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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 20:00:28 +0000Strotz Meets Allais: Diminishing Impatience and the Certainty Effect: Comment
https://resolver.caltech.edu/CaltechAUTHORS:20160113-104604528
Authors: Saito, Kota
Year: 2011
DOI: 10.1257/aer.101.5.2271
Halevy (2008) states the equivalence between diminishing impatience (i.e., quasi-hyperbolic discounting) and the common ratio effect. The present paper shows that one way of the equivalence is false and shows the correct and general relationships: diminishing impatience is equivalent to the certainty effect and that strong diminishing impatience (i.e., hyperbolic discounting) is equivalent to the common ratio effect.https://authors.library.caltech.edu/records/77r2n-ghs71Social Preferences under Risk: Equality of Opportunity versus Equality of Outcome
https://resolver.caltech.edu/CaltechAUTHORS:20140109-084611692
Authors: Saito, Kota
Year: 2013
DOI: 10.1257/aer.103.7.3084
This paper axiomatizes a utility function for social preferences under risk. In the model, a single parameter captures a preference for equality of opportunity (i.e., equality of exante expected payoffs) relative to equality of outcome (i.e., equality of ex-post payoffs). In a deterministic environment, the model reduces to the model of Fehr and Schmidt (1999). The model is consistent with recent experiments on probabilistic dictator games.https://authors.library.caltech.edu/records/83q4w-j9967Preferences for Flexibility and Randomization under Uncertainty
https://resolver.caltech.edu/CaltechAUTHORS:20150423-100723008
Authors: Saito, Kota
Year: 2015
DOI: 10.1257/aer.20131030
An uncertainty-averse agent prefers betting on an event whose probability is known, to betting on an event whose probability is unknown. Such an agent may randomize his choices to eliminate the effects of uncertainty. For what sort of preferences does a randomization eliminate the effects of uncertainty? To answer this question, we investigate an agent's preferences over sets of acts. We axiomatize a utility function, through which we can identify the agent's subjective belief that a randomization eliminates the effects of uncertainty.https://authors.library.caltech.edu/records/ttmn5-crw76Savage in the Market
https://resolver.caltech.edu/CaltechAUTHORS:20150828-084210399
Authors: Echenique, Federico; Saito, Kota
Year: 2015
DOI: 10.3982/ECTA12273
We develop a behavioral axiomatic characterization of subjective expected utility (SEU) under risk aversion. Given is an individual agent's behavior in the market: assume a finite collection of asset purchases with corresponding prices. We show that such behavior satisfies a "revealed preference axiom" if and only if there exists a SEU model (a subjective probability over states and a concave utility function over money) that accounts for the given asset purchases.https://authors.library.caltech.edu/records/54sfk-nxr44Impure altruism and impure selfishness
https://resolver.caltech.edu/CaltechAUTHORS:20150820-111031910
Authors: Saito, Kota
Year: 2015
DOI: 10.1016/j.jet.2015.05.003
Altruism refers to a willingness to benefit others, even at one's own expense. In contrast, selfishness refers to prioritizing one's own interests with no consideration for others. However, even if an agent is selfish, he might nevertheless act as if he were altruistic out of selfish concerns triggered when his action is observed; that is, he might seek to feel pride in acting altruistically and to avoid the shame of acting selfishly. We call such behavior impurely altruistic. Alternatively, even if an agent is altruistic, he might nevertheless give in to the temptation to act selfishly. We call such behavior impurely selfish. This paper axiomatizes a model that distinguishes altruism from impure altruism and selfishness from impure selfishness. In the model, unique real numbers separately capture altruism and the other forces of pride, shame, and the temptation to act selfishly. We show that the model can describe recent experiments on dictator games with an exit option. In addition, we describe an empirical puzzle that government spending only partially crowds out consumers' donations, contrary to the prediction based on standard consumer theory.https://authors.library.caltech.edu/records/cm6tm-43f24Average Choice
https://resolver.caltech.edu/CaltechAUTHORS:20160321-140851688
Authors: Ahn, David; Echenique, Federico; Saito, Kota
Year: 2015
This is an investigation of stochastic choice when only the average of the choices is observable. For example when one observes aggregate sales numbers from a store, but not the frequency with which each item was purchased. The focus of our work is on the Luce model, also known as the Logit model. We show that a simple path independence property of average choice uniquely characterizes the Luce model. We also characterize the linear Luce mode, using similar tools. A linear version of the Luce model is used most frequently in empirical work by applied economists.
Our characterization is based on the property of path independence, which runs counter to early impossibility results on path independent choice. From an empirical perspective, our results provide a small-sample advantage over the tests of Luce's model that rely on estimating choice frequencies.https://authors.library.caltech.edu/records/gnvny-ekt49Testable Implications of Translation Invariance and Homotheticity: Variational, Maxmin, CARA and CRRA preferences
https://resolver.caltech.edu/CaltechAUTHORS:20160308-151531257
Authors: Chambers, Christopher P.; Echenique, Federico; Saito, Kota
Year: 2016
We provide revealed preference axioms that characterize models of translation invariant preferences. In particular, we characterize the models of variational, maxmin, CARA and CRRA utilities. In each case we present a revealed preference axiom that is satisfied by a dataset if and only if the dataset is consistent from the corresponding utility representation. Our results complement traditional exercises in decision theory that take preferences as primitive.https://authors.library.caltech.edu/records/wfa41-5z837The Perception-Adjusted Luce Model
https://resolver.caltech.edu/CaltechAUTHORS:20160321-134531088
Authors: Echenique, Federico; Saito, Kota; Tserenjigmid, Gerelt
Year: 2016
We develop an axiomatic model that builds on Luce's (1959) model to incorporate a role for perception. We identify agents' "perception priorities" from their violations of Luce's axiom of independence from irrelevant alternatives. Using such perception priorities, we adjust choice probabilities to account for the effects of perception. Our axiomatization requires that the agents' adjusted random choice conforms to Luce's model. Our model can explain the attraction, compromise, and similarity effects, which are very well-documented behavioral phenomena in individual choice.https://authors.library.caltech.edu/records/spc2v-qkd82Testable Implications of Exponential Discounting
https://resolver.caltech.edu/CaltechAUTHORS:20160321-132448266
Authors: Echenique, Federico; Saito, Kota
Year: 2016
We develop a behavioral axiomatic characterization of exponentially discounted utility (EDU) over consumption streams. Given is an individual agent's behavior in the market: assume a finite collection of purchases across periods. We show that such behavior satisfies a "revealed preference axiom" if and only if there exists a EDU model (a discount rate per period and a concave utility function over money) that accounts for the given intertemporal consumption.https://authors.library.caltech.edu/records/4jx8b-zxb60Response Time and Utility
https://resolver.caltech.edu/CaltechAUTHORS:20160321-135336141
Authors: Echenique, Federico; Saito, Kota
Year: 2016
Response time is the time an agent needs to make a decision. One fundamental finding in psychology and neuroscience is that, in a binary choice, the response time is shorter as the difference between the utilities of the two options becomes larger. We consider situations in which utilities are not observed, but rather inferred from revealed preferences: meaning they are inferred from subjects' choices. Given data on subjects' choices, and the time to make those choices, we give conditions on the data that characterize the property that response time is decreasing in utility differences.https://authors.library.caltech.edu/records/fbmth-qyz68General Luce Model
https://resolver.caltech.edu/CaltechAUTHORS:20160323-155625592
Authors: Echenique, Federico; Saito, Kota
Year: 2016
We extend the Luce model of discrete choice theory to satisfactorily handle zero-probability choices. The Luce model (or the Logit model) is the most widely applied and used model in stochastic choice, but it struggles to explain choices that are never made. The Luce model requires that if an alternative y is never chosen when x is available, then there is no set of alternatives from which y is chosen with positive probability: y cannot be chose, if from sets of alternatives that exclude x. We relax this assumption. In our model, if an alternative y is never chosen when x is available, then we infer that y is dominated by x. While dominated by x, y may still be chosen with positive probability - even with high probability - when grouped with a comparable set of alternatives.https://authors.library.caltech.edu/records/s6fev-wn796Testing theories of financial decision making
https://resolver.caltech.edu/CaltechAUTHORS:20160329-072655949
Authors: Chambers, Christopher P.; Echenique, Federico; Saito, Kota
Year: 2016
DOI: 10.1073/pnas.1517760113
PMCID: PMC4839416
We describe the observable content of some of the most widely used models of decision under uncertainty: models of translation invariant preferences. In particular, we characterize the models of variational, maxmin, constant absolute risk aversion, and constant relative risk aversion utilities. In each case we present a revealed preference axiom that is satisfied by a dataset if and only if the dataset is consistent with the corresponding utility representation. We test our axioms using data from an experiment on financial decisions.https://authors.library.caltech.edu/records/rgkh0-c6377Response time and utility
https://resolver.caltech.edu/CaltechAUTHORS:20170628-110121372
Authors: Echenique, Federico; Saito, Kota
Year: 2017
DOI: 10.1016/j.jebo.2017.04.008
Response time is the time an agent needs to make a decision. One fundamental finding in psychology and neuroscience is that, in a binary choice, there is a monotonic relationship between the response time and the difference between the utilities of the two options. We consider situations in which utilities are not observed, but rather inferred from revealed preferences: meaning they are inferred from subjects' choices. Given data on subjects' choices, and the time to make those choices, we give conditions on the data that characterize the property that response time is a monotonic function of utility differences.https://authors.library.caltech.edu/records/g1ktf-cwk77Testable Implications of Quasi-Hyperbolic and Exponential Time Discounting
https://resolver.caltech.edu/CaltechAUTHORS:20170726-142753192
Authors: Echenique, Federico; Imai, Taisuke; Saito, Kota
Year: 2017
We present the first revealed-preference characterizations of the models of exponential time discounting, quasi-hyperbolic time discounting, and other time-separable models of consumers' intertemporal decisions. The characterizations provide non-parametric revealed-preference tests, which we take to data using the results of a recent experiment conducted by Andreoni and Sprenger (2012). For such data, we find that less than half the subjects are consistent with exponential discounting, and only a few more are consistent with quasi-hyperbolic discounting.https://authors.library.caltech.edu/records/je1j3-any80Social Preferences under Uncertainty: Equality of Opportunity vs. Equality of Outcome
https://resolver.caltech.edu/CaltechAUTHORS:20170727-093210149
Authors: Saito, Kota
Year: 2017
This paper introduces a model of inequality aversion that captures a preference for equality of ex-ante expected payoff relative to a preference for equality of ex-post payoff by a single parameter. On deterministic allocations, the model reduces to the model of Fehr and Schmidt (1999). The model provides a unified explanation for recent experiments on probabilistic dictator games and dictator games under veil of ignorance. Moreover, the model can describe experiments on a preference for effciency, which seem inconsistent with inequality aversion. We also apply the model to the optimal tournament. Finally, we provide a behavioral foundation of the model.https://authors.library.caltech.edu/records/efypn-vdn31Savage in the Market
https://resolver.caltech.edu/CaltechAUTHORS:20170726-151722235
Authors: Echenique, Federico; Saito, Kota
Year: 2017
We develop a behavioral axiomatic characterization of Subjective Expected Utility (SEU) under risk aversion. Given is an individual agent's behavior in the market: assume a finite collection of asset purchases with corresponding prices. We show that such behavior satisfies a "revealed preference axiom" if and only if there exists a SEU model (a subjective probability over states and a concave utility function over money) that accounts for the given asset purchases.https://authors.library.caltech.edu/records/wx589-46888On path independent stochastic choice
https://resolver.caltech.edu/CaltechAUTHORS:20180215-100647346
Authors: Ahn, David S.; Echenique, Federico; Saito, Kota
Year: 2018
DOI: 10.3982/TE2653
We investigate stochastic choice when only the average and not the entire distribution of choices is observable, focusing attention on the popular Luce model. Choice is path independent if it is recursive, in the sense that choosing from a menu can be broken up into choosing from smaller submenus. While an important property, path independence is known to be incompatible with continuous choice. The main result of our paper is that a natural modification of path independence, which we call partial path independence, is not only compatible with continuity, but ends up characterizing the ubiquitous Luce (or logit) rule.https://authors.library.caltech.edu/records/ghmyn-3w557The perception-adjusted Luce model
https://resolver.caltech.edu/CaltechAUTHORS:20180228-092329391
Authors: Echenique, Federico; Saito, Kota; Tserenjigmid, Gerelt
Year: 2018
DOI: 10.1016/j.mathsocsci.2018.02.004
We develop an axiomatic theory of random choice that builds on Luce's (1959) model to incorporate a role for perception. We capture the role of perception through perception priorities; priorities that determine whether an object or alternative is perceived sooner or later than other alternatives. We identify agents' perception priorities from their violations of Luce's axiom of independence from irrelevant alternatives (IIA). The direction of the violation of IIA implies an orientation of agents' priority rankings. We adjust choice probabilities to account for the effects of perception, and impose that adjusted choice probabilities satisfy IIA. So all violations of IIA are accounted for by the perception order. The theory can explain some very well-documented behavioral phenomena in individual choice.https://authors.library.caltech.edu/records/6z9wj-yap11Random Intertemporal Choice
https://resolver.caltech.edu/CaltechAUTHORS:20180817-103430657
Authors: Lu, Jay; Saito, Kota
Year: 2018
DOI: 10.1016/j.jet.2018.08.005
We provide a theory of random intertemporal choice. Agents exhibit stochastic choice over consumption due to preference shocks to discounting attitudes. We first demonstrate how the distribution of these preference shocks can be uniquely identified from random choice data. We then provide axiomatic characterizations of some common random discounting models, including exponential and quasi-hyperbolic discounting. In particular, we show how testing for exponential discounting under stochastic choice involves checking for both a stochastic version of stationarity and a novel axiom characterizing decreasing impatience.https://authors.library.caltech.edu/records/gsvfv-60e39Axiomatizations of the Mixed Logit Model
https://resolver.caltech.edu/CaltechAUTHORS:20191018-102413328
Authors: Saito, Kota
Year: 2019
A mixed logit function, also known as a random-coefficients logit function, is an integral of logit functions. The mixed logit model is one of the most widely used models in the analysis of discrete choice. Observed behavior is described by a random choice function, which associates with each choice set a probability measure over the choice set. I obtain several necessary and sufficient conditions under which a random choice function becomes a mixed logit function. One condition is easy to interpret and another condition is easy to test.https://authors.library.caltech.edu/records/0vm8e-qys12Approximate Expected Utility Rationalization
https://resolver.caltech.edu/CaltechAUTHORS:20191018-101812371
Authors: Echenique, Federico; Imai, Taisuke; Saito, Kota
Year: 2019
We propose a new measure of deviations from expected utility, given data on economic choices under risk and uncertainty. In a revealed preference setup, and given a positive number e, we provide a characterization of the datasets whose deviation (in beliefs, utility, or perceived prices) is within e of expected utility theory. The number e can then be used as a distance to the theory.
We apply our methodology to three recent large-scale experiments. Many subjects in those experiments are consistent with utility aximization, but not expected utility maximization. The correlation of our measure with demographics is also interesting, and provides new and intuitive findings on expected utility.https://authors.library.caltech.edu/records/9ch8a-m6d21General Luce model
https://resolver.caltech.edu/CaltechAUTHORS:20191114-143959789
Authors: Echenique, Federico; Saito, Kota
Year: 2019
DOI: 10.1007/s00199-018-1145-5
We extend the Luce model of discrete choice theory to satisfactorily handle zero-probability choices. The Luce mode struggles to explain choices that are not made. The model requires that if an alternative y is never chosen when x is available, then there is no set of alternatives from which y is chosen with positive probability. In our model, if an alternative y is never chosen when x is available, then we infer that y is dominated by x. While dominated by x, y may still be chosen with positive probability, when grouped with a comparable set of alternatives.https://authors.library.caltech.edu/records/tc3d5-etn39Repeated Choice: A Theory of Stochastic Intertemporal Preferences
https://resolver.caltech.edu/CaltechAUTHORS:20200106-083810655
Authors: Saito, Kota; Lu, Jay
Year: 2020
We provide a repeated-choice foundation for stochastic choice. We obtain necessary and sufficient conditions under which an agent's observed stochastic choice can be represented as a limit frequency of optimal choices over time. In our model, the agent repeatedly chooses today's consumption and tomorrow's continuation menu, aware that future preferences will evolve according to a subjective ergodic utility process. Using our model, we demonstrate how not taking into account the intertemporal structure of the problem may lead an analyst to biased estimates of risk preferences. Estimation of preferences can be performed by the analyst without explicitly modeling continuation problems (i.e. stochastic choice is independent of continuation menus) if and only ifthe utility process takes on the standard additive and separable form. Applications include dynamic discrete choice models when agents have non-trivial intertemporal preferences, such as Epstein-Zin preferences. We provide a numerical example which shows the significance of biases caused by ignoring the agent's Epstein-Zin preferences.https://authors.library.caltech.edu/records/02s07-vq636Testable Implications of Models of Intertemporal Choice: Exponential Discounting and Its Generalizations
https://resolver.caltech.edu/CaltechAUTHORS:20201214-070708764
Authors: Echenique, Federico; Imai, Taisuke; Saito, Kota
Year: 2020
DOI: 10.1257/mic.20180028
We present revealed-preference characterizations of the most common models of intertemporal choice: the model of exponentially discounted concave utility, and some of its generalizations. Our characterizations take consumption data as primitives, and provide nonparametric revealed-preference tests. We apply our tests to data from two recent experiments and find that our axiomatization delivers new insights and perspectives on datasets that had been analyzed by traditional parametric methods.https://authors.library.caltech.edu/records/y5yt1-aq411Approximate Expected Utility Rationalization
https://resolver.caltech.edu/CaltechAUTHORS:20210303-132639472
Authors: Echenique, Federico; Imai, Taisuke; Saito, Kota
Year: 2021
DOI: 10.48550/arXiv.2102.06331
We propose a new measure of deviations from expected utility theory. For any positive number e, we give a characterization of the datasets with a rationalization that is within e (in beliefs, utility, or perceived prices) of expected utility theory. The number e can then be used as a measure of how far the data is to expected utility theory. We apply our methodology to data from three large-scale experiments. Many subjects in those experiments are consistent with utility maximization, but not with expected utility maximization. Our measure of distance to expected utility is correlated with subjects' demographic characteristics.https://authors.library.caltech.edu/records/5xqrp-hsz23Decision Making under Uncertainty: An Experimental Study in Market Settings
https://resolver.caltech.edu/CaltechAUTHORS:20210303-155349183
Authors: Echenique, Federico; Imai, Taisuke; Saito, Kota
Year: 2021
We design and implement a novel experimental test of subjective expected utility theory and its generalizations. Our experiments are implemented in the laboratory with a student population and pushed out through a large-scale panel to a general sample of the U.S. population. We find that a majority of subjects' choices are consistent with the maximization of some utility function, but not with subjective utility theory. The theory is tested by gauging how subjects respond to price changes. A majority of subjects respond to price changes in the direction predicted by the theory, but not to a degree that makes them fully consistent with subjective expected utility. Surprisingly, maxmin expected utility adds no explanatory power to subjective expected utility.
Our findings remain the same regardless of whether we look at laboratory data or the panel survey, even though the two subject populations are very different. The degree of violations of subjective expected utility theory is not affected by age nor cognitive ability, but it is correlated with financial literacy.https://authors.library.caltech.edu/records/dhmkk-yq437Mixed Logit and Pure Characteristics Models
https://resolver.caltech.edu/CaltechAUTHORS:20220209-001237339
Authors: Lu, Jay; Saito, Kota
Year: 2022
Mixed logit or random coefficients logit models are used extensively in empirical work while pure characteristic models feature in much of theoretical work. We provide a theoretical analysis of the relationship between the two classes of models. First, we show an approximation theorem that precisely characterizes the extent and limitations of mixed logit approximations of pure characteristic models. Second, we present two conditions that highlight novel behavioral differences. The first is a substitutability condition that is satisfied by many pure characteristic models (including models of horizontal differentiation such as Hotelling) but is violated by almost all mixed logit models. The second is a continuity condition that is satisfied by all pure characteristic models but is violated by all mixed logit models. Both conditions pertain to choice patterns when product characteristics change or new products are introduced and illustrate the limitations of using mixed logit models for counterfactual analysis.https://authors.library.caltech.edu/records/bn2y3-w1g29Approximating Choice Data by Discrete Choice Models
https://resolver.caltech.edu/CaltechAUTHORS:20220707-204049764
Authors: Chang, Haoge; Narita, Yusuke; Saito, Kota
Year: 2022
DOI: 10.48550/arXiv.arXiv.2205.01882
We obtain a necessary and sufficient condition under which parametric random-coefficient discrete choice models can approximate the choice behavior generated by nonparametric random utility models. The condition turns out to be very simple and tractable. For the case under which the condition is not satisfied (and hence, where some stochastic choice data are generated by a random utility model that cannot be approximated), we provide algorithms to measure the approximation errors. After applying our theoretical results and the algorithm to real data, we found that the approximation errors can be large in practice.https://authors.library.caltech.edu/records/ksvwk-dgd06Adjacencies on random ordering polytopes and flow polytopes
https://resolver.caltech.edu/CaltechAUTHORS:20230425-487625200.7
Authors: Doignon, Jean-Paul; Saito, Kota
Year: 2023
DOI: 10.1016/j.jmp.2023.102768
The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak(1960). Fishburn(1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). To derive a more enlightening proof of Falmagne Theorem, Fiorini(2004) assimilates the MCP with the flow polytope of some acyclic network. However, apart from a recognition of the facets by Suck(2002), the geometric structure of the MCP was apparently not much investigated. We characterize the adjacency of vertices and the adjacency of facets. Our characterization of the edges of the MCP helps understand recent findings in economics papers such as Chang, Narita and Saito(2022) and Turansick(2022). Moreover, our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).https://authors.library.caltech.edu/records/f5swp-ab196Axiomatization of Random Utility Model with Unobservable Alternatives
https://authors.library.caltech.edu/records/b1wrd-7vh30
Authors: Kono, Haruki; Saito, Kota; Sandroni, Alec
Year: 2023
DOI: 10.1145/3580507.3597792
<p>The random utility model is one of the most fundamental models in discrete choice analysis in economics. Although Falmagne (1978) obtained an axiomatization of the random utility model, his characterization requires strong observability of choices, i.e., the frequencies of choices must be observed from all subsets of the set of alternatives. Little is known, however, about the axiomatization when the frequencies on some choice sets are not observable. In fact, the problem of obtaining a tight characterization appears to be out of reach in most cases in view of a related NP-hard problem. We consider the following incomplete dataset. Let <i>X</i> be a finite set of alternatives. Let <i>X</i>* ⊆ <i>X</i> bea set of unobservable alternatives. Let <i>D ⊆</i> 2ˣ be the set of choice sets. We assume that the choice frequency <i>ρ</i>(<i>D, x</i>) is unobservable (i.e., not defined) if and only if <i>x ∈ X</i>* or <i>D</i> ∉ <i>D.</i> Let <i>M</i>* ≡ {(<i>D,x)|x ∈ D ∈</i> 2ˣ and [<i>x</i> ∈ <i>X</i>* or <i>D</i> ∉ <i>D</i>]} be the set of all pairs (<i>D,x</i>) such that <i>ρ</i>(<i>D, x</i>) is not observable. To state our theorem, for any <i>ρ</i> and (<i>D, x</i>) ∈ <i>M ≡</i> {(<i>D, x</i>) <i>∈ D</i> × <i>X</i> | <i>x</i> ∈ <i>D</i>}, define a Block-Marschak polynomial by <i>K</i>(<i>ρ, D, x</i>) = ΣE:E⊇<i>D</i>(−1)|ᴱ\ᴰ|<i>ρ</i>(<i>E,x</i>).</p>https://authors.library.caltech.edu/records/b1wrd-7vh30Approximate Expected Utility Rationalization
https://resolver.caltech.edu/CaltechAUTHORS:20230717-55915200.35
Authors: Echenique, Federico; Imai, Taisuke; Saito, Kota
Year: 2023
DOI: 10.1093/jeea/jvad028
We propose a new measure of deviations from expected utility theory. For any positive number e, we give a characterization of the datasets with a rationalization that is within e (in beliefs, utility, or perceived prices) of expected utility (EU) theory, under the assumption of risk aversion. The number e can then be used as a measure of how far the data is to EU theory. We apply our methodology to data from three large-scale experiments. Many subjects in these experiments are consistent with utility maximization, but not with EU maximization. Our measure of distance to expected utility is correlated with the subjects' demographic characteristics.https://authors.library.caltech.edu/records/ekcgk-93m11