Approximate Expected Utility Rationalization
AApproximate Expected Utility Rationalization
Federico Echenique Taisuke Imai Kota Saito ∗ This draft: February 15, 2021First draft: June 22, 2018
Abstract
We propose a new measure of deviations from expected utility theory. For any positivenumber 𝑒 , we give a characterization of the datasets with a rationalization that is within 𝑒 (inbeliefs, utility, or perceived prices) of expected utility theory. The number 𝑒 can then be usedas a measure of how far the data is to expected utility theory. We apply our methodologyto data from three large-scale experiments. Many subjects in those experiments are consis-tent with utility maximization, but not with expected utility maximization. Our measure ofdistance to expected utility is correlated with subjects’ demographic characteristics. ∗ Echenique: Division of the Humanities and Social Sciences, California Institute of Technology, [email protected] . Imai: Department of Economics, LMU Munich, [email protected] . Saito:Division of the Humanities and Social Sciences, California Institute of Technology, [email protected] . We arevery grateful to Nicola Persico, who posed questions to us that led to some of the results in this paper, and toJose Apesteguía, Miguel Ballester, Georoy de Clippel, Dan Friedman, Yves Le Yaouanq, Pietro Ortoleva, MatthewPolisson, John Quah and Kareen Rozen for helpful comments. We are also grateful for the feedback provided bynumerous audience at FUR 2018, ESA World Meetings 2018, 4th Hitotsubashi Summer Institute: MicroeconomicTheory, 2018 European Summer Meeting of the Econometric Society, CESifo Area Conference on BehaviouralEconomics 2018, Measuring Individual Well-Being Workshop, and 2019 European Summer Symposium in EconomicTheory. This research is supported by Grant SES-1558757 from the National Science Foundation. The authors alsoacknowledge nancial support by the NSF through the grants CNS-1518941 (Echenique) and SES-1919263 (Saito),and the Deutsche Forschungsgemeinschaft through CRC TRR 190 (Imai). a r X i v : . [ ec on . GN ] F e b Introduction
Revealed preference theory has traditionally, through its 80-year history, dealt with the empiricalcontent of general utility maximization. Recent research has, in contrast, turned to the empiri-cal content of specic utility theories. Mostly the focus has been on expected utility (EU): recenttheoretical work seeks to characterize the observable choice behaviors that are consistent with ex-pected utility maximization. At the same time, a number of recent empirical revealed-preferencestudies use data on choices under risk and uncertainty, in which participants make a series ofchoices from budget sets. We seek to bridge the gap between the theoretical understanding ofexpected utility theory, and the machinery needed to analyze experimental data on choices underrisk and uncertainty.Imagine an agent making economic decisions, choosing contingent consumption given mar-ket prices and income. Revealed preference theory studies the consistency of such choices withutility maximization. Consistency, however, is a black or white question. The choices are eitherconsistent with EU or they are not. Our contribution is to provide a way to describe the de-gree to which choices are consistent with EU. We propose a measure of the degree of a dataset’sconsistency with EU.Revealed preference theory has developed measures of consistency with general utility maxi-mization. The most widely used measure is the Critical Cost Eciency Index (CCEI) proposed byAfriat (1972). The basic idea in the CCEI is to ctitiously decrease an agent’s budget so that feweroptions are revealed preferred to a given choice. The CCEI has been widely used to analyze ex-perimental data on choices from budget sets. See, for example, Choi et al. (2007), Ahn et al. (2014),Choi et al. (2014), Carvalho et al. (2016), Carvalho and Silverman (2019), and Halevy et al. (2018).All of these experimental studies involve subjects making decisions under risk or uncertainty,and CCEI was proposed as a measure of consistency with general utility maximization, not EU,the most commonly-used theory to explain choices under risk or uncertainty.Of course, there is nothing wrong with studying general utility maximization in environmentswith risk and uncertainty, but the data is ideally suited to studying theories of choice underrisk and uncertainty, and it should be of great interest to evaluate EU using this data. We shallargue (on both theoretical and empirical grounds) that our method provides a more accurate andintuitive measure of consistency with EU than using CCEI.Our main contribution is to propose a measure of how far a dataset is from being consistentwith EU. The measure is dierent from CCEI: we explain theoretically why our measure, and notCCEI, best captures the distance of a dataset to EU theory. We also argue on empirical groundsthat our measure passes “smell tests” that CCEI fails. For example, CCEI ignores the manifestviolations of EU where subjects make rst-order stochastically dominated choices. And CCEI2oes not correlate well with the property of downward-sloping demand, a property that is impliedby EU maximization. We also provide a revealed preference axiomatization of the measure basedon observed prices and consumption.In the sequel, we rst lay out the implications of EU that cannot be captured by CCEI, andgive an overview of our approach. After a theoretical discussion of our measure of consistency(with objective EU discussed in Section 3 and subjective EU in Section 5), we present an empiricalapplication using data from experiments on choices under risk (Section 4).Our empirical application has two purposes. The rst is to illustrate how our method can beapplied and to argue that our measure of distance to EU is useful and sensible. The second is tooer new insights into existing data. We use data from three large-scale experiments (Choi et al.,2014; Carvalho et al., 2016; Carvalho and Silverman, 2019), each with over 1,000 subjects, thatinvolve choices under risk. Consistency with general utility maximization is well understood inthese studies using CCEI. We test for EU theory using our methodology.There are two main take-away messages from our empirical application. First, the data showsthat there is a gap between consistency with general utility maximization (measured with CCEI)and EU maximization (quantied with our measure). Subjects with CCEI close to one, who arelargely consistent with utility maximization, exhibit diverse degrees of consistency with EU. Ourmeasure detects violations of a basic property of EU that we term downward-sloping demand,and violations of monotonicity with respect to rst-order stochastic dominance. CCEI, on theother hand, is less sensitive to these features in choice data. Second, the correlation betweencloseness to EU and demographic characteristics yields intuitive results. We nd that youngersubjects, those who have high cognitive abilities, and those who are working, are closer to EUbehavior than older, low cognitive ability, or non-working, subjects. For some of the three experi-ments, we also nd that highly educated, high-income, and male subjects, are closer to EU. Theseobservations suggest that our measure complements CCEI as an empirical toolkit and providesadditional insights on datasets that had been analyzed primarily with CCEI.
The CCEI is meant to test deviations from general utility maximization. If an agent’s behavioris not consistent with utility maximization, then it cannot possibly be consistent with EU maxi-mization. Thus it stands to reason that if an agent’s behavior is far from being rationalizable asmeasured by CCEI, then it is also far from being rationalizable with an EU function. The problemis, of course, that an agent’s behavior may be rationalizable with a general utility function butnot with EU. Roughly speaking, it says that prices and quantities must be inversely related, subject to certain qualications. 𝑎 𝑥 𝑏 𝑥 𝑥 A 𝑥 𝑎 𝑥 𝑏 𝑥 𝑥 B MRS = 𝜇 (cid:8)(cid:8)(cid:8) 𝑢 (cid:48) ( 𝑥 𝑘 ) 𝜇 (cid:8)(cid:8)(cid:8) 𝑢 (cid:48) ( 𝑥 𝑘 ) = 𝜇 𝜇 𝑥 𝑎 𝑥 𝑏 𝑥 𝑥 C Figure 1: (A) A violation of WARP. (B) A violation of EU: 𝑥 𝑎 > 𝑥 𝑎 , 𝑥 𝑏 > 𝑥 𝑏 , and 𝑝 𝑏 / 𝑝 𝑏 < 𝑝 𝑎 / 𝑝 𝑎 . (C) Achoice pattern consistent with EU. Broadly speaking, the CCEI proceeds by “amending” inconsistent choices through the deviceof changing income. This works for general utility maximization, but it is the wrong way toamend choices that are inconsistent with EU. Since EU is about getting marginal rates of substi-tution right, prices, not incomes, need to be changed. The problem is illustrated with a simpleexample in Figure 1.Suppose that there are two states of the world, labeled 1 and 2. An agent purchases a state-contingent asset 𝑥 = ( 𝑥 , 𝑥 ) , given Arrow-Debreu prices 𝑝 = ( 𝑝 , 𝑝 ) and her income. Pricesand income dene a budget set. In Figure 1A, we are given two choices for the agent, 𝑥 𝑎 and 𝑥 𝑏 ,for two dierent budgets. The choices in Figure 1A are inconsistent with utility maximization:they violate the weak axiom of revealed preference (WARP). When 𝑥 𝑏 ( 𝑥 𝑎 ) was chosen, 𝑥 𝑎 ( 𝑥 𝑏 ,respectively) was strictly inside of the budget set. This violation of WARP can be resolved byshifting down the budget line associated with choice 𝑥 𝑏 to the dashed green line passing through 𝑥 𝑎 . Alternatively, the violation can be resolved by shifting down the budget line associated withchoice 𝑥 𝑎 to the dashed blue line passing through 𝑥 𝑏 . CCEI is the smallest of the two shifts thatare needed: the smallest proportion of shifting down a budget line to resolve WARP violation.Therefore, the CCEI of this dataset is given by the dashed green line passing through 𝑥 𝑎 . That is,the CCEI is ( 𝑝 𝑏 · 𝑥 𝑎 )/( 𝑝 𝑏 · 𝑥 𝑏 ) .Now consider the example in Figure 1B. There are again two choices, 𝑥 𝑎 and 𝑥 𝑏 , for twodierent budgets. These choices do not violate WARP, and comply with the theory of utilitymaximization with CCEI =
1. The choices in the panel are not , however, compatible with EU.To see why, assume that the dataset were rationalized by an expected utility: 𝜇 𝑢 ( 𝑥 𝑘 ) + 𝜇 𝑢 ( 𝑥 𝑘 ) ,where ( 𝜇 , 𝜇 ) are the probabilities of the two states, and 𝑢 is a (smooth) concave utility functionover money. Note that the slope of a tangent line to the indierence curve at a point 𝑥 𝑘 is equalto the marginal rate of substitution (MRS): 𝜇 𝑢 (cid:48) ( 𝑥 𝑘 )/ 𝜇 𝑢 (cid:48) ( 𝑥 𝑘 ) . Moreover, at the 45-degree line4i.e., when 𝑥 𝑘 = 𝑥 𝑘 ), the slope must be equal to 𝜇 (cid:8)(cid:8)(cid:8)(cid:8) 𝑢 (cid:48) ( 𝑥 𝑘 )/ 𝜇 (cid:8)(cid:8)(cid:8)(cid:8) 𝑢 (cid:48) ( 𝑥 𝑘 ) = 𝜇 / 𝜇 . This is a contradictionbecause in Figure 1B, the two tangent lines (green dashed lines) associated with 𝑥 𝑎 and 𝑥 𝑏 crosseach other. Figure 1C shows an example of choices that are consistent with EU. Note that tangentlines at the 45-degree line are parallel in this case.Importantly, the violation in Figure 1B cannot be resolved by shifting budget lines up or down,or more generally by adjusting agents’ expenditures. The reason is that the empirical content ofexpected utility is captured by the relation between prices and marginal rates of substitution. Theslope, not the level, of the budget line, is what matters. The basic insight comes from the equalityof marginal rates of substitution and relative prices: 𝜇 𝑢 (cid:48) ( 𝑥 𝑘 ) 𝜇 𝑢 (cid:48) ( 𝑥 𝑘 ) = 𝑝 𝑘 𝑝 𝑘 . (1)Since marginal utility is decreasing, equation (1) imposes a negative relation between prices andquantities. The distance to EU is directly related to how far the data is to complying with sucha negative relation between prices and quantities. The formal connection is established in The-orem 2. Empirically, as we shall see, the degree of compliance of a subject’s choices with this“downward sloping demand” property, goes a long way to capturing the degree of compliance ofthe subject’s choices with EU.We propose a measure of how close the data is to being consistent with EU maximization.Our measure is based on the idea that marginal rates of substitution have to conform to EUmaximization: whether data conform to equation (1). If one “perturbs” marginal utility enough,then a dataset is always consistent with expected utility. Our measure is simply a measure ofhow large of a perturbation is needed to rationalize the data. Perturbations of marginal utilitycan be interpreted in three dierent, but equivalent, ways: as measurement error on prices, asrandom shocks to marginal utility in the fashion of random utility theory (McFadden, 1974), oras perturbations to agents’ beliefs. For example, if the data in Figure 1B is “ 𝑒 away” from beingconsistent with expected utility given a positive number 𝑒 , then one can nd beliefs 𝜇 𝑎 and 𝜇 𝑏 ,one for each observation so that EU is maximized for these observation-specic beliefs, and thedegree of perturbation of beliefs is bounded by 𝑒 .Our measure can be applied in settings where probabilities are known and objective, for whichwe develop a theory in Section 3, and an application to experimental data in Section 4. It can alsobe applied to settings where probabilities are not known, and therefore subjective (Section 5).Finally, we propose a statistical methodology for testing the null hypothesis of consistencywith EU (Section 4.3). Our test relies on a set of auxiliary assumptions. The test indicates moderatelevels of rejection of the EU hypothesis. 5 .2 Related Literature Revealed preference theory has developed tests for consistency with general utility maximization.The seminal papers include Samuelson (1938), Afriat (1967), and Varian (1982). See Chambers andEchenique (2016) for an exposition of the basic theory.More recent work has explored the testable implications of EU theory. This work includesGreen and Srivastava (1986), Chambers et al. (2016), Kübler et al. (2014), Echenique and Saito(2015), and Polisson et al. (2020). The rst four papers focus, as we do here, on rationalizabilityfor risk-averse agents. Green and Srivastava (1986) and Chambers et al. (2016) allow for manygoods in each state, which our methodology cannot accommodate. Polisson et al. (2020) present ageneral approach to testing that allows for a test of EU in isolation, not jointly with risk aversion.Our assumptions are the same as in Kübler et al. (2014) and Echenique and Saito (2015).Compared to most of the existing revealed preference literature on EU, our focus is on mea-suring consistency with EU, not on providing a test. Our assumption of monetary payos andrisk aversion is restrictive but consistent with how EU theory is used in economics. Many eco-nomic models assume EU together with risk aversion. Our results speak directly to the empiricalrelevance of such models. A further motivation for focusing on risk aversion is empirical: in thedata we have looked at, corner choices are very rare. This would rule out risk-seeking behav-ior in the context of EU. Thus, arguably, EU and risk-loving behavior would not be a candidateexplanation of the experimental data we examine in this paper.As mentioned, the CCEI was proposed by Afriat (1972). Varian (1990) proposes a modication,and Echenique et al. (2011) and Dean and Martin (2016) propose alternative measures. Dziewulski(2020) provides a foundation for CCEI based on the model in Dziewulski (2016), which seeks torationalize violations of utility-maximizing behavior with a model of just-noticeable dierences.Compared to the literature based on the CCEI, we present an explicit model of the errors thatwould explain the deviation from EU. As a consequence, our measure of consistency with EU isbased on a “story” for why choices are inconsistent with EU. And, as we have explained above,the nature of EU-consistent choices is poorly reected in the CCEI’s budget adjustments.Apesteguia and Ballester (2015) propose a general method to measure the distance betweentheory and data in revealed preference settings. For each possible preference relation, they cal-culate the swaps index , which counts the number of alternatives that must be swapped with thechosen alternative in order for the preference relation to rationalize the data. Then, Apesteguiaand Ballester (2015) consider the preference relation that minimizes the total number of swaps inall the observations, weighted by their relative occurrence in the data. Apesteguia and Ballester(2015) assume that there is a nite number of alternatives, and thus a nite number of preferencerelations over the set of alternatives. Because of the niteness, they can calculate the swaps index6or each preference relation and nd the preference relation that minimizes the swaps index. Thismethod by Apesteguia and Ballester (2015) is not directly applicable to our setup because in oursetup, a set of alternatives is a budget set and contains innitely many elements; moreover, thenumber of expected utility preferences relation is innite. There are many other studies of revealed preference that are based on a notion of distancebetween the theory and the data. For example, Halevy et al. (2018) uses such distances as a guidein estimating parametric functional forms for the utility function.Polisson et al. (2020) develop a general method called the Generalized Restriction of InniteDomain (GRID) for testing consistency with models of choice under risk and uncertainty. UsingGRID, they provide a way to calculate CCEI for departures from EU. Importantly, and in contrastwith our measure, their approach does not rely on risk aversion. They present measures of de-parture from EU and risk-averse EU. We compare empirically our measure to theirs in Section 4.2(the Online Appendix has additional details). Suce it to say here that the measures are similar,but distinct, when applied to the data, and that the dierences cannot be attributed to risk aver-sion. Theoretically, our approach has the advantage of modeling a specic source of deviationsfrom EU, and our results connect the measure to certain observable behavioral patterns. Theseinclude exact behavioral patterns described by the theorems, but also an empirically motivatedobservation that our measure captures compliance with downward-sloping demand.Finally, de Clippel and Rozen (2020) measure consistency with utility maximization by way ofdepartures from rst-order conditions, an approach similar to ours. Their FOC-Departure Index(FDI) can be computed for dierent classes of utility functions. In particular, their FDI measurefor risk-averse expected utility is equivalent to our measure, except for the use of dierent scaling(their measure 𝜀 ∈ [ , ] is the same as a transformation of our measure 𝑒 ≥
0, with 𝜀 = 𝑒 /( + 𝑒 ) ).Their axiomatization is dierent from ours in that their primitives are weak orderings on pairsof price and utility gradient (derivatives of utility function). On the other hand, we provide anaxiomatization based on the observed prices and chosen allocations. The result in their Proposi-tion 8 is perhaps closest in spirit to our exercise, where they show that computing the measurereduces to checking a set of inequalities. See Remarks C.1 and C.2 in Online Appendix C of ourpaper. De Clippel and Rozen’s work is independent and contemporaneous to ours. In Appendix D.1 of Apesteguia and Ballester (2015), they consider the swaps index for expected utility prefer-ences while assuming the niteness of the set of alternatives. In their Appendix D.3, without axiomatization, theyconsider the swaps index for an innite set of alternatives using the Lebesgue measure to “count” the number ofswaps. However, they do not study the case where the number of alternatives is innite and the preference relationsare expected utility. In their paper, ( 𝑝, 𝑔 ) (cid:23) ( 𝑝 , 𝑔 ) means that “the utility gradient 𝑔 is farther apart from the price vector 𝑝 than 𝑔 is from 𝑝 .” Model
Let 𝑆 be a nite set of states . We occasionally use 𝑆 to denote the number | 𝑆 | of states. Let Δ ++ ( 𝑆 ) = { 𝜇 ∈ R 𝑆 ++ | (cid:205) 𝑆𝑠 = 𝜇 𝑠 = } denote the set of strictly positive probability measures on 𝑆 .In our model, the objects of choice are state-contingent monetary payos, or monetary acts . Amonetary act is a vector in R 𝑆 + . Denition 1. A dataset is a nite collection of pairs ( 𝑥, 𝑝 ) ∈ R 𝑆 + × R 𝑆 ++ . The interpretation of a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is that it describes 𝐾 purchases of a state-contingentpayo 𝑥 𝑘 at some given vector of prices 𝑝 𝑘 , and income 𝑝 𝑘 · 𝑥 𝑘 = (cid:205) 𝑠 ∈ 𝑆 𝑝 𝑘𝑠 𝑥 𝑘𝑠 . We sometimes use 𝐾 to denote the set { , . . . , 𝐾 } . For any prices 𝑝 ∈ R 𝑆 ++ and positive number 𝐼 >
0, the set 𝐵 ( 𝑝, 𝐼 ) = { 𝑦 ∈ R 𝑆 + | 𝑝 · 𝑦 ≤ 𝐼 } is the budget set dened by 𝑝 and 𝐼 .Expected utility theory requires a decision maker to solve the problemmax 𝑥 ∈ 𝐵 ( 𝑝,𝐼 ) ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝑢 ( 𝑥 𝑠 ) , (2)when faced with prices 𝑝 ∈ R 𝑆 ++ and income 𝐼 >
0, where 𝜇 ∈ Δ ++ ( 𝑆 ) is a belief and 𝑢 is a concaveutility function over money. We are interested in concave 𝑢 ; an assumption that corresponds torisk aversion.The belief 𝜇 will have two interpretations in our model. First, in Section 3, we shall focus ondecisions taken under risk . The belief 𝜇 will be a known “objective” probability measure 𝜇 ∗ ∈ Δ ++ ( 𝑆 ) . Then, in Section 5, we study choice under uncertainty . Consequently, The belief 𝜇 willbe a subjective beliefs, which is unobservable to us as outside observers.The following denition formalizes the concept of as-if choices (Echenique and Saito, 2015). Denition 2.
A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is Objective Expected Utility rational if there exists a concaveand strictly increasing function 𝑢 : R + → R such that, for all 𝑘 , 𝑦 ∈ 𝐵 ( 𝑝 𝑘 , 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝑢 ( 𝑥 𝑘𝑠 ) , where 𝜇 ∗ ∈ Δ ++ ( 𝑆 ) is an objective probability. A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is Subjective Expected Utilityrational if there exist 𝜇 ∈ Δ ++ ( 𝑆 ) and a concave and strictly increasing function 𝑢 : R + → R suchthat, for all 𝑘 , 𝑦 ∈ 𝐵 ( 𝑝 𝑘 , 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝑢 ( 𝑥 𝑘𝑠 ) . When imposed on a dataset, expected utility maximization (2) may be too demanding. Weare interested in situations where the model in (2) holds approximately . As a result, we shall8elax (2) by “perturbing” some elements of the model. The exercise will be to see if a dataset isconsistent with the model in which some elements have been perturbed. Specically, we shallperturb beliefs, utilities, or prices.First, consider a perturbation of utility 𝑢 . We allow 𝑢 to depend on the choice problem 𝑘 andthe realization of the state 𝑠 . We suppose that the utility of consumption 𝑥 𝑠 in state 𝑠 is givenby 𝜀 𝑘𝑠 𝑢 ( 𝑥 𝑠 ) , with 𝜀 𝑘𝑠 being a (multiplicative) perturbation in utility. To sum up, given price 𝑝 andincome 𝐼 , a decision maker solves the problemmax 𝑥 ∈ 𝐵 ( 𝑝,𝐼 ) ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝜀 𝑘𝑠 𝑢 ( 𝑥 𝑠 ) , when faced with prices 𝑝 ∈ R 𝑆 ++ and income 𝐼 >
0. Here { 𝜀 𝑘𝑠 } 𝑠 ∈ 𝑆,𝑘 ∈ 𝐾 is a set of perturbations, and 𝑢 is, as before, a concave utility function over money.In the second place, consider a perturbation of beliefs. We allow 𝜇 to be dierent for eachchoice problem 𝑘 . That is, given price 𝑝 and income 𝐼 , a decision maker solves the problemmax 𝑥 ∈ 𝐵 ( 𝑝,𝐼 ) ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑘𝑠 𝑢 ( 𝑥 𝑠 ) , (3)when faced with prices 𝑝 ∈ R 𝑆 ++ and income 𝐼 >
0, where { 𝜇 𝑘 } 𝑘 ∈ 𝐾 ⊂ Δ ++ ( 𝑆 ) is a set of beliefs and 𝑢 is a concave utility function over money.Finally, consider a perturbation of prices. Our consumer faces perturbed prices ˜ 𝑝 𝑘𝑠 = 𝜀 𝑘𝑠 𝑝 𝑘𝑠 ,with a perturbation 𝜀 𝑘𝑠 that depends on the choice problem 𝑘 and the state 𝑠 . Given price 𝑝 andincome 𝐼 , a decision maker solves the problemmax 𝑥 ∈ 𝐵 ( ˜ 𝑝,𝐼 ) ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝑢 ( 𝑥 𝑠 ) , when faced with income 𝐼 > 𝑝 𝑘𝑠 = 𝜀 𝑘𝑠 𝑝 𝑘𝑠 for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 .Observe that our three sources of perturbations have dierent interpretations, each can betraced back to a long-standing tradition for how errors are introduced in economic models. Per-turbed prices can be thought of a prices subject to measurement error, measurement error beinga very common source of perturbations in econometrics (Griliches, 1986). Perturbed utility isan instance of random utility models (McFadden, 1974). Finally, perturbations of beliefs can bethought of as a kind of random utility, or as an inability to exactly use probabilities. Note that weperturb one source at a time and do not consider combinations of perturbations. In this section, we discuss choice under risk: there exists a known “objective” belief 𝜇 ∗ ∈ Δ ++ ( 𝑆 ) that determines the realization of states. The experiments we discuss in Section 4 are all on choice9nder risk.As mentioned above, we go through each of the sources of perturbation: beliefs, utility, andprices. We seek to understand how large a perturbation has to be in order to rationalize a dataset.It turns out that, for this purpose, all sources of perturbations are equivalent. Deviations from EU are accommodated by allowing a dierent belief at each observation. So weassume a belief 𝜇 𝑘 for each choice 𝑘 , and allow 𝜇 𝑘 to dier from the objective 𝜇 ∗ . We seek tounderstand how much the belief 𝜇 𝑘 deviates from the objective belief 𝜇 ∗ by evaluating how farthe ratio, 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 , where 𝑠 ≠ 𝑡 , diers from 1. If the ratio is larger (smaller) than one, then it means that in choice 𝑘 , the decision maker believes the relative likelihood of state 𝑠 with respect to state 𝑡 is larger(smaller, respectively) than what he should believe, given the objective belief 𝜇 ∗ .Given a non-negative number 𝑒 , we say that a dataset is 𝑒 -belief-perturbed objective expectedutility (OEU) rational, if it can be rationalized using expected utility with perturbed beliefs forwhich the relative likelihood ratios do not dier by more than 𝑒 from their objective equivalents.Formally: Denition 3.
Let 𝑒 ∈ R + . A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -belief-perturbed OEU rational if there exist 𝜇 𝑘 ∈ Δ ++ ( 𝑆 ) for each 𝑘 ∈ 𝐾 , and a concave and strictly increasing function 𝑢 : R + → R , such that,for all 𝑘 , 𝑦 ∈ 𝐵 ( 𝑝 𝑘 , 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑘𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑘𝑠 𝑢 ( 𝑥 𝑘𝑠 ) , and for each 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , + 𝑒 ≤ 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 ≤ + 𝑒 . (4)When 𝑒 = 𝑒 -belief-perturbed OEU rationality requires that 𝜇 𝑘𝑠 = 𝜇 ∗ 𝑠 for all 𝑠 and 𝑘 , sothe case of exact consistency with expected utility is obtained with a zero bound of belief per-turbations. Moreover, it is easy to see that by taking 𝑒 to be large enough, any dataset can be 𝑒 -belief-perturbed rationalizable.We should note that 𝑒 bounds belief perturbations for all states and observations. As such,it can be sensitive to extreme observations and outliers (the CCEI is also subject to this critique:see Echenique et al., 2011). In our empirical application, we carry out a robustness analysis toaccount for such sensitivity (see Online Appendix F.3).10inally, we mention a potential relationship with models of nonexpected utility. One couldthink of rank-dependent utility, for example, as a way of allowing agent’s beliefs to adapt tohis observed choices. However, unlike 𝑒 -belief-perturbed OEU, the nonexpected utility theoryrequires some consistencies on the dependency. For example, for the case of rank-dependentutility, the agent’s belief over the states is aected by the ranking of the outcomes across states. We now turn to perturbed prices: think of them as prices measured with error. The perturbationis a multiplicative noise term 𝜀 𝑘𝑠 to the Arrow-Debreu state price 𝑝 𝑘𝑠 . Thus, perturbed state pricesare 𝜀 𝑘𝑠 𝑝 𝑘𝑠 . Note that if 𝜀 𝑘𝑠 = 𝜀 𝑘𝑡 for all 𝑠, 𝑡 , then introducing the noise does not aect anythingbecause it only changes the scale of prices. In other words, what matters is how perturbationsaect relative prices, that is 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 .We can measure how much the noise 𝜀 𝑘 perturbs relative prices by evaluating how much theratio, 𝜀 𝑘𝑠 𝜀 𝑘𝑡 , where 𝑠 ≠ 𝑡 , diers from 1. Denition 4.
Let 𝑒 ∈ R + . A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -price-perturbed OEU rational if there exists aconcave and strictly increasing function 𝑢 : R + → R , and 𝜀 𝑘 ∈ R 𝑆 + for each 𝑘 ∈ 𝐾 such that, for all 𝑘 , 𝑦 ∈ 𝐵 ( ˜ 𝑝 𝑘 , ˜ 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝑢 ( 𝑥 𝑘𝑠 ) , where for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 ˜ 𝑝 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 and for each 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 + 𝑒 ≤ 𝜀 𝑘𝑠 𝜀 𝑘𝑡 ≤ + 𝑒 . (5)It is without loss of generality to add an additional restriction that ˜ 𝑝 𝑘 · 𝑥 𝑘 = 𝑝 𝑘 · 𝑥 𝑘 for each 𝑘 ∈ 𝐾 because what matters are the relative prices.The idea is illustrated in Figure 2. The gure shows how the perturbations to relative pricesaect budget lines, under the assumption that | 𝑆 | =
2. For each value of 𝑒 ∈ { . , . , } and 𝑘 ∈ 𝐾 , the blue area represents the set (cid:40) 𝑥 ∈ R 𝑆 + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ 𝑝 · 𝑥 = 𝑝 𝑘 · 𝑥 𝑘 for some ˜ 𝑝 ∈ R 𝑆 ++ such that ∀ 𝑠, 𝑡 ∈ 𝑆, + 𝑒 ≤ ˜ 𝑝 𝑠 / 𝑝 𝑘𝑠 ˜ 𝑝 𝑡 / 𝑝 𝑘𝑡 ≤ + 𝑒 (cid:41) State 1 S t a t e e = 0.10 A State 1 S t a t e e = 0.25 B State 1 S t a t e e = 1.00 C Figure 2: Illustration of the set of possible perturbed budget sets with 𝑒 ∈ { . , . , } . Notes : Panel Cpresents an example of price-perturbed OEU rationalization. The solid blue line represents the perturbedbudget set and the green line represents the indierence curve. of perturbed budget lines. The dataset in the gure is the same as in Figure 1B, which is notrationalizable with any expected utility function as we discussed.Figure 2C illustrates how we rationalize the dataset in Figure 1B. The blue bold lines areperturbed budget lines and the green bold curves are (xed) indierence curves passing througheach of the 𝑥 𝑘 in the data. The blue shaded areas are the sets of perturbed budget lines boundedby 𝑒 =
1. Perturbed budget lines needed to rationalize the choices are indicated with blue boldlines. Since they are inside the shaded areas, the dataset is price-perturbed OEU rational with 𝑒 = Finally, we turn to perturbed utility. As explained above, perturbations are multiplicative andtake the form 𝜀 𝑘𝑠 𝑢 ( 𝑥 𝑘𝑠 ) . It is easy to see that this method is equivalent to belief perturbation. Asfor price perturbations, we seek to measure how much the 𝜀 𝑘 perturbs utilities at choice problem 𝑘 by evaluating how much the ratio, 𝜀 𝑘𝑠 𝜀 𝑘𝑡 , where 𝑠 ≠ 𝑡 , diers from 1. Denition 5.
Let 𝑒 ∈ R + . A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -utility-perturbed OEU rational if there exists a We consider state-contingent perturbations. As such, perturbed utilities fall outside of the domain of EU theory.We thank Jose Apesteguía and Miguel Ballester for pointing this out to us. oncave and strictly increasing function 𝑢 : R + → R and 𝜀 𝑘 ∈ R 𝑆 + for each 𝑘 ∈ 𝐾 such that, for all 𝑘 , 𝑦 ∈ 𝐵 ( 𝑝 𝑘 , 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝜀 𝑘𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝜀 𝑘𝑠 𝑢 ( 𝑥 𝑘𝑠 ) , and for each 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 + 𝑒 ≤ 𝜀 𝑘𝑠 𝜀 𝑘𝑡 ≤ + 𝑒 . (6) The rst observation we make is that the three sources of perturbations are equivalent, in thesense that for any 𝑒 a dataset is 𝑒 -perturbed rationalizable according to one of the sources if andonly if it is also rationalizable according to any of the other sources with the same 𝑒 . By virtueof this result, we can interpret our measure of deviations from OEU in any of the ways we haveintroduced. Theorem 1.
Let 𝑒 ∈ R + , and 𝐷 be a dataset. The following are equivalent: • 𝐷 is 𝑒 -belief-perturbed OEU rational; • 𝐷 is 𝑒 -price-perturbed OEU rational; • 𝐷 is 𝑒 -utility-perturbed OEU rational. The proof appears in Appendix A. In light of Theorem 1, we shall simply say that a datasetis 𝑒 -perturbed OEU rational if it is 𝑒 -belief-perturbed OEU rational, and this will be equivalent tobeing 𝑒 -price-perturbed OEU rational, and 𝑒 -utility-perturbed OEU rational. We proceed to give a characterization of the dataset that are 𝑒 -perturbed OEU rational. Speci-cally, given 𝑒 ∈ R + , we propose a revealed preference axiom and prove that a dataset satises theaxiom if and only if it is 𝑒 -perturbed OEU rational.Before we state the axiom, we need to introduce some additional notation. In the currentmodel, where 𝜇 ∗ is known and objective, what matters to an expected utility maximizer is not thestate price itself, but instead the risk-neutral price. Denition 6.
For any dataset ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = , the risk neutral price 𝜌 𝑘𝑠 ∈ R 𝑆 ++ in choice problem 𝑘 atstate 𝑠 is dened by 𝜌 𝑘𝑠 = 𝑝 𝑘𝑠 𝜇 ∗ 𝑠 .
13s in Echenique and Saito (2015), the axiom we propose involves a sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ofpairs satisfying certain conditions. Denition 7.
A sequence of pairs ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = is called a test sequence if(i) 𝑥 𝑘 𝑖 𝑠 𝑖 > 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 for all 𝑖 ;(ii) each 𝑘 appears as 𝑘 𝑖 (on the left of the pair) the same number of times it appears as 𝑘 (cid:48) 𝑖 (on theright). Echenique and Saito (2015) provide an axiom for OEU rationalization, termed the Strong Ax-iom for Revealed Objective Expected Utility (SAROEU), which states that for any test sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = , we have 𝑛 (cid:214) 𝑖 = 𝜌 𝑘 𝑖 𝑠 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ . (7)SAROEU is equivalent to the axiom provided by Kübler et al. (2014).It is easy to see why SAROEU is necessary for OEU rationalization. Assuming (for simplicityof exposition) that 𝑢 is dierentiable, the rst-order condition of the maximization problem (2)for choice problem 𝑘 is 𝜆 𝑘 𝑝 𝑘𝑠 = 𝜇 ∗ 𝑠 𝑢 (cid:48) ( 𝑥 𝑘𝑠 ) , or equivalently, 𝜌 𝑘𝑠 = 𝑢 (cid:48) ( 𝑥 𝑘𝑠 ) 𝜆 𝑘 , where 𝜆 𝑘 > 𝑛 (cid:214) 𝑖 = 𝜌 𝑘 𝑖 𝑠 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝑛 (cid:214) 𝑖 = 𝜆 𝑘 (cid:48) 𝑖 𝜆 𝑘 𝑖 · 𝑛 (cid:214) 𝑖 = 𝑢 (cid:48) ( 𝑥 𝑘 𝑖 𝑠 𝑖 ) 𝑢 (cid:48) ( 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) ≤ . To see that this term is smaller than 1, note that the rst term of the product of the 𝜆 -ratios isequal to one because of the condition (ii) of the test sequence: all 𝜆 𝑘 must cancel out. The secondterm of the product of 𝑢 (cid:48) -ratio is less than one because of the concavity of 𝑢 , and the condition (i)of the test sequence (i.e., 𝑢 (cid:48) ( 𝑥 𝑘 𝑖 𝑠 𝑖 )/ 𝑢 (cid:48) ( 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) ≤ 𝑒 -perturbed OEU rationality allows the decision maker to use dierent beliefs 𝜇 𝑘 ∈ Δ ++ ( 𝑆 ) for each choice problem 𝑘 . Consequently, SAROEU is not necessary for 𝑒 -perturbed OEUrationality. To see that SAROEU can be violated, note that the rst-order condition of the max-imization (3) for choice 𝑘 is as follows: there exists a positive number (Lagrange multiplier) 𝜆 𝑘 𝑠 ∈ 𝑆 , 𝜆 𝑘 𝑝 𝑘𝑠 = 𝜇 𝑘𝑠 𝑢 (cid:48) ( 𝑥 𝑘𝑠 ) , or equivalently, 𝜌 𝑘𝑠 = 𝜇 𝑘𝑠 𝜇 ∗ 𝑠 𝑢 (cid:48) ( 𝑥 𝑘𝑠 ) 𝜆 𝑘 . Suppose that 𝑥 𝑘𝑠 > 𝑥 𝑘𝑡 . Then ( 𝑥 𝑘𝑠 , 𝑥 𝑘𝑡 ) is a test sequence (of length one) according to Deni-tion 7. We have 𝜌 𝑘𝑠 𝜌 𝑘𝑡 = (cid:18) 𝜇 𝑘𝑠 𝜇 ∗ 𝑠 𝑢 (cid:48) ( 𝑥 𝑘𝑠 ) 𝜆 𝑘 (cid:19) (cid:30) (cid:32) 𝜇 𝑘𝑡 𝜇 ∗ 𝑡 𝑢 (cid:48) ( 𝑥 𝑘𝑡 ) 𝜆 𝑘 (cid:33) = 𝑢 (cid:48) ( 𝑥 𝑘𝑠 ) 𝑢 (cid:48) ( 𝑥 𝑘𝑡 ) 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 . Even though 𝑥 𝑘𝑠 > 𝑥 𝑘𝑡 implies the rst term of the ratio of 𝑢 (cid:48) is less than one, the second term canbe strictly larger than one. When 𝑥 𝑘𝑠 is close enough to 𝑥 𝑘𝑡 , the rst term is almost one while thesecond term can be strictly larger than one. Consequently, SAROEU can be violated.However, by (4), we know that the second term is bounded by 1 + 𝑒 . So we must have 𝜌 𝑘𝑠 𝜌 𝑘𝑡 ≤ + 𝑒 . In general, for a sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = of pairs, one may suspect that the bound is calculated as ( + 𝑒 ) 𝑛 . This is not true because if 𝑥 𝑘𝑠 appears both as 𝑥 𝑘 𝑖 𝑠 𝑖 for some 𝑖 (on the left of the pair) andas 𝑥 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 for some 𝑗 (on the right of the pair), then all 𝜇 𝑘𝑠 can be canceled out. What matters is thenumber of times 𝑥 𝑘𝑠 appears without being canceled out. This number can be dened as follows. Denition 8.
Consider any sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = of pairs. Let ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ≡ 𝜎 . For any 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , 𝑑 ( 𝜎, 𝑘, 𝑠 ) = { 𝑖 | 𝑥 𝑘𝑠 = 𝑥 𝑘 𝑖 𝑠 𝑖 } − { 𝑖 | 𝑥 𝑘𝑠 = 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 } , and 𝑚 ( 𝜎 ) = ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 : 𝑑 ( 𝜎,𝑘,𝑠 ) > 𝑑 ( 𝜎, 𝑘, 𝑠 ) . Note that, if 𝑑 ( 𝜎, 𝑘, 𝑠 ) is positive, then 𝑑 ( 𝜎, 𝑘, 𝑠 ) is the number of times 𝜇 𝑘𝑠 appears as a nu-merator without being canceled out. If it is negative, then 𝑑 ( 𝜎, 𝑘, 𝑠 ) is the number of times 𝜇 𝑘𝑠 appears as a denominator without being canceled out. So 𝑚 ( 𝜎 ) is the “net” number of terms suchas 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 that are present in the numerator. Thus the relevant bound is ( + 𝑒 ) 𝑚 ( 𝜎 ) .Given the discussion above, it is easy to see that the following axiom is necessary for 𝑒 -perturbed OEU rationality. Axiom 1 ( 𝑒 -Perturbed Strong Axiom for Revealed Objective Expected Utility ( 𝑒 -PSAROEU)) . Forany test sequence of pairs ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ≡ 𝜎 , we have 𝑛 (cid:214) 𝑖 = 𝜌 𝑘 𝑖 𝑠 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ ( + 𝑒 ) 𝑚 ( 𝜎 ) . Theorem 2.
Given 𝑒 ∈ R + , and let 𝐷 be a dataset. The following are equivalent: • 𝐷 is 𝑒 -belief-perturbed OEU rational. • 𝐷 satises 𝑒 -PSAROEU. The proof appears in Appendix A.Axioms like 𝑒 -PSAROEU can be interpreted as a statement about downward-sloping demand(see Echenique et al., 2020). For example, ( 𝑥 𝑘𝑠 , 𝑥 𝑘𝑠 (cid:48) ) with 𝑥 𝑘𝑠 > 𝑥 𝑘𝑠 (cid:48) is a test sequence. If risk neutralprices satisfy 𝜌 𝑘𝑠 > 𝜌 𝑘𝑠 (cid:48) , then the dataset violates downward-sloping demand. Now 𝑒 -PSAROEUmeasures the extent of the violation by controlling the size of 𝜌 𝑘𝑠 / 𝜌 𝑘𝑠 (cid:48) .In its connection to downward-sloping demand, Theorem 2 formalizes the idea of testingOEU through the correlation of risk-neutral prices and quantities: see Friedman et al. (2018) andour discussion in Section 4.2. Theorem 2 and the axiom 𝑒 -PSAROEU give the precise form thatthe downward-sloping demand property takes in order to characterize OEU, and provide a non-parametric justication to the practice of analyzing the correlation of prices and quantities.As mentioned, 0-PSAROEU is equivalent to SAROEU. When 𝑒 = ∞ , the 𝑒 -PSAROEU alwaysholds because ( + 𝑒 ) 𝑚 ( 𝜎 ) = ∞ .Given a dataset, we shall calculate the smallest 𝑒 for which the dataset satises 𝑒 -PSAROEU.It is easy to see that such a minimal level of 𝑒 exists. We explain in Online Appendices C and Dhow it is calculated in practice.
Denition 9.
Minimal 𝑒 , denoted 𝑒 ∗ , is the smallest 𝑒 (cid:48) ≥ for which the data satises 𝑒 (cid:48) -PSAROEU. The number 𝑒 ∗ is a crucial component of our empirical analysis. Importantly, it is the basis ofa statistical procedure for testing the null hypothesis of OEU rationality.As mentioned above, 𝑒 ∗ is a bound that has to hold across all observations, and therefore maybe sensitive to extreme outliers. It is, however, easy to check the sensitivity of the calculated 𝑒 ∗ to an extreme observation. One can, for example, re-calculate 𝑒 ∗ after dropping one or twoobservations, and look for large changes.Finally, 𝑒 ∗ depends on the prices and the objective probability which a decision maker faces. Inparticular, it is clear from 𝑒 -PSAROEU that 1 + 𝑒 is bounded by the maximum ratio of risk-neutralprices (i.e., max 𝑘,𝑘 (cid:48) ∈ 𝐾,𝑠,𝑠 (cid:48) ∈ 𝑆 𝜌 𝑘𝑠 / 𝜌 𝑘 (cid:48) 𝑠 (cid:48) ).We should mention that Theorem 2 is similar in spirit to some of the results in Allen andRehbeck (2020), who consider approximate rationalizability of quasilinear utility. They present In Online Appendix C, we show that 𝑒 ∗ can be obtained as a solution of minimization of a continuous functionon a compact space. Hence, the minimum exists.
16 revealed preference characterization with a measure of error “built in” to the axiom, similarto ours, which they then use as an input to a statistical test. The two papers were developedindependently, and since the models in question are very dierent, the results are unrelated.
We apply our methodology to data from three large-scale online experiments. The experimentswere implemented through representative surveys, and the task involved objective risk, not un-certainty. The data are taken from Choi et al. (2014, hereafter CKMS), Carvalho et al. (2016,hereafter CMW), and Carvalho and Silverman (2019, hereafter CS). All three experiments sharea common experimental structure, the portfolio allocation task introduced by Loomes (1991) andChoi et al. (2007).It is worth mentioning again that the three studies focus on CCEI as a measure of violation ofbasic rationality. We shall instead look at OEU, and use 𝑒 ∗ as our measure of violations of OEU.The procedure for calculating 𝑒 ∗ is explained in Online Appendices C and D. In the experiments, subjects were presented with a sequence of decision problems under riskin a graphical illustration of a two-dimensional budget line. They were asked to select a point ( 𝑥 , 𝑥 ) , an “allocation,” by clicking on the budget line (subjects were therefore forced to exhaustthe income). The coordinates of the selected point represent an allocation of points between“accounts” 1 and 2. They received the points allocated to one of the accounts, determined atrandom with an equal chance ( 𝜇 ∗ = 𝜇 ∗ = . 𝑥 -intercept is larger than the 𝑥 -intercept,points in the account 2 are “cheaper” than those in the account 1.Choi et al. (2014) implemented the task using the instrument of the CentERpanel, randomlyrecruiting subjects from the entire panel sample in the Netherlands. Carvalho et al. (2016) admin-istered the task using the GfK KnowledgePanel, a representative panel of the adult U.S. popula-tion. Carvalho and Silverman (2019) used the Understanding America Study panel. The numberof subjects who completed the task in each study is 1,182 in CKMS, 1,119 in CMW, and 1,423 inCS. 17 Account ( x ) A cc oun t ( x ) Figure 3: Sample budget lines. A set of 25 budgets from one real subject in Choi et al. (2014).
The survey instruments in these studies allowed them to collect a wide variety of individualdemographic and economic information from the respondents. The main demographic infor-mation they obtained include gender, age, education level, household income, occupation, andhousehold composition.The selection of 25 budget lines was independent across subjects in CKMS (i.e., the subjectswere given dierent sets of budget lines), xed in CMW (i.e., all subjects saw the same set ofbudgets), and semi-randomized across subjects in CS (i.e., each subject drew one of the preparedsets of 25 budgets).
Summary statistics.
We exclude ve subjects who are “exactly” OEU rational, leaving us atotal of 3,719 subjects in the three experiments. About 76% of subjects never chose corners ofthe budget lines, and there is only two percent of the entire sample who chose corners in morethan half of the 25 questions. Finally, no subjects chose corners in all 25 questions. Given theseobservations, our focus on risk aversion does not seem to be too restrictive in these datasets.We calculate 𝑒 ∗ for each individual subject. The distributions of 𝑒 ∗ are displayed in Fig-ure 4A. , The CKMS sample has a mean 𝑒 ∗ of 3.034, and a median of 2.729. The CMW subjectshave a mean of 2.487 and a median of 2.533. The CS sample has a mean of 2.494 and a median Earlier drafts of the paper (posted before summer 2019) reported log ( + 𝑒 ∗ ) , not 𝑒 ∗ itself. The empirical CDF for the CMW data has several “steps” since all subjects faced with the same set of 25 budgetlines. For example, there are 172 subjects with 𝑒 ∗ = . 𝑒 -perturbed OEU rational is on the budget line, with prices ( 𝑝 , 𝑝 ) = ( , . ) . .000.250.500.751.000.0 2.5 5.0 7.5 10.0 Minimal e CD F CKMS (1182)CMW (1116)CS (1421) A Minimal e CD F CKMS (270)CMW (207)CS (313) B Figure 4: Empirical CDFs of 𝑒 ∗ . (A) All subjects. (B) The subsample of subjects with CCEI = Notes : Thenumber of observations in each dataset is presented in parentheses. of 2.088. Recall that the smaller a subject’s 𝑒 ∗ is, the closer are her choices to OEU rationality. Itis, however, hard to exactly interpret the magnitude of 𝑒 ∗ . We turn to this issue in Section 4.3. Downward-sloping demand and 𝑒 ∗ . Perturbations in beliefs, prices, or utility, seek to ac-commodate a dataset so that it is OEU rationalizable. The accommodation can be seen as cor-recting a mismatch of relative prices and marginal rates of substitution: recall our discussionin the introduction. Another way to see the accommodation is through the relation betweenprices and quantities. Our revealed preference axiom, 𝑒 -PSAROEU, bounds certain deviationsfrom downward-sloping demand. The minimal 𝑒 is therefore a measure of the kinds of deviationsfrom downward-sloping demand that are crucial to OEU rationality.Figure 5 illustrates this idea. We calculate the Spearman’s correlation coecient betweenlog ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) for each subject in the datasets. Roughly speaking, downward-slopingdemand corresponds to the correlation between changes in quantities log ( 𝑥 / 𝑥 ) , and changes inprices log ( 𝑝 / 𝑝 ) , being negative. The idea is that if a subject properly responds to price changes,then as log ( 𝑥 / 𝑥 ) becomes larger, log ( 𝑥 / 𝑥 ) should become lower. The correlation is close tozero if subjects do not respond to price changes.The top panels of Figure 5 conrms that 𝑒 ∗ and the correlation between prices and quantitiesare closely related. This means that subjects with smaller 𝑒 ∗ tend to exhibit downward-slopingdemand, while those with larger 𝑒 ∗ are insensitive to price changes. Across all three datasets, 𝑒 ∗ and downward-sloping demand are strongly and positively related.The CCEI, on the other hand, is not clearly related to downward-sloping demand. As illus- Since 𝑒 ∗ depends on the design of set(s) of budgets, comparing 𝑒 ∗ across studies requires caution. Note that log ( 𝑥 / 𝑥 ) is not dened at the corners. We thus adjust corner choices (less than 5% of all choices) bya small constant, 0.1% of the budget in each choice, in calculation of the correlation coecient. .02.55.07.5 −1.0 −0.5 0.0 0.5 1.0 M i n i m a l e A1 B1 C1 Correlation CC E I A2 Correlation B2 Correlation C2 Figure 5: Correlation between log ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) and measures of rationality. Panels: (A) CKMS,(B) CMW, (C) CS. Notes : The vertical dashed line indicates the threshold below which Spearman’s cor-relation is signicantly negative (one-sided, at the 1% level). Black curves represent LOESS smoothingwith 95% condence bands. trated in the bottom panels of Figure 5, the relation between CCEI and the correlation betweenprices and quantities is not monotonic. Agents who are closer to complying with utility maxi-mization do not necessarily display a stronger negtive correlation between prices and quantities.The nding is consistent with our comment about CCEI, 𝑒 ∗ , and OEU rationality: CCEI measuresthe distance from utility maximization, which is related to parallel shifts in budget lines, while 𝑒 ∗ and OEU are about the slope of the budget lines, and about a negative relation between quantitiesand prices.We should mention the practice by some authors, notably, Friedman et al. (2018), to evaluatecompliance with OEU by looking at the correlation between risk-neutral prices and quantities.Our 𝑒 ∗ is related to that idea, and the empirical results presented in this section can be read as avalidation of the correlational approach. Friedman et al. (2018) use their approach to estimate aparametric functional form, using experimental data in which they vary objective probabilities,not just prices. Our approach is non-parametric, and focused on testing OEU itself, not estimatingany particular utility specication. First-order stochastic dominance and 𝑒 ∗ . In the experiments we consider, choosing ( 𝑥 , 𝑥 ) at prices ( 𝑝 , 𝑝 ) violates monotonicity with respect to rst-order stochastic dominance (hereafter FOSD-monotonicity ) when either (i) 𝑝 > 𝑝 and 𝑥 > 𝑥 or (ii) 𝑝 > 𝑝 and 𝑥 > 𝑥 . Since the20 .02.55.07.5 0.00 0.25 0.50 0.75 1.00 M i n i m a l e A1 B1 C1 Frac. FOSD−monotonicity violation CC E I A2 Frac. FOSD−monotonicity violation B2 Frac. FOSD−monotonicity violation C2 Figure 6: Violation of FOSD-monotonicity and measures of rationality. Black curves represent LOESSsmoothing with 95% condence bands. Panels: (A) CKMS, (B) CMW, (C) CS. two states have the same objective probability in our datasets, choosing a greater payo in themore expensive state violates FOSD-monotonicity. Violations of FOSD-monotonicity are relatedto downward-sloping demand, as they involve consuming more in the more expensive state.Choices that violate FOSD-monotonicity are not uncommon in the data (see Online Appendix F.1).Since OEU-rational choices must satisfy FOSD-monotonicity, 𝑒 ∗ = 𝑒 ∗ is a good indicator of FOSD-monotonicity viola-tions. See the positive relationship between the fraction of FOSD-monotonicity violations and 𝑒 ∗ in the top row of Figure 6: subjects who frequently made choices violating FOSD-monotonicitytend to have larger 𝑒 ∗ compared to those with fewer such violations.The relation between 𝑒 ∗ and violations of FOSD-monotonicity stands in sharp contrast withCCEI. First, choices that violate FOSD-monotonicity can be consistent with GARP. Our data ex-hibits subjects that pass GARP while making choices that violate FOSD-monotonicity (an empir-ical fact that was rst pointed out by Choi et al., 2014). The bottom panels of Figure 6 show thata substantial number of subjects with perfect compliance with GARP (CCEI =
1) make at leastone violation of FOSD-monotonicity. The existence of these subjects generates a nonmonotonicrelationship between CCEI and the frequency of violation of FOSD-monotonicity.
Typical patterns of choices.
We can gain some insights into the data by considering “typical”patterns of choice. Figure 7 presents choice patterns from selected subjects with CCEI = 𝑒 ∗ . Panels A-F plot observed choices and panels a-f plot the relationshipbetween log ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) associated with each choice pattern. As discussed above,panels a-f should exhibit a negative relationship (downward-sloping demand) for the subject tobe OEU rational.Panel A presents a choice pattern that is “almost” consistent with OEU. The relation betweenlog ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) ts close to a line with negative slope, but there is a small deviationaround log ( 𝑝 / 𝑝 ) = − 𝑒 ∗ nonzero. Panel B also shows a patternthat does not involve any FOSD-monotonicity violations but is not OEU rational due to smalldeviations from the downward-sloping demand (see panel b). The pattern in panel C exhibitslarger deviations from the downward-sloping demand (panel c), which push its 𝑒 ∗ higher thanthe previous two subjects.The subject’s choices in panel D are close to the 45-degree line. At rst glance, such choicesmight seem to be rationalizable by a very risk-averse expected utility function. However, aspanel d shows, the subject’s choices deviate from the downward-sloping demand property, andhence cannot be rationalized by any risk-averse expected utility function. Note that the “size” ofthe deviation from the downward-sloping demand is small (see the scale of the 𝑦 -axis in panel d).One might be able to rationalize the choices made in panel D with some models of errors inchoices, but not with the types of errors captured by our model. We will discuss other twosubjects (panels E and F) below.Figure 7 also illustrates how 𝑒 ∗ operates in practice when there are two states. Under the price-perturbation interpretation, it measures how big of an adjustment of prices would be needed tosatisfy downward-sloping demand. Such adjustments will be represented as “horizontal shifts”of points in the bottom panels of the gure (since we x the chosen bundle and rotate the budgetline), and the largest adjustment corresponds to 𝑒 ∗ . A scatterplot of log ( 𝑥 / 𝑥 ) versus log ( 𝑝 / 𝑝 ) ,as in panels a-f of Figure 7, works as a graphical tool to get a sense of whether a subject’s 𝑒 ∗ is big or small. Online Appendix F.5 discusses this idea, and illustrates 𝑒 -price-perturbed OEUrationalization using the choice data presented in Figure 7. Relationship between 𝑒 ∗ , CCEI, and EU-CCEI. CCEI serves a dierent purpose than 𝑒 ∗ ; it ismeant to capture deviations from general utility maximization, and not OEU. Nevertheless, it is The patterns in Figure 7 are not an exhaustive list by any means. See Online Appendix F.7 for more examples. This is, in our opinion, a strength of our approach. We do not ex-post seek to invent a model of errors that mightrescue EU. Instead we have written down what we think are natural sources of errors and perturbation (randomutility, beliefs, and measurement errors). Our results deal with what can be rationalized when these sources oferrors, and only those, are used to explain the data. A general enough model of errors will, of course, render thetheory untestable. State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 1.000e = 0.060 A State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 0.990e = 0.310 B State 1 S t a t e CCEI = 1.000F−GARP = 0.960EU−CCEI = 0.960cEU−CCEI = 0.960e = 2.490 C −1.000.001.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.99f = 0.00 a −2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.96f = 0.00 b −0.500.000.50 −1 0 1 log ( p p ) l og ( x x ) r = −0.70f = 0.28 c State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 1.000e = 4.390 D State 1 S t a t e CCEI = 1.000F−GARP = 0.700EU−CCEI = 0.700cEU−CCEI = 0.700e = 0.620 E State 1 S t a t e CCEI = 1.000F−GARP = 0.660EU−CCEI = 0.660cEU−CCEI = 0.660e = 0.780 F −0.01−0.010.000.00 −1 0 1 log ( p p ) l og ( x x ) r = 0.13f = 0.64 d −2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.91f = 0.04 e −3.000.003.006.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.81f = 0.12 f Figure 7: Choice patterns from six subjects in the CMW data with CCEI = 𝑒 ∗ . (A-F) Observedchoices. (a-f) The relation between log ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) . Notes : Choices in shaded areas violateFOSD-monotonicity. 𝑟 indicates the Spearman’s correlation coecient and 𝑓 indicates the fraction ofchoices violating FOSD-monotonicity. In this data, median CCEI is 0.889, median EU-CCEI is 0.730, andmedian 𝑒 ∗ is 2.533. F-GARP, EU-CCEI, cEU-CCEI are calculated with the GRID method of Polisson et al.(2020). .000.250.500.751.00 0.0 2.5 5.0 7.5 10.0 Minimal e CC E I A1 Minimal e CC E I B1 Minimal e CC E I C1 Minimal e E U − CC E I A2 Minimal e E U − CC E I B2 Minimal e E U − CC E I C2 Figure 8: Correlation between 𝑒 ∗ and CCEI (top panels) and EU-CCEI of Polisson et al. (2020) (bottompanels). Panels: (A) CKMS, (B) CMW, (C) CS. informative to understand the relationship between these measures in the data. We also commenton the recent proposal by Polisson et al. (2020) of an adaptation of CCEI to test for OEU.We observe, in Figure 4, that the distribution of 𝑒 ∗ among subjects whose CCEI is equal toone (panel B) varies as much as in thw whole population (panel A). Many subjects have CCEIequal to one, but their 𝑒 ∗ ’s can be far from zero. This means that consistency with general utilitymaximization is not necessarily a good indication of consistency with OEU.That said, the measures are clearly correlated. Figure 8, top panels, plot the relation betweenCCEI and 𝑒 ∗ . As we expect from their denitions ( larger CCEI and smaller 𝑒 ∗ correspond to higherconsistency), there is a negative and signicant relation between them (Spearman’s correlationcoecient: 𝑟 = − .
18 for CKMS, 𝑟 = − .
11 for CMW, 𝑟 = − .
35 for CS, all 𝑝 < . 𝑒 ∗ (they have 𝑒 ∗ =
0) must exhibit CCEI = 𝑒 ∗ becomes larger. Obviously, subjects witha small 𝑒 ∗ are close to being consistent with general utility maximization, and therefore have aCCEI that is close to one. However, subjects with large 𝑒 ∗ seem to have dispersed values of CCEI.Polisson et al. (2020) propose a version of CCEI meant to measure departures from EU usingtheir GRID method. We term this measure EU-CCEI. In contrast with our measure 𝑒 ∗ , whichassumes risk aversion and is based on rotating budget lines, EU-CCEI does not impose risk aver-sion and uses the same idea of shrinking budget lines as in standard CCEI. The bottom panels ofFigure 8 exhibit the relationship between 𝑒 ∗ and EU-CCEI. It is clear that the relation between 𝑒 ∗ and EU-CCEI is similar to that between 𝑒 ∗ and CCEI. The two measures are strongly correlated,24ut they also provide dierent conclusions for many subjects.There are many subjects that EU-CCEI deems consistent with OEU, but have high levels of 𝑒 ∗ . This could be attributed to the more restrictive theory being tested by 𝑒 ∗ . Subjects with EU-CCEI close to one and large 𝑒 ∗ could simply be non-risk-averse OEU maximizers. Perhaps morepuzzling is the existence of subjects that 𝑒 ∗ sees as close to OEU while EU-CCEI does not: subjectswith small values of both 𝑒 ∗ and EU-CCEI.It is hard to investigate the dierences between EU-CCEI and 𝑒 ∗ methodologically. EU-CCEIdoes not specify a source of deviations from OEU, so we cannot say that one measure emphasizesone source of errors and the other a dierent source. Instead, we look at some of the patterns inthe data that gives rise to dierences. An example of a choice pattern in which 𝑒 ∗ and EU-CCEIdier is provided by Figure 7, panel D. The subject in question exhibits CCEI = EU-CCEI = 𝑒 ∗ is large and indicates a violation of OEU. (The pattern involves choices close to the 45-degree line, but with a clear violation of downward sloping demand, see panel d.) Panels E and Fexhibit subjects that 𝑒 ∗ says are close to (risk-averse) OEU, but EU-CCEI deems far from OEU. Wesee in panels e and f that the conclusion using 𝑒 ∗ can be understood by the subjects’ compliancewith downward sloping demand. The subjects in panels E and F make a few FOSD-monotonicityviolations, which might explain the behavior of EU-CCEI, but that cannot be the end of the storybecause the subject in panel D makes substantial FOSD-monotonicity violations and exhibits theopposite behavior of 𝑒 ∗ and EU-CCEI. Finally, we should say that there are many other patterns forwhich the conclusions of 𝑒 ∗ and EU-CCEI dier: see Online Appendices F.2 and F.7 for additionalexamples.In Online Appendix F.6, we examine the relationship between 𝑒 ∗ and modied CCEI indices fortwo additional models considered in Polisson et al. (2020): stochastically monotone utility max-imization and risk-averse EU. We call these indices F-GARP and cEU-CCEI, respectively. Theirvalues are reported for the patterns in Figure 7; see Figures F.15-F.17 in the Online Appendix forpairwise scatter plots of ve indices (CCEI, F-GARP, EU-CCEI, cEU-CCEI, and 𝑒 ∗ ). The modiedCCEI measures provide a more rened index for consistency for EU than CCEI, but dierenceswith 𝑒 ∗ persist. In fact, the basic conclusions outlined in the comparison between 𝑒 ∗ and EU-CCEIhold true for these indices. Correlation with demographic characteristics.
We investigate the correlation between ourmeasure of consistency with OEU, 𝑒 ∗ , and various demographic variables available in the data.The exercise is analogous to ndings in Choi et al. (2014) that use CCEI.We nd that younger subjects, those who have high cognitive abilities, and those who areworking, are closer to being consistent with OEU than older, low ability, or non-working, subjects.For some of the three experiments we also nd that highly educated, high-income subjects, and25 .252.502.753.00 Female Male Gender M i n i m a l e A Age M i n i m a l e B Education level M i n i m a l e C Working M i n i m a l e D CRT score M i n i m a l e E1 Stroop RT M i n i m a l e E2 Monthly income (Euro) M i n i m a l e F1 Annual income (USD) M i n i m a l e F2 Annual income (USD) M i n i m a l e F3 CKMS CMW CS
Figure 9: Correlation between 𝑒 ∗ and demographic variables. Notes : Bars represent standard errors ofmeans. males, are closer to OEU. Figure 9 summarizes the mean 𝑒 ∗ (along with the standard error ofmean) across several socioeconomic categories. We use the same categorization as in Choi et al.(2014) to compare our results with their Figure 3.We observe statistically signicant (at the 5% level) gender dierences in CS (Welch’s 𝑡 = − . df = . 𝑝 = . 𝑡 = − . df = . 𝑝 = . 𝑡 = − . df = . 𝑝 = . 𝑡 -tests give 𝑝 < . 𝑒 ∗ (panel C). Subjects with higher education are on The low, medium, and high education levels correspond to primary or prevocational secondary education, pre-university secondary education or senior vocational training, and vocational college or university education, respec- 𝑡 = . df = . 𝑝 = . 𝑡 = . df = . 𝑝 = .
155 in CMW; Welch’s 𝑡 = . df = . 𝑝 = .
295 in CS).Panel D shows that subjects who were working at the time of the survey are on averagecloser to OEU than those who were not (Wlech’s 𝑡 = . df = . 𝑝 = .
043 in CKMS;Welch’s 𝑡 = . df = . 𝑝 = .
042 in CMW; Welch’s 𝑡 = . df = . 𝑝 = .
005 in CS).In panels E1 and E2, we classify subjects according to their Cognitive Reection Test score(CRT; Frederick, 2005) or average reaction times in the numerical Stroop task. The average 𝑒 ∗ for those who correctly answered two questions or more of the CRT is lower than the average forthose who answered at most one question (Welch’s 𝑡 = − . df = . 𝑝 = . 𝑒 ∗ (Welch’s 𝑡 = − . df = . 𝑝 = . 𝑒 ∗ and household income, there is a negative trend but the dierences across income brackets arenot statistically signicant (bracket “0-2.5k” vs. “5k+”, Welch’s 𝑡 = . df = . 𝑝 = . 𝑡 = . df = . 𝑝 = . 𝑒 ∗ is signicantlysmaller for the latter sample (Welch’s 𝑡 = . df = . 𝑝 = . Robustness of the results.
The measure 𝑒 ∗ is a bound that has to hold across all observationsand states (see conditions (4), (5), and (6) in the denitions of 𝑒 -perturbed OEU). One may wonderhow sensitive 𝑒 ∗ is to a small number of “bad” choices. Online Appendix F.3 presents two robust-ness checks. In the rst robustness check, we recalculate 𝑒 ∗ using subsets of observed choicesafter dropping one or two “critical mistakes”. More precisely, for each subject, we calculate 𝑒 ∗ forall combinations of 25 − 𝑚 ( 𝑚 = ,
2) choices and pick the smallest 𝑒 ∗ among them. In the secondrobustness check, we calculate the “average” perturbation necessary to rationalize the data tomitigate the eect of extreme mistakes. These alternative ways of calculating 𝑒 ∗ do not changethe general pattern of correlation between 𝑒 ∗ and CCEI or 𝑒 ∗ and demographic variables. The tively. CRT consists of three questions, all of which have an intuitive and spontaneous, but incorrect, answers, and adeliberative and correct answer. In the numerical Stroop task, subjects are presented with a number, such as 888, andare asked to identify the number of times the digit is repeated (in this example the answer is “3”, while an “intuitive”response is “8”). It has been shown that response times in this task capture the subject’s cognitive control ability.
Our discussion so far has sidestepped one issue: How are we to interpret the absolute magnitudeof 𝑒 ∗ ? When can we say that 𝑒 ∗ is large enough to “reject” consistency with OEU rationality? Toanswer this question, we present a statistical test of the hypothesis that an agent is OEU rational.The test needs some assumptions, but it gives us a threshold level (a critical value) for 𝑒 ∗ . Anyvalue of 𝑒 ∗ that exceeds the threshold indicates inconsistency with OEU at some given statisticalsignicance level.Our approach follows the methodology laid out in Echenique et al. (2011) and Echenique et al.(2016). First, we adopt the price perturbation interpretation of 𝑒 in Section 3.2, that is we consideran agent who may misperceive prices. The advantage of doing so is that we can use the observedvariability in price to get a handle on the assumptions we need to make on perturbed prices. Tothis end, let 𝐷 true = ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = denote a dataset and 𝐷 pert = ( ˜ 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = denote an “perturbed”dataset, where ˜ 𝑝 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 and 𝜀 𝑘𝑠 > 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 . Prices ˜ 𝑝 𝑘 are prices 𝑝 𝑘 measuredwith error, or misperceived.If the variance of 𝜀 is large, it will be easy to accommodate a dataset as OEU rational. The largeris the variance of 𝜀 , the larger the magnitudes of 𝑒 that can rationalize a dataset as consistent withOEU. In other words, we can attribute the agent’s large 𝑒 as his misperception of prices ratherthan his violation of EU rationality. Our procedure is thus sensitive to the assumptions we makeabout the variance of 𝜀 .To get a handle on the variance of 𝜀 , our approach is to assume that an agent mistakes trueprices 𝑝 with perturbed prices ˜ 𝑝 . The distributions of 𝑝 and ˜ 𝑝 should be similar enough thatthe agent might plausibly confuse the two. To make this operational, we imagine an agent whoconducts a statistical test for the variance of prices. If the true variance of 𝑝 is 𝜎 and the impliedvariance of ˜ 𝑝 is 𝜎 > 𝜎 , then the agent would conduct a test for the null of 𝜎 = 𝜎 against thealternative of 𝜎 = 𝜎 . We want the variances to be close enough that the agent might reasonablyget inconclusive results from such a test (i.e., the agent may reasonably mistake true prices 𝑝 withperturbed prices ˜ 𝑝 , as we assumed). Specically, we assume the sum of probabilities of type I andtype II errors in this test is relatively large . The details of how we design the test are presentedin Online Appendix E.The main results are summarized in Figure 10. The probability of a type I error is 𝜂 𝐼 and the The problem of variance is pervasive in statistical implementations of revealed preference tests, see Varian(1990), Echenique et al. (2011), and Echenique et al. (2016) for example. The use of the sum of type I and type II errorsto calibrate a variance, is new to the present paper. .01 0.02 0.06 0.08 0.06 0.02 0.02 0.02 00.02 0.04 0.08 0.09 0.09 0.1 0.07 0.07 0.02 00.09 0.17 0.050.04 0.1 0.11 0.14 0.13 0.14 0.1 00.13 0.16 0.21 0.22 0.21 0.21 0.21 0.17 0.14 0.10.06 0.020.26 0.28 0.25 0.26 0.26 0.230.12 0.2 0.24 0.3 0.22 0.12 0.05 P ( Type error ) P ( T y pe e rr o r ) A P ( Type error ) P ( T y pe e rr o r ) B P ( Type error ) P ( T y pe e rr o r ) C Figure 10: Rejection rates under each combination of type I and type II error probabilities ( 𝜂 𝐼 , 𝜂 II ) . Panels:(A) CKMS, (B) CMW, (C) CS. probability of a type II error is 𝜂 II . Recall that we focus on situations where 𝜂 𝐼 + 𝜂 II is relativelylarge, as we want our consumer to plausibly mistake the distributions of 𝑝 and ˜ 𝑝 . Consider, forexample, our results for CKMS. The outermost numbers assume that 𝜂 𝐼 + 𝜂 II = .
7. For suchnumbers, the rejection rates range from 5% to 30%. This means that if prices 𝑝 and ˜ 𝑝 are closeenough so that the agent may misperceive the prices and make type I and type II errors withprobability 70%, then we can reject the hypothesis that the agent is an OEU maximizer at most30% of the cases.Overall, it is fair to say that rejection rates of the hypothesis that the decision maker is anOEU miximizer are modest. Notice also that smaller values of 𝜂 𝐼 + 𝜂 II corresponds to smallerrejection rates. This is because when values of 𝜂 𝐼 + 𝜂 II are smaller (i.e., the decision maker doesnot misperceive prices much), the dierence between 𝑝 and ˜ 𝑝 should be large, which correspondsto larger variances of 𝜀 . Larger variance, in turn, leads to smaller rejection rates. The gure alsoillustrates that the conclusions of the test are very sensitive to what one assumes about variances,through the assumptions about 𝜂 𝐼 and 𝜂 II . But if we look at the largest rejection rates, for thelargest values of 𝜂 𝐼 + 𝜂 II , we get 30% for CKMS, 11% for CMW, and 21% for CS. Hence, while manysubjects in the experiments are inconsistent with OEU, for most of these subjects, our statistical29ests would attribute such inconsistency to misperception of prices and do not reject that thesubjects are OEU maximizers. We now turn to the model of subjective expected utility (SEU), in which beliefs are not known.Instead, beliefs are subjective and unobservable. The analysis will be analogous to what we did forOEU, and therefore proceed at a faster pace. In particular, all the denitions and results parallelthose of the section on OEU. The proof of the main result (the axiomatic characterization) issubstantially more challenging here because both beliefs and utilities are unknown: there is aclassical problem in disentangling beliefs from utility. The technique for solving this problem wasintroduced in Echenique and Saito (2015). The proofs of the theorems are in Online Appendix B.
Denition 10.
Let 𝑒 ∈ R + . A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -belief-perturbed SEU rational if there exist 𝜇 𝑘 ∈ Δ ++ ( 𝑆 ) for each 𝑘 ∈ 𝐾 and a concave and strictly increasing function 𝑢 : R + → R such that,for all 𝑘 , 𝑦 ∈ 𝐵 ( 𝑝 𝑘 , 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑘𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑘𝑠 𝑢 ( 𝑥 𝑘𝑠 ) and for each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 ≤ + 𝑒 . (8)Note that the denition of 𝑒 -belief-perturbed SEU rationality diers from the denition of 𝑒 -belief-perturbed OEU rationality, only in condition (8), establishing bounds on perturbations.Here there is no objective probability from which we can evaluate the deviation of the set { 𝜇 𝑘 } of beliefs. Thus we evaluate perturbations among beliefs, as in (8). Remark 1.
The constraint on the perturbation applies for each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , so it impliesfor each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 + 𝑒 ≤ 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 ≤ + 𝑒 . Hence, when 𝑒 = , it must be that 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 = 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 . This implies that 𝜇 𝑘 = 𝜇 𝑙 for a dataset that is -belief perturbed SEU rational. Next, we propose perturbed SEU rationality with respect to prices.
Denition 11.
Let 𝑒 ∈ R + . A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -price-perturbed SEU rational if there exist 𝜇 ∈ Δ ++ ( 𝑆 ) and a concave and strictly increasing function 𝑢 : R + → R and 𝜀 𝑘 ∈ R 𝑆 + for each 𝑘 ∈ 𝐾 uch that, for all 𝑘 , 𝑦 ∈ 𝐵 ( ˜ 𝑝 𝑘 , ˜ 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝑢 ( 𝑥 𝑘𝑠 ) , where for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 ˜ 𝑝 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 , and for each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 ≤ + 𝑒 . (9)Again, the denition diers from the corresponding denition of price-perturbed OEU ratio-nality only in condition (9), establishing bounds on perturbations. In condition (9), we measurethe size of the perturbations by 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 , not 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 as in (5). This change is necessary to accommodate the existence of subjective beliefs. Bychoosing subjective beliefs appropriately, one can neutralize the perturbation in prices if 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 = 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 for all 𝑘, 𝑙 ∈ 𝐾 . That is, as long as 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 = 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 for all 𝑘, 𝑙 ∈ 𝐾 , if we can rationalize thedataset by introducing the noise with some subjective belief 𝜇 , then without using the noise, wecan rationalize the dataset with another subjective belief 𝜇 (cid:48) such that 𝜀 𝑘𝑠 𝜇 (cid:48) 𝑠 / 𝜀 𝑘𝑡 𝜇 (cid:48) 𝑡 = 𝜇 𝑠 / 𝜇 𝑡 .Finally, we dene utility-perturbed SEU rationality. Denition 12.
Let 𝑒 ∈ R + . A dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -utility-perturbed SEU rational if there exist 𝜇 ∈ Δ ++ ( 𝑆 ) , a concave and strictly increasing function 𝑢 : R + → R , and 𝜀 𝑘 ∈ R 𝑆 + for each 𝑘 ∈ 𝐾 suchthat, for all 𝑘 , 𝑦 ∈ 𝐵 ( 𝑝 𝑘 , 𝑝 𝑘 · 𝑥 𝑘 ) = ⇒ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝜀 𝑘𝑠 𝑢 ( 𝑦 𝑠 ) ≤ ∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝜀 𝑘𝑠 𝑢 ( 𝑥 𝑘𝑠 ) , and for each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 ≤ + 𝑒 . As in the previous section, given 𝑒 , we can show that these three concepts of rationality areequivalent. Theorem 3.
Let 𝑒 ∈ R + and 𝐷 be a dataset. The following are equivalent: • 𝐷 is 𝑒 -belief-perturbed SEU rational; • 𝐷 is 𝑒 -price-perturbed SEU rational; • 𝐷 is 𝑒 -utility-perturbed SEU rational.
31n light of Theorem 3, we shall speak simply of 𝑒 -perturbed SEU rationality to refer to any ofthe above notions of perturbed SEU rationality.Echenique and Saito (2015) prove that a dataset is SEU rational if and only if it satises arevealed-preference axiom termed the Strong Axiom for Revealed Subjective Expected Utility(SARSEU). SARSEU states that, for any test sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = , if each 𝑠 appears as 𝑠 𝑖 (on theleft of the pair) the same number of times it appears as 𝑠 (cid:48) 𝑖 (on the right), then 𝑛 (cid:214) 𝑖 = 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ . SARSEU is no longer necessary for perturbed SEU-rationality. This is easy to see, as we allowthe decision maker to have a dierent belief 𝜇 𝑘 for each choice 𝑘 , and reason as in our discussionof SAROEU. Analogous to our analysis of OEU, we introduce a perturbed version of SARSEU tocapture perturbed SEU rationality. Let 𝑒 ∈ R + . Axiom 2 ( 𝑒 -Perturbed SARSEU ( 𝑒 -PSARSEU)) . For any test sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ≡ 𝜎 , if each 𝑠 appears as 𝑠 𝑖 (on the left of the pair) the same number of times it appears as 𝑠 (cid:48) 𝑖 (on the right), then 𝑛 (cid:214) 𝑖 = 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ ( + 𝑒 ) 𝑚 ( 𝜎 ) . We can easily see the necessity of 𝑒 -PSARSEU by reasoning from the rst-order conditions,as in our discussion of 𝑒 -PSAROEU. The main result of this section shows that 𝑒 -PSARSEU is notonly necessary for 𝑒 -perturbed SEU rationality, but also sucient. Theorem 4.
Let 𝑒 ∈ R + and 𝐷 be a dataset. The following are equivalent: • 𝐷 is 𝑒 -perturbed SEU rational; • 𝐷 satises 𝑒 -PSARSEU. It is easy to see that 0-PSARSEU is equivalent to SARSEU, and that by choosing 𝑒 to be ar-bitrarily large it is possible to rationalize any dataset. As a consequence, we shall be interestedin nding a minimal value of 𝑒 that rationalizes a dataset. Echenique et al. (2019) apply the ideato datasets of choice under uncertainty collected in the laboratory as well as on the large-scaleonline survey of the general U.S. population. 32 Conclusion
We present a measure of deviations from expected utility theory, called minimal 𝑒 (or 𝑒 ∗ ), that isbased on a revealed-preference characterization of the “perturbed” version of the model.We start from an observation that the empirical content of EU is captured by the relationbetween prices and marginal rates of substitution. We measure the deviations from EU by thesmallest amount of perturbations one needs to add in order to get the “right” relation betweenprices and marginal rates of substitution. There are three components of the EU model, beliefs,prices, and utilities, which we can perturb, but we can interpret the measure in any of the ways(Theorem 1).We apply our method to data from three large-scale experiments and nd that the measuredelivers additional insights on datasets that had been analyzed with CCEI, a measure of consis-tency with general utility maximization. Our measure can be used as an additional toolkit fordata analysis in empirical studies employing choices from linear budgets. Appendix A Proofs of Theorems 1 and 2
A.1 Proof of Theorem 1
First we prove a lemma that implies Theorem 1, and is useful for the suciency part of Theorem 2.The lemma provides “Afriat inequalities” for the problem at hand.
Lemma 1.
Given 𝑒 ∈ R + , and let ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = be a dataset. The following statements are equivalent.(a) ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -belief-perturbed OEU rational.(b) There are strictly positive numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑘𝑠 , for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 𝑘𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝑝 𝑘𝑠 , and 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ 𝑣 𝑘𝑠 ≤ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , (10) and for all 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , + 𝑒 ≤ 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 ≤ + 𝑒 . (11) (c) ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -price-perturbed OEU rational.(d) There are strictly positive numbers ˆ 𝑣 𝑘𝑠 , ˆ 𝜆 𝑘 , and 𝜀 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 ∗ 𝑠 ˆ 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 , and 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ ˆ 𝑣 𝑘𝑠 ≤ ˆ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , nd for all 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , + 𝑒 ≤ 𝜀 𝑘𝑠 𝜀 𝑘𝑡 ≤ + 𝑒 . (e) ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -utility-perturbed OEU rational.(f) There are strictly positive numbers ˆ 𝑣 𝑘𝑠 , ˆ 𝜆 𝑘 , and ˆ 𝜀 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 ∗ 𝑠 ˆ 𝜀 𝑘𝑠 ˆ 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝑝 𝑘𝑠 , and 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ ˆ 𝑣 𝑘𝑠 ≤ ˆ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , and for all 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , + 𝑒 ≤ ˆ 𝜀 𝑘𝑠 ˆ 𝜀 𝑘𝑡 ≤ + 𝑒 . Proof.
The equivalence between (a) and (b), the equivalence between (c) and (d), and the equiva-lence between (e) and (f) follow from arguments in Echenique and Saito (2015). The equivalencebetween (d) and (f) with 𝜀 𝑘𝑠 = / ˆ 𝜀 𝑘𝑠 for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 is straightforward. Thus, to show theresult, it suces to show that (b) and (d) are equivalent.To show that (d) implies (b), dene 𝑣 = ˆ 𝑣 and 𝜇 𝑘𝑠 = 𝜇 ∗ 𝑠 𝜀 𝑘𝑠 / (cid:16)(cid:205) 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝜀 𝑘𝑠 (cid:17) for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 and 𝜆 𝑘 = ˆ 𝜆 𝑘 / (cid:16)(cid:205) 𝑠 ∈ 𝑆 𝜇 ∗ 𝑠 𝜀 𝑘𝑠 (cid:17) for each 𝑘 ∈ 𝐾 . Then, 𝜇 𝑘 ∈ Δ ++ ( 𝑆 ) . Since 𝜇 ∗ 𝑠 ˆ 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 , we have 𝜇 𝑘𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝑝 𝑘𝑠 .Moreover, for each 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , 𝜀 𝑘𝑠 𝜀 𝑘𝑡 = 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 . Hence, + 𝑒 ≤ 𝜀 𝑘𝑠 𝜀 𝑘𝑡 ≤ + 𝑒 .To show that (b) implies (d), for all 𝑠 ∈ 𝑆 dene ˆ 𝑣 = 𝑣 and for all 𝑘 ∈ 𝐾 , ˆ 𝜆 𝑘 = 𝜆 𝑘 . For all 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , dene 𝜀 𝑘𝑠 = 𝜇 ∗ 𝑠 𝜇 𝑘𝑠 . For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , since 𝜇 𝑘𝑠 𝑢 𝑘𝑠 = 𝜆 𝑘 𝑝 𝑘𝑠 , we have 𝜇 ∗ 𝑠 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 .Finally, for each 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , 𝜀 𝑘𝑠 𝜀 𝑘𝑡 = 𝜇 ∗ 𝑠 / 𝜇 𝑘𝑠 𝜇 ∗ 𝑡 / 𝜇 𝑘𝑡 = 𝜇 𝑘𝑡 / 𝜇 𝑘𝑠 𝜇 ∗ 𝑡 / 𝜇 ∗ 𝑠 . Therefore, we obtain + 𝑒 ≤ 𝜀 𝑘𝑠 𝜀 𝑘𝑡 ≤ + 𝑒 . (cid:3) A.2 Proof of the Necessity Direction of Theorem 2
Lemma 2.
Given 𝑒 ∈ R + , if a dataset is 𝑒 -belief-perturbed OEU rational, then the dataset satises 𝑒 -PSAROEU.Proof. Fix any sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ≡ 𝜎 of pairs that satises conditions (i) and (ii) in Denition 7.By Lemma 1, there exist 𝑣 𝑘 𝑖 𝑠 𝑖 , 𝑣 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 , 𝜆 𝑘 𝑖 , 𝜆 𝑘 (cid:48) 𝑖 , 𝜇 𝑘 𝑖 𝑠 𝑖 , 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 such that 𝑣 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≥ 𝑣 𝑘 𝑖 𝑠 𝑖 and 𝑣 𝑘 𝑖 𝑠 𝑖 = 𝜇 ∗ 𝑠𝑖 𝜇 𝑘𝑖𝑠𝑖 𝜆 𝑘 𝑖 𝜌 𝑘 𝑖 𝑠 𝑖 , and 𝑣 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝜇 ∗ 𝑠 (cid:48) 𝑖 𝜇 𝑘 (cid:48) 𝑖𝑠 (cid:48) 𝑖 𝜆 𝑘 (cid:48) 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 . Thus, we have 1 ≥ 𝑛 (cid:214) 𝑖 = 𝜆 𝑘 𝑖 ( 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 / 𝜇 ∗ 𝑠 (cid:48) 𝑖 ) 𝜌 𝑘 𝑖 𝑠 𝑖 𝜆 𝑘 (cid:48) 𝑖 ( 𝜇 𝑘 𝑖 𝑠 𝑖 / 𝜇 ∗ 𝑠 𝑖 ) 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 / 𝜇 ∗ 𝑠 (cid:48) 𝑖 𝜇 𝑘 𝑖 𝑠 𝑖 / 𝜇 ∗ 𝑠 𝑖 𝑛 (cid:214) 𝑖 = 𝜌 𝑘 𝑖 𝑠 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 , 𝑛 (cid:214) 𝑖 = 𝜌 𝑘 𝑖 𝑠 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 𝑖 𝑠 𝑖 / 𝜇 ∗ 𝑠 𝑖 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 / 𝜇 ∗ 𝑠 (cid:48) 𝑖 . In the following, we evaluate the right hand side. For each ( 𝑘, 𝑠 ) , we rst cancel out all theterms 𝜇 𝑘𝑠 that can be canceled out. Then, the number of 𝜇 𝑘𝑠 ’s that remain in the numerator is 𝑑 ( 𝜎, 𝑘, 𝑠 ) , as in Denition 8. Since the number of terms in the numerator and the denominatormust be the same, the number of remaining fractions is 𝑚 ( 𝜎 ) ≡ (cid:205) 𝑠 ∈ 𝑆 (cid:205) 𝑘 ∈ 𝐾 : 𝑑 ( 𝜎,𝑘,𝑠 ) > 𝑑 ( 𝜎, 𝑘, 𝑠 ) . Soby relabeling the index 𝑖 to 𝑗 if necessary, we obtain 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 𝑖 𝑠 𝑖 / 𝜇 ∗ 𝑠 𝑖 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 / 𝜇 ∗ 𝑠 (cid:48) 𝑖 = 𝑚 ( 𝜎 ) (cid:214) 𝑗 = 𝜇 𝑘 𝑗 𝑠 𝑗 / 𝜇 ∗ 𝑠 𝑗 𝜇 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 / 𝜇 ∗ 𝑠 (cid:48) 𝑗 . Consider the corresponding sequence ( 𝑥 𝑘 𝑗 𝑠 𝑗 , 𝑥 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 ) 𝑚 ( 𝜎 ) 𝑗 = . Since the sequence is obtained by can-celing out 𝑥 𝑘𝑠 from the rst element and the second element of the pairs, and since the originalsequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = satises condition (ii), it follows that ( 𝑥 𝑘 𝑗 𝑠 𝑗 , 𝑥 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 ) 𝑚 ( 𝜎 ) 𝑗 = satises condition (ii).By condition (ii), we can assume without loss of generality that 𝑘 𝑗 = 𝑘 (cid:48) 𝑗 for each 𝑗 . Therefore,by the condition on the perturbation, 𝑚 ( 𝜎 ) (cid:214) 𝑗 = 𝜇 𝑘 𝑗 𝑠 𝑗 / 𝜇 ∗ 𝑠 𝑗 𝜇 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 / 𝜇 ∗ 𝑠 (cid:48) 𝑗 ≤ ( + 𝑒 ) 𝑚 ( 𝜎 ) . In conclusion, we obtain that (cid:206) 𝑛𝑖 = ( 𝜌 𝑘 𝑖 𝑠 𝑖 / 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) ≤ ( + 𝑒 ) 𝑚 ( 𝜎 ) . (cid:3) A.3 Proof of the Suciency Direction of Theorem 2
We need three lemmas to prove the suciency direction. The idea behind the argument is thesame as in Echenique and Saito (2015). We know from Lemma 1 that it suces to nd a solution tothe relevant system of Afriat inequalities. We take logarithms to linearize the Afriat inequalitiesin Lemma 1. Then we set up the problem to nd a solution to the system of linear inequalities.The rst lemma, Lemma 3, shows that 𝑒 -PSAROEU is sucient for 𝑒 -belief-perturbed OEUrationality under the assumption that the logarithms of the prices are rational numbers. The as-sumption of rational logarithms comes from our use of a version of the theorem of the alternative(see Lemma 12 in Appendix B.4): when there is no solution to the linearized Afriat inequalities,a rational solution to the dual system of inequalities exists. Then we construct a violation of 𝑒 -PSAROEU from the given solution to the dual.35he second lemma, Lemma 4, establishes that we can approximate any dataset satisfying 𝑒 -PSAROEU with a dataset for which the logarithms of prices are rational, and for which 𝑒 -PSAROEU is satised.The last lemma, Lemma 5, establishes the result by using another version of the theorem ofthe alternative, stated as Lemma 11.The rest of the section is devoted to the statement of these lemmas. Lemma 3.
Given 𝑒 ∈ R + , let a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝑘𝑘 = satisfy 𝑒 -PSAROEU. Suppose that log ( 𝑝 𝑘𝑠 ) ∈ Q for all 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , log ( 𝜇 ∗ 𝑠 ) ∈ Q for all 𝑠 ∈ 𝑆 , and log ( + 𝑒 ) ∈ Q . Then there are numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 satisfying (10) and (11) in Lemma 1. Lemma 4.
Given 𝑒 ∈ R + , let a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝑘𝑘 = satisfy 𝑒 -PSAROEU with respect to 𝜇 ∗ . Then forall positive numbers 𝜀 , there exist a positive real numbers 𝑒 (cid:48) ∈ [ 𝑒, 𝑒 + 𝜀 ] , 𝜇 (cid:48) 𝑠 ∈ [ 𝜇 ∗ 𝑠 − 𝜀, 𝜇 ∗ 𝑠 + 𝜀 ] , and 𝑞 𝑘𝑠 ∈ [ 𝑝 𝑘𝑠 − 𝜀, 𝑝 𝑘𝑠 ] for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 such that log 𝑞 𝑘𝑠 ∈ Q for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , log ( 𝜇 (cid:48) 𝑠 ) ∈ Q for all 𝑠 ∈ 𝑆 , and log ( + 𝑒 (cid:48) ) ∈ Q , 𝜇 (cid:48) ∈ Δ ++ ( 𝑆 ) , and the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝑘𝑘 = satisfy 𝑒 (cid:48) -PSAROEU withrespect to 𝜇 (cid:48) . Lemma 5.
Given 𝑒 ∈ R + , let a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝑘𝑘 = satisfy 𝑒 -PSAROEU with respect to 𝜇 . Then thereare numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 satisfying (10) and (11) in Lemma 1. A.3.1 Proof of Lemma 3
The proof is similar to the proof of the main result in Echenique and Saito (2015), which corre-sponds to the case 𝑒 =
0. By log-linearizing the equation in system (10) and the inequality (11) inLemma 1, we have for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such thatlog 𝜇 𝑘𝑠 + log 𝑣 𝑘𝑠 = log 𝜆 𝑘 + log 𝑝 𝑘𝑠 , (12) 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log 𝑣 𝑘𝑠 ≤ log 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , (13)and for all 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , − log ( + 𝑒 ) + log 𝜇 ∗ 𝑠 − log 𝜇 ∗ 𝑡 ≤ log 𝜇 𝑘𝑠 − log 𝜇 𝑘𝑡 ≤ log ( + 𝑒 ) + log 𝜇 ∗ 𝑠 − log 𝜇 ∗ 𝑡 . (14)We are going to write the system of inequalities (12)-(14) in matrix form, following Echeniqueand Saito (2015) with some modications.Let 𝐴 be a matrix with 𝐾 × 𝑆 rows and 2 ( 𝐾 × 𝑆 ) + 𝐾 + ( 𝑘, 𝑠 ) , two columns for every pair ( 𝑘, 𝑠 ) , one columns for each 𝑘 ,and one last column. In the row corresponding to ( 𝑘, 𝑠 ) , the matrix has zeroes everywhere with36he following exceptions: it has 1’s in columns for ( 𝑘, 𝑠 ) ; it has a − 𝑘 ; it has − log 𝑝 𝑘𝑠 in the very last column. Matrix 𝐴 looks as follows: ··· 𝑣 𝑘𝑠 𝑣 𝑘𝑡 𝑣 𝑙𝑠 𝑣 𝑙𝑡 ··· ··· 𝜇 𝑘𝑠 𝜇 𝑘𝑡 𝜇 𝑙𝑠 𝜇 𝑙𝑡 ··· ··· 𝜆 𝑘 𝜆 𝑙 ··· 𝑝 ... ... ... ... ... ... ... ... ... ... ... ( 𝑘,𝑠 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑘𝑠 ( 𝑘,𝑡 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑘𝑡 ( 𝑙,𝑠 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑙𝑠 ( 𝑙,𝑡 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑙𝑡 ... ... ... ... ... ... ... ... ... ... ... . Next, we write the system of inequalities (13) and (14) in a matrix form. There is one row inmatrix 𝐵 for each pair ( 𝑘, 𝑠 ) and ( 𝑘 (cid:48) , 𝑠 (cid:48) ) for which 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) . In the row corresponding to 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) ,we have zeroes everywhere with the exception of a − ( 𝑘, 𝑠 ) and a 1 in thecolumn for ( 𝑘 (cid:48) , 𝑠 (cid:48) ) . Matrix 𝐵 has additional rows, that capture the system of inequalities (14), asfollows: ··· 𝑣 𝑘𝑠 𝑣 𝑘𝑡 𝑣 𝑙𝑠 𝑣 𝑙𝑡 ··· ··· 𝜇 𝑘𝑠 𝜇 𝑘𝑡 𝜇 𝑙𝑠 𝜇 𝑙𝑡 ··· ··· 𝜆 𝑘 𝜆 𝑙 ··· 𝑝 ... ... ... ... ... ... ... ... ... ... ... · · · · · · · · · − · · · · · · · · · log ( + 𝑒 ) − log 𝜇 ∗ 𝑠 + log 𝜇 ∗ 𝑡 · · · · · · · · · − · · · · · · · · · log ( + 𝑒 ) + log 𝜇 ∗ 𝑠 − log 𝜇 ∗ 𝑡 · · · · · · · · · − · · · · · · · · · log ( + 𝑒 ) + log 𝜇 ∗ 𝑠 − log 𝜇 ∗ 𝑡 · · · · · · · · · − · · · · · · · · · log ( + 𝑒 ) − log 𝜇 ∗ 𝑠 + log 𝜇 ∗ 𝑡 ... ... ... ... ... ... ... ... ... ... ... . Finally, we have a matrix 𝐸 which has a single row and has zeroes everywhere except for 1 inthe last column.To sum up, there is a solution to the system (12)-(14) if and only if there is a vector 𝑢 ∈ R ( 𝐾 × 𝑆 )+ 𝐾 + that solves the system of equations and linear inequalities 𝑆 𝐴 · 𝑢 = ,𝐵 · 𝑢 ≥ ,𝐸 · 𝑢 > . The entries of 𝐴 , 𝐵 , and 𝐸 are either 0, 1 or −
1, with the exception of the last column of 𝐴 and 𝐵 . Under the hypotheses of the lemma we are proving, the last column consists of rationalnumbers. By Motzkin’s theorem, then, there is such a solution 𝑢 to 𝑆 ( 𝜃, 𝜂, 𝜋 ) that solves the system of equations and linear inequalities 𝑆 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = ,𝜂 ≥ ,𝜋 > . In the following, we shall prove that the non-existence of a solution 𝑢 implies that the datasetmust violate 𝑒 -PSAROEU. Suppose then that there is no solution 𝑢 and let ( 𝜃, 𝜂, 𝜋 ) be a rationalvector as above, solving system 𝑆 ( 𝜃, 𝜂, 𝜋 ) are rational vectors, by multiplying a large enough integer, we can make the vectorsintegers. Then we transform the matrices 𝐴 and 𝐵 using 𝜃 and 𝜂 . (i) If 𝜃 𝑟 >
0, then creat 𝜃 𝑟 copiesof the 𝑟 th row; (ii) omitting row 𝑟 when 𝜃 𝑟 =
0; and (iii) if 𝜃 𝑟 <
0, then 𝜃 𝑟 copies of the 𝑟 th rowmultiplied by − 𝐵 and 𝜂 𝑟 copies of eachrow (and thus omitting row 𝑟 when 𝜂 𝑟 =
0; recall that 𝜂 𝑟 ≥ 𝑟 ).By using the transformed matrices and the fact that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = 𝜂 ≥
0, we canprove the following claims:
Claim.
There exists a sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛 ∗ 𝑖 = ≡ 𝜎 of pairs that satises conditions (i) and (ii) inDenition 7. Proof.
We can construct a sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛 ∗ 𝑖 = in a similar way to the proof of Lemma 11 ofEchenique and Saito (2015). By construction, the sequence satises condition (i) that 𝑥 𝑘 𝑖 𝑠 𝑖 > 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 for all 𝑖 .In the following, we show that the sequence satises condition (ii) that each 𝑘 appears as 𝑘 𝑖 thesame number of times it appears as 𝑘 (cid:48) 𝑖 . Let 𝑛 ( 𝑥 𝑘𝑠 ) ≡ { 𝑖 | 𝑥 𝑘𝑠 = 𝑥 𝑘 𝑖 𝑠 𝑖 } and 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) ≡ { 𝑖 | 𝑥 𝑘𝑠 = 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 } .It suces to show that for each 𝑘 ∈ 𝐾 , (cid:205) 𝑠 ∈ 𝑆 (cid:2) 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) (cid:3) = 𝐵 . We have a constraint for each triple ( 𝑘, 𝑠, 𝑡 ) with 𝑠 < 𝑡 . Denote the weight on the rows capturing 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 ≤ + 𝑒 by 𝜂 ( 𝑘, 𝑠, 𝑡 ) and 1 + 𝑒 ≤ 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 ∗ 𝑠 / 𝜇 ∗ 𝑡 by 𝜂 ( 𝑘, 𝑡, 𝑠 ) .For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , in the column corresponding to 𝜇 𝑘𝑠 in matrix 𝐴 , remember that wehave 1 if we have 𝑥 𝑘𝑠 = 𝑥 𝑘 𝑖 𝑠 𝑖 for some 𝑖 and − 𝑥 𝑘𝑠 = 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 for some 𝑖 . This is because a rowin 𝐴 must have 1 ( −
1) in the column corresponding to 𝑣 𝑘𝑠 if and only if it has 1 ( −
1, respectively)in the column corresponding to 𝜇 𝑘𝑠 . By summing over the column corresponding to 𝜇 𝑘𝑠 , we have 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) . 38ow we consider matrix 𝐵 . In the column corresponding to 𝜇 𝑘𝑠 , we have 1 in the row multipliedby 𝜂 ( 𝑘, 𝑡, 𝑠 ) and − 𝜂 ( 𝑘, 𝑠, 𝑡 ) . By summing over the column correspondingto 𝜇 𝑘𝑠 , we also have − (cid:205) 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑠, 𝑡 ) + (cid:205) 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑡, 𝑠 ) .For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , the column corresponding to 𝜇 𝑘𝑠 of matrices 𝐴 and 𝐵 must sum upto zero; so we have 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) + ∑︁ 𝑡 ≠ 𝑠 [− 𝜂 ( 𝑘, 𝑠, 𝑡 ) + 𝜂 ( 𝑘, 𝑡, 𝑠 )] = . (15)Hence for each 𝑘 ∈ 𝐾 for each 𝑘 ∈ 𝐾 (cid:205) 𝑠 ∈ 𝑆 (cid:2) 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) (cid:3) = (cid:3) Claim. (cid:206) 𝑛 ∗ 𝑖 = ( 𝜌 𝑘 𝑖 𝑠 𝑖 / 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) > ( + 𝑒 ) 𝑚 ( 𝜎 ∗ ) . Proof.
By (15), So for each 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 [ 𝜂 ( 𝑘, 𝑠, 𝑡 ) − 𝜂 ( 𝑘, 𝑡, 𝑠 )] log 𝜇 ∗ 𝑠 = ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 (cid:104) 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) (cid:105) log 𝜇 ∗ 𝑠 = 𝑛 ∗ ∑︁ 𝑖 = log 𝜇 ∗ 𝑠 𝑖 𝜇 ∗ 𝑠 (cid:48) 𝑖 , where the last equality holds by the denition of 𝑛 and 𝑛 (cid:48) . Moreover, since 𝑑 ( 𝜎 ∗ , 𝑘, 𝑠 ) = 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) = (cid:205) 𝑡 ≠ 𝑠 [ 𝜂 ( 𝑘, 𝑠, 𝑡 ) − 𝜂 ( 𝑘, 𝑡, 𝑠 )] ≤ (cid:205) 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑠, 𝑡 ) , we have 𝑚 ( 𝜎 ∗ ) ≡ ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 : 𝑑 ( 𝜎 ∗ ,𝑘,𝑠 ) > 𝑑 ( 𝜎 ∗ , 𝑘, 𝑠 ) = ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 min { 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) , } ≤ ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑠, 𝑡 ) . By the equality and the inequality above and by the fact that the last column must sum up to zeroand 𝐸 has one at the last column, we have0 > 𝑛 ∗ ∑︁ 𝑖 = log 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝑝 𝑘 𝑖 𝑠 𝑖 + log ( + 𝑒 ) ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑠, 𝑡 ) + ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 ( 𝜂 ( 𝑘, 𝑠, 𝑡 ) − 𝜂 ( 𝑘, 𝑡, 𝑠 )) log 𝜇 ∗ 𝑠 = 𝑛 ∗ ∑︁ 𝑖 = log 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝑝 𝑘 𝑖 𝑠 𝑖 − 𝑛 ∗ ∑︁ 𝑖 = log 𝜇 ∗ 𝑠 𝑖 𝜇 ∗ 𝑠 (cid:48) 𝑖 + log ( + 𝑒 ) ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑠, 𝑡 ) = 𝑛 ∗ ∑︁ 𝑖 = log 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝜌 𝑘 𝑖 𝑠 𝑖 + log ( + 𝑒 ) ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑠, 𝑡 ) ≥ 𝑛 ∗ ∑︁ 𝑖 = log 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝜌 𝑘 𝑖 𝑠 𝑖 + log ( + 𝑒 ) 𝑚 ( 𝜎 ∗ ) . That is, (cid:205) 𝑛 ∗ 𝑖 = log ( 𝜌 𝑘 𝑖 𝑠 𝑖 / 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) > 𝑚 ( 𝜎 ∗ ) log ( + 𝑒 ) . This is a contradiction. (cid:3) A.3.2 Proof of Lemma 4
Let X = { 𝑥 𝑘𝑠 | 𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 } . Consider the set of sequences that satisfy conditions (i) and (ii) inDenition 7: Σ = (cid:40) ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ⊂ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = satises conditions (i) and (ii)in Denition 7 for some 𝑛 (cid:41) . 𝜎 ∈ Σ , we dene a vector 𝑡 𝜎 ∈ N 𝐾 𝑆 . For each pair ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) , we shall identifythe pair with ( ( 𝑘 𝑖 , 𝑠 𝑖 ) , ( 𝑘 (cid:48) 𝑖 , 𝑠 (cid:48) 𝑖 )) . Let 𝑡 𝜎 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) be the number of times that the pair ( 𝑥 𝑘𝑠 , 𝑥 𝑘 (cid:48) 𝑠 (cid:48) ) appears in the sequence 𝜎 . One can then describe the satisfaction of 𝑒 -PSAROEU by means of thevectors 𝑡 𝜎 . Observe that 𝑡 depends only on ( 𝑥 𝑘 ) 𝐾𝑘 = in the dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = . It does not dependon prices.For each ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) such that 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) , dene 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = log ( 𝑝 𝑘𝑠 / 𝑝 𝑘 (cid:48) 𝑠 (cid:48) ) . And dene 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = 𝑥 𝑘𝑠 ≤ 𝑥 𝑘 (cid:48) 𝑠 (cid:48) . Then, 𝛿 is a 𝐾 𝑆 -dimensional real-valued vector. If 𝜎 = ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = , then 𝛿 · 𝑡 𝜎 = ∑︁ (( 𝑘,𝑠 ) , ( 𝑘 (cid:48) ,𝑠 (cid:48) ))∈( 𝐾𝑆 ) 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) 𝑡 𝜎 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = log (cid:169)(cid:173)(cid:173)(cid:171) 𝑛 (cid:214) 𝑖 = 𝜌 𝑘 𝑖 𝑠 𝑖 𝜌 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 (cid:170)(cid:174)(cid:174)(cid:172) . So the dataset satises 𝑒 -PSAROEU with respect to 𝜇 if and only if 𝛿 · 𝑡 𝜎 ≤ 𝑚 ( 𝜎 ) log ( + 𝑒 ) for all 𝜎 ∈ Σ .Enumerate the elements in X in increasing order: 𝑦 < 𝑦 < · · · < 𝑦 𝑁 , and x an arbitrary 𝜉 ∈ ( , ) . We shall construct by induction a sequence {( 𝜀 𝑘𝑠 ( 𝑛 ))} 𝑁𝑛 = , where 𝜀 𝑘𝑠 ( 𝑛 ) is dened for all ( 𝑘, 𝑠 ) with 𝑥 𝑘𝑠 = 𝑦 𝑛 .By the denseness of the rational numbers, and the continuity of the exponential function, foreach ( 𝑘, 𝑠 ) such that 𝑥 𝑘𝑠 = 𝑦 , there exists a positive number 𝜀 𝑘𝑠 ( ) such that log ( 𝜌 𝑘𝑠 𝜀 𝑘𝑠 ( )) ∈ Q and 𝜉 < 𝜀 𝑘𝑠 ( ) <
1. Let 𝜀 ( ) = min { 𝜀 𝑘𝑠 ( ) | 𝑥 𝑘𝑠 = 𝑦 } .In second place, for each ( 𝑘, 𝑠 ) such that 𝑥 𝑘𝑠 = 𝑦 , there exists a positive 𝜀 𝑘𝑠 ( ) such thatlog ( 𝜌 𝑘𝑠 𝜀 𝑘𝑠 ( )) ∈ Q and 𝜉 < 𝜀 𝑘𝑠 ( ) < 𝜀 ( ) . Let 𝜀 ( ) = min { 𝜀 𝑘𝑠 ( ) | 𝑥 𝑘𝑠 = 𝑦 } .In third place, and reasoning by induction, suppose that 𝜀 ( 𝑛 ) has been dened and that 𝜉 < 𝜀 ( 𝑛 ) . For each ( 𝑘, 𝑠 ) such that 𝑥 𝑘𝑠 = 𝑦 𝑛 + , let 𝜀 𝑘𝑠 ( 𝑛 + ) > ( 𝜌 𝑘𝑠 𝜀 𝑘𝑠 ( 𝑛 + )) ∈ Q , and 𝜉 < 𝜀 𝑘𝑠 ( 𝑛 + ) < 𝜀 ( 𝑛 ) . Let 𝜀 ( 𝑛 + ) = min { 𝜀 𝑘𝑠 ( 𝑛 + ) | 𝑥 𝑘𝑠 = 𝑦 𝑛 } .This denes the sequence ( 𝜀 𝑘𝑠 ( 𝑛 )) by induction. Note that 𝜀 𝑘𝑠 ( 𝑛 + )/ 𝜀 ( 𝑛 ) < 𝑛 . Let¯ 𝜉 < 𝜀 𝑘𝑠 ( 𝑛 + )/ 𝜀 ( 𝑛 ) < ¯ 𝜉 .For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , let ˆ 𝜌 𝑘𝑠 = 𝜌 𝑘𝑠 𝜀 𝑘𝑠 ( 𝑛 ) , where 𝑛 is such that 𝑥 𝑘𝑠 = 𝑦 𝑛 . Choose 𝜇 (cid:48) ∈ Δ ++ ( 𝑆 ) such that for all 𝑠 ∈ 𝑆 log 𝜇 (cid:48) 𝑠 ∈ Q and 𝜇 (cid:48) 𝑠 ∈ [ ¯ 𝜉 𝜇 𝑠 , 𝜇 𝑠 / ¯ 𝜉 ] for all 𝑠 ∈ 𝑆 . Such 𝜇 (cid:48) exists by the densenessof the rational numbers. Now for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , dene 𝑞 𝑘𝑠 = ˆ 𝜌 𝑘𝑠 𝜇 (cid:48) 𝑠 . (16)Then, log 𝑞 𝑘𝑠 = log ˆ 𝜌 𝑘𝑠 − log 𝜇 (cid:48) 𝑠 ∈ Q .We claim that the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = satises 𝑒 (cid:48) -PSAROEU with respect to 𝜇 (cid:48) . Let 𝛿 ∗ be denedfrom ( 𝑞 𝑘 ) 𝐾𝑘 = in the same manner as 𝛿 was dened from ( 𝜌 𝑘 ) 𝐾𝑘 = .40or each pair ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) with 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) , if 𝑛 and 𝑚 are such that 𝑥 𝑘𝑠 = 𝑦 𝑛 and 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = 𝑦 𝑚 ,then 𝑛 > 𝑚 . By denition of 𝜀 , 𝜀 𝑘𝑠 ( 𝑛 ) 𝜀 𝑘 (cid:48) 𝑠 (cid:48) ( 𝑚 ) < 𝜀 𝑘𝑠 ( 𝑛 ) 𝜀 ( 𝑚 ) < ¯ 𝜉 < . Hence, 𝛿 ∗ ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = log 𝜌 𝑘𝑠 𝜀 𝑘𝑠 ( 𝑛 ) 𝜌 𝑘 (cid:48) 𝑠 (cid:48) 𝜀 𝑘 (cid:48) 𝑠 (cid:48) ( 𝑚 ) < log 𝜌 𝑘𝑠 𝜌 𝑘 (cid:48) 𝑠 (cid:48) + log ¯ 𝜉 < log 𝜌 𝑘𝑠 𝜌 𝑘 (cid:48) 𝑠 (cid:48) = 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) . Now, we choose 𝑒 (cid:48) such that 𝑒 (cid:48) ≥ 𝑒 and log ( + 𝑒 (cid:48) ) ∈ Q .Thus, for all 𝜎 ∈ Σ , 𝛿 ∗ · 𝑡 𝜎 ≤ 𝛿 · 𝑡 𝜎 ≤ 𝑚 ( 𝜎 ) log ( + 𝑒 ) ≤ 𝑚 ( 𝜎 ) log ( + 𝑒 (cid:48) ) as 𝑡 · ≥ ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = satises 𝑒 -PSAROEU with respect to 𝜇 .Thus the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = satises 𝑒 (cid:48) -PSAROEU with respect to 𝜇 (cid:48) . Finally, note that 𝜉 < 𝜀 𝑘𝑠 ( 𝑛 ) < 𝑛 and each 𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 . So that by choosing 𝜉 close enough to 1, we can take ˆ 𝜌 to be as close to 𝜌 as desired. By the denition, we also can take 𝜇 (cid:48) to be as close to 𝜇 as desired.Consequently, by (16), we can take ( 𝑞 𝑘 ) 𝐾𝑘 = to be as close to ( 𝑝 𝑘 ) 𝐾𝑘 = as desired. We also can take 𝑒 (cid:48) to be as close to 𝑒 as desired. A.3.3 Proof of Lemma 5
We use the following notational convention: For a matrix 𝐷 with 2 ( 𝐾 × 𝑆 ) + 𝐾 + 𝐷 for the submatrix of 𝐷 corresponding to the rst 𝐾 × 𝑆 columns; let 𝐷 be the submatrixcorresponding to the following 𝐾 × 𝑆 columns; 𝐷 correspond to the next 𝐾 columns; and 𝐷 tothe last column. Thus, 𝐷 = [ 𝐷 𝐷 𝐷 𝐷 ] .Consider the system comprised by (12), (13), and (14) in the proof of Lemma 3. Let 𝐴 , 𝐵 , and 𝐸 be constructed from the dataset as in the proof of Lemma 3. The dierence with respect toLemma 3 is that now the entries of 𝐴 and 𝐵 may not be rational. Note that the entries of 𝐸 , 𝐵 ,and 𝐴 𝑖 , 𝑖 = , , 𝑆 F = R implies that there is a real vector ( 𝜃, 𝜂, 𝜋 ) such that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = 𝜂 ≥ , 𝜋 >
0. Recall that 𝐸 =
1, so we obtain that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 = ( 𝑞 𝑘 ) 𝐾𝑘 = , 𝜇 (cid:48) , and 𝑒 (cid:48) be such that the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = satises 𝑒 (cid:48) -PSAROEU withrespect to 𝜇 (cid:48) , and log 𝑞 𝑘𝑠 ∈ Q for all 𝑘 and 𝑠 , log 𝜇 (cid:48) 𝑠 ∈ Q for all 𝑠 ∈ 𝑆 , and log ( + 𝑒 (cid:48) ) ∈ Q . (Such ( 𝑞 𝑘 ) 𝐾𝑘 = , 𝜇 (cid:48) , and 𝑒 (cid:48) exist by Lemma 4.) Construct matrices 𝐴 (cid:48) , 𝐵 (cid:48) , and 𝐸 (cid:48) from this dataset in thesame way as 𝐴 , 𝐵 , and 𝐸 is constructed in the proof of Lemma 3. Note that only the prices, theobjective probabilities, and the bounds are dierent. So 𝐸 (cid:48) = 𝐸 and 𝐴 (cid:48) 𝑖 = 𝐴 𝑖 and 𝐵 (cid:48) 𝑖 = 𝐵 𝑖 for 𝑖 = , ,
3. Only 𝐴 (cid:48) and 𝐵 (cid:48) may be dierent from 𝐴 and 𝐵 , respectively.41y Lemma 4, we can choose 𝑞 𝑘 , 𝜇 (cid:48) , and 𝑒 (cid:48) such that | ( 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) ) − ( 𝜃 · 𝐴 + 𝜂 · 𝐵 ) | < 𝜋 /
2. Wehave shown that 𝜃 · 𝐴 + 𝜂 · 𝐵 = − 𝜋 , so the choice of 𝑞 𝑘 , 𝜇 (cid:48) , and 𝑒 (cid:48) guarantees that 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) < 𝜋 (cid:48) = − 𝜃 · 𝐴 (cid:48) − 𝜂 · 𝐵 (cid:48) > 𝜃 · 𝐴 (cid:48) 𝑖 + 𝜂 · 𝐵 (cid:48) 𝑖 + 𝜋 (cid:48) 𝐸 𝑖 = 𝑖 = , ,
3, as ( 𝜃, 𝜂, 𝜋 ) solves system 𝑆 𝐴 , 𝐵 and 𝐸 , and 𝐴 (cid:48) 𝑖 = 𝐴 𝑖 , 𝐵 (cid:48) 𝑖 = 𝐵 𝑖 and 𝐸 𝑖 = 𝑖 = , ,
3. Finally, 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) + 𝜋 (cid:48) 𝐸 = 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) + 𝜋 (cid:48) = 𝜂 ≥ 𝜋 (cid:48) >
0. Therefore 𝜃 , 𝜂 , and 𝜋 (cid:48) constitute a solution to 𝑆 𝐴 (cid:48) , 𝐵 (cid:48) , and 𝐸 (cid:48) .Lemma 11 then implies that there is no solution to system 𝑆 𝐴 (cid:48) , 𝐵 (cid:48) , and 𝐸 (cid:48) . Sothere is no solution to the system comprised by (12), (13), and (14) in the proof of Lemma 3.However, this contradicts Lemma 3 because the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) satises 𝑒 (cid:48) -PSAROEU with 𝜇 (cid:48) ,log ( + 𝑒 (cid:48) ) ∈ Q , log 𝜇 (cid:48) 𝑠 ∈ Q for all 𝑠 ∈ 𝑆 , and log 𝑞 𝑘𝑠 ∈ Q for all 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 . References
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B Omitted Proofs
B.1 Proof of Theorem 3
First, we prove a lemma which establishes Theorem 3 and proves useful for the suciency partof Theorem 4. This lemma provides “Afriat inequalities” for the problem at hand.
Lemma 6.
Given 𝑒 ∈ R + , and let ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = be a dataset. The following statements are equivalent.(a) ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -belief-perturbed SEU rational.(b) There are strictly positive numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑘𝑠 , for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 𝑘𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝑝 𝑘𝑠 , 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ 𝑣 𝑘𝑠 ≤ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , (B.1) and for each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 ≤ + 𝑒 . (B.2) (c) ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -price-perturbed SEU rational.(d) There are strictly positive numbers ˆ 𝑣 𝑘𝑠 , ˆ 𝜆 𝑘 , 𝜇 𝑠 , and 𝜀 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 𝑠 ˆ 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 , 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ ˆ 𝑣 𝑘𝑠 ≤ ˆ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , and for all 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 ≤ + 𝑒 . (e) ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -utility-perturbed SEU rational.(f) There are strictly positive numbers ˆ 𝑣 𝑘𝑠 , ˆ 𝜆 𝑘 , 𝜇 𝑠 , and ˆ 𝜀 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 𝑠 ˆ 𝜀 𝑘𝑠 ˆ 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝑝 𝑘𝑠 , 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ ˆ 𝑣 𝑘𝑠 ≤ ˆ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , and for all 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , ˆ 𝜀 𝑘𝑠 / ˆ 𝜀 𝑘𝑡 ˆ 𝜀 𝑙𝑠 / ˆ 𝜀 𝑙𝑡 ≤ + 𝑒 . roof. The equivalence between (a) and (b), the equivalence between (c) and (d), and the equiv-alence between (e) and (f) follow from standard arguments: see Echenique and Saito (2015) fordetails. Moreover, it is easy to see the equivalence between (d) and (f) with 𝜀 𝑘𝑠 = / ˆ 𝜀 𝑘𝑠 for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 . Hence, to prove the result, it suces to show that (b) and (d) are equivalent.To show that (d) implies (b), dene 𝑣 = ˆ 𝑣 and 𝜇 𝑘𝑠 = 𝜇 𝑠 𝜀 𝑘𝑠 (cid:44) (cid:32)∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝜀 𝑘𝑠 (cid:33) for each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 and 𝜆 𝑘 = ˆ 𝜆 𝑘 (cid:44) (cid:32)∑︁ 𝑠 ∈ 𝑆 𝜇 𝑠 𝜀 𝑘𝑠 (cid:33) for each 𝑘 ∈ 𝐾 . Then, 𝜇 𝑘 ∈ Δ ++ ( 𝑆 ) . Since 𝜇 𝑠 ˆ 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 , we have 𝜇 𝑘𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝑝 𝑘𝑠 . Moreover, foreach 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 = 𝜀 𝑘𝑡 / 𝜀 𝑘𝑠 𝜀 𝑙𝑡 / 𝜀 𝑙𝑠 ≤ + 𝑒 . To show (b) implies (d), for all 𝑠 ∈ 𝑆 dene ˆ 𝑣 = 𝑣 and 𝜇 𝑠 = ∑︁ 𝑘 ∈ 𝐾 𝜇 𝑘𝑠 | 𝐾 | . Then, 𝜇 ∈ Δ ++ ( 𝑆 ) . For all 𝑘 ∈ 𝐾 , ˆ 𝜆 𝑘 = 𝜆 𝑘 . For all 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , dene 𝜀 𝑘𝑠 = 𝜇 𝑠 𝜇 𝑘𝑠 . For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , since 𝜇 𝑘𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝑝 𝑘𝑠 , we have 𝜇 𝑠 𝑣 𝑘𝑠 = ˆ 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 . Finally, for each 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 , 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 𝜀 𝑙𝑠 / 𝜀 𝑙𝑡 = 𝜇 𝑘𝑡 / 𝜇 𝑘𝑠 𝜇 𝑙𝑡 / 𝜇 𝑙𝑠 ≤ + 𝑒 . (cid:3) B.2 Proof of the Necessity Direction of Theorem 4
Lemma 7.
Given 𝑒 ∈ R + , if a dataset is 𝑒 -belief-perturbed SEU rational then the dataset satises 𝑒 -PSARSEU.Proof. Fix any sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ≡ 𝜎 of pairs that satises conditions (i) and (ii) in Denition 7and another condition that each 𝑠 appears as 𝑠 𝑖 (on the left of the pair) the same number of times2t appears as 𝑠 (cid:48) 𝑖 (on the right), which we refer to as condition (iii) throughout this section. By thestandard argument using the concavity of 𝑢 , for each 𝑖 , there exist 𝑣 𝑘 𝑖 𝑠 𝑖 , 𝑣 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 , 𝜆 𝑘 𝑖 , 𝜆 𝑘 (cid:48) 𝑖 , 𝜇 𝑘 𝑖 𝑠 𝑖 , 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 such that 𝑣 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≥ 𝑣 𝑘 𝑖 𝑠 𝑖 and 𝑣 𝑘 𝑖 𝑠 𝑖 = 𝜆 𝑘𝑖 𝑝 𝑘𝑖𝑠𝑖 𝜇 𝑘𝑖𝑠𝑖 , and 𝑣 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝜆 𝑘 (cid:48) 𝑖 𝑝 𝑘 (cid:48) 𝑖𝑠 (cid:48) 𝑖 𝜇 𝑘 (cid:48) 𝑖𝑠 (cid:48) 𝑖 . Thus, we have1 ≥ 𝑛 (cid:214) 𝑖 = 𝜆 𝑘 𝑖 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝑝 𝑘 𝑖 𝑠 𝑖 𝜆 𝑘 (cid:48) 𝑖 𝜇 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝜇 𝑘 𝑖 𝑠 𝑖 𝑛 (cid:214) 𝑖 = 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 , where the second equality holds by condition (ii). See the proof of Lemma 10 of Echenique andSaito (2015) for detail. Thus, 𝑛 (cid:214) 𝑖 = 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 𝑖 𝑠 𝑖 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 . In the following, we evaluate the right hand side. For each ( 𝑘, 𝑠 ) , we rst cancel out the same 𝜇 𝑘𝑠 as much as possible both from the denominator and the numerator. Then, the number of 𝜇 𝑘𝑠 remained in the numerator is 𝑑 ( 𝜎, 𝑘, 𝑠 ) as dened in Denition 8. Since the number of termsin the numerator and the denominator must be the same, the number of remaining fraction is 𝑚 ( 𝜎 ) ≡ (cid:205) 𝑠 ∈ 𝑆 (cid:205) 𝑘 ∈ 𝐾 : 𝑑 ( 𝜎,𝑘,𝑠 ) > 𝑑 ( 𝜎, 𝑘, 𝑠 ) . So by relabeling the index 𝑖 to 𝑗 if necessary, we obtain 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 𝑖 𝑠 𝑖 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝑚 ( 𝜎 ) (cid:214) 𝑗 = 𝜇 𝑘 𝑗 𝑠 𝑗 𝜇 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 . Consider the corresponding sequence ( 𝑥 𝑘 𝑗 𝑠 𝑗 , 𝑥 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 ) 𝑚 ( 𝜎 ) 𝑗 = . Since the sequence is obtained by can-celing out 𝑥 𝑘𝑠 from the rst element and the second element of the pairs the same number oftimes; and since the original sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = satises conditions (ii) and (iii), it follows that ( 𝑥 𝑘 𝑗 𝑠 𝑗 , 𝑥 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 ) 𝑚 ( 𝜎 ) 𝑗 = satises conditions (ii) and (iii).By condition (iii), we can assume without loss of generality that 𝑠 𝑗 = 𝑠 (cid:48) 𝑗 for each 𝑗 . Fix 𝑠 ∗ ∈ 𝑆 .Then by condition (8) of 𝑒 -belief perturbed SEU, for each 𝑗 ∈ { , . . . , 𝑚 ( 𝜎 )} , 𝜇 𝑘 𝑗 𝑠 𝑗 𝜇 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 = 𝜇 𝑘 𝑗 𝑠 𝑗 𝜇 𝑘 (cid:48) 𝑗 𝑠 𝑗 ≤ ( + 𝑒 ) 𝜇 𝑘 (cid:48) 𝑗 𝑠 ∗ 𝜇 𝑘 𝑗 𝑠 ∗ . Moreover by condition (ii), 𝑚 ( 𝜎 ) (cid:214) 𝑗 = 𝜇 𝑘 (cid:48) 𝑗 𝑠 ∗ 𝜇 𝑘 𝑗 𝑠 ∗ = . 𝑛 (cid:214) 𝑖 = 𝜇 𝑘 𝑖 𝑠 𝑖 𝜇 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 = 𝑚 ( 𝜎 ) (cid:214) 𝑗 = 𝜇 𝑘 𝑗 𝑠 𝑖 𝜇 𝑘 (cid:48) 𝑗 𝑠 (cid:48) 𝑗 ≤ ( + 𝑒 ) 𝑚 ( 𝜎 ) 𝑛 (cid:214) 𝑗 = 𝜇 𝑘 (cid:48) 𝑗 𝑠 ∗ 𝜇 𝑘 𝑗 𝑠 ∗ = ( + 𝑒 ) 𝑚 ( 𝜎 ) , and hence, 𝑛 (cid:214) 𝑖 = 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ≤ ( + 𝑒 ) 𝑚 ( 𝜎 ) . (cid:3) B.3 Proof of the Suciency Direction in Theorem 4
The outline of the argument is the same as the proof of Theorem 2 and Echenique and Saito(2015). As in the proof of Theorem 2, we need three lemmas to prove the suciency direction.We know from Lemma 6 that it suces to nd a solution to the Afriat inequalities (actuallyrst-order conditions). So we set up the problem to nd a solution to a system of linear inequal-ities obtained from using logarithms to linearize the Afriat inequalities in Lemma 6.The rst lemma, Lemma 8, establishes that 𝑒 -PSARSEU is sucient for e-belief-perturbed SEUrationality when the logarithms of the prices are rational numbers.The second lemma, Lemma 9, establishes that we can approximate any dataset satisfying 𝑒 -PSARSEU with a dataset for which the logarithms of prices are rational, and for which 𝑒 -PSARSEUis satised.Finally, Lemma 10 establishes the result by using another version of the theorem of the alter-native, stated as Lemma 11 above.The statement of the lemmas follow. The rest of the section is devoted to the proof of theselemmas. Lemma 8.
Given 𝑒 ∈ R + , let a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝑘𝑘 = satisfy 𝑒 -PSARSEU. Suppose that log ( 𝑝 𝑘𝑠 ) ∈ Q forall 𝑘 and 𝑠 and log ( + 𝑒 ) ∈ Q . Then there are numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 satisfying (B.1) and (B.2) in Lemma 6. Lemma 9.
Given 𝑒 ∈ R + , let a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝑘𝑘 = satisfy 𝑒 -PSARSEU. Then for all positive numbers 𝜀 , there exist a positive real number 𝑒 (cid:48) ∈ [ 𝑒, 𝑒 + 𝜀 ] and 𝑞 𝑘𝑠 ∈ [ 𝑝 𝑘𝑠 − 𝜀, 𝑝 𝑘𝑠 ] for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 suchthat log 𝑞 𝑘𝑠 ∈ Q and the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝑘𝑘 = satisfy 𝑒 (cid:48) -PSARSEU. Lemma 10.
Given 𝑒 ∈ R + , let a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝑘𝑘 = satisfy 𝑒 -PSARSEU. Then there are numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 satisfying (B.1) and (B.2) in Lemma 6. .3.1 Proof of Lemma 8 The proof is similar to the proof of Echenique and Saito (2015), which corresponds to the casewith 𝑒 =
0. By log-linearizing system (B.1), and inequality (B.2) in Lemma 6, we have for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that log 𝜇 𝑘𝑠 + log 𝑣 𝑘𝑠 = log 𝜆 𝑘 + log 𝑝 𝑘𝑠 , (B.3) 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log 𝑣 𝑘𝑠 ≤ log 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , (B.4)and for all 𝑘, 𝑙 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 ,log 𝜇 𝑘𝑠 − log 𝜇 𝑘𝑡 − log 𝜇 𝑙𝑠 + log 𝜇 𝑙𝑡 ≤ log ( + 𝑒 ) . (B.5)We are going to write the system of inequalities (B.3)-(B.5) in matrix form. The formulationfollows Echenique and Saito (2015), with some modications.Let 𝐴 be a matrix with 𝐾 × 𝑆 rows and 2 ( 𝐾 × 𝑆 ) + 𝐾 + ( 𝑘, 𝑠 ) , two columns for every pair ( 𝑘, 𝑠 ) , one column for each 𝑘 , andone last column. In the row corresponding to ( 𝑘, 𝑠 ) , the matrix has zeroes everywhere with thefollowing exceptions: it has 1’s in columns for ( 𝑘, 𝑠 ) ; it has a − 𝑘 ; it has − log 𝑝 𝑘𝑠 in the very last column. Matrix 𝐴 looks as follows: ··· 𝑣 𝑘𝑠 𝑣 𝑘𝑡 𝑣 𝑙𝑠 𝑣 𝑙𝑡 ··· ··· 𝜇 𝑘𝑠 𝜇 𝑘𝑡 𝜇 𝑙𝑠 𝜇 𝑙𝑡 ··· ··· 𝜆 𝑘 𝜆 𝑙 ··· 𝑝 ... ... ... ... ... ... ... ... ... ... ... ( 𝑘,𝑠 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑘𝑠 ( 𝑘,𝑡 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑘𝑠 ( 𝑙,𝑠 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑙𝑠 ( 𝑙,𝑡 ) · · · · · · · · · · · · · · · − · · · − log 𝑝 𝑙𝑠 ... ... ... ... ... ... ... ... ... ... ... . Next, we write the system of inequalities (B.4) and (B.5) in matrix form. There is one row inmatrix 𝐵 for each pair ( 𝑘, 𝑠 ) and ( 𝑘 (cid:48) , 𝑠 (cid:48) ) for which 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) . In the row corresponding to 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) ,we have zeroes everywhere with the exception of a − ( 𝑘, 𝑠 ) and a 1 in thecolumn for ( 𝑘 (cid:48) , 𝑠 (cid:48) ) . Matrix 𝐵 has additional rows, that capture the system of inequalities (B.5):We do not need a constraint for each quadruple ( 𝑘, 𝑙, 𝑠, 𝑡 ) , as some of them would be redundant.Specically, we need the constraints 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 ≤ + 𝑒 , and 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 ≤ + 𝑒 , which is equivalent to 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 ≥ /( + 𝑒 ) . But note that 𝜇 𝑙𝑡 / 𝜇 𝑙𝑠 𝜇 𝑘𝑡 / 𝜇 𝑘𝑠 ≤ + 𝑒 is redundant, as 𝜇 𝑙𝑡 / 𝜇 𝑙𝑠 𝜇 𝑘𝑡 / 𝜇 𝑘𝑠 = 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 . So for each ( 𝑠, 𝑡 ) with 𝑠 < 𝑡 , and each 𝑘 ≠ 𝑙 we are going to have the constraint ( 𝑘, 𝑙, 𝑠, 𝑡 ) . For each such ( 𝑘, 𝑙, 𝑠, 𝑡 ) we The inequality 𝑠 < 𝑡 is simply a devise to ensure that we choose only one of the two ordered pairs of 𝑠 and 𝑡 . 𝜇 𝑘𝑠 and 𝜇 𝑙𝑡 , a − 𝜇 𝑘𝑡 and 𝜇 𝑙𝑠 , and log ( + 𝑒 ) in the very last column; one of these rows has a 1 in the column for 𝜇 𝑘𝑡 and 𝜇 𝑙𝑠 , a − 𝜇 𝑘𝑠 and 𝜇 𝑙𝑡 , and log ( + 𝑒 ) in the very last column. So this part of matrix 𝐵 is as follows: ··· 𝑣 𝑘𝑠 𝑣 𝑘𝑡 𝑣 𝑙𝑠 𝑣 𝑙𝑡 ··· ··· 𝜇 𝑘𝑠 𝜇 𝑘𝑡 𝜇 𝑙𝑠 𝜇 𝑙𝑡 ··· ··· 𝜆 𝑘 𝜆 𝑙 ··· 𝑝 ... ... ... ... ... ... ... ... ... ... ... · · · · · · · · · − − · · · · · · · · · log ( + 𝑒 )· · · · · · · · · − − · · · · · · · · · log ( + 𝑒 ) ... ... ... ... ... ... ... ... ... ... ... . Finally, we have a matrix 𝐸 which has a single row and has zeroes everywhere except for 1 inthe last column.To sum up, there is a solution to the system (B.3)-(B.5) if and only if there is a vector 𝑢 ∈ R ( 𝐾 × 𝑆 )+ 𝐾 + that solves the system of equations and linear inequalities 𝑆 𝐴 · 𝑢 = ,𝐵 · 𝑢 ≥ ,𝐸 · 𝑢 > . The entries of 𝐴 , 𝐵 , and 𝐸 are either 0, 1 or −
1, with the exception of the last column of 𝐴 and 𝐵 . Under the hypotheses of the lemma we are proving, the last column consists of rationalnumbers. By Motzkin’s theorem, then, there is such a solution 𝑢 to 𝑆 ( 𝜃, 𝜂, 𝜋 ) that solves the system of equations and linear inequalities 𝑆 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = ,𝜂 ≥ ,𝜋 > . In the following, we shall prove that the non-existence of a solution 𝑢 implies that the datasetmust violate 𝑒 -PSARSEU. Suppose then that there is no solution 𝑢 and let ( 𝜃, 𝜂, 𝜋 ) be a rationalvector as above, solving system 𝑆 ( 𝜃, 𝜂, 𝜋 ) arerational vectors, by multiplying all of their entries by a large enough integer, we can without lossof generality assume that ( 𝜃, 𝜂, 𝜋 ) are integer vectors.Then we transform the matrices 𝐴 and 𝐵 using 𝜃 and 𝜂 . (i) If 𝜃 𝑟 >
0, then create 𝜃 𝑟 copiesof the 𝑟 th row; (ii) omitting row 𝑟 when 𝜃 𝑟 =
0; and (iii) if 𝜃 𝑟 <
0, then 𝜃 𝑟 copies of the 𝑟 th rowmultiplied by −
1. 6imilarly, we create a new matrix by including the same columns as 𝐵 and 𝜂 𝑟 copies of eachrow (and thus omitting row 𝑟 when 𝜂 𝑟 =
0; recall that 𝜂 𝑟 ≥ 𝑟 ).By using the transformed matrices and the fact that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = 𝜂 ≥
0, we canprove the following claims:
Claim.
There exists a sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛 ∗ 𝑖 = of pairs that satises conditions (i) and (ii) in Deni-tion 7. Proof.
The proof is the same as in the proof of Lemma 11 in Echenique and Saito (2015). (cid:3)
Claim.
In the sequence ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛 ∗ 𝑖 = ≡ 𝜎 ∗ , each 𝑠 appears as 𝑠 𝑖 (on the left of the pair) the samenumber of times it appears as 𝑠 (cid:48) 𝑖 (on the right). Proof.
Recall our construction of the matrix 𝐵 . We have a constraint for each quadruple ( 𝑘, 𝑙, 𝑠, 𝑡 ) with 𝑠 < 𝑡 . Denote the weight on the rows capturing 𝜇 𝑘𝑠 / 𝜇 𝑘𝑡 𝜇 𝑙𝑠 / 𝜇 𝑙𝑡 ≤ + 𝑒 by 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) . Let 𝑛 ( 𝑥 𝑘𝑠 ) ≡ { 𝑖 | 𝑥 𝑘𝑠 = 𝑥 𝑘 𝑖 𝑠 𝑖 } and 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) ≡ { 𝑖 | 𝑥 𝑘𝑠 = 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 } . For notational convenience, dene 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) = ( 𝑘, 𝑙, 𝑠, 𝑡 ) with 𝑡 < 𝑠 .For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , in the column corresponding to 𝜇 𝑘𝑠 in matrix 𝐴 , remember that wehave 1 if we have 𝑥 𝑘𝑠 = 𝑥 𝑘 𝑖 𝑠 𝑖 for some 𝑖 and − 𝑥 𝑘𝑠 = 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 for some 𝑖 . This is because a rowin 𝐴 must have 1 ( −
1) in the column corresponding to 𝑣 𝑘𝑠 if and only if it has 1 ( −
1, respectively)in the column corresponding to 𝜇 𝑘𝑠 . By summing over the column corresponding to 𝜇 𝑘𝑠 , we have 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) .Now we consider matrix 𝐵 . In the column corresponding to 𝜇 𝑘𝑠 and 𝑠 < 𝑡 , we have − 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) and 1 in the row multiplied by 𝜂 ( 𝑙, 𝑘, 𝑠, 𝑡 ) . By summing over thecolumn corresponding to 𝜇 𝑘𝑠 , we also have − (cid:205) 𝑙 ≠ 𝑘 (cid:205) 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) + (cid:205) 𝑙 ≠ 𝑘 (cid:205) 𝑡 ≠ 𝑠 𝜂 ( 𝑙, 𝑘, 𝑠, 𝑡 ) .For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , the column corresponding to 𝜇 𝑘𝑠 of matrices 𝐴 and 𝐵 must sum upto zero; so we have 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) − ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) + ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑙, 𝑘, 𝑠, 𝑡 ) = . Therefore, for each 𝑠 , ∑︁ 𝑘 ∈ 𝐾 (cid:16) 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) (cid:17) = ∑︁ 𝑘 ∈ 𝐾 (cid:34)∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) − ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑙, 𝑘, 𝑠, 𝑡 ) (cid:35) = ∑︁ 𝑡 ≠ 𝑠 (cid:34)∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑙 ≠ 𝑘 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) − ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑙 ≠ 𝑘 𝜂 ( 𝑙, 𝑘, 𝑠, 𝑡 ) (cid:35) = . 𝑠 appears as 𝑠 𝑖 (on the left of the pair) the same number of times it appearsas 𝑠 (cid:48) 𝑖 (on the right). (cid:3) Claim. (cid:206) 𝑛 ∗ 𝑖 = ( 𝑝 𝑘 𝑖 𝑠 𝑖 / 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) > ( + 𝑒 ) 𝑚 ( 𝜎 ∗ ) . Proof.
By the fact that the last column must sum up to zero and 𝐸 has one at the last column, wehave 𝑛 ∗ ∑︁ 𝑖 = log 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 𝑝 𝑘 𝑖 𝑠 𝑖 + (cid:32)∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) (cid:33) log ( + 𝑒 ) = − 𝜋 < . Hence, by multiplying −
1, we have 𝑛 ∗ ∑︁ 𝑖 = log 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 − (cid:32)∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) (cid:33) log ( + 𝑒 ) > . Remember that for all 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) = + ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) − ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑙, 𝑘, 𝑠, 𝑡 ) ≤ ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) . Since 𝑑 ( 𝜎 ∗ , 𝑘, 𝑠 ) = 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) , we have 𝑚 ( 𝜎 ∗ ) ≡ ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 : 𝑑 ( 𝜎 ∗ ,𝑘,𝑠 ) > 𝑑 ( 𝜎 ∗ , 𝑘, 𝑠 ) = ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 max { 𝑛 ( 𝑥 𝑘𝑠 ) − 𝑛 (cid:48) ( 𝑥 𝑘𝑠 ) , }≤ ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) . Therefore, 𝑛 ∗ ∑︁ 𝑖 = log 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 > (cid:32)∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑙 ≠ 𝑘 ∑︁ 𝑠 ∈ 𝑆 ∑︁ 𝑡 ≠ 𝑠 𝜂 ( 𝑘, 𝑙, 𝑠, 𝑡 ) (cid:33) log ( + 𝑒 ) ≥ 𝑚 ( 𝜎 ∗ ) log ( + 𝑒 ) . This is a contradiction. (cid:3)
B.3.2 Proof of Lemma 9
Let X = { 𝑥 𝑘𝑠 | 𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 } . Consider the set of sequences that satisfy conditions (i) and (ii) inDenition 7, and (iii) in 𝑒 -PSARSEU: Σ = (cid:40) ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = ⊂ X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = satises conditions (i) and (ii)in Denition 7 and (iii) for some 𝑛 (cid:41) . 𝜎 ∈ Σ , we dene a vector 𝑡 𝜎 ∈ N 𝐾 𝑆 . For each pair ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) , we shall identifythe pair with ( ( 𝑘 𝑖 , 𝑠 𝑖 ) , ( 𝑘 (cid:48) 𝑖 , 𝑠 (cid:48) 𝑖 )) . Let 𝑡 𝜎 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) be the number of times that the pair ( 𝑥 𝑘𝑠 , 𝑥 𝑘 (cid:48) 𝑠 (cid:48) ) appears in the sequence 𝜎 . One can then describe the satisfaction of 𝑒 -PSARSEU by means of thevectors 𝑡 𝜎 . Observe that 𝑡 depends only on ( 𝑥 𝑘 ) 𝐾𝑘 = in the dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = . It does not dependon prices.For each ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) such that 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) , dene 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = log ( 𝑝 𝑘𝑠 / 𝑝 𝑘 (cid:48) 𝑠 (cid:48) ) . And dene 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = 𝑥 𝑘𝑠 ≤ 𝑥 𝑘 (cid:48) 𝑠 (cid:48) . Then, 𝛿 is a 𝐾 𝑆 -dimensional real-valued vector. If 𝜎 = ( 𝑥 𝑘 𝑖 𝑠 𝑖 , 𝑥 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 ) 𝑛𝑖 = , then 𝛿 · 𝑡 𝜎 = ∑︁ (( 𝑘,𝑠 ) , ( 𝑘 (cid:48) ,𝑠 (cid:48) ))∈( 𝐾 × 𝑆 ) 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) 𝑡 𝜎 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = log (cid:169)(cid:173)(cid:173)(cid:171) 𝑛 (cid:214) 𝑖 = 𝑝 𝑘 𝑖 𝑠 𝑖 𝑝 𝑘 (cid:48) 𝑖 𝑠 (cid:48) 𝑖 (cid:170)(cid:174)(cid:174)(cid:172) . So the dataset satises 𝑒 -PSARSEU if and only if 𝛿 · 𝑡 𝜎 ≤ 𝑚 ( 𝜎 ) log ( + 𝑒 ) for all 𝜎 ∈ Σ .Enumerate the elements in X in increasing order: 𝑦 < 𝑦 < · · · < 𝑦 𝑁 , and x an arbitrary 𝜉 ∈ ( , ) . We shall construct by induction a sequence {( 𝜀 𝑘𝑠 ( 𝑛 ))} 𝑁𝑛 = , where 𝜀 𝑘𝑠 ( 𝑛 ) is dened for all ( 𝑘, 𝑠 ) with 𝑥 𝑘𝑠 = 𝑦 𝑛 .By the denseness of the rational numbers, and the continuity of the exponential function, foreach ( 𝑘, 𝑠 ) such that 𝑥 𝑘𝑠 = 𝑦 , there exists a positive number 𝜀 𝑘𝑠 ( ) such that log ( 𝑝 𝑘𝑠 𝜀 𝑘𝑠 ( )) ∈ Q and 𝜉 < 𝜀 𝑘𝑠 ( ) <
1. Let 𝜀 ( ) = min { 𝜀 𝑘𝑠 ( ) | 𝑥 𝑘𝑠 = 𝑦 } .In second place, for each ( 𝑘, 𝑠 ) such that 𝑥 𝑘𝑠 = 𝑦 , there exists a positive 𝜀 𝑘𝑠 ( ) such thatlog ( 𝑝 𝑘𝑠 𝜀 𝑘𝑠 ( )) ∈ Q and 𝜉 < 𝜀 𝑘𝑠 ( ) < 𝜀 ( ) . Let 𝜀 ( ) = min { 𝜀 𝑘𝑠 ( ) | 𝑥 𝑘𝑠 = 𝑦 } .In third place, and reasoning by induction, suppose that 𝜀 ( 𝑛 ) has been dened and that 𝜉 < 𝜀 ( 𝑛 ) . For each ( 𝑘, 𝑠 ) such that 𝑥 𝑘𝑠 = 𝑦 𝑛 + , let 𝜀 𝑘𝑠 ( 𝑛 + ) > ( 𝑝 𝑘𝑠 𝜀 𝑘𝑠 ( 𝑛 + )) ∈ Q , and 𝜉 < 𝜀 𝑘𝑠 ( 𝑛 + ) < 𝜀 ( 𝑛 ) . Let 𝜀 ( 𝑛 + ) = min { 𝜀 𝑘𝑠 ( 𝑛 + ) | 𝑥 𝑘𝑠 = 𝑦 𝑛 } .This denes the sequence ( 𝜀 𝑘𝑠 ( 𝑛 )) by induction. Note that 𝜀 𝑘𝑠 ( 𝑛 + )/ 𝜀 ( 𝑛 ) < 𝑛 . Let¯ 𝜉 < 𝜀 𝑘𝑠 ( 𝑛 + )/ 𝜀 ( 𝑛 ) < ¯ 𝜉 .For each 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 , let 𝑞 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 ( 𝑛 ) , where 𝑛 is such that 𝑥 𝑘𝑠 = 𝑦 𝑛 . We claim that thedataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = satises 𝑒 -PSARSEU. Let 𝛿 ∗ be dened from ( 𝑞 𝑘 ) 𝐾𝑘 = in the same manner as 𝛿 was dened from ( 𝑝 𝑘 ) 𝐾𝑘 = .For each pair ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) with 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) , if 𝑛 and 𝑚 are such that 𝑥 𝑘𝑠 = 𝑦 𝑛 and 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = 𝑦 𝑚 ,then 𝑛 > 𝑚 . By denition of 𝜀 , 𝜀 𝑘𝑠 ( 𝑛 ) 𝜀 𝑘 (cid:48) 𝑠 (cid:48) ( 𝑚 ) < 𝜀 𝑘𝑠 ( 𝑛 ) 𝜀 ( 𝑚 ) < ¯ 𝜉 < . Hence, 𝛿 ∗ ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) = log 𝑝 𝑘𝑠 𝜀 𝑘𝑠 ( 𝑛 ) 𝑝 𝑘 (cid:48) 𝑠 (cid:48) 𝜀 𝑘 (cid:48) 𝑠 (cid:48) ( 𝑚 ) < log 𝑝 𝑘𝑠 𝑝 𝑘 (cid:48) 𝑠 (cid:48) + log ¯ 𝜉 < log 𝑝 𝑘𝑠 𝑝 𝑘 (cid:48) 𝑠 (cid:48) = 𝛿 ( ( 𝑘, 𝑠 ) , ( 𝑘 (cid:48) , 𝑠 (cid:48) )) . 𝑒 (cid:48) such that 𝑒 (cid:48) ≥ 𝑒 and log ( + 𝑒 (cid:48) ) ∈ Q .Thus, for all 𝜎 ∈ Σ , 𝛿 ∗ · 𝑡 𝜎 ≤ 𝛿 · 𝑡 𝜎 ≤ 𝑚 ( 𝜎 ) log ( + 𝑒 ) ≤ 𝑚 ( 𝜎 ) log ( + 𝑒 (cid:48) ) as 𝑡 · ≥ ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = satises 𝑒 -PSARSEU.Therefore, the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = satises 𝑒 (cid:48) -PSARSEU. Finally, note that 𝜉 < 𝜀 𝑘𝑠 ( 𝑛 ) < 𝑛 and each 𝑘 ∈ 𝐾, 𝑠 ∈ 𝑆 . So that by choosing 𝜉 close enough to 1 we can take ( 𝑞 𝑘 ) 𝐾𝑘 = to be asclose to ( 𝑝 𝑘 ) 𝐾𝑘 = as desired. We also can take 𝑒 (cid:48) to be as close to 𝑒 as desired. B.3.3 Proof of Lemma 10
Consider the system comprised by (B.3), (B.4), and (B.5) in the proof of Lemma 8. Let 𝐴 , 𝐵 , and 𝐸 be constructed from the dataset as in the proof of Lemma 8. The dierence with respect toLemma 8 is that now the entries of 𝐴 and 𝐵 may not be rational. Note that the entries of 𝐸 , 𝐵 𝑖 ,and 𝐴 𝑖 , for 𝑖 = , , 𝑆 F = R implies that there is a real vector ( 𝜃, 𝜂, 𝜋 ) such that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = 𝜂 ≥ , 𝜋 >
0. Recall that 𝐸 =
1, so we obtain that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 = ( 𝑞 𝑘 ) 𝐾𝑘 = vectors of prices and a positive real number 𝑒 (cid:48) be such that the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = satises 𝑒 (cid:48) -PSARSEU and log 𝑞 𝑘𝑠 ∈ Q for all 𝑘 and 𝑠 and log ( + 𝑒 (cid:48) ) ∈ Q . (Such ( 𝑞 𝑘 ) 𝐾𝑘 = and 𝑒 (cid:48) existby Lemma 9.) Construct matrices 𝐴 (cid:48) , 𝐵 (cid:48) , and 𝐸 (cid:48) from this dataset in the same way as 𝐴 , 𝐵 , and 𝐸 is constructed in the proof of Lemma 8. Since only prices 𝑞 𝑘 and the bound 𝑒 (cid:48) are dierent inthis dataset, only 𝐴 (cid:48) and 𝐵 (cid:48) may be dierent from 𝐴 and 𝐵 , respectively. So 𝐸 (cid:48) = 𝐸 , 𝐵 (cid:48) 𝑖 = 𝐵 𝑖 and 𝐴 (cid:48) 𝑖 = 𝐴 𝑖 for 𝑖 = , , 𝑞 𝑘 and 𝑒 (cid:48) such that | ( 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) ) − ( 𝜃 · 𝐴 + 𝜂 · 𝐵 ) | < 𝜋 / 𝜃 · 𝐴 + 𝜂 · 𝐵 = − 𝜋 , so the choice of prices 𝑞 𝑘 and 𝑒 (cid:48) guarantees that 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) <
0. Let 𝜋 (cid:48) = − 𝜃 · 𝐴 (cid:48) − 𝜂 · 𝐵 (cid:48) > 𝜃 · 𝐴 (cid:48) 𝑖 + 𝜂 · 𝐵 (cid:48) 𝑖 + 𝜋 (cid:48) 𝐸 𝑖 = 𝑖 = , ,
3, as ( 𝜃, 𝜂, 𝜋 ) solves system 𝑆 𝐴 , 𝐵 and 𝐸 , and 𝐴 (cid:48) 𝑖 = 𝐴 𝑖 , 𝐵 (cid:48) 𝑖 = 𝐵 𝑖 and 𝐸 𝑖 = 𝑖 = , ,
3. Finally, 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) + 𝜋 (cid:48) 𝐸 = 𝜃 · 𝐴 (cid:48) + 𝜂 · 𝐵 (cid:48) + 𝜋 (cid:48) = . We also have that 𝜂 ≥ 𝜋 (cid:48) >
0. Therefore 𝜃 , 𝜂 , and 𝜋 (cid:48) constitute a solution to 𝑆 𝐴 (cid:48) , 𝐵 (cid:48) , and 𝐸 (cid:48) .Lemma 11 then implies that there is no solution to system 𝑆 𝐴 (cid:48) , 𝐵 (cid:48) , and 𝐸 (cid:48) . Sothere is no solution to the system comprised by (B.3), (B.4), and (B.5) in the proof of Lemma 8.However, this contradicts Lemma 8 because the dataset ( 𝑥 𝑘 , 𝑞 𝑘 ) satises 𝑒 (cid:48) -PSARSEU, log ( + 𝑒 (cid:48) ) ∈ Q , and log 𝑞 𝑘𝑠 ∈ Q for all 𝑘 ∈ 𝐾 and 𝑠 ∈ 𝑆 . 10 .4 Theorem of the Alternative We shall use the following lemma, which is a version of the Theorem of the Alternative. This isTheorem 1.6.1 in Stoer and Witzgall (1970). We shall use it here in the cases where 𝐹 is either thereal or the rational number eld. Lemma 11.
Let 𝐴 be an 𝑚 × 𝑛 matrix, 𝐵 be an 𝑙 × 𝑛 matrix, and 𝐸 be an 𝑟 × 𝑛 matrix. Suppose thatthe entries of the matrices 𝐴 , 𝐵 , and 𝐸 belong to a commutative ordered eld F . Exactly one of thefollowing alternatives is true.1. There is 𝑢 ∈ F 𝑛 such that 𝐴 · 𝑢 = , 𝐵 · 𝑢 ≥ , 𝐸 · 𝑢 (cid:29) .2. There is 𝜃 ∈ F 𝑟 , 𝜂 ∈ F 𝑙 , and 𝜋 ∈ F 𝑚 such that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = ; 𝜋 > and 𝜂 ≥ . The next lemma is a direct consequence of Lemma 11. See Lemma 12 in Chambers andEchenique (2014) for a proof.
Lemma 12.
Let 𝐴 be an 𝑚 × 𝑛 matrix, 𝐵 be an 𝑙 × 𝑛 matrix, and 𝐸 be an 𝑟 × 𝑛 matrix. Suppose thatthe entries of the matrices 𝐴 , 𝐵 , and 𝐸 are rational numbers. Exactly one of the following alternativesis true.1. There is 𝑢 ∈ R 𝑛 such that 𝐴 · 𝑢 = , 𝐵 · 𝑢 ≥ , and 𝐸 · 𝑢 (cid:29) .2. There is 𝜃 ∈ Q 𝑟 , 𝜂 ∈ Q 𝑙 , and 𝜋 ∈ Q 𝑚 such that 𝜃 · 𝐴 + 𝜂 · 𝐵 + 𝜋 · 𝐸 = ; 𝜋 > and 𝜂 ≥ . Computing 𝑒 ∗ We demonstrate how to calculate 𝑒 ∗ given a dataset of choice under risk. To calculate the value,it is easier to use price-perturbed OEU rationality, rather than belief-perturbed OEU rationality.Formally, for a given data set ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = , we want to compute 𝑒 ∗ such that the data set is price per-turbed OEU rational given the number 𝑒 . We can transform this problem into an easier problemwith the following remark. Remark C.1.
Given 𝑒 ∈ R + , a data set ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = is 𝑒 -price-perturbed OEU rational if and only ifthere are strictly positive numbers 𝑣 𝑘𝑠 , 𝜆 𝑘 , 𝜇 𝑠 , and 𝜀 𝑘𝑠 for 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 , such that 𝜇 ∗ 𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 , 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ 𝑣 𝑘𝑠 ≤ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , (C.1) and for all 𝑘 ∈ 𝐾 and 𝑠, 𝑡 ∈ 𝑆 + 𝑒 ≤ 𝜀 𝑘𝑠 𝜀 𝑘𝑡 ≤ + 𝑒 . By the remark, the 𝑒 ∗ can be obtained by solving the following problem:min ( 𝜇 𝑠 ,𝑣 𝑘𝑠 ,𝜆 𝑘 ,𝜀 𝑘𝑠 ) 𝑘,𝑠 max 𝑘 ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 𝜀 𝑘𝑠 𝜀 𝑘𝑡 s.t. 𝜇 ∗ 𝑠 𝑣 𝑘𝑠 = 𝜆 𝑘 𝜀 𝑘𝑠 𝑝 𝑘𝑠 ,𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ 𝑣 𝑘𝑠 ≤ 𝑣 𝑘 (cid:48) 𝑠 (cid:48) . We then substitute 𝜀 𝑘𝑠 in the objective function by using the equality constraint in (C.1). Bycanceling out 𝜆 𝑘 and log-linearizing, we obtain the following:min ( 𝑣 𝑘𝑠 ) 𝑘,𝑠 max 𝑘 ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 ( log 𝜇 ∗ 𝑠 + log 𝑣 𝑘𝑠 − log 𝑝 𝑘𝑠 ) − ( log 𝜇 ∗ 𝑡 + log 𝑣 𝑘𝑡 − log 𝑝 𝑘𝑡 ) s.t. 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log 𝑣 𝑘𝑠 ≤ log 𝑣 𝑘 (cid:48) 𝑠 (cid:48) . ( ★ )By the discussion above, we have the following result: Remark C.2.
For any data set ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = , 𝑒 ∗ is the solution of the problem ( ★ ) , which always exists. By using ( ★ ) and the peculiarities of the experiments, we can simplify the problem: we have | 𝑆 | = 𝜇 ∗ 𝑠 = / 𝑠 ∈ 𝑆 . Hence, the problem simplies to the following:min ( 𝑣 𝑘𝑠 ) 𝑘,𝑠 max 𝑘 ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 ( log 𝑣 𝑘𝑠 − log 𝑝 𝑘𝑠 ) − ( log 𝑣 𝑘𝑡 − log 𝑝 𝑘𝑡 ) s.t. 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log 𝑣 𝑘𝑠 ≤ log 𝑣 𝑘 (cid:48) 𝑠 (cid:48) . ( (cid:5) )12 Implementation Details
In order to calculate 𝑒 ∗ for each subject’s data, we solve problem ( ★ ) using Matlab R2017b (Math-Works).For each subject, the decision in every trial is characterized by a tuple ( 𝑎 , 𝑎 , 𝑥 , 𝑥 ) where 𝑎 𝑖 represents the intercept of the budget line on each axis (here we call the 𝑥 -axis “account 1” and the 𝑦 -axis “account 2”), and 𝑥 𝑖 represents the subject’s allocation to account 𝑖 . In order to rewrite thechoice data in a price-consumption format as in the theory, we set prices 𝑝 = 𝑝 = 𝑎 / 𝑎 . This gives us a dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = .Remember that the problem we are going to solve is:min ( 𝑣 𝑘𝑠 ) 𝑘,𝑠 max 𝑘 ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 ( log 𝜇 ∗ 𝑠 + log 𝑣 𝑘𝑠 − log 𝑝 𝑘𝑠 ) − ( log 𝜇 ∗ 𝑡 + log 𝑣 𝑘𝑡 − log 𝑝 𝑘𝑡 ) s.t. 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log 𝑣 𝑘𝑠 ≤ log 𝑣 𝑘 (cid:48) 𝑠 (cid:48) . ( ★ )Our main task is to express this problem in a matrix notation.Let 𝒛 be a vector of length 𝐾 × 𝑆 + 𝐾 × 𝑆 + 𝑆 , whose rst 𝐾 × 𝑆 entries correspond to each of ( log 𝑣 𝑘𝑠 ) 𝑠,𝑘 and the last 𝐾 × 𝑆 + 𝑆 entries are all 1. This vector corresponds to the control variablesof the problem. The reason why we have 𝐾 × 𝑆 additional rows of 1 in the vector will becomeclear shortly.We construct two matrices 𝐴 and 𝐵 . The rst matrix 𝐴 has 𝐾 × 𝑆 rows and 𝐾 × 𝑆 + 𝐾 × 𝑆 + 𝑆 columns, and looks as follows: ··· 𝑣 𝑘𝑠 𝑣 𝑘𝑡 𝑣 𝑙𝑠 𝑣 𝑙𝑡 ··· ··· 𝑝 𝑘𝑠 𝑝 𝑘𝑡 𝑝 𝑙𝑠 𝑝 𝑙𝑡 ··· ··· 𝜇 ∗ 𝑠 𝜇 ∗ 𝑡 ··· ... ... ... ... ... ... ... ... ... ... ... ( 𝑘,𝑠,𝑡 ) · · · − · · · · · · − log 𝑝 𝑘𝑠 log 𝑝 𝑘𝑡 · · · · · · − · · · ( 𝑘,𝑡,𝑠 ) · · · − · · · · · · log 𝑝 𝑘𝑠 − log 𝑝 𝑘𝑡 · · · · · · − · · · ( 𝑙,𝑠,𝑡 ) · · · − · · · · · · − log 𝑝 𝑙𝑠 log 𝑝 𝑙𝑡 · · · · · · − · · · ( 𝑙,𝑡,𝑠 ) · · · − · · · · · · 𝑝 𝑙𝑠 − log 𝑝 𝑙𝑡 · · · · · · − · · · ... ... ... ... ... ... ... ... ... ... ... . Similarly, the second matrix 𝐵 has 𝐾 × 𝑆 + 𝐾 × 𝑆 + 𝑆 columns. There is one row for every pair ( 𝑘, 𝑠 ) and ( 𝑘 (cid:48) , 𝑠 (cid:48) ) with 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) . In the row corresponding to ( 𝑘, 𝑠 ) and ( 𝑘 (cid:48) , 𝑠 (cid:48) ) we have zeroeseverywhere with the exception of a − 𝑣 𝑘𝑠 and a 1 in the column for 𝑣 𝑘 (cid:48) 𝑠 (cid:48) .We use the function fmincon to nd a solution 𝒛 ∗ and the value of the problem (i.e., 𝑒 ∗ ), withmax 𝐴 · 𝒛 being the objective function we are going to minimize and 𝐵 · 𝒛 ≥ Minimum Perturbation Test
Rationale behind the test.
We provide a detailed exposition of how we derive our test. Let 𝐻 and 𝐻 denote the null hypothesis that the true dataset 𝐷 true = ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = is OEU rationaland the alternative hypothesis that 𝐷 true is not OEU rational. To construct our test, consider anumber E ∗ , which is the result of the following optimization problem given a dataset 𝐷 true :min ( 𝑣 𝑘𝑠 ,𝜆 𝑘 ,𝜀 𝑘𝑠 ) 𝑠,𝑘 max 𝑘 ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 𝜀 𝑘𝑠 𝜀 𝑘𝑡 s.t. log 𝜇 ∗ 𝑠 + log 𝑣 𝑘𝑠 − log 𝜆 𝑘 − log 𝑝 𝑘𝑠 − log 𝜀 𝑘𝑠 = 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log 𝑣 𝑘𝑠 ≤ log 𝑣 𝑘 (cid:48) 𝑠 (cid:48) . (E.1)Under 𝐻 , the true dataset 𝐷 true = ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = is OEU rational. A slight modication of Lemma 7in Echenique and Saito (2015) implies that there exist strictly positive numbers (cid:101) 𝑣 𝑘𝑠 , and (cid:101) 𝜆 𝑘 for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 such thatlog 𝜇 ∗ 𝑠 + log (cid:101) 𝑣 𝑘𝑠 − log (cid:101) 𝜆 𝑘 − log 𝑝 𝑘𝑠 = 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log (cid:101) 𝑣 𝑘𝑠 ≤ log (cid:101) 𝑣 𝑘 (cid:48) 𝑡𝑠 . Substituting the relationship ˜ 𝑝 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 for all 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 yieldslog 𝜇 ∗ 𝑠 + log (cid:101) 𝑣 𝑘𝑠 − log (cid:101) 𝜆 𝑘 − log ˜ 𝑝 𝑘𝑠 = log 𝜀 𝑘𝑠 and 𝑥 𝑘𝑠 > 𝑥 𝑘 (cid:48) 𝑠 (cid:48) = ⇒ log (cid:101) 𝑣 𝑘𝑠 ≤ log (cid:101) 𝑣 𝑘 (cid:48) 𝑠 (cid:48) , which implies that the tuple ( (cid:101) 𝑣 𝑘𝑠 , (cid:101) 𝜆 𝑘 , 𝜀 𝑘𝑠 ) 𝑠,𝑘 satises the constraint in problem (E.1).Letting E ∗ (cid:16) ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = (cid:17) denote the optimal value of the problem (E.1), we have E ∗ (cid:16) ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = (cid:17) ≤ max 𝑘 ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 𝜀 𝑘𝑠 𝜀 𝑘𝑠 = (cid:98) E under the null hypothesis.We construct a test as follows: reject 𝐻 if ∫ ∞E ∗ ( ( 𝑝 𝑘 ,𝑥 𝑘 ) 𝐾𝑘 = ) 𝑓 (cid:98) E ( 𝑧 ) 𝑑𝑧 < 𝛼 accept 𝐻 otherwise , where 𝛼 is the size of the test and 𝑓 (cid:98) E is the density function of the distribution of (cid:98) E = max 𝑘,𝑠,𝑡 𝜀 𝑘𝑠 / 𝜀 𝑘𝑡 .Given a nominal size 𝛼 , we can nd a critical value 𝐶 𝛼 satisfying Pr [ (cid:98) E > 𝐶 𝛼 ] = 𝛼 ; we set 𝐶 𝛼 = 𝐹 − (cid:98) E ( − 𝛼 ) , where 𝐹 (cid:98) E denotes the cumulative distribution function of (cid:98) E . However, because E ∗ (cid:16) ( 𝑝 𝑘 , 𝑥 𝑘 ) 𝐾𝑘 = (cid:17) ≤ (cid:98) E , the true size of the test is better than 𝛼 . Concretely, size = Pr [E ∗ > 𝐶 𝛼 ] ≤ Pr [ (cid:98) E > 𝐶 𝛼 ] = 𝛼 . 14 arameter tuning. In order to perform the test, we need to obtain the distribution of (cid:98) E andits critical value 𝐶 𝛼 given a signicance level 𝛼 . We obtain the distribution of (cid:98) E by assuming that 𝜀 follows a log-normal distribution 𝜀 ∼ Λ ( 𝜈, 𝜉 ) . The crucial step in our approach is the selection of parameters ( 𝜈, 𝜉 ) . It is natural to choosethese parameters so that there is no price perturbation on average (i.e., E [ 𝜀 ] = ( 𝜀 ) . Therefore,we use variation in (relative) prices observed in the data.We have assumed that ˜ 𝑝 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 for all 𝑠 ∈ 𝑆 , 𝑘 ∈ 𝐾 , and the noise term 𝜀 is independent ofthe random selection of budgets ( 𝑝 𝑘𝑠 ) 𝑠 ∈ 𝑆,𝑘 ∈ 𝐾 . Hence,Var ( ˜ 𝑝 ) = Var ( 𝑝 ) · Var ( 𝜀 ) + Var ( 𝑝 ) · E [ 𝜀 ] + E [ 𝑝 ] · Var ( 𝜀 )⇐⇒ Var ( ˜ 𝑝 ) Var ( 𝑝 ) = E [ 𝜀 ] + (cid:18) + E [ 𝑝 ] Var ( 𝑝 ) (cid:19) Var ( 𝜀 ) . Given the observed variation in ( 𝑝 𝑘𝑠 ) 𝑠 ∈ 𝑆,𝑘 ∈ 𝐾 , Var ( 𝜀 ) determines how much larger (or smaller, inratio) the variation of perturbed prices ( ˜ 𝑝 𝑘𝑠 ) 𝑠 ∈ 𝑆,𝑘 ∈ 𝐾 is relative to actual prices.Let us consider an agent who has trouble telling the two variances apart. More generally,the agent has trouble telling the distributions of prices apart, that is why she is confusing actualand perceived prices, but the distribution depends only on the variance; so we focus on variance.Consider a hypothesis test for the null hypothesis that the variance of a normal random variablewith known mean has variance 𝜎 against the alternative that 𝜎 ≥ 𝜎 . Let ˆ 𝜎 𝑛 be the samplevariance.The agent performs an upper-tailed chi-squared test dened asH : 𝜎 = 𝜎 H : 𝜎 > 𝜎 The test statistic is: 𝑇 𝑛 = ( 𝑛 − ) ˆ 𝜎 𝑛 𝜎 where 𝑛 is the sample size (i.e., the number of budget sets). The sampling distribution of the teststatistic 𝑇 𝑛 under the null hypothesis follows a chi-squared distribution with 𝑛 − 𝜂 𝐼 of rejecting the null hypothesis when it is true, a type I error;and the probability 𝜂 II of failing to reject the null hypothesis when the alternative 𝜎 = 𝜎 > 𝜎 Note that parameters ( 𝜈, 𝜉 ) correspond to the mean and the variance of the random variable in the log-scale.In other words, log 𝜀 ∼ 𝑁 ( 𝜈, 𝜉 ) . The moments of the log-normal distribution 𝜀 ∼ Λ ( 𝜈, 𝜉 ) are then calculated by E [ 𝜀 ] = exp ( 𝜈 + 𝜉 / ) and Var ( 𝜀 ) = exp ( 𝜈 + 𝜉 )( exp ( 𝜉 ) − ) .
15s true, a type II error. The test rejects the null hypothesis that the variance is 𝜎 if 𝑇 𝑛 > 𝜒 − 𝛼,𝑛 − where 𝜒 − 𝛼,𝑛 − is the critical value of a chi-squared distribution with 𝑛 − 𝛼 , dened by Pr [ 𝜒 < 𝜒 − 𝛼,𝑛 − ] = − 𝜂 𝐼 . Under the alternative hypothesis that 𝜎 = 𝜎 > 𝜎 , the statistic ( 𝜎 / 𝜎 ) · 𝑇 𝑛 follows a chi-squared distribution (with 𝑛 − 𝜂 II of making a type IIerror is given by 𝜂 II = Pr [ 𝑇 𝑛 < 𝜒 − 𝛼,𝑛 − | H : 𝜎 > 𝜎 is true ] = Pr (cid:20) 𝜎 𝜎 · 𝑇 𝑛 < 𝜎 𝜎 · 𝜒 − 𝛼,𝑛 − (cid:21) = Pr (cid:20) 𝜒 < 𝜎 𝜎 · 𝜒 − 𝛼,𝑛 − (cid:21) . Let 𝜒 𝛽,𝑛 − be the value that satises Pr [ 𝜒 < 𝜒 𝛽,𝑛 − ] = 𝜂 II . Then, given 𝜂 𝐼 and 𝜂 II , we obtainPr (cid:20) 𝜒 < 𝜎 𝜎 · 𝜒 − 𝛼,𝑛 − (cid:21) = 𝜂 II ⇐⇒ 𝜎 𝜎 · 𝜒 − 𝛼,𝑛 − = 𝜒 𝛽,𝑛 − ⇐⇒ 𝜎 𝜎 = 𝜒 − 𝛼,𝑛 − 𝜒 𝛽,𝑛 − . As a consequence, given a measured variance 𝜎 , calculated from observed prices, and as-sumed values for 𝜂 𝐼 and 𝜂 II , we can back out the minimum “detectable” value of the variance 𝜎 .From this variance of prices, we obtain Var ( 𝜀 ) . An alternative approach, without assuming that a distribution for 𝑇 𝑛 , and based on a large sample approximationto the distribution of 𝑇 𝑛 , yields very similar results. Calculations and empirical ndings are available from the authorsupon request. Supplementary Empirical Analysis
F.1 First-Order Stochastic Dominance
In the portfolio allocation environment studied in the three studies we looked at, choosing anallocation ( 𝑥 , 𝑥 ) from a budget line dened by prices ( 𝑝 , 𝑝 ) violates monotonicity with respectto rst-order stochastic dominance (FOSD-monotonicity) when either (i) 𝑝 > 𝑝 and 𝑥 > 𝑥 or (ii) 𝑝 > 𝑝 and 𝑥 > 𝑥 (i.e., the choice involves more allocation toward more-expensive security).Table F.1 presents the average fraction (out of 25) of choices violating FOSD-monotonicityand the number of subjects without FOSD-monotonicity violations. On average, subjects made24-34% violations of FOSD-monotonicity. The number of subjects who made no FOSD-violatingchoices is less than 10% for all datasets. As discussed in Choi et al. (2014), choices can be consistentwith GARP even with violations of FOSD-monotonicity. The average fraction of FOSD-violatingchoices calculated from the subsample of GARP-compliant (CCEI =
1) subjects is close to the onewe obtain from the whole sample. The entire distributions are presented in Figure F.1.
Table F.1: FOSD violation.
All subjects CCEI = 1CKMS CMW CS CKMS CMW CSNumber of subjects 1,182 1,116 1,421 270 207 313Average fraction of FOSD-mon. violations 0.335 0.320 0.239 0.364 0.312 0.221Fraction of subjects without FOSD-mon. violations 0.025 0.047 0.066 0.066 0.164 0.153
Frac. FOSD−mon violation CD F CKMSCMWCS A Frac. FOSD−mon violation CD F CKMSCMWCS B Figure F.1: Empirical CDFs of fraction of choices that violate FOSD-monotonicity. (A) All subjects. (B)Subjects with CCEI = .2 Choices on the 45-Degree Line In the experiments, subjects made choices of allocations ( 𝑥 , 𝑥 ) by clicking on the budget linegraphically presented on the screen. Note that points on the 45-degree line correspond to equalallocations between the two accounts ( 𝑥 = 𝑥 ) and therefore involve no risk (i.e., the 45-degreeline is the “full insurance” line). If a subject’s all choices are on the 45-degree line (call suchpattern diagonal allocations ), we can rationalize the data with EU and hence 𝑒 ∗ = 𝑒 ∗ (through violations of the downward-sloping demand) whileCCEI and EU-CCEI stay close to 1 (see Figure 7, panel D). In this section, we examine how muchof the disagreement between 𝑒 ∗ and CCEI or EU-CCEI are driven by small deviations from thediagonal allocations.To this end, we rst re-dene diagonal allocations. Instead of requiring all choices to beexactly on he 45-degree line, we call a data almost diagonal allocations if all choices are insidesmall balls (with xed radius 𝑟 ) drawn around the intersections of budget lines and the 45-degreeline. We can control the size of acceptable deviations by changing the radius 𝑟 of the ball. Theidea is shown in Figure F.2. In this example, chosen allocations (black dots) are not exactly on the45-degree line, but they are inside the balls around the diagonal allocations (red circles). 𝑥 𝑥 Figure F.2: Almost diagonal allocations. These choices also violate FOSD-monotonicity. We would expect relatively large 𝑒 ∗ from this choice pattern,but its CCEI is 1 because it satises GARP. 𝑟 . Between 6% and 12% of subjects made such choice pattern whenthe radius is set to 𝑟 = Table F.2: Fraction of subjects who made almost diagonal allocations.
Radius of the ball ( 𝑟 )Study 𝑁 .
05 0 .
20 0 .
50 1 . 𝑒 ∗ and CCEI as well as EU-CCEI, as inFigure 8 (Section 4.2). Bottom panels in each gure focus on subjects who made almost diagonalallocations (the radius of the ball is set to 𝑟 =
1) in all 25 questions, and top panels present therest of the subjects.Bottom panels in each gure conrm that almost diagonal allocations yield values of CCEIand EU-CCEI that are close to 1. The same subjects have dispersed values of 𝑒 ∗ , including thehighest value in each experiment.It does not meant that the disagreement between 𝑒 ∗ and CCEI-based measures come mainlyfrom slight deviations from the diagonal allocations. Top panels in each gure show that thereare choice patterns, other than almost diagonal allocations, that have CCEI/EU-CCEI ≈ 𝑒 ∗ . 19 .000.250.500.751.00 0.0 2.5 5.0 7.5 10.0 Minimal e CC E I A Minimal e CC E I B Minimal e CC E I C Minimal e CC E I Minimal e CC E I Minimal e CC E I Figure F.3: Correlation between 𝑒 ∗ and CCEI. Top panels show subjects who did not choose almost di-agonal allocations and bottom panels show those who selected almost diagonal allocations (with 𝑟 = Minimal e E U − CC E I A Minimal e E U − CC E I B Minimal e E U − CC E I C Minimal e E U − CC E I Minimal e E U − CC E I Minimal e E U − CC E I Figure F.4: Correlation between 𝑒 ∗ and EU-CCEI. Top panels show subjects who did not choose almostdiagonal allocations and bottom panels show those who selected almost diagonal allocations (with 𝑟 = .3 Sensitivity As is clear from the denition, our measure 𝑒 ∗ is a bound that has to hold across all observationsand states (see conditions (4), (5), and (6) in the denitions of 𝑒 -perturbed OEU in Section 3). It ispossible that a couple of “bad” choices signicantly inuence the measure. This section presentsseveral robustness checks for the main empirical result. Dropping critical mistakes.
In this robustness check, we recalculate 𝑒 ∗ using subsets of ob-served choices that exclude outliers. More precisely, for each subject, we calculate 𝑒 ∗ for allcombinations of 25 − 𝑚 choices and pick the smallest 𝑒 ∗ . We do this for 𝑚 = , 𝑒 ∗ and CCEI (Fig-ure F.6) nor between 𝑒 ∗ and demographic characteristics (Figures F.7 and F.8). In this sense, themain empirical results are robust to the presence of small number of bad choices. Minimal e CD F AllDrop 1Drop 2 A1 Minimal e
AllDrop 1Drop 2 B1 Minimal e
AllDrop 1Drop 2 C1 CCEI CD F AllDrop 1Drop 2 A2 CCEI
AllDrop 1Drop 2 B2 CCEI
AllDrop 1Drop 2 C2 Figure F.5: Empirical CDFs of 𝑒 ∗ and CCEI, using all observations or subsets of observations dropping oneor two critical mistakes. Panels: (A) CKMS, (B) CMW, (C) CS. .250.500.751.00 0.0 2.5 5.0 7.5 10.0 CC E I CC E I Minimal e CC E I Minimal e
Minimal e
Figure F.6: Correlation between 𝑒 ? and CCEI. (Top panels) All 25 observations. (Middle panels) Drop onecritical mistake. (Bottom panels) Drop two critical mistakes. .61.71.81.92.02.1 Female Male Gender M i n i m a l e ( d r op ) A Age M i n i m a l e ( d r op ) B Education level M i n i m a l e ( d r op ) C Working M i n i m a l e ( d r op ) D CRT score M i n i m a l e ( d r op ) E1 Stroop RT M i n i m a l e ( d r op ) E2 Monthly income (Euro) M i n i m a l e ( d r op ) F1 Annual income (USD) M i n i m a l e ( d r op ) F2 Annual income (USD) M i n i m a l e ( d r op ) F3 CKMS CMW CS
Figure F.7: Robustness of demographic correlations in Figure 9. For each subject, 𝑒 ∗ is recalculated afterdropping one critical mistake. .21.31.41.5 Female Male Gender M i n i m a l e ( d r op ) A Age M i n i m a l e ( d r op ) B Education level M i n i m a l e ( d r op ) C Working M i n i m a l e ( d r op ) D CRT score M i n i m a l e ( d r op ) E1 Stroop RT M i n i m a l e ( d r op ) E2 Monthly income (Euro) M i n i m a l e ( d r op ) F1 Annual income (USD) M i n i m a l e ( d r op ) F2 Annual income (USD) M i n i m a l e ( d r op ) F3 CKMS CMW CS
Figure F.8: Robustness of demographic correlations in Figure 9. For each subject, 𝑒 ∗ is recalculated afterdropping two critical mistakes. Average” perturbation.
Let ¯ 𝑒 be the solution to the following minimization problem:min ( 𝜀 𝑘𝑠 ) 𝑠,𝑘 ∑︁ 𝑘 ∈ 𝐾 ∑︁ 𝑠 ∈ 𝑆 (cid:12)(cid:12) log 𝜀 𝑘𝑠 (cid:12)(cid:12) 𝐾𝑆 s.t. ( 𝑥 𝑘 , 𝑞 𝑘 ) 𝐾𝑘 = is OEU rational 𝑞 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 for each 𝑠 ∈ 𝑆, 𝑘 ∈ 𝐾 The idea behind this alternative measure is simple. As in the case of 𝑒 -price-perturbed util-ity, we search for sets of multiplicative noises ( 𝜀 𝑘𝑠 ) 𝑠,𝑘 which could rationalize the observed data.Instead of looking at the uniform bound max 𝑠,𝑡,𝑘 ( log 𝜀 𝑘𝑠 − log 𝜀 𝑘𝑡 ) and minimizing it, we take the av-erage of these perturbations and minimize it. A similar idea was applied to quantify the distancefrom several models of time preferences in Echenique et al. (2016).Figure F.9 presents the relationship between ¯ 𝑒 , 𝑒 ∗ , and CCEI. Figure F.10 shows the correlationbetween ¯ 𝑒 and demographic variables. These gures do not show correlational patterns thatare markedly dierent from those presented in the main empirical results (Figures 8 and 9 inSection 4.2). Avg. perturbation M i n i m a l e A1 Avg. perturbation M i n i m a l e B1 Avg. perturbation M i n i m a l e C1 Avg. perturbation CC E I A2 Avg. perturbation CC E I B2 Avg. perturbation CC E I C2 Figure F.9: Correlation between ¯ 𝑒 and 𝑒 ∗ (top panels) and ¯ 𝑒 and CCEI (bottom panels). Panels: (A) CKMS,(B) CMW, (C) CS. .180.190.200.21 Female Male Gender M i n i m a l e ( a v g . ) A Age M i n i m a l e ( a v g . ) B Education level M i n i m a l e ( a v g . ) C Working M i n i m a l e ( a v g . ) D CRT score M i n i m a l e ( a v g . ) E1 Stroop RT M i n i m a l e ( a v g . ) E2 Monthly income (Euro) M i n i m a l e ( a v g . ) F1 Annual income (USD) M i n i m a l e ( a v g . ) F2 Annual income (USD) M i n i m a l e ( a v g . ) F3 CKMS CMW CS
Figure F.10: ¯ 𝑒 and demographic variables. .4 Properties of 𝑒 ∗ 𝑒 ∗ from observed and simulated choices. The statistical approach described in Section 4.3is one way to assess “how big” the observed 𝑒 ∗ ’s are. Another way is to simulate choice dataassuming some behavioral model and calculate 𝑒 ∗ on the simulated dataset. Following Bronars(1987), we randomly select an allocation from each budget line. Since subjects in CKMS andCS faced a randomly selected set of budgets, we rst randomly select one set of budgets (fromthe observed sets of budgets) and then randomly choose allocations on these budgets. We thencalculate 𝑒 ∗ , as well as CCEI, using the simulated choices. We repeat this 10,000 times for each ofthe three datasets.Figure F.11 compares the observed and simulated 𝑒 ∗ . The distribution of observed 𝑒 ∗ lo-cates left of simulated 𝑒 ∗ (all dierences are statistically signicant, according to two-sampleKolmogorov-Smirnov test). The actual subjects’ behavior is thus closer to OEU rationality com-pared to completely random behavior (even though complete random is unrestrictive and maynot be the best benchmark).Figure F.12 looks at the correlation between 𝑒 ∗ and CCEI and compares the pattern in observedand simulated datasets (panels A-C in the top row are same as Figure 8).27 .000.250.500.751.00 2 4 6 8 Minimal e CD F ObservedSimulated A1 Minimal e CD F ObservedSimulated B1 Minimal e CD F ObservedSimulated C1 CCEI CD F ObservedSimulated A2 CCEI CD F ObservedSimulated B2 CCEI CD F ObservedSimulated C2 Figure F.11: Comparison between observed and simulated 𝑒 ∗ (top panels) and CCEI (bottom panels). Pan-els: (A) CKMS, (B) CMW, (C) CS. Minimal e CC E I A1 Minimal e CC E I B1 Minimal e CC E I C1 Minimal e CC E I A2 Minimal e CC E I B2 Minimal e CC E I C2 Figure F.12: Comparison between observed (top panels) and simulated (bottom panels) 𝑒 ∗ and CCEI. Pan-els: (A) CKMS, (B) CMW, (C) CS. Notes : Top panels are identical to those in Figure 8. ound of 𝑒 ∗ . The value of 𝑒 ∗ depends on the structure of the budgets an agent faces. In par-ticular, it is clear from 𝑒 -PSAROEU that 1 + 𝑒 ∗ is bounded by the maximum ratio of risk-neutralprices: 1 + 𝑒 ∗ ≤ max 𝑘, ∈ 𝐾,𝑠,𝑡 ∈ 𝑆 𝜌 𝑘𝑠 𝜌 𝑘𝑡 . Since CKMS, CMW, and CS experiments all used two equally-likely states, the ratio of risk-neutralprices is equal to the ratio of prices. Figure F.13 shows the observed 𝑒 ∗ and (participant-specic)upper bound. (Since all subjects faced the same set of budgets in the CMW study, there is only onevertical line.) About 13% of the subjects (475 / / / / 𝑒 ∗ exactly at the upper bound. Bound M i n i m a l e A Bound M i n i m a l e B Bound M i n i m a l e C Figure F.13: Bound of 𝑒 ∗ . The 𝑥 -axis in each plot is the upper bound of 𝑒 ∗ , given by max 𝑘,𝑠,𝑡 𝑝 𝑘𝑠 / 𝑝 𝑘𝑡 − Notes : There is no variation in bounds in the CMW data (panel B) since all subjects faced the same set ofbudgets. In the CS data (panel C), the 𝑥 -axis is cut at 10 for better visualization. There are 22 additionalobservations in the data with the bounds ranging from 11 to 48. .5 Illustration of 𝑒 -Perturbed OEU In Figure 7, we present typical choice patterns from selected subjects with CCEI = 𝑒 ∗ . Panels A-F plot observed choices and panels a-f plot the relationship betweenlog ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) , which shows how much the dataset conforms to the downward-sloping demand. The measure 𝑒 ∗ , roughly speaking, captures the degree of deviation from thedownward-sloping demand.Consider an observed dataset ( 𝑥 𝑘 , 𝑝 𝑘 ) 𝐾𝑘 = and a perturbed dataset ( 𝑥 𝑘 , ˜ 𝑝 𝑘 ) 𝐾𝑘 = , where ˜ 𝑝 𝑘𝑠 = 𝑝 𝑘𝑠 𝜀 𝑘𝑠 and 𝜀 𝑘𝑠 ≥ 𝑠 ∈ 𝑆 and 𝑘 ∈ 𝐾 . Since we x the chosen bundle ( 𝑥 𝑘 ) 𝐾𝑘 = and rotate the budgetlines around them, price perturbation “moves” points in panels a-f horizontally.To make the dataset 𝑒 -price-perturbed OEU rational (Denition 4), we need to move the pointshorizontally so that they satisfy the downward-sloping demand. Note that the horizontal distancefor each observation 𝑘 is given bylog (cid:32) ˜ 𝑝 𝑘 ˜ 𝑝 𝑘 (cid:33) − log (cid:32) 𝑝 𝑘 𝑝 𝑘 (cid:33) = log (cid:32) ˜ 𝑝 𝑘 / 𝑝 𝑘 ˜ 𝑝 𝑘 / 𝑝 𝑘 (cid:33) = log (cid:32) 𝜀 𝑘 𝜀 𝑘 (cid:33) . We thus need to look at the maximal horizontal adjustment among observations, and the measure 𝑒 ∗ is obtained by minimizing it.Figure F.14 shows the idea behind calculation of 𝑒 ∗ using price perturbation. It plots the samesix subjects as in Figure 7. In panels A-F, red dotted lines represent the original budgets and bluesolid lines represent perturbed budgets. In panels a-f, green circles represent the original datasetand blue triangles represent the perturbed dataset. Red arrows connect points that correspondto the maximal adjustment. The gure shows that 𝑒 ∗ -perturbed datasets satisfy the downward-sloping demand. We can draw several observations about the practical aspect of 𝑒 ∗ . First, observe that the“cheapest” way for correcting choices violating FOSD-monotonicity is to perturb budgets corre-sponding to these observations so that ˜ 𝑝 𝑘 = ˜ 𝑝 𝑘 . Second, the gure provides an intuitive explana-tion of why 𝑒 ∗ can be large for choice patterns like panel D. Since clicking on the point exactlyon the 45-degree line is a challenging task, choices would scatter around the 45-degree line, oc-casionally falling in the region of FOSD-monotonicity. No matter how small these deviationsfrom the 45-degree line are, 𝑒 -price perturbation requires horizontal adjustments to achieve thedownward-sloping demand. If the necessary adjustment is applied on a relatively extreme budgetline, 𝑒 ∗ for such a subject can be very high. Perturbed dataset in each panel is based on one particular set of ( 𝜀 𝑘𝑠 ) 𝑠 ∈ 𝑆,𝑘 ∈ 𝐾 returned by Matlab . There aresmall deviations from the downward-sloping demand (e.g., in panels C and E), but it is possible to correct for thesenumerical deviations without inuencing the value of 𝑒 ∗ . State 1 S t a t e CCEI = 1.000EU−CCEI = 1.000Minimal e = 0.060 A State 1 S t a t e CCEI = 1.000EU−CCEI = 1.000Minimal e = 0.310 B State 1 S t a t e CCEI = 1.000EU−CCEI = 0.960Minimal e = 2.490 C −1.000.001.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.99f = 0.00 a −2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.96f = 0.00 b −0.500.000.50 −1 0 1 log ( p p ) l og ( x x ) r = −0.70f = 0.28 c State 1 S t a t e CCEI = 1.000EU−CCEI = 1.000Minimal e = 4.390 D State 1 S t a t e CCEI = 1.000EU−CCEI = 0.700Minimal e = 0.620 E State 1 S t a t e CCEI = 1.000EU−CCEI = 0.660Minimal e = 0.780 F −0.01−0.010.000.00 −1 0 1 log ( p p ) l og ( x x ) r = 0.13f = 0.64 d −2.000.002.00 −2 −1 0 1 2 log ( p p ) l og ( x x ) r = −0.91f = 0.04 e −3.000.003.006.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.81f = 0.12 f Figure F.14: Illustration of 𝑒 -price-perturbed OEU rationalization. (A-F) Perturbed budgets (blue solidlines) and the original budgets (red dotted lines). (a-f) The relation between log ( 𝑥 / 𝑥 ) and log ( 𝑝 / 𝑝 ) (green circles), and log ( 𝑥 / 𝑥 ) and log ( ˜ 𝑝 / ˜ 𝑝 ) (blue triangles). Red arrows indicate observations requiringthe largest adjustment. .6 Comparing Measures We calculate CCEI at which a subject is consistent with a given model, stochastically monotoneutility maximization (Nishimura et al., 2017), EU, and concave EU, using the GRID method devel-oped in Polisson et al. (2020). We call these measures F-GARP, EU-CCEI, and cEU-CCEI. For agiven dataset, the measures are ordered ascEU-CCEI ≤ EU-CCEI ≤ F-GARP ≤ CCEI , since models we look at are nested in this order. Note that Polisson et al. (2020) calculated andreported CCEI, F-GARP, EU-CCEI, and cEU-CCEI for the CKMS dataset but not for the CMW andthe CS datasets.Figures F.15-F.17 compare 𝑒 ∗ , CCEI, and these three additional measures. Panels on thediagonal show the distribution of each measure. Pairwise scatter plots are presented below diag-onal, and their Spearman’s correlation coecients are shown above the diagonal (all 𝑝 < . 𝑒 ∗ and other measures. Thesecond and the fourth panels in this column ( 𝑒 ∗ vs. CCEI and 𝑒 ∗ vs. EU-CCEI) are identical tothose presented in Figure 8. As we discussed in Section 4.2 of the paper, we see that there are asignicant number of subjects whose CCEI and EU-CCEI are close to one but their 𝑒 ∗ ’s are widelydispersed and further away from zero.This observation is not specic to CCEI and EU-CCEI. In the third and the fth panels of thesame column, we can see a similar pattern between 𝑒 ∗ and F-GARP as well as 𝑒 ∗ and EU-CCEI.The pattern is a general feature that distinguishes the idea behind the measures: 𝑒 ∗ is based onrotating budget lines while the other measures, which are all variants of CCEI, are based onshrinking budget sets. A stohastically monotone utility function gives strictly higher utility to bundle 𝑥 compared to another bundle 𝑦 if 𝑥 rst-order stochastically dominates 𝑦 and gives them the same utility if two bundles are stochastically equivalent. Inthe environment we consider (two states with equally likely objective probabilities), a utility function is stochasticallymonotone if and only if it is symmetric and strictly increasing.Choi et al. (2014) also discuss a similar idea. They propose additional measure, which jointly captures the extentof GARP violations and violations of stochastic dominance, by combining the observed data and its “mirror-image”.More precisely, they assume that if an allocation ( 𝑥 , 𝑥 ) is chosen under the budget constraint 𝑝 𝑥 + 𝑝 𝑥 =
1, then ( 𝑥 , 𝑥 ) would have been chosen under the mirror-image budget constraint 𝑝 𝑥 + 𝑝 𝑥 =
1. They then re-calculateCCEI for the “combined” data consisting of 50 (25 budgets ×
2) choices. We did not compute cEU-CCEI for 23 subjects (8 in CMW, and 15 in CS) since the code spent signicantly longcomputation time. (Polisson et al. (2020) used a high-performance computing facility.) We also treated cEU-CCEIfor six subjects in CS as missing values, since the code incorrectly returned cEU-CCEI =
0. Note that F-GARP andEU-CCEI for these 29 subjects are included in Figures F.15-F.17. orr:−0.178*** Corr:−0.402***Corr:0.715*** Corr:−0.401***Corr:0.717***Corr:0.997*** Corr:−0.386***Corr:0.726***Corr:0.976***Corr:0.983*** Minimal e CCEI F−GARP EU−CCEI cEU−CCEI M i n i m a l e CC E I F − G A R PE U − CC E I c E U − CC E I . . . . .
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75 1 . Figure F.15: Comparing measures of rationality in the CKMS data. orr:−0.119*** Corr:−0.215***Corr:0.844*** Corr:−0.210***Corr:0.843***Corr:0.998*** Corr:−0.193***Corr:0.848***Corr:0.986***Corr:0.989*** Minimal e CCEI F−GARP EU−CCEI cEU−CCEI M i n i m a l e CC E I F − G A R PE U − CC E I c E U − CC E I . . . . . .
25 0 .
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75 1 . Figure F.16: Comparing measures of rationality in the CMW data. orr:−0.352*** Corr:−0.446***Corr:0.831*** Corr:−0.451***Corr:0.830***Corr:0.996*** Corr:−0.445***Corr:0.834***Corr:0.960***Corr:0.966*** Minimal e CCEI F−GARP EU−CCEI cEU−CCEI M i n i m a l e CC E I F − G A R PE U − CC E I c E U − CC E I . . . . . . . . . . .
25 0 .
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75 1 . Figure F.17: Comparing measures of rationality in the CS data. .7 Choice Pattern: Additional Examples Choice data from four subjects presented in Section 4.2, Figure 7, are not meant to be representa-tive of the entire dataset consisting of more than 3,000 subjects. In this section, we present moreexamples to understand the similarity and dierences between 𝑒 ∗ , CCEI, and EU-CCEI.We pick subjects from the CMW experiment, where all the subjects faced with the same setof 25 budget lines. This feature of the design makes the variation of 𝑒 ∗ smaller than in the otherdatasets (we observe several “jumps” in the empirical CDF of 𝑒 ∗ in Figure 4), but the comparisonacross choice patterns becomes easier.Figure F.18 is the scatterplot of 𝑒 ∗ and EU-CCEI in the CMW data. Dashed lines represent the25th, 50th, and 75th percentiles of 𝑒 ∗ and EU-CCEI. Two shaded areas represent combinationsof 𝑒 ∗ and EU-CCEI that “disagree”, in the sense that one measure says the subject is close to EU(relative to the median subject) but the other measure says the same subject far from EU (again,relative to the median subject). Each subject’s choice pattern is shown below. E U − CC E I Figure F.18: 𝑒 ∗ and EU-CCEI in CMW data. Notes : Vertical dashed lines represent the 25th, 50th, and 75thpercentiles of 𝑒 ∗ . Horizontal dashed lines represent the 25th, 50th, and 75th percentiles of EU-CCEI. State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 1.000e = 0.060 State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 0.980e = 0.240 State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 0.990e = 0.990 −1.000.001.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.99f = 0.00 −4.00−2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.98f = 0.04 −2.00−1.000.001.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.62f = 0.00 State 1 S t a t e CCEI = 0.960F−GARP = 0.940EU−CCEI = 0.940cEU−CCEI = 0.900e = 0.680 State 1 S t a t e CCEI = 0.950F−GARP = 0.900EU−CCEI = 0.900cEU−CCEI = 0.900e = 0.670 State 1 S t a t e CCEI = 0.940F−GARP = 0.940EU−CCEI = 0.940cEU−CCEI = 0.920e = 1.460 −10.00−5.000.005.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.93f = 0.04 −2.00−1.000.001.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.85f = 0.20 −4.00−2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.81f = 0.28 State 1 S t a t e CCEI = 1.000F−GARP = 0.960EU−CCEI = 0.960cEU−CCEI = 0.960e = 2.720 State 1 S t a t e CCEI = 1.000F−GARP = 0.990EU−CCEI = 0.960cEU−CCEI = 0.950e = 3.590 State 1 S t a t e CCEI = 1.000F−GARP = 1.000EU−CCEI = 1.000cEU−CCEI = 1.000e = 4.390 −0.500.000.50 −1 0 1 log ( p p ) l og ( x x ) r = −0.69f = 0.32 −0.80−0.60−0.40−0.200.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.43f = 0.44 −0.01−0.010.000.00 −1 0 1 log ( p p ) l og ( x x ) r = 0.13f = 0.64 State 1 S t a t e CCEI = 0.980F−GARP = 0.930EU−CCEI = 0.930cEU−CCEI = 0.860e = 3.090 State 1 S t a t e CCEI = 1.000F−GARP = 0.930EU−CCEI = 0.930cEU−CCEI = 0.930e = 3.590 State 1 S t a t e CCEI = 0.960F−GARP = 0.930EU−CCEI = 0.930cEU−CCEI = 0.910e = 4.390 −1.000.001.002.003.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.79f = 0.20 log ( p p ) l og ( x x ) r = −0.37f = 0.32 −1.00−0.500.000.50 −1 0 1 log ( p p ) l og ( x x ) r = −0.50f = 0.36 State 1 S t a t e CCEI = 1.000F−GARP = 0.700EU−CCEI = 0.700cEU−CCEI = 0.700e = 0.620 State 1 S t a t e CCEI = 1.000F−GARP = 0.660EU−CCEI = 0.660cEU−CCEI = 0.660e = 0.780 State 1 S t a t e CCEI = 0.900F−GARP = 0.680EU−CCEI = 0.680cEU−CCEI = 0.680e = 0.990 −2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.91f = 0.04 −3.000.003.006.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.81f = 0.12 −4.00−2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.90f = 0.16 State 1 S t a t e CCEI = 0.940F−GARP = 0.690EU−CCEI = 0.690cEU−CCEI = 0.690e = 1.460 State 1 S t a t e CCEI = 0.990F−GARP = 0.700EU−CCEI = 0.700cEU−CCEI = 0.700e = 1.590 State 1 S t a t e CCEI = 0.950F−GARP = 0.590EU−CCEI = 0.590cEU−CCEI = 0.590e = 1.320 −2.000.002.004.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.78f = 0.20 −4.00−2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.76f = 0.16 −3.00−2.00−1.000.001.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.74f = 0.24 State 1 S t a t e CCEI = 0.970F−GARP = 0.770EU−CCEI = 0.770cEU−CCEI = 0.700e = 2.720 State 1 S t a t e CCEI = 0.910F−GARP = 0.740EU−CCEI = 0.740cEU−CCEI = 0.650e = 3.340 State 1 S t a t e CCEI = 0.940F−GARP = 0.700EU−CCEI = 0.700cEU−CCEI = 0.700e = 2.270 −3.00−2.00−1.000.001.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.74f = 0.28 −2.00−1.000.001.002.003.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.42f = 0.36 −2.000.002.004.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.70f = 0.24 State 1 S t a t e CCEI = 0.900F−GARP = 0.670EU−CCEI = 0.670cEU−CCEI = 0.670e = 2.080 State 1 S t a t e CCEI = 0.900F−GARP = 0.510EU−CCEI = 0.510cEU−CCEI = 0.510e = 2.030 State 1 S t a t e CCEI = 0.950F−GARP = 0.370EU−CCEI = 0.370cEU−CCEI = 0.370e = 2.360 −2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.70f = 0.20 −2.000.002.004.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.43f = 0.32 −4.00−2.000.002.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.58f = 0.32 State 1 S t a t e CCEI = 0.940F−GARP = 0.450EU−CCEI = 0.450cEU−CCEI = 0.450e = 2.780 State 1 S t a t e CCEI = 1.000F−GARP = 0.300EU−CCEI = 0.300cEU−CCEI = 0.300e = 2.900 State 1 S t a t e CCEI = 0.950F−GARP = 0.240EU−CCEI = 0.240cEU−CCEI = 0.240e = 4.390 −2.500.002.505.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.36f = 0.44 −5.00−2.500.002.50 −1 0 1 log ( p p ) l og ( x x ) r = −0.53f = 0.32 −2.500.002.505.00 −1 0 1 log ( p p ) l og ( x x ) r = −0.25f = 0.40 eferences Bronars, S. 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