also want to choose it from that choice set if she thenexpected that lottery, and it subsequently acted as her reference point.What implications does the Reference Lottery Bias axiom have? K?szegi-Rabinpreferences do not satisfy Reference Lottery Bias; recall the example in Section 2.2.2in which v(p|p) > v(r|r) but v(r|p) > v(p|p). This suggests a conflict between thepsychology of reference-dependent loss aversion captured by the K?szegi-Rabin modeland the notion of Reference Lottery Bias defined in the axiom. No experimentalevidence to my knowledge sheds light on this matter.282.4. Special cases of PPE representationsProposition 2.7. A PPE representation satisfies Reference Lottery Bias if and onlyif c(D) = m(D,?L).Proposition 2.7 implies (recalling Proposition 2.3) that under Reference LotteryBias, reference-dependent behaviour in a PPE representation is tightly connected tonon-expected utility behaviour in ?L.The non-expected utility literature has provided numerous models of decision-making under risk based on complete and transitive preferences that, motivated bythe Allais paradox, satisfy a relaxed version of the Mixture Independence axiom (e.g.Quiggin (1982); Chew (1983); Dekel (1986); Gul (1991)). The model of expectations-based reference-dependence based on the Reference Lottery Bias axiom is based ona dynamically consistent implementation of non-expected utility preferences (as inMachina (1989)). I offer two examples of PPE representations that satisfy Reference-Lottery Bias and capture expectations-based reference-dependence.Example (Disappointment Aversion). Suppose ?L satisfies Gul?s (1991) disap-pointment aversion; that is (letting u(x) denote u(?x, 1?)),u(p) = 11+??i pi (u(xi) + ? min[u(xi), u(p)]) represents ?L for some ? ? 0. ThenReference Lottery Bias implies:vDA(p|r) =11 + ??ipi (u(xi) + ? min[u(xi), u(r)]) (2.5)In cases of lotteries over multidimensional choice objects, it is not hard to see howto extend (2.5) via additive separability across dimensions. The resulting functionalform captures loss aversion relative to past expectations (as in K?szegi-Rabin) butdoes not generate IIA violations.?Example (Mixture Symmetry). Suppose?L satisfies Chew, Epstein, and Segal?s(1991) mixture symmetric utility; that is, there is a symmetric function ? such thatu(p) =?i?j?(xi, xj) represents ?L. Then Reference Lottery Bias implies:vMS(p|r) =?i?jpirj?(xi, xj) (2.6)292.5. Alternative models of expectations-based reference-dependence: analysis of PE and CPE representationsWhile the functional form for vMS in 2.6 does capture the K?szegi-Rabin functionalform in (2.3), but the ? function corresponding to vKR is generally not symmetric.?2.5 Alternative models of expectations-basedreference-dependence: analysis of PE and CPErepresentations2.5.1 Characterization of PEIn addition to the PPE representation in (2.2) which is used in most applicationsof expectations-based reference-dependence, K?szegi and Rabin (2006) also discussthe PE as a solution concept as in (2.1). The analysis below shows that the PErepresentation can be axiomatized similar to the PPE representation, by replacingWeak RARP with Sen?s ?, changing the continuity assumptions, and modifying IIAIndependence.Sen?s ?. p ? D? ? D and p ? c(D) implies p ? c(D?)Sen?s ? requires that if an item p is choosable in a larger set D, then it is also deemedchoosable in any subset D? of D where p is available. Sen?s ? is strictly weaker thanIIA.16The Upper Hemicontinuity axiom is the continuity property satisfied by contin-uous versions of the standard model, in which choice is determined by a continuousbinary relation.UHC. c(D) = cU(D)Proposition 2.8 (i) ?? (ii) provides an axiomatic characterizating of PE decision-making that does not make use of the structure of environments with risk; (ii) ??(iii) is a continuous version of Gul and Pesendorfer?s (2008) result (Proposition 2.1 inthis paper).1716Sen?s ? and Sen?s ? are jointly equivalent to IIA; see Sen (1971) and Arrow (1959).17The result (i) ?? (iii) is a continuous version of Theorem 9 in Sen (1971).302.5. Alternative models of expectations-based reference-dependence: analysis of PE and CPE representationsProposition 2.8. (i)-(iii) are equivalent: (i) c satisfies Expansion, Sen?s ?, andUHC, (ii) c has a continuous PE representation, (iii) c is induced by a continuousbinary relation.IIA Independence 2 modifies the antecedent in the IIA Independence axiom to PE.Under PE, a lottery q is revealed to block p if there is a D such that p ? c(D) butp /? c(D ? q). IIA Independence 2 has a different antecedent from IIA Independencethat reflects the differences in how constraining lottery pairs are revealed in the twomodels. IIA Independence 2 also embeds a continuity requirement.IIA Independence 2. If p ? c(D) and ?? ? (0, 1] such that p /? c(D?(1??)p+?q)),then ?? > 0 such that ??? ? (0, 1], ?p? ? N ?p, ?q? ? N?q , and ?D? ? (1???)p?+??q?,p? /? c(D?).Theorem 2.3 provides a characterization of a continuous EU-PE representation.Theorem 2.3. c satisfies Expansion, Sen?s ?, UHC, IIA Independence 2, and Tran-sitive Limit if and only if c has a continuous EU-PE representation. These axiomsjointly imply that Induced Reference Lottery Bias holds as well.2.5.2 Characterization of CPEK?szegi and Rabin (2007) also introduce the choice-acclimating personal equilibrium(CPE) concept:CPEv(D) = argmaxp?Dv(p|p) (2.7)While most applications of expectations-based reference-dependence use the PPEsolution concept, many use CPE. Theorem 2.4 clarifies the revealed preference foun-dations of CPE decision-making.Theorem 2.4. (i)-(iii) are equivalent. (i) c satisfies IIA and UHC, (ii) c has acontinuous EU-CPE representation in which v is continuous, (iii) there is a complete,transitive, and continuous binary relation ? such that c(D) = m(D, ?) ?D.312.5. Alternative models of expectations-based reference-dependence: analysis of PE and CPE representationsTheorem 2.4 appears to be a negative result - it suggests that expectations-basedreference-dependence combined with CPE has no testable implications beyond thestandard model of preference maximization! However, CPE decision-making canfail the Mixture Independence Axiom in ways that are consistent with expectations-based reference-dependent behaviour. This raises the question of what restrictionsthe Induced Reference Lottery Bias impose on the representation. Say that a binaryrelation ? is quasiconvex if p ? q =? p ? (1? ?)p + ?q ?? ? (0, 1).Proposition 2.9. Suppose ? ?, v such that c(D) = m(D, ?) = CPEv(D). (i)-(iii)are equivalent: (i) c satisfies Induced Reference Lottery Bias, (ii) ? is quasiconvex,(iii) v(p|p) ? v(q|q) =? v(p|p) ? v((1? ?)p + ?q|(1? ?)p + ?q) ?? ? (0, 1).Remark 2.4. Proposition 2.7 and Theorem 2.4 establish that if c has a PPE represen-tation that satisfies the Reference Lottery Bias axiom, then PPEv(D) = CPEv(D).Example (K?szegi-Rabin and Mixture Symmetry). Under CPE concept, therequirement that ? in 2.6 be symmetric is without loss of generality. Thus the K?szegi-Rabin functional form in 2.3 corresponds to a special case of the mixture symmetricutility functional form in 2.6 under CPE.?32Chapter 3Calibration without Reduction forNon-Expected UtilityRecent calibration critiques of Rabin (2000) and Safra and Segal (2008) show thatwhenever expected utility (EU) and non-expected utility (non-EU) define utility overfinal wealth states, they cannot simultaneously exhibit nonnegligible risk aversionover small stakes and can only exhibit moderate risk aversion over large stakes. In-trospection and empirical evidence suggest that even if the stakes are small, mostpeople would rather not take a small risk with a positive expected value if it couldinvolve a loss of money. Yet most people still take substantial risks over large stakes,for instance, by investing in stocks. As a result, these calibration critiques havebeen widely understood as suggesting the demise of descriptive theories that defineutility over final wealth except as a normative benchmark?further suggesting thatdescriptive models must define utility over gains and losses. But defining utility overfinal wealth gives non-EU theories a tractability and modelling discipline that morepsychologically based theories such as prospect theory lack.This paper shows that non-expected utility can generate both nonnegligible small-stakes risk aversion as well as the moderate large-stakes risk aversion. The crucialassumption made here is that a decision maker (DM) who faces preexisting riskstreats a gamble that is offered as the first stage of a two-stage compound lottery,which is then not treated as equivalent to the one-stage lottery that gives the sameprobability distribution over final wealth but is evaluated recursively (Segal, 1990).This contrasts sharply with existing proposals for solving the Rabin critique, whichhave all relied on abandoning consequentialism?the assumption that utility is definedover final wealth levels?assuming instead that utility is also evaluated over gains andlosses or lab income.33Chapter 3. Calibration without Reduction for Non-Expected UtilityThe intuition for the rank-dependent utility (RDU) (Quiggin, 1982) case of themain results of the paper is as follows: without background risk, RDU can pro-duce descriptively reasonable risk aversion at a range of stakes through probabil-ity weighting even if utility is defined over wealth levels. Now suppose DM facesbackground risk w?, and the utility-for-wealth function u is linear. Under recur-sive RDU, a compound lottery is evaluated by a folding back procedure, and DMevaluates the offered gamble (?L, .5;+G, .5) by folding back the compound lottery[w??L, .5; w?+G, .5] to [c(w??L), .5; c(w?+G), .5] where c is DM?s certainty-equivalentfunction. When u is linear, c(w?+x) = c(w?)+x; this compound lottery is evaluated as[c(w?)?L, .5; c(w?)+G, .5], and probability weighting produces small-stakes risk aver-sion over the offered gamble the way it would without background risk. With a linearu, DM turns down (?L, .5;+G, .5) if and only if DM turns down (?tL, .5;+tG, .5)for all t > 0; therefore, small-stakes risk aversion due to probability weighting iscompatible with reasonable large stakes risk aversion.Relation to previous literature Since EU maximizers are approximately riskneutral over small stakes, they would only be willing to pay a trivial amount to avoidsmall risks. Popular alternatives to EU that define utility over final wealth levels areeither (i) ?smooth? as in (Machina, 1982), locally risk neutral, and subject to the samecriticism as EU (Safra and Segal, 2008), or (ii) obtain nonnegligible risk aversion oversmall stakes because they weigh probabilities nonlinearly (as suggested by the Allaisparadox) and are hence immune from Rabin?s critique.However, most people face substantial lifetime wealth risk (e.g., employment-income risk and ownership of risky assets). The combination of a gamble offered inthe lab in the presence of preexisting wealth risk is naturally viewed as a two-stagecompound lottery in which the offered gamble resolves first. When DM reduces com-pound lotteries to single-stage lotteries by multiplying out probabilities, any small-stakes gamble offered adds only minimally to lifetime wealth risk; therefore, prob-ability weighting is mostly determined by preexisting lifetime wealth risk and doesnot produce substantial risk aversion over offered small-stakes gambles (Safra and Se-gal, 2008; Barberis, Huang, and Thaler, 2006). This argument relies crucially on theassumption that DM satisfies reduction of compound lotteries?an assumption that34Chapter 3. Calibration without Reduction for Non-Expected Utilityis not consistent with experimental evidence. Instead, this paper assumes recursivepreferences over compound lotteries.Recursive non-EU (RNEU) preferences over compound lotteries are used in thispaper as a descriptive model of decision making, following Segal (1990). The theo-retical distinction between compound versus single-stage lotteries was first suggestedby Samuelson (1952). RNEU preferences have been applied by Segal (1987b) to ex-plain ambiguity aversion (see also Dillenberger and Segal (2012)) and by Dillenberger(2010) to explain preferences for one-shot resolution of uncertainty. Dillenberger alsoremarks that an RNEU DM behaves as if they bracket narrowly; section 1.4 of thispaper makes a precise connection between RNEU and narrow bracketing.The theoretical tradition of RNEU preferences following Segal (1990) is relatedto but distinct from the use of recursive utility due to preferences over the timing ofresolution of uncertainty (Kreps and Porteus, 1978; Epstein and Zin, 1989). Whenrecursive preferences are used only because of preferences over the resolution of un-certainty then DM applies reduction of compound lotteries to an offered delayed riskcombined with income risk that resolves at the same time, and will not demonstratesmall-stakes risk aversion over such gambles (Barberis, Huang, and Thaler, 2006).Existing theoretical approaches that avoid a calibration critique (Kahneman andTversky (1979); Cox and Sadiraj (2006); Barberis, Huang, and Thaler (2006)) directlyincorporate narrow bracketing by assuming that the value function is defined overthe outcomes of a gamble as well as (possibly) over final wealth states. RNEU isformally very different from these nonconsequentialist models in that in RNEU theutility function is only defined over final wealth states and not directly over theoutcomes of a gamble. While RNEU does not assume narrow bracketing and isfully consequentialist, Theorems 3.1 and 3.2 precisely establish a connection betweenRNEU and models of narrow bracketing: an RNEU DM behaves as-if she engages ina form of narrow bracketing, at least over small-stakes or if her lottery preferencessatisfy constant absolute risk aversion.The RDU special case of the RNEU preferences studied in this paper are stillsubject to calibration arguments by Neilson (2001) and by Safra and Segal (2008,Theorem 1)18; however, the assumed risk-averse choice patterns behind these critiques18See also a related critique of RDU by Sadiraj (2012)353.1. Theory: RNEU risk preferences with background wealth riskhave limited experimental evidence and lack field evidence. For a literature review ofcalibration critiques, see Section 8.6 of Wakker (2010).Experimental evidence on compound lotteries Halevy (2007) finds that 80percent of subjects violate reduction of compound lotteries, while 59 percent of sub-jects? choices are best explained by RNEU. Previous experimental work also foundsubstantial violations of reduction of compound lotteries that suggest the use of RNEUpreferences; for example, Carlin (1992); Camerer and Ho (1994). Recursive prefer-ences over compound lotteries are also consistent with experimental findings thatrandomly picking one of the subject?s many decisions to determine payment is anincentive compatible mechanism for eliciting preferences (Cubitt and Sugden (1998)).3.1 Theory: RNEU risk preferences withbackground wealth risk3.1.1 Non-expected utility over lotteriesNotation Let W = ?+ denote the set of feasible final wealth levels, and consider apreference over one-stage lotteries, V : ?(W ) ? ? with utility-for-wealth function u :W ? ? and associated with certainty equivalent function c. A one-stage lottery overW can be written as q = [w1, q1; ...;wm, qm] ? ?(W ) whenever q has finite support,where qi denotes the probability of receiving prize wi. Assume V is increasing in thesense of first-order stochastic dominance. Adopt the convention that w1 ? ... ? wm.Say that V is risk averse if it is averse to mean-preserving spreads.Popular models The two most commonly used non-EU theories are RDU (Quiggin(1982), Yaari (1987), Segal (1990)), and disappointment aversion (DA) (Gul, 1991).Table 3.1 reviews these and EU; it notes conditions under which RDU and DA demon-strate the Allais paradox and small-stakes risk aversion not present under EU. DApreferences are a special case of the larger class of betweenness-satisfying preferences(Dekel (1986), Chew (1989)).363.1. Theory: RNEU risk preferences with background wealth riskTable 3.1: Non-expected utility theoriesTheory V ([w1, q1; ...;wm, qm]) Allais? Small-stakes risk averse?EUm?i=1qiu(wi) No Not if u? existsRDUm?i=1[g(i?j=1qj)? g(i?1?j=1qj)]u(wi) g concave g concaveDAm?i=11+?Iu(wi)<V (q)1+?n?i=1qjIu(wj)<V (q)qiu(wi) ? > 0 ? > 0Sources: Gul (1991); Segal and Spivak (1990); Segal (1987a)3.1.2 Recursive non-expected utilityRNEU extends non-EU preferences over single-stage lotteries to the domain of com-pound lotteries.Define a compound lottery as a finite lottery over lotteries over final wealth levels;a compound lottery can be written as Q = [q1, p1; ...; qn, pn] where qi ? ?(W ) and piis the probability of receiving lottery qi; let ?(?(W )) denote the set of compoundlotteries. The utility function U is defined over compound lotteries over final wealthlevels. Without loss of generality, adopt the convention that for a compound lotteryQ as above, V ([q1, 1]) ? ... ? V ([qn, 1]).19A RNEU maximizer evaluates a compound lottery Q via a simple two step pro-cedure:1. Compute the certainty equivalent of each lottery qi that is a possible prize ofQ:c(qi) = u?1 ? V (qi)2. Recursively compute the value of the compound lottery as the non-expected19Readers familiar with Segal (1990) will note that I assume time neutrality here. This is notessential for the conclusions.373.1. Theory: RNEU risk preferences with background wealth riskutility of the one-step lottery [c(q1), p1; ...; c(qn), pn]:U(Q) = c([c(q1), p1; ...; c(qn), pn]) (3.1)An alternative to the recursivity assumption is that a DM immediately reduces thecompound lottery to a single-stage lottery, which is then evaluated according to V .Such a reduction of compound lotteries assumption is not consistent with evidencethat subjects fail to reduce compound lotteries to one-stage lotteries presented earlier.If a non-EU DM reduces compound lotteries, compound lottery Q is evaluated asequivalent to the one-stage lottery QR = [w1,n?i=1piqi1; ...;wK ,n?i=1piqiK ]. For purposes ofcomparison, a non-expected utility with reduction DM evaluates a compound lotteryp by:UROCL(Q) = c(QR) (3.2)3.1.3 Wealth risk as a compound lotteryA one-time choice is never the only thing going on in a DM?s life. Empirical workshows that people face substantial risks in their lives (Guiso, Jappelli, and Pistaferri,2002). If we want to retain the modeling discipline of defining utility over final wealthlevels, then we have to make a choice about how to model a DM?s attitude to risk froma one-time gamble and from everything else in life. The combination of a one-timegamble offered (like those offered in lab experiments) and background wealth riskconstitutes a compound lottery composed of two distinct and independent sources ofrisk in which the one-time gamble resolves first, and the rest of life?s uncertaintiesresolve in due course.Consider a DM who faces background wealth risk described by the random variablew? = [w1, q1; ...;wm, qm], which is not the subject of choice, and who is offered thegamble over prizes p? = (y1, p1; ...; yn, pn) where yi ? Y ? ? is a monetary prize addedto or taken away from the DM?s final wealth after lottery p? resolves. Let p?? w? denotethe compound lottery formed by simple gamble over prizes p?, which resolves first, andindependent background risk w?, which resolves second. The compound lottery p?? w?383.1. Theory: RNEU risk preferences with background wealth riskis given by:p?? w? = [w? + y1, p1; ...; w? + yn, pn] (3.3)where w?+ yi = [w1 + yi, q1; ...;wm + yi, qm] denotes the lottery over final wealth statesthat the DM faces if prize yi is won in the gamble p?. The compound lottery p?? w? iswell defined whenever w + yi ? W for each w in the support of w? and each yi in thesupport of p?.20Say that a DM with utility function U defined on ?(?(W )) treats an offeredgamble p? in the presence of background risk w? as a compound lottery in which p?resolves first if for any offered gambles p? ? ?(Y ), DM evaluates the utility of p?according to U(p?? w?).3.1.4 Nonreduction and narrow bracketingSegal (1990) first suggested replacing the reduction of compound lotteries axiom withrecursivity as a consequentialist alternative to prospect theory that captures Kahne-man and Tversky?s (1979) ?isolation effect? (a particular example of a failure of thereduction of compound lotteries axiom). Rabin (2000) noted that prospect theory isimmune to his calibration critique. Existing theoretical approaches that avoid a cal-ibration critique (Kahneman and Tversky (1979); Cox and Sadiraj (2006); Barberis,Huang, and Thaler (2006)) directly incorporate narrow bracketing by assuming thatthe value function is defined over the outcomes of a gamble as well as (possibly) overfinal wealth states. RNEU is formally very different from these nonconsequentialistmodels in that the value function is only defined over final wealth states and notdirectly over the outcomes of a gamble.While RNEU does not assume that a gamble is framed narrowly, Theorem 3.1demonstrates that when c satisfies constant absolute risk aversion, an RNEU DMbehaves as if she brackets narrowly: that is, her choices among offered gambles areindependent of the background risk she faces. Formally, c satisfies constant absolute20An alternative but less intuitive assumption is that DM distinguishes between one-stage andcompound risks but treats the gamble p? as resolving in the second stage. The main results of thepaper are not sensitive to this assumption.393.1. Theory: RNEU risk preferences with background wealth riskrisk aversion if for any w? ? ?(W ) and y ? ? such that w? + y ? ?(W ), c(w? + y) =c(w?) + y.Theorem 3.1. Suppose a recursive non-EU decision maker treats an offered gamblein the presence of background risk as a compound lottery as in (3.3), and has lotterypreferences that satisfy constant absolute risk aversion. Then c has a unique extensionto ?(?), c?, such that, U(p? ? w?) = c?(p?) + c(w?) represents preferences whenenverp?? w? ? ?(?(W )).Proof. Extend c to the set of the set ?(?) of lotteries on ?(?) with support boundedfrom below by the identification that for any q ? ?(?) define w = ? inf{supportq},and extend c to ?(?) by c?(q) = c(q +w)?w. Since c satisfies constant absolute riskaversion, this extension is unique. Under RNEU,U(p?? w?) = c([c(w? + y1), p1; ...; c(w? + yn), pn])Applying constant absolute risk aversion at the second stage, and then at the firststage making use of c?,= c([c(w?) + y1, p1; ...; c(w?) + yn, pn])= c?(p?) + c(w?)3.1.5 Non-reduction and small-stakes risk aversionTheorem 3.1 suggests that demonstrating small-stakes risk aversion under RNEUdoes not lead to calibration implications for risk aversion over larger stakes gambles.Say that c satisfies constant relative risk aversion if for any w? ? ?(W ) and t ? ?+,c([tw1, q1; ...; twm, qm]) = tc(w?). Preferences that satisfy both constant absolute riskaversion and constant relative risk aversion have been termed constant risk averse bySafra and Segal (1998). Special cases of constant risk averse preferences include linearu cases of RDU and DA. Corollary 3.1 shows that for the class of non-EU preferencessatisfying constant risk aversion, fairly tight calibration implications can be drawnfrom turning a gamble p?, but these implications seem reasonable.Corollary 3.1. Suppose c satisfies constant risk aversion. For p?t = (ty1, p1; ...; tyn, pn),U(p?t? w?) = tc?(p?)+ c(w?) represents preferences whenenver p?t? w? ? ?(?(W )), wherec? is the extension of c from Theorem 3.1.403.1. Theory: RNEU risk preferences with background wealth riskAn implication of the corollary is that whenever we know that lottery preferencessatisfy constant risk aversion, then DM turns down p? at w? at which U(p?? w?) is welldefined if and only if she turns down p?t for all w? and for all t > 0.Typical applications of non-EU preferences do not assume constant risk aversion,but rather allow diminishing marginal utility of wealth. However, one might ex-pect that over small stakes, non-expected utility preferences behave like constant riskaverse preferences since u is almost linear locally. I define a notion of dual differentia-bility, as a way of approximating preferences by a derivative taken with respect to adual mixture of lotteries in the sense of Yaari (1987), but that is only defined aroundmixtures in which one of the lotteries is degenerate. In the Appendix (PropositionB.1), I show that RDU, semi-weighted utility preferences including DA (Chew, 1989),and Fr?chet differentiable preferences are all weakly dually differentiable in the sensedescribed below. Theorem 3.2 shows that for weakly dually differentiable preferences,behavior over small-stakes gambles given a fixed distribution of background wealthrisk is well approximated by a constant risk averse certainty equivalent function.Formally, say that c is dually differentiable at wealth w if for each p? ? ?(Y ) andw ? W if there is a linear-in-money c?(?) such that c(p?t + w) = w + tc?(p?) + o(t). Saythat c is weakly dually differentiable given wealth risk w? if there are c?y+(w?), c?y?(w?)such that c(w? + ty) = c(w?) + tyc?ysign(y)(w?) + o(t). Additionally, say that a certaintyequivalent function c is first-order risk averse at wealth w if for any p?t with anexpected value of zero, dc(w+p?t)dt |t=0+ < 0 (Segal and Spivak, 1990).Theorem 3.2. If c is weakly dually differentiable at w? with c?y+(w?) ? c?y?(w?), and isalso dually differentiable at w = c(w?), then for any a ? ?, U(p?t ? w?) = tc?(p? + a) ?ta + c(w?) + o(t). Moreover, if c is first-order risk averse then so is c?.3.1.6 The Rabin critique?An immediate corollary of Theorem 3.1 in the spirit of Rabin?s Theorem is that ifc is constant absolute risk averse, U(p? ? w?), DM turns down gamble p? when herbackground wealth distribution is w? if and only if she turns down p? for any w?. Thisgives no sense of DM?s attitude towards larger stakes gambles, leading to the followingremark:413.1. Theory: RNEU risk preferences with background wealth riskRemark 3.1. Suppose a recursive non-EU decision maker treats an offered gamble inthe presence of background risk as a compound lottery as in (3.3), and always turnsdown an actuarially favorable gamble p? for any distribution of background risk w?.Knowing only that V is globally risk averse, the strongest conclusion that can bedrawn is that DM will always turn down any mean-preserving spread of p?.Remark 3.1 shows that weak assumptions lead to weak conclusions, but what ifwe made stronger assumptions about c? Theorem 3.2 and Corollary 3.1 give strongintuition for what attitudes towards a small-stakes gamble p? will imply about attitudestowards gambles at larger stakes for preferences for classes of preferences that includea constant risk averse subclass. Theorem 3.3 offers an counterpart to Rabin?s (2000)calibration theorem, assuming instead that DM has risk averse RDU/DA RNEUpreferences and faces background risk.Theorem 3.3. Suppose a recursive non-EU decision maker treats an offered gamblein the presence of background risk as a compound lottery as in (3.3), and alwaysturns down a gamble p? = (y1, p1; ...; yn, pn) for any distribution of background riskw?. Knowing only that c is risk averse and is RDU, then the strongest restriction onlarge-stakes gambles that can be drawn without further assumptions is that DM willturn down any gamble p?t = (ty1, p1; ...; tyn, pn) for all t > 1 and for all w?. The sameresult applies if ?RDU? is replaced with ?DA?.Proof. DM turns down p?t if U(p?t?w?)?c(w?) < 0. Turning down p? implies U(p??w?)?c(w?) < 0. If u is linear, then under DA and RDU this implies that DM turns down p?tfor all t but cannot rule out accepting more favorable gambles. If u is concave, thenit can be shown that under risk aversion and either DA or RDU that (u ?U)(p?t ? w?)is concave in t; therefore, DM will still turn down p?t for any t > 1. However, EUremains a special case of DA and RDU, and if DM had EU preferences, p?t would beaccepted for a sufficiently small t whenever 0 /? support(w?).Under DA and RDU, if V is globally risk averse u must be weakly concave, so thestrongest calibration result possible comes from the case where u is linear for t > 1.That is, DM will turn down p?t for all t > 1.Since recursive DA is a special case of the more general class of betweenness pref-erences (Dekel, 1986; Chew, 1989), if recursive DA is immune to calibration critiques,423.2. Calibrationthen the class of recursive betweenness preferences is immune to Rabin-style calibra-tion critiques, as are more general classes of preferences. Thus, recursive versions ofa wide range of non-EU theories are not susceptible to calibration critiques, and arepotentially suitable for modeling risk preferences over both small and large stakes.3.2 CalibrationWhat constitutes descriptively reasonable risk aversion is a quantitative question.This section calibrates a version of recursive RDU and shows that this calibrationcan produce descriptively reasonable risk aversion, while EU and RDU with reductioncannot.Convenient functional forms for g and u should have as few parameters as possibleto calibrate and should be easily comparable to commonly used models. I adopt thestandard power utility-for-wealth function:u(w) = w1??1??and the probability weighting function:g(p) = p?axiomatized by Grant and Kajii (1998) and used in Safra and Segal (2008) sinceit is only one parameter richer than EU, is consistent with small-stakes risk aversionand Allais-type choices when 0 < ? < 1, and captures expected utility as a specialcase when ? = 1.Chetty (2006) points out that the curvature of the utility-for-wealth functionalso governs how an individual makes trade-offs between labour and leisure. I use? = .71, suggested by Chetty based on previous studies of labor supply responsesto wage changes.21 I then calibrate ? to match modal choices in Holt and Laury?s(2002) experimental data on small-stakes risk aversion to the extent possible. WhileEU cannot avoid mispredicting the modal choice in their data when most subjectsdemonstrate risk aversion, if ? ? [.5, .64], the calibrated recursive RDU model fits thedata reasonably but not perfectly.21While Chetty assumes expected utility in his calculations, the approach he takes fully carriesthrough to RDU in the case where utility is separable in consumption and leisure; I use Chetty?sestimates from this case.433.2. CalibrationTable 3.2: Calibration results - small and large stakes risk aversionLoss ? = .5 ? = .64 ? = .5, reduction ? = 1 (EU)10 24.14 17.91 10.10 10.00100 241.60 179.21 103.36 100.03200 483.59 358.62 209.61 200.13500 1211.84 898.12 538.77 500.831000 2433 1801 1112 10032000 4904 3623 2328 20135000 12555 9219 6403 508410000 26133 18992 14364 1034325000 73931 52052 46738 2727050000 185392 123239 137678 60105Gain required for DM to take (-Loss, .5;Gain, .5)While the risk in w? only has a second-order effect on decisions among offeredgambles in recursive RDU, the risk in w? reduces risk aversion RDU with reduction.To allow for comparison, take w? = U [$100000, $500000] to capture background wealthrisk.22Table 3.2 summarizes how different calibrated models discussed above would pre-dict that a DM would make choices in (?L, .5; G, .5) gambles. In each row of thetable, L is fixed at the level in the left-hand column, while the entry in the table liststhe G at which a DM would be indifferent to either taking or turning down the listedgamble.Table 3.2 (Columns 1 and 2) indicates that for ? = .5, .64 recursive RDU canproduce descriptively reasonable risk aversion over both small and large stakes. RDUwith reduction produces barely any risk aversion over small stakes (Column 3). Evenfor stakes into the thousands of dollars, EU induces preferences over gambles thatare extremely close to expected value maximization (Column 4). Even with a highervalue for ?, EU would induce preferences over gambles that are extremely close toexpected value maximization over stakes of hundreds of dollars. These quantitative22I derive quantitative results using a discrete approximation of the uniform distribution. I assumethat lifetime wealth has an expected present value of $300,000, since this figure is emphasized inRabin (2000), but the quantitative results are not particularly sensitive to this assumption.443.3. Conclusionresults are not sensitive to the choice of a distribution for background wealth risk.3.3 ConclusionThis paper has shown that RNEU can produce non-negligible small-stakes risk aver-sion without implying ridiculous large-stakes risk aversion, and generate a form of ?as-if? narrow bracketing over small-stakes gambles. A calibration exercise demonstratedthat recursive RDU can be calibrated to provide descriptively reasonable levels of riskaversion in the small and in the large. The non-expected utility theories studied inthis paper are attractive. Non-expected utility theories, rank-dependent utility anddisappointment aversion in particular, have clear axiomatic foundations, have beenwell studied, and have proven tractable in applications. The RNEU approach to ap-plying non-expected utility preserves this tractability for a decision-maker who facesmultiple risks. Furthermore, the two departures this model does make from expectedutility theory are each well supported by experimental evidence on the Allais paradoxand nonreduction of compound lotteries.45Chapter 4List Elicitation of Risk PreferencesAn ideal experimental study of individual decision-making would elicit precise datawhile providing subjects with monetary incentives that allow for unambiguous infer-ences about individual preferences. List elicitation with the random incentive scheme(RIS) is a standard technique to achieve this goal and has become the workhorsemethod for experimental economists. In list elicitation, subjects respond to a seriesof binary choices lined up as a list. List elicitation is typically combined with theRIS, in which one of these choices is randomly selected to determine the subject?spayment. The unambiguous interpretation of choice behavior in terms of individualpreferences relies on the assumption that the list elicitation combined with the RISdoes not impact subjects? choices.A common hypothesis is that subjects behave as-if they make each choice inisolation from all other choices and independent of the details of the incentive schemethey face. Experimental data obtained from list elicitation with the RIS is frequentlyinterpreted under the assumption that the isolation hypothesis holds. But when isthe isolation hypothesis appropriate?The theoretical literature provides conflicting guidance as to whether we shouldexpect list elicitation combined with the RIS to impact subjects? choices. Formally,isolation holds if and only if the subjects? preferences over compound lotteries inducedby her choices and the external randomization device satisfy Segal?s (1990) compoundindependence axiom.23 In particular, isolation does not hold if subjects have non-expected utility preferences and obey the reduction of compound lotteries axiom(Karni and Safra, 1987). An earlier literature showed that the Becker, DeGroot,and Marschak (1964) (BDM) elicitation mechanism can lead to preference reversalsGrether and Plott (1979) in the direction predicted by Karni and Safra (1987). This23Cox and Epstein (1990) elaborate on this point.46Chapter 4. List Elicitation of Risk PreferencesTable 4.1: Binary choice versus list elicitationBinary choice List p-value($y, 1) 23% 50% <.001($y, p) 30% 45% .02Percent choosing the riskier option.p-values are for a Fisher?s exact test of association.suggests that the possibility of a failure of isolation due to a failure of the independenceaxiom is more than a theoretical curiousity.The experimental literature provides only partial guidance on whether the theo-retical possibility suggested by Karni and Safra is empirically relevant in the contextof list elicitation. Experimentalists often invoke the evidence for isolation from binarychoice experiments (Starmer and Sugden, 1991; Cubitt and Sugden, 1998) as supportfor isolation in the context of list elicitation. However, no previous study examinedwhether this interpretation is appropriate. If list presentation invokes reduction andsubjects have non-expected utility preferences, then we should not expect subjects toisolate each choice in a list of binary choices.Whether isolation holds under list elicitation is ultimately an empirical question.We answer this question by using a between-subjects design to compare behaviourunder list elicitation to behaviour in binary choice tasks. In one group of treatments,subjects respond to a list (or two) of binary choices. In a second group of treat-ments, subjects make a single (or two) binary choice(s); the binary choice questionscorrespond exactly to the questions faced in line 11 of the lists.Table 4.1 demonstrates our main finding. Since the binary choice tasks correspondto unique lines in each list, we can compare the frequency of risky choices (withthe higher payment of $x) under the alternative treatments. We find that the listincreases the probability of choosing the risky lottery from 23% to 50% when thesafer alternative is certain, and from 30% to 45% when the safer alternative is risky.This suggests that subjects do not isolate their choices under list elicitation.An additional contribution of this paper is to evaluate the behaviour of a rela-tively new subject pool, that of online workers on Amazon?s Mechanical Turk, whosebehavior has barely been studied using real monetary incentives in standard decision-making tasks often studied by experimental economists. Our paper provides a new474.1. Experimentinterface for managing subjects that allows us to use Mechanical Turk as a virtual re-cruitment tool for running fully incentivized individual decision-making experiments.We also replicate our findings using a standard subject pool of university students.4.1 ExperimentSubjects were recruited from Amazon?s Mechanical Turk online labour market. InAppendix A (to be written), we outline our procedures and relevant aspects of Me-chanical Turk in detail. To recruit subjects, we released an ad for a task (?HIT? forHuman Intelligence Task) that could be seen by online workers on Amazon?s Mechan-ical Turk site (?turkers?). Any turker with a US based account and who had at least95% of their past HITs approved was eligible to view a brief description of our studythat mentioned the fixed payment of one dollar (which worked like the show-up fee inlab experiments) and the possibility of a bonus (which corresponded to the incentivescheme). From the description, interested turkers could accept a HIT and proceed toour website with a unique identifying code from the HIT. Each HIT would disappearonce accepted by a turker.On our website, turkers input their Mechanical Turk ID as well as the HIT ID, thencompleted a standard consent form, followed by a description of how the experimentwould work and how payment would be determined, which was accompanied by amultiple choice comprehension quiz which subjects had to answer correctly beforeproceeding.The main experiment consisted of sixteen different treatments, based on varyingthe payment mechanism and the order of the two questions (Q1 and Q2). Table 4.2shows the questions and subquestions when using list elicitation, and Table 4.3 showsthe sixteen different treatments. In treatments B1 and B2, subjects only answered onequestion (a binary choice) and were paid based on that question while in B12 and B21subjects answered two binary choice questions, one of which was randomly selected todetermine payment; the questions corresponded to line 11 of the lists in Table 4.2. Intreatments beginning with L, each question consisted of a list of subquestions, eachsubquestion involved a binary choice, where the probability of winning the prize inthe right hand side option decreased each line as subjects proceeded down the list. In484.1. Experimenttreatments L1 and L2, subjects answered only one list question while in treatmentsLO12, LO21, LA12, and LA21, subjects answered two list questions. In treatmentsLO12 and LO21, the instructions made it clear that one of the two list questionswould be randomly selected to determine payment. In treatments LA12 and LA21,the instructions made it clear that both of the two list questions would determinepayment. Subjects were informed that whenever a list was used to determine payment(the sole list in L1 and L2, both lists in LA12 and LA21, and a randomly chosen listin LO12 and LO21), one line from the list would be randomly selected to be playedout to determine the subject?s bonus payment. Treatments LO21 and LA21 reversedthe order of the list questions of treatments LO12 and LA12 respectively. In the Ltreatments, we allowed subjects to switch from Option A to Option B at any numberof points on the list, but used a javascript pop-up to warn subjects who switchedfrom Option B to Option A and then from Option A to Option B. The S (separatescreens) treatments mirror the L treatments, except that before completing each listsubjects responded to a sequence of (non-incentivized) binary choices that appearedon separate screens. In the S treatments, the binary choices tasks are chosen so asto spiral towards finding the switching point for a monotone subject. Subjects thenresponded to an incentivized list that was already filled in using their responses tothe binary choice tasks but was otherwise identical to that in the corresponding Ltreatment (crucially, subjects were free to change their answers in the list).Subjects completed the HIT by submitting a completion code generated by ourwebsite to the Mechanical Turk interface. A random number generator was usedto determine the outcomes of all risk automatically, and subjects were informed ofhow much of a bonus would be paid after completing the study. Subject paymentswere credited to subjects? Mechanical Turk accounts within 30 minutes of completingthe HIT. Our $1 base payment is somewhat high compared to other experimentsusing Mechanical Turk given that our experiment should take at most 15 minutes(e.g. Horton, Rand, and Zeckhauser (2011)). Bonus payments of $3 or $4 providedrelatively high stakes for this subject pool.494.1. ExperimentTable 4.2: QuestionsQ1 Q2Line Option A Option B Option A Option B1 (3, 1) (4, 1) (3, .5) (4, .50)2 (3, 1) (4, .98) (3, .5) (4, .49)3 (3, 1) (4, .96) (3, .5) (4, .48)4 (3, 1) (4, .94) (3, .5) (4, .47)5 (3, 1) (4, .92) (3, .5) (4, .46)6 (3, 1) (4, .90) (3, .5) (4, .45)7 (3, 1) (4, 88) (3, .5) (4, .44)8 (3, 1) (4, .86) (3, .5) (4, .43)9 (3, 1) (4, .84) (3, .5) (4, .42)10 (3, 1) (4, .82) (3, .5) (4, .41)11 (3, 1) (4, .80) (3, .5) (4, .40)12 (3, 1) (4, .78) (3, .5) (4, .39)13 (3, 1) (4, .76) (3, .5) (4, .38)14 (3, 1) (4, .74) (3, .5) (4, .37)15 (3, 1) (4, .72) (3, .5) (4, .36)16 (3, 1) (4, .70) (3, .5) (4, .35)17 (3, 1) (4, .68) (3, .5) (4, .34)18 (3, 1) (4, .66) (3, .5) (4, .33)19 (3, 1) (4, .64) (3, .5) (4, .32)20 (3, 1) (4, .62) (3, .5) (4, .31)21 (3, 1) (4, .60) (3, .5) (4, .30)22 (3, 1) (4, .58) (3, .5) (4, .29)23 (3, 1) (4, .56) (3, .5) (4, .28)24 (3, 1) (4, .54) (3, .5) (4, .27)25 (3, 1) (4, .52) (3, .5) (4, .26)26 (3, 1) (4, .50) (3, .5) (4, .25)Table 4.3: TreatmentsOrder Binary choice One list Pay one list Pay both listsQ1 only B1 L1, S1Q2 only B2 L2, S2Q1 then Q2 B12 LO12, SO12 LA12, SA12Q2 then Q1 B21 LO21, SO21 LA21, SA21S, L, and B respectively denote separate screen, standard list, and binary choice treatments504.2. ResultsTable 4.4: Subjects by treatmentB1 B2 B12 B21 L1 L2 LO12 LO21n 39 41 20 22 47 45 36 35n monotone 43 41 27 29n regular 32 27 18 21LA12 LA21 S1 S2 SO12 SO21 SA12 SA21n 36 33 48 49 37 32 26 25n monotone 31 26 46 48 36 28 25 25n regular 18 17 34 47 27 24 17 20Table 4.5: Answers to line 11Binary choice List elicitationOne choice Two choices One list Two listsQ1 23% 24% 47% 51%Q2 27% 33% 46% 44%n 39/41 42 66/74 162Treatments B1,B2 B12, B21 L1,L2,S1,S2 All O and AFraction choosing the riskier option4.2 ResultsAll analysis of results focuses solely on monotone subjects: those who exhibit single-switching in each list and who do not choose a dominated option in the first line of alist. Except where specified otherwise, our results focus on regular subjects: subjectswho never exhibit extreme risk seeking by sticking with B throughout the list, norexhibit extreme risk aversion by switching immediately to A on the second line.4.2.1 List elicitation versus binary choiceTable 4.5 shows the distribution of answers in binary choice for line 11 of the list,grouped by the incentives provided.There are no significant differences between asking one question or asking twoquestions shown on separate screens when binary choice is used (p = .92 for Q1 andp = .53 for Q2), a finding consistent with the literature supporting the incentivecompatibility of the RIS.514.2. ResultsTable 4.6: Fanning in vs outSCR EU RCRL,S 34% 20% 46%B 19% 71% 10%Fisher?s test: p < .001The most obvious difference in Table 4.5 is that in Q1 23% of subjects choosethe risky option in binary choice, but 50% choose the risky option in list elicitation,a significant difference (p < .001, exact test). In Q2, 30% of subjects choose therisky option in binary choice, but 45% choose the risky option in list elicitation, asignificant difference (p = .02). Comparable results would hold up if we included allmonotone subjects or only focused on subsets of list treatments.4.2.2 The independence axiomUnder both list elicitation and binary choice, responses are close to expected utility,though exhibiting a slight reverse common ratio effect when using list elicitation anda slight common ratio effect with binary choice elicitation. In binary choice, theviolation of expected utility is not significant (p = .80 for an exact test for B1 vs B2,.47 for an exact aggregate test for B12 and B21). However, since these two questionsonly look in a very particular region of the Marschak-Machina triangle, we do notview this as providing strong evidence in favour of the independence axiom.Pooling all the list treatments, the median choice pattern is ($4, .82) ? ($3, 1) ?($4, .80) and ($4, .41) ? ($3, .5) ? ($4, .40), consistent with EU. A rank-sum test forequality of distributions of Q1 and Q2 has p = .06, suggesting an aggregate leveldeviation from EU that is borderline statistically significant (and is in the oppositedirection of the standard common-ratio effect). A within-subject test suggests thatviolations of the independence axiom are significant (p = .01, signed-rank test).Aggregate analysis of behaviour masks substantial heterogeneity of individual de-cisions that list elicitation picks up: under list elicitation, we detect violations of theindependence axiom for 79% of subjects split between standard common ratio andreverse common ratio violations with the latter type of violation being slightly morefrequent (Table 4.6). Binary choice data only detects violations of the independence524.2. ResultsTable 4.7: Tests for payment mechanism and order effectsQ1 Q2Order effect (12=21) .97 .32Payment mechanism effect (O=A) .06 .58Separate screens effect (L=S) .47 .38p-values reported for a rank-sum test of equality of distributionaxiom (AB and BA choice patterns) for 29% of subjects, and using only data fromline 11 would detect a similar fraction of EU violations for the L and S treatments.4.2.3 Treatment effectsBy using list elicitation, the possibility of within-list contamination is equally presentin all L and S treatments. Our treatments allow us to test whether any cross-list con-tamination occurs. The 2x2x2 design embedded in the treatments ({L,S}?{O,A}?{12,21})allows us to separately test for the presence of separate screen effects, payment mech-anism effects, and order effects in each question.Applying non-parametric analysis, we do find a mildly significant payment mech-anism effect in Q1 (Table 4.7), however this effect would not be significant if wecorrected for multiple hypothesis tests using a Bonferroni correction.24So far, our results have focused on regular subjects. However, the most strikingtreatment effect is that the proportion of regular subjects is much higher in the Streatments than in the L treatments (78% vs. 57%, p < .001, see Table 4.4). As wemight expect, there are relatively more (79% vs. 62%, p = .01) regular subjects in thetreatments in which subjects faced only one list (as opposed to two). Neither order norpayment mechanism significantly affect the proportion of regular subjects (p = .31, .52respectively, exact tests). We suspected that a combination of isolation in binarychoice, a lack of influence of hypothetical versus real incentives, and a status-quobias when the actual list was displayed filled in, might bridge the difference betweenstandard list elicitation and binary choice. Table 4.7 shows that the incentivizedchoice data do not tend to support this view.24The payment mechanism effect is driven entirely by the O12 treatments. With enough speci-fication searching this may appear significant in some tests, but would not be after correcting formultiple hypothesis tests.534.3. Theory: binary choice versus list elicitationTable 4.8: mTurk vs. student subjectsBinary choice List elicitationOne choice Students mTurk StudentsQ1 23% 33% 50% 52%n 81 27 228 21Treatments All B All L,S,O,AFraction choosing the riskier option4.2.4 Comparison to a student subject poolA limited pair of treatments confirms that student subjects exhibit behaviour that issimilar to turkers. Students were recruited through the UBC Economics Lab subjectpool using ORSEE (Greiner, 2004), and offered the opportunity to do an experimentonline for payment by Interac money transfer. Since student subjects tend to be paidmuch higher amounts per hour than turkers typically earn, we changed the higherpayoffs to $13, $10, and $0 from $4, $3, and $0.25 Table 4.8 shows the results of thetwo student treatments, one corresponds to Q1 under binary choice (B1), and onecorresponds to Q1 under list elicitation (L1).The student data demonstrates exactly the same pattern as the data from turkers- students are more likely to choose the safer lottery in a binary choice task than whenthe choice is embedded in a list. Students? responses to Q1 in binary choice were notsignificantly different from turkers? answers to Q1 under binary choice (p = .32, exacttest), nor were students? responses to Q1 in the list significantly different from turkers?responses to Q1 under list elicitation (p = .82, exact test).4.3 Theory: binary choice versus list elicitationHow is it that subjects? preferences demonstrate more risk aversion when we asksubjects one question at a time? How is it that subject preferences appear closeto consistent with EU when we use list elicitation, in spite of its precise data beingideally-suited to test the Independence Axiom?25All major Canadian banks have a $10 minimum transfer, there was no show-up fee, which wasmade clear to subjects in advance.544.3. Theory: binary choice versus list elicitationA subject who chooses option B for the last time at line i of the list version of Q1receives the two-stage compound lottery:?($4, 1) ,126; ...; ($4, 1.02? .02i) ,126; ($3, 1) ,26? i26?(4.1)One possible explanation for our results is that there is some cross-question con-tamination within the elicitation list, which biased the answers of non-EU subjectsunder list elicitation. A subject satisfies Segal?s (1990) Compound Independence ax-iom and evaluates the compound lottery in (4.1) by folding back, if and only if thebinary choice at each ?branch? of the compound lottery is unaffected by the otherbranches (see Segal (1988); Starmer and Sugden (1991)). Karni and Safra (1987)show that if instead subjects reduces the compound lottery formed by her choicesin the list and has non-EU preferences, then list elicitation will distort her choices.Segal (1988) shows that if subjects takes an alternative view of the compound lotteryformed by her choices in the list, then list elicitation will distort her choices even ifshe satisfies Compound Independence. We sketch these two approaches below in thecontext of our experiment.4.3.1 Karni and Safra (1987): Reduction of CompoundLotteriesIf the subject satisfies the Reduction of Compound Lotteries axiom, then she evaluates(4.1) as equivalent to the single-stage lottery:?$4,1.01i? .01i226; $3,26? i26?(4.2)While a binary choice between ($3, 1) and ($4, .8) involves a choice between acertain and a risky option, the choice of a switching line in a list involves choiceamong multiple risky alternatives and a dominated certain one (for i = 0) when thesubject views the list according to (4.2). The logic of the certainty effect suggeststhat a subject may rank ($3, 1) ? ($4, .8) and choose accordingly in a binary choicetask, yet choose ($4, .8) over ($3, 1) on line 11 of a list since this conditional choiceinvolves a choice between a riskier and a less risky (but not certain) alternatives. This554.3. Theory: binary choice versus list elicitationexact logic could be extended, but does not directly apply to the reversal observed inQ2, although the reversal in Q2 is empirically weaker.Some non-EU theories predict, or can accommodate this type of behaviour. Forexample, the Negative Certainty Independence (NCI) axiom of Cerreia-Vioglio, Dil-lenberger, and Ortoleva (2013) allows exactly the type of reversal observed in Q1,while disallowing the opposite pattern of behaviour. NCI does not make a directprediction that relates to Q2. Some (but not all) functional forms for rank-dependentutility can also capture the observed reversals. For example, the power weightingfunction:f(p) = p?, ? > 1and the neo-additive weighting function (Chateauneuf, Eichberger, and Grant,2007; Webb and Zank, 2011):f(p) =?????????1 if p = 1ap + b if p ? (0, 1)0 if p = 0f(p) = ap + bare both able to accommodate observed reversal behaviour for reasonable param-eter values.26 In the Appendix, we sketch how these weighting functions can be usedto rationalize the data.4.3.2 Segal (1988): non-standard view of the compoundlottery formed by the listSegal (1988) assumes that a subject satisfies Compound Independence, but views aBDM certainty equivalent elicitation scheme as the two-stage lottery with two prizesafter the first stage (i) lottery at all ?lines? where it is chosen, (ii) the uniform lotteryover all $ amounts for ?lines? where the dollar amount is chosen.The version of list elicitation used in this paper corresponds to BDM with varyingprobabilities rather than the traditional BDM. To adapt Segal?s (1988) approach tothis setting, suppose a subject views the list elicitation with the RIS for Q1 according26Some other weighting functions are less successful at accommodating the observed behaviour.For example, the Prelec (1998) weighting function does not predict the reversal observed in Q1 forstandard parameter values.564.4. Discussionto the three-stage compound lottery:??($4, 1) ,1i; ...; ($4, 1.02? .02i) ,1i?,i26; [$3, 1] ,26? i26?(4.3)A subject who evaluates her choices in the list by applying the Compound Inde-pendence axiom to (4.3) will tend to violate isolation unless her preferences are EU.As in the Karni and Safra (1987) explanation for the preference reversal, some (butnot all) versions of rank-dependent utility, including some parameter values for thepower weighting function, are able to account for the preference reversal observed inour data. The Appendix shows how RDU with the power weighting function can beused to rationalize our main results.4.4 Discussion4.4.1 Related literature on elicitation mechanisms andincentive-compatibilityIncentivized list elicitation was pioneered by Becker, DeGroot, and Marschak (1964),who introduced it as a way of eliciting certainty equivalents. Unfortunately, use ofBDM induced preference reversals in which subjects? reported certainty equivalentsof two lotteries were inconsistent with their choices in a binary choice task (Gretherand Plott, 1979). Holt (1986), Karni and Safra (1987), and Segal (1988) showed thatunless subjects evaluate their choices by applying the compound independence axiomto evaluate the compound lottery generated by the external randomizing device andtheir choices, BDM will not elicit subjects? true certainty equivalents if subjects violatethe independence axiom. Safra, Segal, and Spivak (1990) show that assumptionsthat generate the common-ratio effect will tend to generate the observed preferencereversals.Starmer and Sugden (1991) and Cubitt and Sugden (1998) are widely interpretedas providing evidence in favour of the use of the RIS in experiments. They havesome subjects answer multiple questions, with one randomly selected for payment,and compare their responses to those of subjects who answer only one incentivized574.4. Discussionquestion. They find only weak evidence for any ?contamination effect? of RIS. Inrecent work, Cox, Sadiraj, and Schmidt (2011) and Harrison and Swarthout (2012)use comparable designs but consider a larger number of ways of paying subjects whoanswer multiple questions. None of these four papers, however, uses list elicitation.Harrison and Swarthout (2012) suggest that asking 30 questions and paying one doeslead to different structural utility function estimates as compared to asking and payingone question.While recent variants on list elicitation have been seen as avoiding some of theearly problems with BDM, we show that the analysis of Karni and Safra (1987) canbe extended to another variant on list elicitation, with an approach that would alsoapply to other variations on list elicitation. Recent experiments using list elicitationwidely cite Starmer and Sugden (1991) and Cubitt and Sugden (1998) in support ofcombining list elicitation with the RIS; our results suggests that this inference is notwarranted. List elicitation experiments typically have 10 to 100 questions per list ona single screen or sheet of paper at a time, and then subjects respond to multiplelists; this contrasts with experiments by Starmer and Sugden (1991) and Cubitt andSugden (1998) in which each subject completes at most a small number of binarychoice tasks, which are not given together as a list.4.4.2 Discussion of our resultsVariations on list elicitation have been extremely popular in recent years (e.g. Holtand Laury (2002), Andersen, Harrison, Lau, and Rutstr?m (2008), Bruhin, Fehr-Duda, and Epper (2010), see also Andersen, Harrison, Lau, and Rutstr?m (2006)).Probability list elicitation was first used (without incentives) by Davidson, Suppes,and Siegel (1957), revisited by McCord and De Neufville (1986), and was revisited(with the RIS) by Andreoni and Sprenger (2013) and ?.List elicitation supposedly provides an efficient method for collecting precise dataon individual preferences. But its attractiveness relies on the assumption that theisolation hypothesis holds. Without the assumption that isolation holds, one musttake into account the structure of the list and the RIS in order to interpret subjects?choices in terms of their preferences, which complicates the inferences that can be584.4. Discussiondrawn from choice data collected using list elicitation.Our results are clear evidence against the isolation hypothesis. Our results are inthe direction consistent with non-expected utility theory combined with assumptionsabout how subjects view and evaluate the their choices and the external random-izing device as a compound lottery. This theoretical approach is attractive, sinceit already provides a unified understanding of preference reversals in BDM and ofthe Allais paradox. However, our experimental design cannot rule out other possibleexplanations for our findings.We have suggested two different ways in which non-expected utility preferenceswould lead to the biases we observe in our experiment. Each of these explanationshas different implications for how we might recover preferences from list elicitation.Given that there are other possible explanations for our findings, we do not see anyway to draw unambiguous inferences about preferences from list elicitation. Futurework might distinguish between different possible explanations, as Keller, Segal, andWang (1993) do for explanations of preference reversals in BDM.The experimental economics literature has developed many other ways of elicitinginformation about preferences. Alternatives to list elicitation include binary choice(Hey and Orme, 1994) and convex budget sets (Choi, Fisman, Gale, and Kariv,2007). A typical implementation of each of these designs still uses RIS to select oneof multiple questions to determine payment. However, in these designs choice tasksare always displayed on separate screens, with no RIS applying within a screen, whichmight induce subjects to isolate their choices.4.4.3 Experiments on Mechanical TurkAs a large online labour market, Mechanical Turk provides a convenient way to recruitand pay subjects over the internet. Mechanical Turk allows researchers to economizeon costs and experiment on a different population from undergraduates. MechanicalTurk has been advocated as a platform for recruiting subjects by psychologists study-ing judgement and decision-making (Mason and Suri (2011), Paolacci, Chandler, andIpeirotis (2010), Buhrmester, Kwang, and Gosling (2011)), political scientists (Berin-sky, Huber, Lenz, et al., 2012), and economists (Horton, Rand, and Zeckhauser,594.4. Discussion2011). 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Journal of Economic Theory, 61(1), 139?159.Webb, C. S., and H. Zank (2011): ?Accounting for optimism and pessimism inexpected utility,? Journal of Mathematical Economics, 47(6), 706?717.Yaari, M. (1987): ?The dual theory of choice under risk,? Econometrica, 55(1),95?115.69Appendix AProofs for Chapter 2Lemma A.1. For any two sets D,D? and any asymmetric binary relation P , m(D, P )?m(D?, P ) ? m(D ?D?, P ).Proof. Suppose p ? m(D ?D?, P ) ?D.=? ?q ? D ?D? s.t. qPp.=? ?q ? D s.t. qPp=? p ? m(D,P ).If p ? m(D ?D?, P ) ?D?, an analogous result would follow.Thus p ? m(D ?D?, P ) implies p ? m(D,P ) ?m(D?, P ).=? m(D, P ) ?m(D?, P ) ? m(D ?D?, P )Results on IIA Independence and IIA Independence 2.Lemma A.2. Suppose Expansion and Weak RARP hold. If p ? c(D), p /? c(D?q) ?r, and p ??Wr, then ?Dpq such that p ? c(Dpq).Proof. If ?Dpq such that p ? c(Dpq) then by Expansion, p ? c(D ? Dpq). SinceD ? q ? D ? Dpq and r ? c(D ? q) with p ??Wr, it follows by Weak RARP thatp ? c(D ? q), a contradiction. Thus no such Dpq can exist.Lemma A.3. Suppose Expansion and Sen?s ? hold. If p ? c(D), p /? c(D ? q) , then?Dpq such that p ? c(Dpq).Proof. If p ? c(D) ? Dpq then by Expansion, p ? c(D ? Dpq). Then by Sen?s ?,p ? c(D ? q). This proves the claim.70Appendix A. Proofs for Chapter 2Proof of Proposition 2.2.(i) ?? (iii)Let P1, P2 denote the asymmetric part of relations P?1, P?2 that form a transitive short-listing representation. By definition, m(D, Pi) = m(D, P?i) for i = 1, 2 and for anyD.Necessity of Expansion. p ? c(D) and p ? c(D?) implies:(i) p ? m(D,P1) and p ? m(D?, P1)=? ?q ? D s.t. qP1p and ?q ? D? s.t. qP1p=? ?q ? D ?D? s.t. qP1p=? p ? m(D ?D?, P1)(ii) p ? m(m(D,P1), P2) and p ? m(m(D?, P1), P2)=? ?q ? m(D,P1) s.t. qP2p and ?q ? m(D?, P1) s.t. qP2p=? ?q ? m(D,P1) ?m(D?, P1) s.t. qP2pby Lemma A.1,=? ?q ? m(D ?D?, P1) s.t. qP2pBy (i),=? p ? m(m(D ?D?, P1), P2) = c(D ?D?)This implies that Expansion holds.Necessity of Weak RARP. Suppose q ??Wp, and there areD,D? such that: {p, q} ?D ? D? and p ? c(D), q ? cU(D?).By definition of q ??Wp, there is a chain q = r0, r1, ..., rn?1, rn = p such that foreach i ? {1, ..., n}, there are Di, D?i such that {ri?1, ri} ? Di ? D?i, ri ? cU(D?i) andri?1 ? c(Di), or (if not) there is a net {D?i,?, Di,?} ? D?i, Di for which ri ? cU(D?i,,?)and ri?1 ? c(Di,?) ?? > 0.For each i, from the representation, it follows that:=? ri ? m(Di, P1)=? not riP2ri?1.Since the transitive completion of P2 is transitive, it follows that not qP2p.Since q ? cU(D?), by continuity of P1, q ? m(D?, P1).71Appendix A. Proofs for Chapter 2Since q ? D ? D? as well, q ? m(D, P1).Since p ? m(m(D,P1), P2), not pP2q, and P2 has a transitive completion, it followsthat not rP2q ?r ? m(D, P1).Thus, q ? m(m(D, P1), P2) = c(D).Sufficiency. Part of the idea of the proof follows Manzini and Mariotti (2007). Thetwo rationales constructed here are not unique.Define P1 by:qP1p if ?Dpq s.t. p ? cU(Dpq)Define P?2 by:P?2 = ??WDefine P2 as the asymmetric part of P2.First, show that P1 and P2 are appropriately continuous.If pP1q, ? a net {Dp?q?}? ? Dpq with p? ? c(Dp?q?) and max [d(p?, p), d(q?, q)] < ?for each ? > 0, since then we would have p ? cU(Dpq) for some Dpq. Thus, ??? > 0such that ?p? ? N ??p, ?q? ? N ??q , p?P1q?. This implies that P1 has open better and worsethan sets.P2 is continuous by construction.Second, show c(D) ? m(m(D,P1), P2).By definition of P1, p ? c(D) implies p ? m(D,P1).Take any q ? m(D,P1). By the definition of P1, ?r ? D, ?Dqr such that q ?c(Dqr). Successively applying Expansion implies that q ? c( ?r?DDqr). Since D ??r?DDqr and p ? c(D), it follows that p ??Wq, thus pP?2q. Since this implies not qP2p forany arbitrary q ? m(D,P1), it further follows that p ? m(m(D,P1), P2).Third, show m(m(D,P1), P2) ? c(D)Suppose p ? m(m(D,P1), P2).Then, ?r ? D, ?Dpr : p ? c(Dpr). By Expansion, p ? c( ?r?DDpr).Since p ? m(m(D,P1), P2), it p ??Wq ?q ? c(D) by the definition of ??W .Thus by Weak RARP, p ? c(D).72Appendix A. Proofs for Chapter 2(ii) ?? (iii) Consider a continuous PPE representation v that represents c, anda continuous and transitive shortlisting representation P1, P2.Map between v and P1 by:qP1p ?? v(q|p) > v(p|p)Map between v and P2 by:qP2p ?? v(q|q) > v(p|p)Joint continuity of v will map to continuity of P1 and P2.Notice that the mapping from P1 to v only specifies v(?|p) partially; the mappingfrom P2 to v imposes an continuous additive normalization on v.Consider the following construction of v from P1, P2:Let u : ?? ? be a continuous utility function that represents P2. Define v(p|p) =u(p) ?p ? ?. Let I(p) = {q ? ? : (q, p) ? cl{(q?, p?) : q?P1p?}\{(q?, p?) : q?P1p?}}.The following definition of v is consistent with the mapping proposed above:v(q|p) =???u(p) + dH({q}, I(p)) if qP1pu(p)? dH({q}, I(p)) otherwiseIt can be verified that continuity of P1 and u imply that v so constructed satisfiesjoint continuity.?Proof of Theorem 2.1.Notation.Let for p, q ? ?, let Dpq ? D denote an arbitrary choice set that contains p and q.Sufficiency: Lemmas.In the lemmas in this section, assume that c satisfies Expansion, Weak RARP, IIAIndependence, Induced Reference Lottery Bias, and Transitive Limit.Lemma A.4. R?p is complete, transitive, and if there exists a net {p?, q?, r?} ? p, q, rwith q?R?p?r? for each term in the net, then qR?pr.73Appendix A. Proofs for Chapter 2Proof. Transitivity of R?p follows by Transitive Limit.For any net {p?, q?, r?} ? p, q, r, non-emptiness of c implies that the net either hasa convergent subnet p?, q?, r? in which (1??)p?+?q? ? c({(1??)p?+?q?, (1??)p?+?r?}or in which (1? ?)p? + ?r? ? c({(1? ?)p? + ?q?, (1? ?)p? + ?r?} for each term in thesubnet. Thus R?p is complete.Take a net {p?, q?, r?} ? p, q, r, for which q?R?p?r? for each term in the net. Bythe definition of R?p? , for each ? there is a net {p?,?, q?,?, r?,?}? ? p?, q?, r? such that(1? ?)p?,? + ?q?,? ? c((1? ?)p?,? + ?{q?,?, r?,?}) for each term in the net. Let ??? denotethe largest element in the index set for {p?,?, q?,?, r?,?}? and ?? the largest element inthe index set for {p?, q?, r?} . Take ?? := ????. For each ? < ??, define ?? as a decreasingnet such that for each ? < ???? . Then define {p??, q??, r??} := {p?? ,?, q?? ,?, r?? ,?}?. Byconstruction, {p??, q??, r??} establishes that qR?pr.Let Rp denote the strict part of R?p. Lemma A.5 shows that Rp satisfies theIndependence Axiom.For a binary relation R, say that R satisfies the Independence axiom if qRr ??(1? ?)s + ?qR(1? ?)s + ?r ?? ? (0, 1). ?s ? ?.Lemma A.5. Rp satisfies the Independence Axiom if p ? int?.Proof. Part I: suppose qRpr, and take a ? ? (0, 1) and s ? ?.Then,???, ?? > 0 such that ?? ? (0, ??), p?, q?, r? ? N ??p ? N??q ? N??r , {(1 ? ?)p? + ?q?} =c((1? ?)p? + ?{q?, r?}).Define ??? = min????, (1? ?)???.Let p?, s? ? N ???p ?N???s . Since dE(p, q) ? 1, it follows that dE((1??)p?+?s?, p) ? (1??)??? +? by the triangle inequality. Thus if ? ? ??? := ??????1???? , then (1??)p?+?s? ? N??p .Then for any q?, r? ? N ??q ?N??r , ? ? (0, ??), and ? ? (0, ??), {(1? ?) ((1? ?)p? + ?s?) +?q?} = c((1? ?) ((1? ?)p? + ?s?) + ?{q?, r?}). Define ?? := ?? and ??,? := ??1??1?? . Then ???such that ??? ? (0, ??) and ?? 1??1???? ? (0, ??), it follows that {(1? ??)p?+ ?? ((1? ?)s? + ?q?)} =c((1 ? ??)p? + ?? ((1? ?)s? + ?{q?, r?})). Since N ???(1??)s+?q ? (1 ? ?)N??s + ?N??q andN ???(1??)s+?q ? (1? ?)N??s + ?N??qIt follows that (1? ?)s + ?qRp(1? ?)s + ?r.Part II: suppose (1? ?)s + ?qRp(1? ?)s + ?r.Recall that N ??(1??)s+?q ? (1? ?)N??s + ?N??q .74Appendix A. Proofs for Chapter 2Then,???, ?? > 0 such that N ??p ? int? and ?? ? (0, ??), p?, q?, r?, s? ? N??p?N??q ?N??r ?N??s ,{(1? ?)p? + ?((1? ?)s? + ?q?)} = c((1? ?)p? + ?((1? ?)s? + ?{q?, r?})).Fix ? ? (0, 1). Fix p?, q?, r?, s? ? N???p ?N???q ?N???r ?N???s .Given ? ? (0, ??), take ??,? := ?1??1?? . If ? < (1??)??, then p?+??,?(p?? s?) ? N ??p ? ?.Then,(1 ? ?) (p? + ??,?(p?? s?)) + ?((1 ? ?)s? + ?q?) = c((1 ? ?) (p? + ??,?(p?? s?)) + ?((1 ??)s? + ?{q?, r?}))?? (1? ??)p? + ??{q?} = c((1? ??)p? + ??{q?, r?})Since the above holds ?p?, q?, r?, s?, ? ? N???p ?N???q ?N???r ?N???s ? (0, ??) it follows thatqRpr.Lemma A.6. R?p satisfies the Independence Axiom if p ? int?.Proof. I already have a proof that Rp satisfies the Independence Axiom.Suppose that qR?pr and take (1? ?)s + ?q and (1? ?)s + ?r.If it is not the case that (1? ?)s+ ?qR?p(1? ?)s+ ?r, then (1? ?)s+ ?rRp(1??)s + ?q.Then it follows by Lemma A.5 that rRpq, which contradicts that qR?pr.Define qR?pr if either:(i) p ? int? and qR?pr(ii) p /? int?, and ??, s ? (0, 1)?? such that (1? ?)s + ?qR?p(1? ?)s + ?r(iii) ??, s, q?, r? such that q = (1? ?)s + ?q?, r = (1? ?)s + ?r?, and q?R?pr?The relation R?p is the minimal extension of R?p that respects with the IndependenceAxiom for all p ? ?.By construction, R?p satisfies the joint continuity properties in Lemma A.4 as well.Lemma A.7. For each p ? ?, there exists a vector u?p ? ?N such that qR?pr ??q ? u?p ? r ? u?p.Proof. Lemma A.4. shows R?p is complete, and transitive. By construction, R?p satis-fies the Independence axiom. The joint continuity property on R?p in Lemma A.4 thenimplies the notion of mixture continuity required (condition 3) to apply Fishburn?s(1970) Theorem 8.2.75Appendix A. Proofs for Chapter 2Say that a vector up is flat if maxiupi = miniupi . Let F := {p ? ? : up is flat}.Lemma A.8. Suppose up is not flat. Then, there is an ? neighbourhood N ?p of p suchthat ?p? ? N ?p, up? is not flat.Proof. Suppose there is a net p?? such that p?? ? N ?p and up?? is flat. Since up??mustrepresent R?p?? , it follows that qR?p??r ?q, r ? ? and for each p??. By Lemma A.4, itfollows that qR?pr. It follows that up must be flat as well, a contradiction.Let u?p denote a vector that provides an EU representarion for R?p (i.e. q ?u?p ? r ?u?p?? qR?pr ?q, r ? ?). For all p such that p is non-flat, define:up :=dH({p}, F )maxi?u?pi ??u?pjj??u?p ??ju?pj?(A.1)If u?p is flat, define up as the zero vector.By Lemma A.8 and the EU theorem, up provides an EU representation for R?p.Lemma A.9. If p? ? p, then up? ? upProof. If up is flat, then dH({p?}, F ) ? 0 as ? ? 0, thus up? ? up.Now suppose that up is non-flat. Suppose p? ? p but for convergent subnet {p??}of {p?}, up??? u?p ?= up. Since up??represents R?p?? , by the joint continuity propertyin Lemma A.4., it follows that u?p ranks q ? r if and only if up ranks q ? r. Since upand u?p must satisfy the same normalizations, they must coincide by the uniquenessresult of the EU theorem.Define v : ???? ? by v(q|p) := q ? upLemma A.10. v is jointly continuous.Proof. v(q|p) = q ? up =?i qiupi and up is continuous as a function of p, and jointcontinuity of the sum?i qiupi in q and up is a standard exercise.Lemma A.11 shows that Limit Consistency is implied by the axioms assumed inTheorem 2.1.76Appendix A. Proofs for Chapter 2Lemma A.11. The axioms in Theorem 2.1 imply Limit Consistency.Proof. Part 1. Suppose {q} = m(D,Rp) ?= {p} ? c(D).That is, qRpr ?r ? D.Then ?r ? D ???r > 0 such that ?? ? (0, ??r), (1??)p+?q = c((1??)+?{q, r}).Since D is finite, minr?D??r > 0.By Expansion, ?? ? (0,minr?D??r), (1? ?)p + ?q ? c((1? ?) + ?D).By Induced Reference Lottery Bias, ?? ? (0,minr?D??r) p ? c((1? ?) + ?D). Thusp ??W (1 ? ?)p + ?q. Weak RARP then implies that p ? c((1 ? ?)p + ?{p, q}) ?? ?(0,minr?D??r). This implies that pR?pq, a contradiction.Part 2. Suppose there are elements q1, ..., ql ? D such that qiRpp for each i =1, ..., l.Suppose qi ? m(D,Rp), and let D? := D\m(D,Rp).Then by the previous result ?i = 1, ..., l, ???i > 0 such that ?? ? (0, ??i), (1 ??)p + ?qi ? c((1? ?)p + ??D? ? qi?).Since {q1, ..., ql} is finite and each ??i > 0, mini??i > 0.For each ? ? (0, 1), c((1? ?)p + ?{q1, ..., ql}) is non-empty.For q? such that (1??)p+?q? ? c((1??)p+?{q1, ..., ql}), Expansion implies that(1? ?)p + ?q? ? c(?(1? ?)p + ?{q1, ..., ql}???(1? ?)p + ??D? ? q???)= c((1? ?)p + ?D).Thus ?? ? (0,mini??i), ((1? ?)p + ?{q1, ..., ql}) ? c((1? ?)p + ?D) ?= ?.It follows that for at least one q? ? {q1, ..., ql}, ??? ? (0,mini??i), ?? < (0, ??) suchthat ((1? ?)p + ?q? ? c((1? ?)p + ?D).Since p ? c((1 ? ?)p + ?D) ?? ? (0, 1) by Induced Reference Lottery Bias, itfollows that p ??W (1? ?)p + ?q? whenever (1? ?)p + ?q? ? c((1? ?)p + ?D). For such?, it further follows by Weak RARP that p ? c((1??)p+?{p, q?}). This contradictsthat q?Rpp.Define P?E(D) = {p ? D : pR?pq ?q ? D}.Define ?PPE(D) = {p ? P?E(D) : ?q ? P?E(D) s.t. q ??Wp}.Lemma A.9 establishes that p ? c({p, q}) implies p ? ?PPE({p, q}).77Appendix A. Proofs for Chapter 2Lemma A.12. If qR?qp and p ? c({p, q}), then p ??Wq.Proof. If ?Dpq such that q ? cU(Dpq) then the result follows automatically. Similarlyif there exists a chain p = r0, r1, ..., rn = q such that ri?1 ??Wri for i = 1, ..., n.If ?p?, q? that establish qR?qp, then if for some such sequence, p? ? c({p?, q?}) fora convergent subsequence of p?, q?, then p? ??Wq? for such pairs. Then, continuity of ??Wimplies that p ??Wq.So suppose instead that for each sequence p?, q? that establishes that qR?qp, q? =c({p?, q?}) except on a non-convergent subsequence of p?, q?. This implies that q ?cU({p, q}). Then by the definition of ??W , p ??Wq.Lemma A.10 establishes that p ? ?PPE({p, q}) implies p ? c({p, q}).Lemma A.13. If pR?pq and p ??Wq, then p ? c({p, q}).Proof. Since {p} = c({p}), if {q} = c({p, q}) and p ??Wq it would follow by IIA Inde-pendence and the definition of Rp that qRpp. This would contradict the assumptionthat pR?pq. Since c({p, q}) ?= ?, it then follows that p ? c({p, q}).Lemmas A.11-A.12 establish that ?PPE({p, q, r}) = c({p, q, r}) ?p, q, r ? D.Lemma A.14. If p ? c({p, q, r}) and qR?qp then p ??Wq or rR?qq.Proof. Suppose p ? c({p, q, r}) and qR?qp.If p ? c({p, q}), then p ??Wq holds.So suppose instead that q = c({p, q}).Then, if q ? c({q, r}) it would follow by Expansion that q ? c({p, q, r}). Sincep ? c({p, q, r}) as well, it follows that p ??Wq; by Weak RARP, it follows that p ?c({p, q}), a contradiction. Thus r = c({q, r}).By Lemma A.5, it follows that either rRqq or r ??Wq; in the former case we?re done,so suppose r ??Wq and that it is not the case that rRqq.If p ? c({p, r}), then it follows that either pRrr or p ??Wr. In the latter case,transitivity of ??W implies p ??Wq and we?re done, so suppose we have that pRrr. Thenby Limit Consistency, p = c({p, r}).78Appendix A. Proofs for Chapter 2To summarize, we now have that q = c({p, q}), p = c({p, r}) = c({p, q, r}), andr = c({q, r}). Then, by IIA Independence, it follows that ?? > 0 : ?? ? (0, 1), ?q? ?N ?q , ?r? ? N?r , ?D? ? {q?, (1??)q?+?r?}, q? /? c(D?). It follows that rRqq, a contradiction.It follows that either rRqq or p ??Wq.Lemma A.15. If p ? ?PPE({p, q, r}) then p ? c({p, q, r}).Proof. Suppose p ? ?PPE({p, q, r}).We know that c({p, q, r}) ?= ?. So it is sufficient to prove that q ? c({p, q, r}) =?p ? c({p, q, r}) and similarly if r ? c({p, q, r}).Suppose q ? c({p, q, r}); the argument starting from r ? c({p, q, r}) is symmet-ric.Then, qR?qp and qR?qr by Limit Consistency. Since p ? ?PPE({p, q, r}) and q ?P?E({p, q, r}), it follows that p ??Wq. Then by Lemma A.10, since pR?pq as well, p ?c({p, q}).If r ? c({p, q, r}) then a similar argument implies p ? c({p, r}). Then by Expan-sion, p ? c({p, q, r}).If instead r /? c({p, q, r}), we have (recalling Lemma A.6) that either p ? c({p, r})or r = c({p, r}). In the former case, Expansion implies p ? c({p, q, r}). In the lattercase, r = c({p, r}). Recall that p ? c({p, q}). If p /? c({p, q, r}) then q = c({p, q, r});by IIA Independence and the definition of Rp, it follows that rRpp, a contradictionof the assumption that p ? ?PPE({p, q, r}).It follows that p ? ?PPE({p, q, r}) =? p ? c({p, q, r}).Remark. ?PPE(D) = ?PPE(P?E(D))Lemma A.16. Suppose we have established that ?PPE(D) = c(D) whenever |D| < n.If P?E(D) = D and |D| ? n, then c(D) = ?PPE(D).Proof. First, suppose P?E(D) = D.Take p ? ?PPE(D). Then p ? ?PPE(D\r) ? ? D\p. Take any distinct r, r? ? D\p,and then since |D\r| = |D\r?| = n? 1 < n, p ? c(D\r) ? c(D\r?). By Expansion, itfollows that p ? c(D).79Appendix A. Proofs for Chapter 2In the reverse, suppose p ? c(D). Then if q ??Wr ?r ? D, since P?E(D) = D, itfollows that q ? c({q, r}) ?r ? D. By Expansion, it follows that q ? c(D). Thensince p ? c(D) and q ? c(D), p ??Wq by definition. Thus p ? ?PPE(D).Lemma A.14 establishes by induction that c(D) = ?PPE(D) for any D ? D.Lemma A.17. Suppose c(D) = ?PPE(D) whenever |D| < n. Then, c(D) = ?PPE(D)whenever |D| ? n as well.Proof. Consider D with |D| = n and P?E(D) ?= D. Partition D into P?E(D) andD\P?E(D). The case where P?E(D) = D was proven in Lemma A.9.Since |P?E(D)| ? n? 1 < n, c(P?E(D)) = ?PPE(P?E(D)) = ?PPE(D).Say that q0, q1, ..., qm form a chain if qiRqi?1qi?1 for i = 1, ...,m. Notice thatif q0, ..., qm form a chain, Limit Consistency implies that qm = c({q0, ..., qm}) =P?E(D) = ?PPE(D). So if the longest chain in D contains all elements of D, thenc(D) = ?PPE(D).Now suppose p ? ?PPE(D).First, further suppose the longest chain in D has length n ? 1; denote the chainq0, q1, ..., qn?1. Then, qn?1 = c({q0, q1, ..., qn?1}) and since q0, q1, ..., qn?1 is the longestchain in D and p ? P?E(D), {p, qn?1} = P?E(D). Since p ? ?PPE(D), it follows thatp ??Wqn?1; Lemmas A.8 and A.10, p ? c({p, qn?1}). Suppose p ? c({p, qk, ..., qn?1}) forsome k ? n?1. Then, since if p /? c({p, qk?1, ..., qn?1}) it follows by IIA Independenceand the definition of Rp that qk?1Rpp, which contradicts that p ? P?E(D). Thus itfollows by induction that p ? c(D).Take an arbitrary chain q0, ..., qm that cannot be extended further as a chain usingelements of D. Since q0, ..., qm cannot be extended, qm ? P?E(D). Since p ? ?PPE(D),p ??Wqm and by Lemma A.8, p ? c({p, qm}). Suppose p ? c({p, qk, ..., qm}) for somek ? m. Then if p /? c({p, qk?1, ..., qm}) it follows by IIA Independence and thedefinition of Rp that qk?1Rpp; this would which contradicts that p ? P?E(D). Thusit follows by induction that p ? c({p, q0, ..., qm}).Notice that any element of D\P?E(D) is in a chain in D. Let D? is the choice setformed by taking the union of {p} and of the all of the choice sets formed by chainsin D. Since for any chain q0, ..., qm in D, p ? c({p, q0, ..., qm}), p ? c(D?) follows by80Appendix A. Proofs for Chapter 2Expansion. Since p ? c(P?E(D)) as well follows (because |P?E(D)| < n or LemmaA.13 applies), it follows by Expansion that p ? c(D). Thus ?PPE(D) ? c(D).In the reverse direction, now suppose p ? c(D). By Limit Consistency, p ?P?E(D). Since c(D) ? ?PPE(D) = ?PPE(P?E(D)) = c(P?E(D)) ?= ?, ?q ? c(D) ??PPE(D). Since p, q ? c(D), p ??Wq. Thus p ? ?PPE(D).Lemma A.15 relates the dislike of mixtures property to the Induced ReferenceLottery Bias axiom.Lemma A.18. Induced Reference Lottery Bias implies that v dislikes mixtures.Proof. By the representation thus far, c(D) = ?PPE(D).If p ? ?PPE({p, q}) then v(p|p) ? v(q|p) and either v(p|p) ? v(q|q) or v(p|q) >v(q|q). Thus v(p|p) ? v(q|p) and v(q|q) ? max [v(p|p), v(q|p)]. Then the InducedReference Lottery Bias axiom implies that then p ? c((1 ? ?)p + ?D) = ?PPE((1 ??)p + ?D), thus v(p|p) ? v((1 ? ?)p + ?q|p) and v((1 ? ?)p + ?q|(1 ? ?)p + ?q) ?max [v(p|p), v((1? ?)p + ?q|p)].Remark. ?PPE(D) = PPEv(D)Necessity.Proposition 2.2 implies that Expansion and Weak RARP are necessary conditionsfor any PPE representation.Lemma A.19. Suppose v represents c by a PPE representation. Then p ??Wr impliesthat v(p|p) ? v(r|r).Proof. Suppose p ??Wr. If ?D, D? with {p, r} ? D ? D? and p ? c(D) and r ? c(D?)then it follows that v(p|p) ? v(r|r) since r ? PE(D?) ? D ? PE(D) follows by therepresentation.If instead there is a chain such that pi?1 ??Wpi for i = 1, ..., n and p0 = p, pn = r,then it follows that v(pi?1|pi?1) ? v(pi|pi) for each i. Chaining these inequalitiestogether, it follows that v(p|p) ? v(r|r).81Appendix A. Proofs for Chapter 2Necessity of IIA Independence. Suppose p ??Wr. Then by Lemma A.12, v(p|p) ?v(r|r). If p ? PPE(D) and p /? PPE(D?q) ? r, then it follows that v(q|p) > v(p|p).Since v is jointly continuous, ?? > 0 such that ?p? ? N ?p, ?q? ? N?q , v(q?|p?) > v(q?|p?).Since v is expected utility, it follows that for all such p?, q? pairs and ?? ? [0, 1),v((1 ? ?)p? + ?q?|p?) > v(p?|p?). It follows that for all such p?, q? pairs and for any such? ? [0, 1), whenever (1 ? ?)p? + ?q? ? D? it follows that p? /? PPE(D?) = c(D?). ThusIIA Independence holds.Necessity of Transitive Limit. First, I show that the antecedent of TransitiveLimit has bite in the presence of, and only in the presence of, a strict preference. Tobe precise, suppose (1? ?)p? + ?q? = c({(1? ?)p? + ?q?, (1? ?)p? + ?r?}) for all small?, and p?, q?, r? sufficiently close to p, q, r. By the representation, this holds only if forall p? close to p, q? close to q, r? close to r, and ? close to zero, v(q?|(1? ?)p? + ?q?) ?v(r?|(1? ?)p? + ?q?), thus v(q?|p?) ? v(r?|p?) for all p?, q?, r?. If v(q|p) = v(r|p), thenfor every q? near q, v(q?|p) ? v(q|p) and for every r? near r, v(r|p) ? v(r?|p); thiscontradicts local strictness of v(?|p) in the representation. Thus when the antecedentof Transitive Limit holds, v(q|p) > v(r|p) must hold.Now take a continuous EU-PE representation and suppose v(q|p) > v(r|p). Then,joint continuity implies that v((1 ? ?)s + ?q?|p?) > v((1 ? ?)s + ?r?|p?) for anys ? ?, ? > 0, and ? close to zero. It follows that v((1? ?)p? + ?q?|(1? ?)p? + ?r?) >v((1 ? ?)p? + ?r?|(1 ? ?)p? + ?r?) for all ?, ? sufficiently small. Thus for sufficientlysmall ?, ?, (1? ?)p? + ?q? = c({(1? ?)p? + ?q?, (1? ?)p? + ?r?}). Thus the antecedentof Transitive Limit holds when v(q|p) > v(r|p).Since v(q|p) > v(r|p) and v(r|p) > v(s|p) imply v(q|p) > v(s|p), the analysis aboveimplies that qRpr and rRps implies qRps, so Transitive Limit must hold.Necessity of Induced Reference Lottery Bias. In the representation, v(p|p) ?v(q|p) and v(q|q) ? max [v(p|p), v(p|q)] imply that ?? ? (0, 1), v((1? ?)p + ?q|(1??)p + ?q) ? max [v(p|p), v(p|(1? ?)p + ?q)].Suppose p ? c(D). Then, v(p|p) ? v(q|p) ?q ? D, and v(p|p) ? v(q|q) ?q ?PE(D). It follows that v(p|p) ? v(q|p) and v(q|q) ? max [v(p|p), v(p|q)]. Sincev(?|p) satisfies expected utility, p ? PE((1 ? ?)p + ?D) ?? ? (0, 1). Since v((1 ?82Appendix A. Proofs for Chapter 2?)p + ?q|(1 ? ?)p + ?q) ? max [v(p|p), v(p|(1? ?)p + ?q)] ?q ? D, it follows thatv(p|p) ? v((1??)p+?q|(1??)p+?q) ?q : (1??)p+?q ? PE((1??)p+?D). Thusp ? PPE((1 ? ?)p + ?D) = c((1 ? ?)p + ?D) ?? ? (0, 1). Thus Induced ReferenceLottery Bias holds.?Proof of Proposition 2.3.Suppose that v(?|p) and v(?|q) are not ordinally equivalent. Then ?r?, s? ? ? suchthat v(r?|p) > v(s?|p) but v(r?|q) ? v(s?|q). By local strictness, ?r, s ? ? that areclose to r?, s? such that v(r|p) > v(s|p) but v(r|q) < v(s|q). By EU of v(?|p) andcontinuity of v, this implies that ???, ?? > 0 such that ?? ? (0, ??), ?r? ? N ?r , ?s? ? N ?s ,v((1 ? ?)p + ?r?|(1 ? ?)p + ?s?) > v((1 ? ?)p + ?s?|(1 ? ?)p + ?s?) but v((1 ? ?)q +?s?|(1??)q+?r?) > v((1??)q+?r?|(1??)q+?r?). By the representation, this impliesthat for such ?, r?, s?, (a) (1? ?)p + ?r? = c({(1? ?)p + ?r?, (1? ?)p + ?s?}) and (b)(1 ? ?)q + ?s? = c({(1 ? ?)q + ?r?, (1 ? ?)q + ?s?}). Thus if v(?|p) and v(?|q) are notordinally equivalent, c strictly exhibits expectations-dependence.Now suppose that c exhibits expectations-dependence at D,?, p, q, r. That is,??? > 0 such that ?r? ? N ?r , ?D? ? r? such that dH(D?, D) < ?, (1 ? ?)p + ?r? ?c((1 ? ?)p + ?D?) but (1 ? ?)q + ?r? /? c((1 ? ?)q + ?D?). Since (1 ? ?)q + ?r? /?c((1 ? ?)q + ?D?), it follows that for each D?, ?s?? ? D?, v(s??|(1 ? ?)p + ?s??) ?v(r?|(1??)p+?s??). Local strictness then implies that for each such s??, r? pair, thereis an arbitrarily close pair s??, r?? such that v(s??|(1??)p+?s??) > v(r??|(1??)p+?s??).By the representation, (1 ? ?)p + ?r? ? c((1 ? ?)p + ?D?) implies that for each r?,?s? ? D?, v(r?|(1 ? ?)p + ?r?) ? v(s?|(1 ? ?)p + ?r?); thus v(r??|(1 ? ?)p + ?r??) ?v(s??|(1??)p+?r??). Thus v exhibits strict expectations-dependence. This proves thefirst part of the proposition.Now suppose c violates IIA. Then there are D,D? such that D? ? D and c(D) ?D? ?= ? but c(D?) ?= c(D) ?D?. This implies that either (a) or (b) holds:(a) ?p ? c(D?) such that p /? c(D). Then by the representation, this impliesthat v(p|p) = v(q|q) for q ? c(D?), so for some r ? D, v(r|p) > v(p|p) ? v(q|p) butv(q|q) ? v(r|q)83Appendix A. Proofs for Chapter 2(b) ?p ? c(D) ? D? with p /? c(D?). Since PE(D) ? D? ? PE(D?), this impliesthat there is a q ? c(D?) with v(q|q) > v(p|p). Thus q /? c(D) =? q /? PE(D),which implies that ?r ? D\D? such that v(r|q) > v(q|q) ? v(p|q) but v(p|p) ? v(r|p).In either case (a) or (b), by the first part of the proposition, c exhibits strictexpectations-dependence.?Proof of Proposition 2.5First prove that K?szegi-Rabin preferences with linear loss aversion satisfy the limited-cycle inequalities.Start with a finite set X with |X| = n + 1 and assume (for now) that there is asingle hedonic dimension. Without loss of generality, assume m(x1) > m(x2) > ... >m(xn+1)Define the matrix V according to:[V ]ij = m(xi) + ?[m(xi)?m(xj)] + ?[?? 1]min[0, m(xi)?m(xj)] (A.2)Observe that v(p|r) = pTV r. Let ?, ? ? ?n+1 denote vectors with?n+1i=1 ?i =?n+1i=1 ?i = 0. By matrix multiplication,?TV ? = ?[?? 1]?[(m(x1)?m(x2))?1?1 + (m(x2)?m(x3))(?1 + ?2)(?1 + ?2)+ (A.3)... + (m(xn)?m(xn+1))(n?i=1?i)(n?i=1?i)]Take a cycle pi+1 = pi + ?i with v(pi+1|pi) > v(pi|pi) for i = 0, ...,m. Then:v(pm|pm)? v(p0|pm) = (p +?ml=1 ?l)TV (p +?ml=1 ?l)? pTV (p +?ml=1 ?l)= (?ml=1 ?l)TV (?ml=1 ?l) + (?ml=1 ?l)TV pRearranging the second term,= (?ml=1 ?l)TV (?ml=1 ?l)+(?m?1l=1 ?l)TV p+(?m)TV (p+?m?1l=1 ?l)?(?m)TV (?m?1l=1 ?l)84Appendix A. Proofs for Chapter 2= (?ml=1 ?l)TV (?ml=1 ?l)+(?m?2l=1 ?l)TV p+(?m?1)TV (p+?m?2l=1 ?l)?(?m?1)TV (?m?2l=1 ?l)+(?m)TV (p +?m?1l=1 ?l)? (?m)TV (?m?1l=1 ?l)= ... = (?ml=1 ?l)TV (?ml=1 ?l) +?i(?i)TV (p +?i?1l=1 ?l)??mi=2 ?iV (?i?1l=1 ?l)By the definition of the cycle, (?i)TV (p +?i?1l=1 ?l) > 0 for each i, thus:> (?ml=1 ?l)TV (?ml=1 ?l)??mi=2 ?iV (?i?1l=1 ?l)By symmetry with respect to ? and ? in (A.3), it can be shown that?mi=2?i?1l=1(?i)TV ?l =?m?1j=1?ml=j+1(?j)TV ?l. Returning to the previous expression, more algebra estab-lishes:=?ml=1(?l)TV ?l +?mi=2?i?1l=1(?i)TV ?l= 12?ml=1(?l)TV ?l + 12(?ml=1 ?l)TV (?ml=1 ?l)> 0This completes the proof for the case with the case of one hedonic dimension.To extend the argument to K > 1, break up a lottery p into marginals pkin eachdimenion k, and define the matrix Vkas the utility matrix corresponding to V indimension k. we can write vKR(p|r) =?k pkTVkrk. Notice that all of the previously-proven properties of V apply to Vk; following through the previous steps yields thedesired result.Second prove that K?szegi-Rabin preferences with linear loss aversion dislike mix-tures.Suppose v(p|p) ? v(q|p) and v(q|q) ? max [v(p|p), v(p|q)].Then,v((1? ?)p + ?q|(1? ?)p + ?q)= (1? ?)2v(p|p) + ?(1? ?)v(p|q) + ?(1? ?)v(q|p) + ?2v(q|q) (A.4)by bilinearity of v under (2.3) and linear loss aversion.If v(p|p) ? v(p|q), then two substitutions to (A.4) yield? (1? ?)2v(p|p) + ?(1? ?)v(p|q) + ?(1? ?)v(p|p) + ?2v(p|q)= v(p|(1? ?)p + ?q) by bilinearity of v= max [v(p|(1? ?)p + ?q), v(p|p)]If instead v(p|q) ? v(p|p), then two different substitutions to (A.4) yield? (1? ?)2v(p|p) + ?(1? ?)v(p|p) + ?(1? ?)v(p|p) + ?2v(p|p)85Appendix A. Proofs for Chapter 2= v(p|p)= max [v(p|(1? ?)p + ?q), v(p|p)]This proves that v dislikes mixtures.?Proof of Proposition 2.6Gul and Pesendorfer (2008) prove that on a finite set X there is an assignment ofhedonic dimensions such that any reference-dependent utility function v?(x|y) can bewritten as a K?szegi-Rabin preference as in (2.3). Extend v?(x|y) to lotteries by settingv(p|q) =?i?j piqj v?(x|y). The resulting representation over ? is thus consistentwith (2.3).K?szegi (2010, Example 3 and footnote 6) provides an example of v : ???? ?in which the only personal equilibrium involves randomization among elements of achoice set. Mapping the v from K?szegi?s example to a K?szegi-Rabin preference asdescribed provides an example of a K?szegi-Rabin preference that does not satisfythe limited-cycle inequalities.?Proof of Proposition 2.7Take a continuous PPE representation corresponding to ?L, {?p}p??. Take p ? D.Reference Lottery Bias implies that if p ?L q ?q ? D then p ?p q ?q ? D; thus,p ? m(D,?L) =? p ? PE(D), which jointly imply p ? PPE(D) = c(D). Since ?Lis continuous and D is finite, it has a maximizer in D, thus there is a p ? m(D,?L);by the previous argument, for any other q ? c(D) it follows from the representationthat q ?L p thus q ? m(D,?L) as well. It follows that if ?L, {?p}p?? satisfiesReference Lottery Bias, that c(D) = m(D,?L).?86Appendix A. Proofs for Chapter 2Proof of Proposition 2.8.(i) ?? (iii)Let suppose c is induced by the continuous binary relation P .Necessity of Expansion. p ? c(D) ?? ?q ? D such that qPp.Thus, p ? c(D) and p ? c(D?)?? both ?q ? D such that qPp and ?r ? D? such that rPp.?? ?q ? D ?D? such that qPp?? p ? m(D ?D?, P )?? p ? c(D ?D?)Necessity of Sen?s ?. p ? c(D) = m(D,P ) ?? ?q ? D such that qPp=? if D? ? D, then ?q ? D? such that qPp?? p ? m(D?, P ) = c(D?)Necessity of UHC. By contradiction.Suppose p? ? c(D?) = m(D?, P ) for a sequence D? ? D such that dH(D?, D) < ?.If p /? c(D), then ?q ? D such that qPp.Then, since q has open better than and worse than sets, ?? such that ?p? ?N ?p, ?q? ? N ?q , q?Pp?.Since dH(D?, D) < ?, it follows that ?D? in the sequence, ?q? ? D? such thatdE(q?, q) < ?. Thus, ??? > 0 such that ?? < ??, q?Pp?. This contradicts that p? ?m(D?, P ) ?D?. ?Sufficiency. Construct P? by:pP? q if ?Dpq such that p ? c(Dpq)Define P as the asymmetric part of P? .(I) show c(D) ? m(D,P )If p ? c(D), then p ? m(D,P ) by the definition of P .(II) show m(D,P ) ? c(D)87Appendix A. Proofs for Chapter 2Suppose p ? m(D,P ). Then, ?r ? D, ?Dpr : p ? c(Dpr).By Expansion, p ? c( ?r?DDpr).Since D ? ?r?DDpr, by Sen?s ?, p ? c(D) as well.(III) show P is continuous.If p?P? q? for a sequence p?, q? ? p, q then by steps (I) and (II), p? ? c({p?, q?}). ByUHC, this implies p ? c({p, q}) thus pP? q. Thus, P? has closed better and worse thansets. Thus P has strictly open better and worse than sets.?Proof of Theorem 2.3.Necessity. Necessity of Expansion, Sen?s ?, and UHC follows from Proposition 2.8.Necessity of IIA Independence 2 and Transitive Limit are similar to Theorem 2.1.To prove the necessity of Induced Reference Lottery Bias,p ? c(D) = PE(D)?? v(p|p) ? v(q|p) ?q ? D?? v(p|p) ? v((1? ?)p + ?q|p) ?q ? D since v(?|p) satisfies EU?? p ? PE((1? ?)p + ?D) = c((1? ?)p + ?D)Thus the representation implies Induced Reference Lottery Bias.Sufficiency.Lemma A.20. IIA Independence 2 implies Limit Consistency.Proof. Suppose qRpp. Then ??? > 0 such that ?? ? (0, ??), {(1 ? ?) + ?q} = c((1 ??)p+?{p, q}). By IIA Independence 2, it follows that ?? ? (0, 1], ?Dp, (1??)p+?q thatp /? c(Dp, (1??)p+?q). Thus Limit Consistency holds.Take v from Lemma A.7 (from the proof of Theorem 2.1).Define PE(D) := {p ? D : v(p|p) ? v(q|p) ?q ? D}.By Lemma A.13, the axioms for Theorem 2.3 imply Limit Consistency. Sincev(?|p) represents R?p, Limit Consistency implies that c(D) ? PE(D).Suppose p /? c(D) - I will show that p /? PE(D).88Appendix A. Proofs for Chapter 2If ?q ? D, ?Dpq such that p ? c(Dpq), then by Expansion, p ? c( ?q?DDpq); by Sen?s?, it follows that p ? c(D), a contradiction.Thus ?q ? D such that p /? c(Dpq) for any Dpq ? {p, q}. It follows by IIAIndependence 2 that ?? > 0 such that ?? ? (0, 1), Dp, (1??)p+?q, and ?(p?, q?) ? N ?p?N?q ,p /? c(Dp?, (1??)p?+?q?). This implies qRpp. Thus p /? PE(D). It follows that D\c(D) ?D\PE(D), thus PE(D) ? D.This establishes that PE(D) = c(D).?Remark. The proof of Theorem 2.3 makes no use of Induced Reference Lottery Bias.It follows that Induced Reference Lottery Bias is not independent of the remainingaxioms.Proof of Theorem 2.4.Ok (2012, Chapters 5 and 9) proves that IIA and UHC hold if and only if c is inducedby a continuous preference relation, if and only if c has a utility representation (since? is a separable metric space).27For any continuous u : ? ? ?, we can take any v that satisfies v(p|p) = u(p);conversely, for any v we can define u by u(p) := v(p|p). Under this mappingCPE(D) = maxp?Dv(p|p) = maxp?Du(p).?Proof of Proposition 2.9.(i) ?? (ii)Assuming IRLB:p ? c({p, q})?? p ? q=? p ? c((1? ?)p + ?{p, q}) by IRLB?? p ? (1? ?)p + ?qwhich proves that IRLB implies quasiconvexity of ?27Arrow (1959) shows that IIA holds if and only if there exists a complete and transitive binaryrelation R such that c is induced by R.89Appendix A. Proofs for Chapter 2Now assume quasiconvexity of ?:p ? c(D)?? p ? q ?q ? D=? p ? (1? ?)p + ?q ?q ? D by quasiconvexity?? p ? c((1? ?)p + ?D).(ii) ?? (iii)comparing the CPE and preference maximization representations, we see that:p ? q ?? v(p|p) ? v(q|q).Thus the statement ?p ? q =? p ? (1 ? ?)p + ?q? holds if and only if thestatement ?v(p|p) ? v(q|q) =? v(p|p) ? v((1? ?)p + ?q|(1? ?)p + ?q)? holds.?90Appendix BProofs for Chapter 3Proposition B.1. The following are sufficient conditions for c to satisfy dual dif-ferentiability at w and weak dual differentiability at w?: (i) c is RDU with u twicedifferentiable at w, (ii) c is semi-weighted utility with twice differentiable upper andlower weighting functions and u, (iii) c is Frechet differentiable.Proof. Case (i): c(w + p?t) = u?1?n?i=1[g(i?j=1pj)? g(i?1?j=1pj)]u(w + tyi)?; define c?(p?) =n?i=1[g(i?j=1pj)? g(i?1?j=1pj)]yi; if u is twice differentiable then u(w + tyi) = tyiu?(w) + o(t)follows by Taylor?s theorem. Similarly, c(w? + ty) =?1u?(c(w?))?u?(w)dg(Fw?(w))?ty +c(w?) + o(t).Case (ii): Suppose V has a semi-weighted utility functional form with upper andlower weighting functions ?, ?? and which need not coincide. For small t,V (w + p?t) =?i:yi?0pi?(w+tyi)u(w+tyi)+?i:yi>0pi??(w+tyi)u(w+tyi)?i:yi?0pi?(w+tyi)+?i:yi>0pi??(w+tyi)If ??,?, u are all twice differentiable around w, then so is V (w + p?t) as a functionof t. This implies c(w + p?t) ? w = 1u?(w)dV (w+p?t)dt |t=0+ + o(t). Taking the derivativeand reorganizing, dV (w+p?t)dt |t=0+ =?i:yi?0pi?(w)yi+?i:yi>0pi??(w)yi?i:yi?0pi?(w)+?i:yi>0pi??(w)u?(w) and define c?(p?) =?i:yi?0pi?(w)yi+?i:yi>0pi??(w)yi?i:yi?0pi?(w)+?i:yi>0pi??(w).Similarly,V (w? + ty)=???sign(w?c(w?))(w + ty)dFw?(w)??1 ???sign(w?c(w?))(w + ty)u(w + ty)dFw?(w)?.Define c?ysign(y)? limt?0+ 1ty [V (w? + ty)? V (w?)]. Algebra yields:cysign(y)=???sign(w?c(w?))(w)dFw?(w)??1 ???sign(w?c(w?))(w)u?(w)dFw?(w)?Note that if w? has positive mass at its certainty equivalent, then w? + ty may ormay not have positive mass on its certainty equivalent. However, we only need to take91Appendix B. Proofs for Chapter 3a one-sided derivative and can choose whether to assign the upper or lower weightfunction at w = c(w?) in the limit as appropriate, and it follows that c(w? + ty) =c(w?) + tycysign(y)(w?) + o(t). When w? has mass on its c(w?), cysign(y)may indeeddepend on sign(y) but otherwise will not.(iii) To illustrate the link between dual differentiability and variations on Fr?chet-differentiability considered in Wang (1993), suppose that V is differentiable with re-spect to the L? norm for some ? ? 1. Notice that ? [w, 1]?[w+p?t] ??= (?i p?i |tyi|)1? =t1? (?i p?i |yi|)1? = o(t1? ) while similarly ? w? ? [w? + ty] ??= t1? . Applying Wang?s localutility approximation obtained by taking the derivative using the L? norm yields ao(t1? ) approximation of preference, which proves the desired result. Safra and Segal(2002) additionally prove that L1 differentiability implies expected value maximiza-tion.Remark B.1. The non-expected utility literature frequently makes use of a notion ofGateaux-differentiability relative to mixtures of lotteries in probability space whichallow for local expected utility approximations of preferences; for example, see Chewand Safra (1987). The notion of dual differentiability used in this paper is a specialcase of a notion of Gateaux-differentiability relative to (comonotonic) mixtures oflotteries in outcome space in the sense of Yaari (1987).Proof of Theorem 3.2.Suppose c is weakly dually differentiable at w?. Then c(w?+ty) = c(w?)+tyc?ysign(y)(w?)+o(t).Then:U(p?t ? w?)= c([c(w? + ty1), p1; ...; c(w? + tyn), pn])= c([c(w?) + c?ysign(y1)(w?)ty1 + o(t), p1; ...; c(w?) + c?ysign(yn)(w?)tyn + o(t), pn])= c(c(w?) + (c?ysign(y)(w?)ty1 + o(t), p1; ...; c?ysign(y)(w?)tyn + o(t), pn))Since c is dually differentiable at c(w?), there is a c? such that c(c(w?) + p?t + tx)) =c(w?) + tc?(p?) + tx + o(t).Define c?(p?) = c?((c?ysign(y1)(w?)y1, p1; ...; c?ysign(yn)(w?)yn, pn)). Since c? is a linear inmoney, so is c?. By linearity in money, we can take the sup and inf of the o(t)92Appendix B. Proofs for Chapter 3terms in c?(ty1 + o(t), p1; ...; tyn + o(t), pn), and bound from above and below byc?(ty1, p1; ...; tyn, pn)? o(t). Thus U(p?t? w?) = c(w?) + c?(p?t) + o(t) = c(w?) + tc?(p?? a) +at + o(t) for any a ? ?.First-order risk-aversion in c directly implies first-order risk aversion in c?. Takej so that yj < 0 for j < i and yj ? 0 for j ? i. Form a new gamble from p?t, callit p?t, that has the same outcomes and probabilities of each as p?t except that all gainoutcomes are multiplied byc?y+ (w?)c?y? (w?). Notice that c?(p?t) = c?(p?tc?y(w?)). Sincec?y+ (w?)c?y? (w?)? 1,c?(p?tc?y(w?)) ? c?(p?tc?y(w?)). Thus c?(p?t) ? tx ? c?(p?t) ? tx = t [c?(p?)? x]. Thus first-orderrisk aversion in c? implies first-order risk aversion in c?.?Proof of Theorem 3.3.Completing the proof of Theorem 3.3 requires showing that if DM is risk averse andV is RDU/DA, then V (w? + x) is concave in x and U?(p?t ? w?) ? (u ? U)(p?t ? w?) isconcave in t.Under RDU, V (w? + x) =?u(w + x)dg(Fw?(w)) where Fw? is the CDF associatedwith w?. Since u is concave, V (w? + x) is concave in x.Now under RDU, U?(p?t?w?) =?V (w?+ty)dg(Fp?(y)). Since V (w?+x) is concave in x,U?(p?t?w?) is concave in t. This implies that 1t (U?(p?t?w?)?V (w?)) ? U?(p??w?)?V (w?) < 0for t > 1, so DM will also turn down p?t.Under DA, the same argument applies, except that it is messier to prove thatV (w? + x) is concave in x and U(p?t ? w?) is concave in t - this is proven below.Proof that V (w? + x) is concave in x under DA A DA DM is globally riskaverse in the sense of weakly not preferring mean-preserving spreads if and only if uis concave and ? ? 0.I will show that (1 + ?)[V (w? + x)? V (w?)] ? (1 + ?)[V (w?)? V (w? ? x)] to proveconcavity.We can write out the left- and right- hand sides of the above equation as:93Appendix B. Proofs for Chapter 3(1 + ?)[V (w? + x)? V (w?)]=?{u(w + x)? u(w) + ? min[u(w + x), V (w? + x)]? ? min[u(w), V (w?)]}dFw?(w)(B.1)(1 + ?)[V (w?)? V (w? ? x)]=?{u(w)? u(w ? x) + ? min[u(w), V (w?)]? ? min[u(w ? x), V (w? ? x)]}dFw?(w)(B.2)To compare (B.1) and (B.2), compare the term inside the integral for each w.First note u(w+ x)? u(w) ? u(w)? u(w? x). Second, compare the remaining partsof the integrals by working with four different regions/cases that depend on w and V .Case A min[V (w?+x), u(w+x)]?min[V (w?), u(w)] = u(w+x)?u(w) and min[V (w?), u(w)]?min[V (w?? x), u(w? x)] = u(w)? u(w? x). By concavity of u for these w, the (B.1)term is smaller than the (B.2) term.Case B min[V (w? + x), u(w + x)] ? min[V (w?), u(w)] = V (w? + x) ? V (w?) andmin[V (w?), u(w)] ? min[V (w? ? x), u(w ? x)] = V (w?) ? V (w? ? x). We can cancelthese terms from (1 + ?)[V (w? + x)? V (w?)] and (1 + ?)[V (w?)? V (w? ? x)].Case C Suppose neither of the above two cases applies and u(w) ? V (w?). Then,applying concavity of u,min[u(w + x), V (w? + x)]?min[u(w), V (w?)]} = min[u(w + x), V (w? + x)]? u(w)? u(w + x)? u(w)? u(w)? u(w ? x)? u(w)?min[u(w? x), V (w?? x)] = min[u(w), V (w?)]?min[u(w? x), V (w?? x)]}94Appendix B. Proofs for Chapter 3so in this case, this term of the (B.1) is smaller than the corresponding term in (B.2).Case D Suppose neither of the above three cases applies, so u(w) > V (w?). Then,min[u(w + x), V (w? + x)]?min[u(w), V (w?)]} = min[u(w + x), V (w? + x)]? V (w?)? V (w? + x)? V (w?)andmin[u(w), V (w?)]?min[u(w ? x), V (w? ? x)]} = V (w?)?min[u(w ? x), V (w? ? x)]}? V (w?)? V (w? ? x)Plugging in these terms and cancelling out as case B establishes the desired in-equality.Proof that U?(p?t? w?) is concave in t under DA Define IV (y) = 1 if V (w?+ ty)?V (w?) < U?(p?t? w?)? V (w?)] and zero otherwise, and IU(y) = 1 if V (w?+ ty)? V (w?) ?U?(p?t ? w?)? V (w?)] and zero otherwise. For t > 1,1+?t {U?(p?t ? w?)? V (w?)}= 1t?{U?(w? + ty)? V (w?) + ? min[V (w? + ty)? V (w?), U?(p?t ? w?)? V (w?)]}dFp?(y)? 1t?{V (w? + ty) ? V (w?) + ?IV (y)[V (w? + ty) ? V (w?)] + ?IU(y)[U?(p?t ? w?) ?V (w?)]}dFp?(y)Let p? = 1 ??IU(y)dFp?(y). Then, rearranging the above expression yields theinequality:1+?p?t {U?(p?t?w?)?V (w?)} ? 1t?{V (w?+ty)?V (w?)+?IV (y)[V (w?+ty)?V (w?)]}dFp?(y)??{V (w? + y)? V (w?) + ?IV (y)[V (w? + y)? V (w?)]}dFp?(y)= 1+?p?t {U?(p?? w?)? V (w?)}?95Appendix CExamples that rationalize our resultsC.0.4 Rationalizing our data using ROCLIn the analysis below, we show by way of a simple example that if a subject views alist as a compound lottery, and satisfies reduction of compound lotteries, then evenif she would exhibit a certainty effect when given binary choices, she will behave asif she satisfies expected utility when her preferences over lotteries are elicited using aprobability list under the (mistaken) assumption of compound independence.28 Thisexample shows the potential of the Karni and Safra (1987) approach to rationalizeour main findings.For simplicity, approximate the ?discrete? price list with a smooth one. Supposethat we have smooth lists for Q1 and Q2 that ask for an indifference points q ? [12 , 1]and r ? [14 ,12 ], and a number is drawn from U [12 , 1] for Q1 and from U [14 ,12 ] for Q2 todetermine which ?line? is paid out.Suppose the subject views Q1 and Q2 as compound lotteries in which the externalrandomizing device picks a line for payment at the first stage, and she gets her chosenlottery from that line at the second stage, and suppose further that the subject?spreferences have a rank-dependent utility representation (Quiggin, 1982; Yaari, 1987).If the subject applies Compound Independence to evaluate this compound lottery, shewould choose switch points q in Q1 and r in Q2 to satisfy:u(3) = f(q)u(4) (Q1)f(.5)u(3) = f(r)u(4) (Q2)28The results below do not require the full force of reduction: if subjects? evaluation of a compoundlottery was a weighted average of its value when reduced and when evaluated recursively, similarresults would apply.96Appendix C. Examples that rationalize our resultsSuppose instead, following Karni and Safra (1987), the subject satisfies the Re-duction of Compound Lotteries axiom. Then, she views her choice of q in Q1 as givingher the reduced lottery (0, (1? q)2; 3, 2q? 1; 4, 1? q2). Similarly, she evaluates herchoice of r in Q2 based on the reduced lottery (0, 1? 2r+2r2; 3, 2r? 12 ; 4,12 ? 2r2).In the smooth approximation to the list, a subject would choose her switch pointsto solve:maxq?[ 12 ,1]f(1? q2) [u(4)? u(3)] + f(2q ? q2)u(3) (Q1)maxr?[ 14 ,12 ]f(12 ? 2r2) [u(4)? u(3)] + f(2r ? 2r2)u(3) (Q2)First, we can see that under EU (f(p) = p) that subjects will switch at the sameline (q = 2r) in Q1 and Q2. If f is non-linear, then the above analysis suggests thatsubjects may switch at different points, depending on the shape of their probabilityweighting function. The neo-additive probability weighting function (Chateauneuf,Eichberger, and Grant, 2007; Webb and Zank, 2011) is closest to the spirit of thecertainty and possibility effects motivating probability weighting, and accomodatesthem by a piece-wise linear f .For a ? (0, 1), and a + b < 1 this probability weighting function demonstratesa certainty effect, and when b > 0 generates a possibility effect. Thus, for normalparameter values for a and b, f will generate a standard common ratio effect. Thisform for f can alternatively be motivated as a piece-wise linear approximation tomore popular, smooth, probability weighting functions.With a neoadditive f , when responding to the list elicitation, the FOCs for Q1and Q2 reduce to:q [u(4)? u(3)] = [1? q] u(3) (Q1)2r [u(4)? u(3)] = [1? 2r] u(3) (Q2)so q = 2r, and subjects behave as if they were expected utility maximizers. Thisis because some risks in both lists are going to be too attractive for subjects to passup under reasonable parameter values. Thus, subjects will always face some risk inboth lists. A certainty effect would bias subjects towards the certain option if any of97Appendix C. Examples that rationalize our resultsthe lines of Q1 were presented independently as a single binary choice question. But,when facing the list subjects will always choose some risk, so the certainty effect willnot bias subjects towards A in Q1.When f is a power function, then when responding to the list elicitation, theFOCs for Q1 and Q2 reduce to:q(1? q2)??1 [u(4)? u(3)] = [1? q] (2q ? q2)??1u(3) (Q1)2r(12 ? 2r2)??1 [u(4)? u(3)] = [1? 2r] (2r ? 2r2)??1u(3) (Q2)which implies that q = 2r, that is, the subject?s behaviour in the list elicitationexperiment is indistinguishable from expected utility maximization. Numerical sim-ulations or algebra can be used to verify that for reasonable parameters with ? > 1,a subject who switches at q would rank ($3, 1) ? ($4, q) in a binary choice task.Now suppose that instead of satisfying RDU, U falls in the class of NCI-satisfyingpreferences studied by Cerreia-Vioglio, Dillenberger, and Ortoleva (2013). Then,their NCI axiom directly requires that if ($3, 1) ? ($4, .8) that the subject willrank($3, 1) ? ($4, .8) in line 11 of the list. This follows from the fact that a subjectwho decides between switching on either line 11 of line 12 faces the compound lotteryin (4.1). Applying reduction as in (4.2), we this choice can be express as a choicebetween 2526?$4, 9.125 ; $3,1525?+ 126 [$3, 1] and2526?$4, 9.125 ; $3,1525?+ 126 [$4, .8], and thesubject must prefer the latter lottery. Thus NCI predicts more risk aversion underlist elicitation. However, NCI does not have any direct implications for Q2.One limitation of the above analysis is that even when considering reduction ofcompound lotteries within a list, we assume that (consistent with our results) com-pound independence holds at the previous stage of the compound lottery at whichQ1 or Q2 is selected to determine payment. One possible explanation is that subjectsviolate both compound independence and reduction, but the violation of compoundindependence is much less severe, and the violation of reduction much more severe,when different lotteries that form branches of a compound lottery are never displayedon the same visual screen in a way that would facilitate reduction.29 Another limi-tation of the above explanation is that rank-dependent utility with the neo-additive29If instead subjects reduced the compound lottery formed by the entire experiment, then if theycorrectly anticipated the second experimental question subject responses in the two-list treatments(L and S) would be consistent with EU regardless of their weighting function.98Appendix C. Examples that rationalize our resultsweighting function can only explain the observed reversal between choices in binarychoice and list elicitation treatments in Q1, not the comparable (but statisticallyweaker) finding in Q2.C.0.5 Non-standard application of Compound IndependenceIn the analysis below, we show by way of a simple example using RDU with thepower weighting function that if a subject views a list according to a non-standardcompound lottery as in 4.3, and and evaluates this compound lottery recursively, thenthe subject?s behaviour under list elicitation. This example shows the potential ofthe Segal (1988) approach to rationalize our main findings.Applying RDU with the power weighting function to (4.3) yields the conditions:maxi?{1, 2,..., 26}??i26???i?j=1?? ji???? j?1i???(1.02? .02j)??u(4) +?1??i26???u(3)?(Q1)maxi?{1, 2,..., 26}??i26???i?j=1?? ji???? j?1i???(.51? .01j)??u(4) +?1??i26???(.5)? u(3)?(Q2)With some algebra, we can see that the maximand in (Q2) is just .5? times themaximand in (Q1). It follows that the subject would switch on the same line in bothquestions. That is, her behaviour would be indistinguishable from expected utilitymaximization in spite of her non-expected utility preferences.Using a smooth approximation to the compound lottery formed by list along, it ispossible to solve out continuous analogues of (Q1) and (Q2). With the power weight-ing function, the subject would switch at a q? that satisfies q? = 1??1? u(3)u(4)? 1?< u(3)u(4)for ? > 1; in contrast, the same subject would require q? =?u(3)u(4)? 1?> u(3)u(4) to be in-different between ($3, 1) and ($4, q). That is, the subject would display more riskaversion in the binary choice task than under list elicitation.99


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