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On the Origin of Utility, Weighting, and DiscountingFunctions: How They Get Their Shapes and How toChange Their ShapesNeil Stewart, Stian Reimers, Adam J. L. Harris

To cite this article:Neil Stewart, Stian Reimers, Adam J. L. Harris (2014) On the Origin of Utility, Weighting, and Discounting Functions: How TheyGet Their Shapes and How to Change Their Shapes. Management Science

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MANAGEMENT SCIENCEArticles in Advance, pp. 1–19ISSN 0025-1909 (print) ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2013.1853

© 2014 INFORMS

On the Origin of Utility, Weighting, and DiscountingFunctions: How They Get Their Shapes and

How to Change Their ShapesNeil Stewart

Department of Psychology, University of Warwick, Coventry CV4 7AL, United Kingdom,[email protected]

Stian ReimersDepartment of Psychology, City University London, London EC1V 0HB, United Kingdom,

[email protected]

Adam J. L. HarrisDepartment of Cognitive, Perceptual and Brain Sciences, University College London,

London WC1H 0AP, United Kingdom, [email protected]

We present a theoretical account of the origin of the shapes of utility, probability weighting, and tem-poral discounting functions. In an experimental test of the theory, we systematically change the shapeof revealed utility, weighting, and discounting functions by manipulating the distribution of monies, proba-bilities, and delays in the choices used to elicit them. The data demonstrate that there is no stable mappingbetween attribute values and their subjective equivalents. Expected and discounted utility theories, and alsotheir descendants such as prospect theory and hyperbolic discounting theory, simply assert stable mappings todescribe choice data and offer no account of the instability we find. We explain where the shape of the mappingcomes from and, in describing the mechanism by which people choose, explain why the shape depends on thedistribution of gains, losses, risks, and delays in the environment.

Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2013.1853.

Keywords : utility; probability weighting; subjective probability; temporal discounting; delay discounting; stablepreferences; decision by sampling; range frequency theory; risky choice; decision under risk; intertemporalchoice; decision under delay

History : Received September 5, 2012; accepted September 19, 2013, by Yuval Rottenstreich, judgment anddecision making. Published online in Articles in Advance.

1. IntroductionCentral to our economic behavior are the attributesmoney, probability, and time. Our representations ofthese attributes are thought to determine our eco-nomic behavior. Theories of decision under risk anddelay generally assume that we transform money,probability, and delay into subjective equivalents, andthen integrate information across these equivalents.These transformations are typically modeled usingutility (or value), weighting, and discounting func-tions. In this paper we present a theory of choicethat accounts for the previously observed nature ofthese functions. A prediction of this theory is thatthese transformations are not stable, and four experi-mental tests confirm this prediction. We show that bymanipulating the distribution of monies, probabilities,and delays in the question set used to elicit utility,weighting, and discounting functions, we can system-atically change their shape; that is, we show that ifyou ask different questions, the revealed subjective

values of given monies, probabilities, and delays canbe adjusted, to some extent, at the experimenter’swill. This means that shape of utility, weighting, anddiscounting functions is a property of the choice envi-ronment and not just of the individual. We argue thatalthough it is possible to derive utility, weighting, anddiscounting functions from behavioral data (such asa series of choices), these functions do not describethe process of choosing. Our theoretical account ofthe choice process does not involve utility, weighting,and discounting functions, but does explain both thesuccess of the traditional approach and also accountsfor the systematic variation of these functions acrossquestion sets.

2. Decision Under RiskExpected utility is the normative model for risky deci-sion making (von Neumann and Morgenstern 1947).In expected utility theory, wealth is translated intoutility by a utility function. To choose between risky

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Stewart, Reimers, and Harris: On the Origin of Utility, Weighting, and Discounting Functions2 Management Science, Articles in Advance, pp. 1–19, © 2014 INFORMS

Figure 1 Psychoeconomic Functions Convert (a) Probability, (b) Amount, and (c) Delay Into Their Subjective Equivalents

00

0 00

1

0

1W

eigh

t

Val

ue

Wei

ght

1

Probability Amount Delay

(a) Probability weighting (b) Value (c) Temporal discounting

options, the average or expected utility of each avail-able course of action is calculated and the action max-imizing expected utility is selected. Expected utilitytheory is often paired with the assumption of dimin-ishing marginal utility—the utility of my second $100,for example, is just a little bit lower than the utility ofmy first $100. To capture this, a concave utility func-tion is used, which is initially quite steep, but thenlater more flat showing a high sensitivity to initialincreases in wealth but then a lower sensitivity to laterincreases.

Economists and psychologists have created a largeset of models of decision under risk that follow theexpected utility framework. Each model adapts util-ity theory to incorporate some psychological insightbased, often, on observations of people’s choicebehavior. Prospect theory (Kahneman and Tversky1979, Tversky and Kahneman 1992) is perhaps themost famous, but there are many other significanttheories (e.g., Birnbaum 2008, Birnbaum and Chavez1997, Busemeyer and Townsend 1993, Edwards 1962,Loomes and Sugden 1982, Quiggin 1993, Savage1954). In prospect theory terminology, a value func-tion converts money into value (the analogue of util-ity), and a weighting function converts probabilityinto decision weight. Figures 1(a) and 1(b) give exam-ple weighting and value functions.

The emphasis in prospect theory was to providea descriptive account of people’s risky decisions,capturing departures from expected utility theory.There are many important empirical departures (forreviews, see Allais 1953, Birnbaum 2008, Camerer1995, Loomes 2010, Luce 2000, Schoemaker 1982,Starmer 2000), but here we use just two—the find-ing that, in choices about gains, people are riskaverse for gambles involving medium and largeprobabilities, but risk seeking for those involvingsmall probabilities—to illustrate how the shapes ofthe value and weight functions were constructedto account for choice data. Although Tversky andKahneman (1992) did motivate their functional forms

by assuming diminishing sensitivity to changes fur-ther from reference points (zero for money andzero and one for probability), the value and deci-sion weighting functions are essentially descriptive,designed to fit particular choice patterns. Considerthe shape of the value function for gains (the top rightquadrant of Figure 1(b)). In an initial choice between(A) 3,000 for sure and (B) an 80% chance of 4,000otherwise nothing, people are risk averse and preferA. In prospect theory, because the value function isconcave, the value of 3,000 is a bit more than 80%of the value of 4,000, and so people prefer the sure3,000; that is, the value function is concave for gainsto describe risk aversion.

The shape of the weighting function is also con-structed to describe the choices that people make. Ina choice between (C) a 25% chance of 3,000 other-wise nothing and (D) a 20% chance of 4,000 otherwisenothing, people are risk seeking and prefer D. Becausethe C–D choice is derived from the A–B choice bymultiplying probabilities by 1/4, participants shouldprefer the A and C or prefer B and D. Their preferencefor A and D, known as the common ratio effect (Allais1953), is not consistent with expected utility theory.Kahneman and Tversky (1979) account for these datausing the weighting function, in which people aremost sensitive to changes in small or large probabil-ities and are least sensitive to changes in intermedi-ate probabilities. Thus, the decision weights of 80%and 100% are quite different, but the weights of 20%and 25% are quite similar (see Figure 1(a)); that is,the weighting function takes its inverse-S shape todescribe the common ratio effect (and other choicepatterns) that people make.

3. Decision Under DelayThe normative starting point for intertemporalchoices is discounted utility theory (Samuelson 1937).When choosing between a series of delayed outcomes,each outcome is discounted by reducing its value

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Stewart, Reimers, and Harris: On the Origin of Utility, Weighting, and Discounting FunctionsManagement Science, Articles in Advance, pp. 1–19, © 2014 INFORMS 3

by a constant fraction for every unit of time it isdelayed. The value of a delayed outcome thus dimin-ishes exponentially with the length of the delay, andthis leads to preferences that remain consistent overtime.

As with decision under risk, psychologists andeconomists have modified this discounted utilityframework to incorporate psychological insight basedon observation. Perhaps the most significant modelis hyperbolic discounting (Ainslie 1975, Loewensteinand Prelec 1992). Here we select one example behav-ioral finding to illustrate the descriptive model, butthere are many (for reviews, see Loewenstein andPrelec 1992, Scholten and Read 2010). In the commondifference effect people have a tendency to reversetheir preferences when a common interval is added orsubtracted from the delays of each option. For exam-ple, in a choice between 100 in 10 days and 110 in11 days, people might prefer the later larger rewardof 110 in 11 days. However, when the same rewardsare brought forward 9 days, so people are choos-ing between 100 in 1 day and 110 in 2 days, peoplemight prefer the smaller sooner reward; that is, thereis more discounting in moving from 1 to 2 days thanfrom 10 to 11 days (Thaler 1981). To explain find-ings like this, Loewenstein and Prelec (1992) (see alsoMazur 1987) suggested that discounting was hyper-bolic, not exponential (see Figure 1(c)). One propertyof a hyperbolic function is that the discount rate is ini-tially high and then decreases. Thus, large discount-ing from 1 to 2 days makes the smaller sooner optionrelatively more attractive, but the reduced discount-ing from 10 to 11 days makes the later larger optionrelatively more attractive. Note that the motivationfor the choice of discounting function is the same asthe motivations for the choice of utility and weight-ing functions: the goal is to describe the choices thatpeople make.

4. Malleability of Risky andDelayed Choices

Below we review a series of studies that demon-strate that choices and valuations of risky and delayedchoices are affected by the distributions of amounts,probabilities, and delays recently experienced. Thesefindings are not consistent with expected utility, dis-counted utility, or their derivatives. To foreshadowthe later theory section, in all of these studies, peoplebehave as if the subjective value of an amount, risk,or delay is given by its rank position in the contextcreated by other recently experienced amounts, risks,and delays.

In decision under risk, Stewart (2009) demonstratedthat a target choice between a 30% chance of 100points and a 40% chance of 75 points can be reversed

by manipulating the distributions of probabilities andamounts encountered just before the choice. Whenprevious choices contained amounts 25, 50, 75, 100,125, and 150 points and probabilities 30%, 32%, 34%,36%, 38%, and 40%, people preferred a 40% chance of75 points. In making this choice, people are behavingas if the difference in amounts is relatively small (theranks of 75 and 100 points are very similar, ranking4th and 3rd, respectively), but the difference in prob-abilities is relatively large (the ranks of 30% and 40%are very different, ranking 6th and 1st, respectively).If people feel that 40% is much better than 30% butthat 100 points is only slightly better than 75 points, a40% chance of 75 points will be more attractive thana 30% chance of 100 points. In contrast, when pre-vious choices contained amounts 75, 80, 85, 90, 95,and 100 points and probabilities 10%, 20%, 30%, 40%,50%, and 60%, people preferred a 30% chance of 100points. In this new context, the ranks of 30% and 40%are very similar, but the ranks of 75 and 100 pointsare very different.

Extending this design, Ungemach et al. (2011)examined how the distribution of attribute values weexperience every day affects choices. In one study,Ungemach et al. (2011) found that customers leavinga supermarket evaluated the prizes on offer in twosimple lotteries against the cost of purchases madea few minutes earlier inside a supermarket. One lot-tery offered £1.50 with a low probability and the otheroffered £0.50 with a high probability. If most pur-chases were for amounts between £0.50 and £1.50,people behaved as if the difference between £0.50 and£1.50 was larger, and selected the £1.50 lottery. Alter-natively, if most purchases were for less than £0.50 ormore than £1.50, people behaved as if the differencebetween £0.50 and £1.50 was smaller, and selected the£0.50 lottery, which has the higher probability of win-ning. Ungemach et al. (2011) present similar findingsfor probability when people generate a probability fora weather event before making a risky choice andfor delay when people plan for their birthday beforemaking an intertemporal choice (see Matthews 2012for a failure to replicate the birthday experiment andPeetz and Wilson 2013 for a successful study usingbirthdays as temporal landmarks). In both this studyand the previous study, the previously encounteredattribute values were irrelevant to the later choice,but differences in the distributions of the previouslyencountered attribute values were sufficient to reversepreferences.

The distribution of prices also affects valuations.Beauchamp et al. (2012), Birnbaum (1992), andStewart et al. (2003) found that when valuing a riskygamble, people were influenced by the range andskew of the options available as potential valua-tions. For example, Birnbaum (1992) found that the

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valuation of a target gamble was higher when peo-ple selected a valuation from a negatively skewed setthan from a positively skewed set. If perceived valueis, in part, determined by rank position, a particularvaluation will seem larger in the positively skewedset (where it has a high rank position) than in a neg-atively skewed set (where it has a low rank posi-tion). Thus, the value of the target gamble is matchedto a lower valuation in the positively skewed set(because the valuations seem higher) and a highervaluation in the negatively skewed set (because thevaluations seem lower). Using an incentive compat-ible auction, Mazar et al. (2013) found that the dis-tribution of prices from which a sale price was tobe randomly drawn affected the reserve price statedby participants in exactly the same way. With neg-atively skewed sale prices (i.e., more larger prices),stated reserve prices were higher than with posi-tively skewed sale prices (i.e., more smaller prices).An example of range affecting valuations is givenby Vlaev et al. (2009). They used a Becker–DeGroot–Marschak auction to show that doubling the rangeof prices from which a sale price was to be drawndoubled the price people would pay to avoid a seriesof electric shocks. Haggag and Paci (2013) report asimilar effect in a natural experiment where taxi cus-tomers give higher tips when offered 20%, 25%, and30% as potential defaults compared to 15%, 20%, and25%—there could be other causes for this taxi tipeffect, but it is consistent with the rank hypothesis.

In making a single choice, the available options alsoaffect the level of risk demonstrated in that choice.Benartzi and Thaler (2001) examined a natural experi-ment. Employees were asked to allocate their pensionfunds between bonds (relatively safe) and stocks (rel-atively risky). Although all employees were offered atleast one stock option and at least one bond option,and thus all employees could exhibit whatever levelof risk they preferred, employees made, on aver-age, more risky investments if there were more stockoptions available. Benartzi and Thaler (2001) accountfor this pattern with the 1/n heuristic, which assumespeople place equal funds into each option, but theallocation is also compatible with the rank hypothesis.Stewart et al. (2003) examine a laboratory analogue,where people are offered either five risky options orfive safe options. Participants in the different con-ditions behaved as though they had different levelsof risk aversion, even though participants were ran-domly assigned to different groups. It is as if the risk-iness of an option is a function of its rank positionwithin the context against which it is evaluated.

5. A Theoretical AccountTo summarize the above studies, people behave as ifthe subjective value of an amount (or probability or

delay) is determined, at least in part, by its rank posi-tion in the set of values currently in a person’s head.So, for example, $10 has a higher subjective value inthe set $2, $5, $8, and $15 because it ranks 2nd, buthas a lower subjective value in the set $2, $15, $19,and $25 because it ranks 4th.

This suggestion—that subjective value is rankwithin a sample—is consistent with Parducci’s (1965,1995) range-frequency model of magnitudes. In thismodel, the subjective value of a magnitude is, in part,given by its rank position. This descriptive modelbegan as an account of scaling of psychophysicalquantities, but has more recently been applied in eco-nomic contexts such as wage satisfaction (Brown et al.2008, Boyce 2009).

Stewart et al. (2006) (see also Stewart 2009) pro-posed the decision-by-sampling model of decisionmaking in which the subjective value of an attribute,whether money, probability, or delay, is its rank posi-tion in a sample of attributes. (The model is moti-vated by evidence from psychophysical studies of therepresentation of magnitudes.) In decision by sam-pling, rank emerges from the application of three sim-ple cognitive tools: sampling, binary comparison, andfrequency accumulation. Continuing the above exam-ple, $10 is compared to a sample of other amounts inmemory: $2, $5, $8, and $15. In binary comparisons,$10 looks good compared to $2, $5, and $8, but looksbad compared to $15. By accumulating the number offavorable comparisons across the sample, $10 is val-ued at 3/4, because three of the four comparisons arefavorable.

Stewart et al. (2006) show how these assumptionsare sufficient to derive psychoeconomic functions likethose in prospect and hyperbolic discounting the-ories. The left column of Figure 2 shows the dis-tributions of gains, losses, risks, and delays in thereal world. Assuming that people’s memories reflectthe world in which they live, Stewart et al. (2006)used these real-world distributions as approximationsof the contents of people’s memories and estimatedthe value, weighting, and discounting functions thatwould be observed under decision by sampling (rightcolumn). The top left panel is the distribution ofprobability phrases occurring in everyday English.People prefer to use verbal phrases rather than num-bers. In a new analysis, we find that of the first500 sentences in the British National Corpus contain-ing the six-letter string “chance,” 65 are about theprobability of an event and, of these, 60 use ver-bal phrases and 5 use numbers. And for responsesto the open question of Ungemach et al. (2011,p. 256), “How likely do you think it is to raintomorrow?,” 340 involved verbal phrases and 47involved numbers. The top right panel is the result-ing weighting function expected if people compare

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Figure 2 Real-World Attribute-Value Distributions (a) and the Resulting Psychoeconomic Functions (b) for Probability (Top), Money (Middle), andDelay (Bottom)

Source. Adapted from Stewart et al. (2006, Figures 1, 2, 5, 6, 7).Note. See the text for details.

probabilities against those they encounter in every-day English. The function is inverse S-shaped becausehigh- and low-probability phrases are more frequentthan intermediate-probability phrases. The functionis asymmetrical because high-probability phrases aremore frequent than low-probability phrases. The mid-dle row repeats the exercise for value. The real-worlddistributions on the left are for credits and debits tobank accounts and are positively skewed. The rightpanel shows the resulting value functions if peoplecompare gains and losses against these credits anddebits. Notice how, because there are more smallgains in the world than large gains, the value func-tion is concave for gains: Against the distribution ofgains, a £100 increase in a prize from £100 to £200improves the rank position substantially more than a£100 increase from £900 to £1,000. The bottom row

repeats the exercise for time. The left panel showsthe frequency with which different delays occur intext on the Internet. The right panel shows the result-ing weighting function if people compare delays tothe real-world distribution on the Internet. (Becauselarger losses and delays are worse than shorter lossesand delays, they have been plotted with negativeranks.) Notice the resemblance between these func-tions and the descriptive functions from prospect andhyperbolic discounting theories in Figure 1.

In independent work, Kornienko (2011) shows for-mally how the decision-by-sampling tools providea cognitive basis for cardinal utility. Stewart andSimpson (2008) provide details of a decision-by-sampling process model of Kahneman and Tversky’s(1979) data. An online decision by sampling calcu-lator, which gives exact choice probabilities for any

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set of prospects, is available at http://www.stewart.warwick.ac.uk.

Equation (1) formalizes the expression for the sub-jective value of an attribute, although this is hardlynecessary for such a simple theory. The subjectivevalue s4x1Y 5 of a target attribute value x in thecontext of a distribution of n attribute values Y =8y11y21 0 0 0 1 yn9 is given by

s4x1Y 5=

∑

y∈Y c4x1y5

n1 (1)

where

c4x1y5=

1 if x compares favorably to y10 if x does not compare

favorably to y0(2)

The subjective value s4x1Y 5 is the probability of afavorable comparison between x and a randomlyselected member of set Y . Equivalently, s4x1Y 5 isthe rank of x in Y , normalized to lie between zeroand one.

A strong prediction of these rank-based accountsis that if one alters the distribution of attribute val-ues encountered, the subjective value of any givenattribute should alter too. We call this the rankhypothesis. Here we test this prediction for money,probability, and delay. Figure 3 shows hypotheticalpsychoeconomic functions for some of the differentsamples of attribute values used in the present exper-iments. For each attribute, two distributions are con-sidered. The subjective value of the attribute withinthe distribution from which it was drawn is plottedagainst its objective value. Notice how the functions

Figure 3 Hypothetical Psychoeconomic Functions for Probability (Left), Amount (Center), and Delay (Right)

Note. The subjective value of an attribute is its rank within the distribution of other attribute values (shown above each plot), normalized to lie between zeroand one.

are steepest where the distribution of attribute valuesis most dense and are shallowest where the distri-bution of attribute values is least dense. For exam-ple, consider the manipulation of the distribution ofamounts (center panel of Figure 3). For the positivelyskewed amounts, the value function first increasesquickly, as fixed-magnitude increases in amount cor-respond to large increases in rank within the sample,and then later slowly, as fixed-magnitude increasesin amount correspond to small increases in rankwithin the sample. Thus, a positively skewed set ofamounts should give a concave utility function. Forthe negatively skewed amounts, the value functionfirst increases slowly, as fixed-magnitude increasesin amount correspond to a small increase in rankwithin the sample, and then later quickly, as fixed-magnitude increases in amount correspond to largerincreases in rank within the sample. Thus, a nega-tively skewed set of amounts should give a convexutility function. Experiments 1 and 2 explore riskydecisions and investigate how changing the distri-bution of amounts and probabilities used to builda set of choices changes the psychoeconomic func-tions revealed from those choices. Experiments 3 and4 repeat this exercise for intertemporal choices.

We do not expect the experimental results to beas extreme as those in Figure 3 because participantsare likely to bring previous experience with money,risk, and delay into the laboratory. Consider, for exam-ple, the manipulation of the distribution of amountsof money. From Stewart et al. (2006), we know thatthe distributions of money in the world are positivelyskewed, with small amounts being highly frequent

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and larger amounts more rare. Thus, in an experi-mental condition with positively skewed amounts ofmoney, the distribution the participant has in mindwill be a mixture of the positively skewed experimen-tal distribution and the positively skewed real-worlddistribution. Thus, the net distribution participantshave in mind will be positively skewed, and the result-ing utility function, under the rank hypothesis, will beconcave. But in an experimental condition with a neg-atively skewed distribution, the distribution the par-ticipant has in mind will be a mixture of the negativelyskewed experimental distribution and the positivelyskewed real-world distribution. In this case, the netdistribution participants experience will be closer touniform, and thus the resulting utility function, underthe rank hypothesis, will be closer to linear. Thus, thecore prediction is that manipulating the distribution ofamounts will give rise to a utility function that is moreconcave in the positive-skew condition compared tothe negative-skew condition—we predict a relativedifference in concavity between conditions, with theabsolute concavity being determined (in the decision-by-sampling theory) by the unmeasured contributionof the real-world distribution of amounts. More gen-erally, it is the relative differences in the shapes of therevealed utility, weighting, and discounting functionsthat test the rank hypothesis, rather than the absoluteshapes of these functions.

To preempt later results, manipulating the distri-bution of amounts, probabilities, and delays system-atically and predictably alters the pattern of choicesthat people make, and thus alters the psychoeconomicfunctions that best describe the data. In fitting thesefunctions, we do not claim that people are using psy-choeconomic functions inside their heads. Instead wefit functions to demonstrate that, within the standardframeworks of prospect theory (or subjective expectedutility) and of hyperbolic discounting, the shape ofa revealed psychoeconomic function is due not to aperson’s stable risk or intertemporal preferences, butinstead, at least in part, to the experimenter’s choiceof attribute values used in the experiment.

6. Experiments6.1. Experiment 1AParticipants made choices of the form “p chance of x,otherwise nothing” or “q chance of y, otherwise noth-ing.” Each choice was between a smaller probabilityof a larger amount of money or a larger probability ofa smaller amount of money. Between participants, thedistribution of amounts available was manipulated tobe either positively skewed or negatively skewed.

6.1.1. Method. Participants. Forty-one Warwickpsychology first-year undergraduates participated.Course credit was given for attending. In addition,for each participant, the gambles they selected ontwo randomly sampled choices were played via urndraws. After applying an experiment exchange rate toany winnings, participants could win up to £5.

Data from four participants were deleted for violat-ing stochastic dominance on more than 10% of catchtrials (see below), though including these data in theanalysis does not alter the pattern of results. Mostparticipants in this experiment, and the others in thispaper, made no catch-trial errors.

Design. A set of five probabilities was crossed witha set of six amounts to create 30 gambles of theform “p chance of x, otherwise nothing.” All partic-ipants experienced probabilities 0.2, 0.4, 0.6, 0.8, and1.0. Participants were randomly assigned to receiveeither a positively or a negatively skewed set ofamounts (see Figure 3, center panel). The positivelyskewed set contained amounts £10, £20, £50, £100,£200, and £500. The negatively skewed set was themirror image of the positively skewed set with thesame range, and was constructed by subtracting eachamount from £510.

The 30 gambles were used to create a set of all pos-sible pairwise choices between them. Choices betweenidentical gambles and choices where one gamblestochastically dominated the other were dropped toleave 150 choices between a small probability ofa large amount and a large probability of a smallamount. In addition, 30 of the choices where one gam-ble stochastically dominated the other were includedas catch trials to detect participants who were notmaking considered choices. Choices and instructionsfor all experiments are available at http://www.stewart.warwick.ac.uk.

Procedure. Participants were tested individually.Written and spoken instructions explained that par-ticipants would be asked to make a series of choicesbetween pairs of gambles. They were told to thinkof each gamble as an urn draw game in which theurn contained 100 balls, with the percentage of win-ning balls matching the percentage chance of winningthe gamble. It was explained that drawing a winningball would result in receiving the amount in the gam-ble, and that nonwinning balls would result in noth-ing. Participants were told that they would randomlyselect two choices at the end of the experiment, withurn draws made for their selected gambles to deter-mine their winnings. The amounts and probabilitieson offer were displayed in lists at the top of the screento remind participants of the attributes they wouldexperience.

Each choice was presented as two buttons, one foreach gamble. Each button had text describing the

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gamble. For example, for one choice in the positive-skew condition, one button was labeled “60% chanceof £200,” and the other was labeled “100% chance of£10.” The assignment of gamble to button was maderandomly on each trial. Participants clicked theirpreferred gamble with the mouse. The next choiceappeared automatically. The ordering of choices wasset randomly for each participant. A progress bar atthe bottom of the screen tracked the progress of theparticipant through the experiment.

Note that in all of the experiments reported here,payments were incentive compatible (except Experi-ment 1C), and there was no deception.

6.1.2. Results and Discussion. For each partic-ipant, raw data were the 150 choices between arelatively safe gamble offering a “q chance of y, other-wise nothing” and a relatively risky gamble offeringa “p chance of x, otherwise nothing,” where q > p andy < x. To recover the subjective values of the amountsof money, we fitted Equation (3) to the choicedata:

Prob4safe5=biascond6qucond4y57cond

biascond6qucond4y57cond +6pucond4x57cond1 (3)

where q ucond4y5 is the expected utility of the safe gam-ble in condition cond, and pucond4x5 is the expectedutility of the risky gamble in condition cond. ThisLuce (1959)–Shepard (1957) choice formulation is astraightforward way to incorporate a stochastic com-ponent in the expected utility model, giving anincreasing probability of selecting the safe gamble asthe utility of the safe gamble increases or the util-ity of the risky gamble decreases. The parame-ter controls the degree of determinism in the model: = 1 gives choice probabilities proportional to theexpected utilities, and > 1 gives more extremechoice probabilities, so gambles with only slightlyhigher expected utility are very likely to be cho-sen. The bias parameter allows for an overall biastoward safe or risky choices independently of theamounts and probabilities in the gambles. (Thoughwe do find bias differences in some experiments, fix-ing bias across conditions produces almost identicalrevealed utility, weighting, and discounting functionsand the same pattern of statistical significance in forevery experiment in this paper, except in two placeswe note below.) The cond subscripts indicate that theutility function u4 5 can differ between conditions, ascan the and bias parameters. Appendix A explainshow ucond4amounti5 was estimated for each amounti ineach condition as part of an off-the-shelf mixed-effectslogistic regression and reports the full set of estimatedparameters.

The revealed utility functions that maximize thelikelihood of the choice data are shown in Figure 4(a).

Figure 4 Utility Functions for Experiments 1A–1C

Amount

Val

ue

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500

Positive skew

Negative skew

(a) Experiment 1A

(b) Experiment 1B

Amount

Val

ue

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500

Positive skew

Uniform

(c) Experiment 1C

Amount

Val

ue

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500

Positive skew

Uniform

Note. Error bars are standard errors.

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Without loss of generality, the utility of £500 was setat 1. Effectively, the logistic regression adjusted theheights of each point in the utility functions to bestfit the choices participants made. Though they werenot constrained to do so, functions increase mono-tonically. The utility function for the positive-skewcondition is concave, whereas the utility functionfor the negative-skew condition is convex. Parame-ter standard errors indicate that the functions differsignificantly. As a summary test of the difference inconcavity, we found that the best fitting power condfor utility functions ucond4x5= xcond differ significantlyacross conditions (2415 = 6036, p = 00012). Figure B.1in Appendix B reports estimates for bias, , and for each condition and the differences between con-ditions with bootstrapped 95% confidence intervalsand shows significant differences for this experi-ment and the later experiments. And this is the coreresult: Participants who experienced a different dis-tribution of amounts chose as if they had a differentshaped utility function. In particular, participants inthe positive-skew condition value £200 more than par-ticipants in the negative-skew condition value £310,even though £200 is smaller than £310 (2415 = 7005,p = 000079).

Do these effects depend on extensive experiencewith the distributions? We do not think so. Splittingthe data for each participant into the first and secondhalves of the experiment and conducting the analysisseparately for each half (for this and later experi-ments) revealed the same sized effect. This is unsur-prising, given attribute distribution effects can takeplace in as few as 10 trials (Stewart 2009) or after pur-chasing a few items in a shop (Ungemach et al. 2011).

6.2. Experiment 1BExperiments 1B and 1C are systematic replicationsof Experiment 1A. Both follow the design of Experi-ment 1A, except for the differences noted here.

6.2.1. Method. Participants. Fifty-six first-yearundergraduates participated in Experiment 1B. Eachexperiment in this paper used a different set ofparticipants. For each participant, the gamble theyselected on one randomly sampled choice was playedvia an urn draw. After applying an experimentexchange rate to any winnings, participants couldwin up to 5. No participants violated stochasticdominance on more than 10% of catch trials, so alldata were retained.Design. Participants were randomly assigned to

receive either a positive-skew condition with amounts£0, £10, £20, £50, £100, £200, and £500, or uniformcondition with amounts £0, £100, £200, £300, £400,and £500. Otherwise gambles and choices were con-structed as in Experiment 1A. The uniform con-dition replaces the negative-skew condition from

Experiment 1A to make the study more represen-tative of distributions used in other experimentsand so that £100 and £200 are common acrossconditions.

Procedure. This replication omitted the panel ofattribute values listing the amounts and probabilitieson offer, which one reviewer was concerned might bedriving the effects. Instead participants only saw theseries of choices, so any effects of distribution wouldhave to be the effect of the attributes viewed on ear-lier choices affecting later choices.

6.2.2. Results and Discussion. Figure 4(b) showsa concave utility function for a positive-skew con-dition and a linear utility function for a uniformcondition. The difference in concavity is significant(2415= 6099, p = 000082). In this experiment, amounts£100 and £200 are common to both distributions, butthe revealed values are higher in the positive-skewcondition because their rank position in the distribu-tion is higher (2415 = 26096, p < 000001 for the dif-ference at £100, and 2415 = 7016, p = 000074 for thedifference at £200).

6.3. Experiment 1C

6.3.1. Method. Experiment 1C replicates Experi-ments 1A and 1B using an online sample of UKsurvey participants making hypothetical rather thanincentivized choices. One hundred twenty-four par-ticipants completed the experiment in their Webbrowsers. Data from 17 participants were deleted forviolating dominance on more than 10% of catch tri-als, though these deletions do not alter the pattern ofresults. Conditions were as Experiment 1B except the£0 amounts were dropped.

6.3.2. Results and Discussion. Figure 4(c) plotsthe utility functions recovered for the positive-skewand uniform conditions, replicating the concave andlinear utility functions from Experiment 1B. Fits ofpower law utility functions shows the difference inconcavity is marginal (2415 = 3050, p = 0006). (Thismarginal result is significant when bias is held con-stant across conditions.) There are significant differ-ences in values at £100, 2415 = 59079 and p < 000001,and £200, 2415= 50047 and p < 000001.

Finally, we note that Mullett (2012) has repli-cated Experiment 1A using a within-participantsmanipulation where the prices of different categoriesof product (holidays versus mobile phones) dif-fered in skew, and Matthews (2013) has replicatedExperiment 1A using a within-participants manipu-lation where two different gamble sets with differ-ent amount distributions were offered by differentexperimenters.

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6.4. Experiment 2AIn Experiment 2A, the distribution of probabilitieswas manipulated between participants, and the dis-tribution of amounts was held constant. In otherrespects, the method was the same as Experi-ments 1A–1C.

6.4.1. Method. Participants. Thirty-five Warwickpsychology first-year undergraduates participated forcourse credit. In addition, participants knew theycould win up to £5 performance-related pay. No par-ticipants violated stochastic dominance on more than10% of catch trials, so all data were retained.Design. Gambles were made by crossing a set of

probabilities with a set of amounts. The amountswere £100, £200, £300, £400, and £500. The set ofprobabilities was manipulated between participants(see Figure 3, left panel) and was either positivelyskewed (10%, 20%, 30%, 40%, 70%, 90%) or negativelyskewed (10%, 30%, 60%, 70%, 80%, 90%). The neg-atively skewed set is the mirror image of the pos-itively skewed set. Choices were made by crossinggambles. One hundred twenty nonstochastically dom-inated choices were selected at random and combinedwith 30 stochastically dominated choices selected atrandom.

Procedure. Because probabilities were the focus ofthis experiment, we wanted to be sure that partic-ipants understood the probabilities and the methodfor resolving them. In this study, probabilities wereresolved by drawing 1 of 100 numbered chips from abag. To be successful, a number smaller than or equalto the probability (as a percentage) had to be drawn.For example, for a “70% chance £100” gamble, £100

Figure 5 Utility Functions for Experiments 2A and 2B

Probability

Weight

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

Positive skew

Positive skew

Negative skewNegative skew

Probability

Wei

ght

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

(a) Experiment 2A (b) Experiment 2B


was received if one of the numbers 1–70 was drawn;numbers 71–100 led to no prize. This procedure wasexplained to participants before they commenced theexperiment. The experimenter showed participantsthe chips in an ordered 10 × 10 grid, sweeping theirhands over the array to indicate, for several exampleprobabilities, which chips were winning chips. Theparticipant then had the opportunity to ask any ques-tions before the experiment began.

6.4.2. Results and Discussion. The analysis re-peats the procedure from Experiment 1, exceptsubjective probabilities instead of utilities variedbetween conditions; that is, we fitted

Prob4safe5

=biascond6wcond4q5y7cond

biascond6wcond4q5y7cond + 6wcond4p5x7cond1 (4)

where wcond4p5 gives the weighting of probability pin condition cond. Figure 5(a) shows the revealedweighting functions. With weighting functions con-strained to follow power laws, the test of a differencein concavity did not reach significance (2415 = 2018,p = 0013). However, modeling with free weightingsrevealed significant differences across conditions forthe common 30% (2415 = 18018, p < 000001) and 70%(2415= 14031, p = 000002).

6.5. Experiment 2B

6.5.1. Method. Experiment 2B is a systematic rep-lication of Experiment 2A with more highly skeweddistributions, and differs only as noted here. Thirty-six Warwick psychology first-year undergraduates

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participated. The positive-skew condition used proba-bilities 1%, 2%, 5%, 10%, 50%, and 99%. The negative-skew condition used probabilities 1%, 50%, 90%, 95%,98%, and 99%. No participant violated dominancein more than 10% of catch trials, so all data wereretained.6.5.2. Results and Discussion. Figure 5(b) shows

that the weighting function in the positive-skewcondition is more concave than the negative-skewcondition. Modeling using power law weightingfunctions reveals a significant difference between con-ditions (2415 = 18105, p < 000001). Modeling withfree weightings tested whether the weighting for thecommon probability 50% differed across conditions.The weighting of 50% is significantly higher in thepositive-skew condition (2415 = 41072, p < 000001),consistent with the rank hypothesis.

6.6. Experiment 3AIn Experiments 3A, 3B, and 4, we repeat Experi-ments 1 and 2, but using delay instead of risk. Par-ticipants made choices of the form “x at time t” or“y at time u.” Each choice was between a later largeramount and a smaller sooner amount. In Experiments3A and 3B, the distribution of amounts was manip-ulated between participants, and the distribution ofdelays was held constant.6.6.1. Method. Participants. Forty Warwick psy-

chology first-year undergraduates participated forcourse credit. In addition, participants knew that oneparticipant would be drawn at random and paidaccording to their selection on one randomly drawnchoice. Payment was by bank transfer on the day setby their choice of delay. After applying an experimentexchange rate, participants could win up to £20.

Data from four participants were deleted for vio-lating dominance on more than 10% of catch trials,though including these data in the analysis does notalter the pattern of results.

Design. Delayed options were made by crossing aset of delays with a set of amounts. All participantsreceived delays of one day, two days, one week, twoweeks, one month, two months, six months, and oneyear. The distribution of amounts was either posi-tively or negatively skewed, with values from Exper-iment 1 (see Figure 3, center panel). One hundredtwenty nondominated choices were selected at ran-dom and combined with 30 stochastically dominatedchoices selected at random.6.6.2. Results and Discussion. The analysis re-

peats the procedure from Experiment 1, with the sub-jective values of amounts estimated by maximumlikelihood; that is, we fitted

Prob4sooner5

=biascond6vcond4y5/u7cond

biascond6vcond4y5/u7cond + 6vcond4x5/t7cond1 (5)

where y at time u is a smaller sooner offer, and x attime t is a later larger offer; vcond4x5 gives the utilityof amount x in condition cond. Figure 6(a) shows therevealed utility functions. The utility functions dif-fer between conditions. The utility function for thepositive-skew condition is relatively linear, whereasthe utility function for the negative-skew condition isconvex. Crucially, the difference in the utility func-tions is in the direction expected: the function ismore concave in the positive-skew condition than thenegative-skew condition (2415 = 4049, p = 00034). Asin Experiment 1A, £200 receives a higher valuation inthe positive-skew condition than £310 in the negative-skew condition (2415 = 3089, p = 00047). (The signif-icant difference in the valuation of £200 and £310 isonly marginally significant when bias is held constantacross conditions.)

6.7. Experiment 3B6.7.1. Method. Experiment 3B is a replication of

Experiment 3A. Nineteen first-year Warwick under-graduates participated for course credit. The incen-tive for the randomly drawn participant was up to£50. The only other difference from Experiment 3Awas that we used an even more positively skewedset of common delays (one day, two days, one week,one month, and one year). No participants violateddominance on more than 10% of catch trials, so alldata were retained.

6.7.2. Results and Discussion. Figure 6(b) plotsthe utility functions. The results are very similar, witha significant difference in concavity (2415= 6063, p =00010), though the conservative £200–£310 comparisononly approached significance (2415= 3050, p = 00061).

6.8. Experiment 4

6.8.1. Method. In Experiment 4 the distributionof delays was manipulated, and the distribution ofamounts was held constant. In all other respects, themethod was the same as Experiments 3A and 3B.

Participants. Thirty Warwick psychology first-yearundergraduates participated for course credit. Partic-ipants knew that one participant would be drawn atrandom and paid according to one of their choices.Data from five participants were deleted for violatingdominance on more than 10% of catch trials, thoughincluding these data in the analysis does not alter thepattern of results.

Design. Options were made by crossing a set ofdelays with a set of amounts. The amounts were£100, £200, £300, £400, and £500. The set of delayswas manipulated between participants (see Figure 3,right panel) and was either positively skewed (1 day,2 days, 1 week, 2 weeks, 1 month, 2 months, 6 months,and 1 year) or uniformly distributed (1 day, 2 months,4 months, 6 months, 8 months, 10 months, and1 year).

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Figure 6 Utility Functions for Experiments 3A and 3B

Amount

Val

ue

Val

ue

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400 500

Positive skew Positive skew

Negative skew

Negative skew

Amount

0 100 200 300 400 500

(a) Experiment 3A (b) Experiment 3B


6.8.2. Results and Discussion. The analysis re-peats the procedure from earlier experiments, withweightings of delays allowed to vary between condi-tions; that is, we fitted

Prob4sooner5

=biascond6wcond4u5y7cond

biascond6wcond4u5y7cond + 6wcond4t5x7cond1 (6)

where wcond4t5 gives the weighting of delay t in con-dition cond. Figure 7 shows the revealed delay dis-counting functions. The delay discounting function is

Figure 7 The Delay Discounting Functions for Experiment 4

Delay

Wei

ght

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 100 200 300

Positive skew

Uniform


initially much steeper in the positive-skew conditionand much closer to linear in the uniform condition.Modeling with discounting functions constrained tofollow power laws showed a significant differencein curvature (2415 = 22025, p < 000001). Modelingwith free weights revealed that, for the common two-month and six-month delays, people behaved as ifthey weighted delayed amounts less heavily in thepositive-skew condition compared to the uniformcondition, 2415= 15045, p < 000001.

We note here the difference between this discount-ing rank effect and subadditivity. Subadditivity is theempirical finding that discounting measured over alarger interval is less than that estimated by multi-plying discounting over constituent subintervals (e.g.,Scholten and Read 2006). Here we find more dis-counting of delays in conditions where the distribu-tion is more positively skewed. So subadditivity isabout whether discounting within a condition is con-sistent when estimated over smaller or larger inter-vals, whereas the present discounting rank effect isabout how discounting changes across conditions.The theoretical account of subadditivity is also differ-ent. Scholten and Read (2006, 2010) explain subaddi-tivity by assuming that delay differences rather thanabsolute delays are the psychological primitives thatare transformed by a stable discounting function, andoffer no account of these rank effects. The decision bysampling account of rank effects works without anysuch transformation (Vlaev et al. 2011).

7. General DiscussionExperimentally manipulating the distribution ofmoney (Experiments 1 and 3), probabilities (Experi-ment 2), and delays (Experiment 4) people experience

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alters the choices people make, which in turn altersthe psychoeconomic functions constructed to describethose choices. The effect of the distribution of attributevalues on the subjective function for those attributevalues was the same for money, probability, and time:The subjective function was steepest when the dis-tribution of attribute values was most dense, so thatmore positively skewed distributions resulted in con-cave functions. The shape of revealed utility, weight-ing, and discounting functions is, at least in part, aproperty of the question set and not the individual.

7.1. Psychological ImplicationsThe literature on choices between gambles docu-ments many choice set effects and other violations ofexpected utility theory. One fix is to change the psy-chological primitive (e.g., from wealth to gains/losses(Kahneman et al. 1991, Kahneman and Tversky 1979)or from cumulative probability distributions to risk-reward branches (Birnbaum 2008)). Another fix isto assume more complexity in the transformation ofobjective into subjective values (e.g., with the four-fold pattern motivating the inverse-S weighting func-tion, Kahneman and Tversky 1979). The effects wepresent here are different. They are not accounted forby a change of primitive or a change in the trans-formation between objective and subjective values. Soaccounts based around the expected or discountedutility framework do not explain the results. Psycho-logically, the translation between objective and sub-jective values is not well modeled as a lookup processusing some stable function or table. We have pro-posed the decision by sampling model as an expla-nation in which attribute values are evaluated againstother attribute values in working memory. Becauseworking memory reflects the distribution of money,probability, and delay in the environment, valua-tions will vary across experimental or environmentaldistributions.

The process claims about memory embodied indecision by sampling may offer an explanation forthe link between individual differences in memoryand intelligence and decision making. Higher intelli-gence, which is associated with higher working mem-ory capacity, is associated with less risk aversion andless discounting (Benjamin et al. 2013, Burks et al.2009, Dohmen et al. 2010, Shamosh and Gray 2008),and experimentally reducing working memory capac-ity increases discounting and risk aversion (Hinsonet al. 2003, Whitney et al. 2008). As speculation, wepropose that decision by sampling may predict theseeffects because those with lower working memorycapacity will be more dominated by the immediateexperimental context (which typically will induce riskaversion and discounting; see below) and less influ-enced by extraexperiment attribute values.

7.2. Application in EconomicsGul and Pesendorfer (2005) explain that economistsare often only interested in choice behaviour, becauseonly actual choices affect institutions and markets.Thus, evidence from psychology and neuroeconomicsabout brain states or about the difference betweenwhat people actually choose and what they wantedto choose is not relevant. So while the decision-by-sampling explanation may not be of direct interestto economics, our experiments—which are just aboutchoices—should be. The experiments show that, todescribe an individual, the three single curves for theutility of money and the weighting of delays andprobabilities in Figure 1 need to be replaced withthree books of curves with pages for each attribute-distribution context. However, using a separate func-tion for each possible context leaves the relationshipbetween the attribute-value distribution and the func-tion unexplained.

The models of Kőszegi and Rabin (2006, 2007)and Maccheroni et al. (2009a, b) both incorporatethe notion that the subjective value of an attributedepends on the distribution of attribute values. Thesemodels differ from decision by sampling in includinga stable classical utility component and, for Kőszegiand Rabin (2006, 2007), their comparison to the ref-erence distribution is cardinal, not ordinal. But, likedecision by sampling, both of these models involvea comparison to a reference distribution: Kőszegi andRabin (2006, 2007) assume the distribution is of (ratio-nal) expected outcomes, Maccheroni et al. (2009a, b)assume it is the outcomes of peers, and Stewart et al.(2006) assume it is the attribute values in memory—and these values could well include expected out-comes or the outcomes of others.

7.3. Previous Investigations of the Shapes ofUtility, Weighting, and Discounting Functions

Previous investigations tend to find concave utilityfunctions, inverse-S-shaped probability weightingfunctions, and hyperbolic-like discounting functions.Why? The answer, we think, is in the choice ofattribute-value distributions used in these experi-ments. The essential observation here is that, quitesensibly, functions are typically measured in moredetail (i.e., at more closely spaced intervals) wherethey are expected to change most quickly. Take, forexample, the seminal study by Gonzalez and Wu(1999) on the shapes of utility and weighting func-tions. The open circles in Figure 8 plot the empiricalutility and weighting functions that Gonzalez and Wurecovered. The lines on the plots show the functionspredicted by decision by sampling assuming thatthe set of gambles offered to participants providesthe distribution of attribute values against which theamounts and probabilities were compared. For both

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Figure 8 Empirically Derived Utility (Left) and Weighting (Right) Functions from Gonzalez and Wu (1999)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Probability

Wei

ght

0 200 400 600 800

0

10

20

30

40

Amount

Val

ue

Notes. Circles plot the functions that Gonzalez and Wu recovered. The solid line plots the predictions of decision-by-sampling theory.

the utility and weighting functions, the functions pre-dicted under this simple rank hypothesis show thesame qualitative pattern seen in the empirical func-tions: Because there were more small amounts thanlarge amounts, the utility function is steeper for smallamounts and flatter for large amounts, and becausethere were more small and large probabilities thanintermediate probabilities, the weighting function issteeper for small and large probabilities than forintermediate probabilities. Because Gonzalez and Wu(1999) used a distribution of probabilities symmetricalaround 0.5, the rank hypothesis does not predict theasymmetry in the empirical weighting function. But,if participants also recalled probabilities from outsidethe experiment from a distribution with more largerprobabilities, like that in Figure 2, the rank hypothesisdoes predict the asymmetry. Therefore, there is a self-fulfilling result here: By taking more measurementswhere previous research indicates functions changedmost quickly, this will lead to steeper functions wheremore measurements are taken.

The same logic can be applied to classic studiesin intertemporal choice. Rachlin et al. (1991) usedhypothetical choices between an immediately avail-able sum of money and a delayed $1,000. The opencircles in Figure 9 plot the discounted value as a func-tion of delay. The line on the plot shows the functionpredicted by decision by sampling assuming that theset of delays presented provides the comparison set.As for the Gonzalez and Wu (1999) data, the functionpredicted under this simple rank hypothesis showsthe same qualitative pattern seen in the empiricalfunction. The discounting function is roughly hyper-bolic because of the geometrically spaced distributionof delays chosen by Rachlin et al. (1991). Virtually alldiscounting studies use this kind of distribution ofdelay.

Figure 9 An Empirically Derived Discounting Function fromRachlin et al. (1991), Showing the Discounted Valueof a Delayed $1,000

0 100 200 300 400 500 600

0

200

400

600

800

1,000

Delay/months

Dis

coun

ted

valu

e/$

Notes. Circles plot the function that Rachlin et al. recovered. The solid lineplots the predictions of decision-by-sampling theory.

7.4. Neuroeconomic EvidenceOur experimental results are consistent with recentevidence of coding of relative value within the brain(for a review, see Seymour and McClure 2008). Forexample, Tremblay and Schultz (1999) recorded fromsingle cells in the macaque orbitofrontal cortex. Cellsfired more strongly on presentation of a piece ofapple as a reward when the other available rewardwas a piece of cereal (which monkeys do not likethat much) compared to when the other availablereward was a raisin (which monkeys love): The valueof the apple was coded relative to the other rewardavailable on that choice. In contrast, Padoa-Schioppaand Assad (2008) find choice-invariant responding.

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With intermixed choices between quantities of threejuices, macaque orbitofrontal cortex neurons respondto the absolute quantity of juice available, irrespec-tive of which other juice is presented in the choice.The value of a juice was coded independently ofthe other juice available on that choice. These twofindings are reconciled by noting that Tremblay andSchultz (1999) blocked their choice pairs, but choicepairs were randomly intermixed for Padoa-Schioppaand Assad (2008). Therefore, the findings of relativevalue responding by Tremblay and Schultz (1999) andchoice-invariant responding by Padoa-Schioppa andAssad (2008) are consistent with orbitofrontal cortexneurons encoding the value of rewards relative toother rewards within the current block, rather thanjust within the immediate choice.

Mullett and Tunney (2013) find exactly this blockdependency using functional magnetic resonanceimaging in humans. In some blocks, participants sawprizes of £0.10, £0.20, and £0.30, and in other blocksparticipants saw prizes of £5.00, £7.00, and £10.00.The activity in the ventral striatum and thalamus wassensitive to the relative value of a prize within ablock. The activity in the ventral medial prefrontalcortex and the anterior cingulate cortex was sensi-tive to the rank of the prize across the whole exper-iment. Mullett and Tunney (2013) can discriminatebetween sensitivity to value and sensitivity to rankbecause across the whole experiment the distribu-tion of prizes is not uniform. As Mullett and Tunney(2013) conclude, this pattern is as predicted by deci-sion by sampling: prizes from the experiment entermemory and form the context against which a cur-rent prize is judged, with representations proportionalto the rank position of the prize within the exper-iment context rather than the absolute value of theprize.

7.5. ConclusionManipulation of the distributions of gains, risks, anddelays people experience systematically changes theutility, weighting, and discounting functions revealedfrom people’s choices. So, if we can measure lots ofutility, weighting, or discounting functions just bychanging the choices used to estimate them (whetherin the lab or the field), which functions are the “true”functions? It may be that there is no stability withina person and that some process like decision by sam-pling is all there is, with the apparent stability comingonly from stability of the distribution of attribute val-ues over time. In this case, there are no “true” func-tions. Or it may be that people do have some stableunderlying mapping between objective and subjectivevalues. This is an empirical matter, but, at the least,this mapping must be quite malleable: In our datathe effect of the attribute-value distributions is about

as large as the individual differences between people.Decision by sampling provides a common accountof the origin of utility, weighting, and discountingfunctions—and the explanation of how we were ableto change their shapes.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2013.1853.

AcknowledgmentsThe authors thank James Adelman for valuable help withthe logistic regression analysis and Gordon D. A. Brown,Emina Canic, Tatiana Kornienko, Graham Loomes, WillMatthews, Timothy L. Mullett, Dani Navarro-Martinez,Takao Noguchi, Ganna Pogrebna, and Christoph Ungemachfor insightful discussion. This research was supportedby the Economic and Social Research Council [GrantsRES-062-23-0952, RES-000-22-3339, ES/K002201/1]; and theLeverhulme Trust [Grant RP2012-V-022].

Appendix A. Estimation of Utilities and WeightsIn Experiments 1A–1C, we fitted Equation (A1) to estimatethe utilities of each amount:

log[

Prob4safe51 − Prob4safe5

]

= + cond+ log(

q

p

)

+ cond log(

q

p

)

+∑

i

iXi0 (A1)

The first term, , is an intercept. In the second term, is the coefficient for a dummy variable cond; cond indi-cates condition, with 0 for positive skew and 1 for negativeskew or uniform. In the third term, is the coefficientfor log4q/p5, the log of the ratio of the safe and riskyprobabilities. In the fourth term, is the coefficient forthe interaction of cond and log4q/p5. In the last term, theXi’s are dummy variables indicating the presence of eachamounti; Xi is +1 if amounti appears in the safe gamble, −1 ifamounti appears in the risky gamble, or 0 if amounti does notappear in the choice. The i’s are coefficients for the amountdummies.

Though it is perhaps not obvious, Equation (A1) is a rear-rangement of Equation (3) if we set

log4biascond5= + cond 1 (A2)

cond =+ cond 1 (A3)

anducond4amounti5= exp4i/cond50 (A4)

The advantage of Equation (A1) is that this is the standardform for a logistic regression with the choice on each trialas the dependent variables and cond, log4q/p5, and the Xi’sas independent variables. If one treats the data as comingfrom one single participant and fits Equation (3) by maxi-mum likelihood, exactly the same parameter estimates areobtained as when Equation (A1) is fitted as a logistic regres-sion within any standard statistics package.

Of course, it is not appropriate to ignore the within-subjects nature of the design. Responses will be correlated

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within each participant, and this means that responses arenot independent. But because we can use the logistic regres-sion form in Equation (A1), we are able to use off-the-shelfmethods to deal with the repeated observations for eachparticipant by fitting the model as a mixed effect logis-tic regression. The fixed effects part of the model is givenby Equation (A1). The random effects part of the modelincluded random intercepts and full random slopes for eachparticipant. It is the ucond4amounti5 values that are of pri-mary concern because these indicate the utilities of theamounts in each condition. These are the estimates plottedin Figure 4.

For Experiments 2A and 2B, where probability weightsare estimated, amounts are swapped with probabilities sothat the roles of p and x and of q and y are exchanged inEquation (A1) to give

log[


]

= + cond+ log(

y

x

)

+ cond log(

y

x

)

+∑

i

iXi 1(A5)

where the Xi’s are now dummies for the probabilities andwcond4p5= exp4i/cond5.

For Experiments 3A and 3B, where utilities are estimatedfrom intertemporal choices, p is exchanged with 1/t, wheret is the longer delay, and q is exchanged with 1/u where, uis the shorter delay, to give

log[


]

= + cond+ log(

t

u

)

+ cond log(

t

u

)

+∑

i

iXi0 (A6)

For Experiment 4, where delay weights are estimated, weused

log[


]

= + cond+ log(

y

x

)

+ cond log(

y

x

)

+∑

i

iXi0 (A7)

Appendix B. Using Parametric Forms toTest CurvatureIn Experiments 1A–1C, we fitted Equation (B1) to test fordifferences in the curvature of the utility functions:

log[

Prob4safe51−Prob4safe5

]

= + cond+log(

q

p

)

+ condlog(

q

p

)

+ log(

y

x

)

+ condlog(

y

x

)

0 (B1)

Here we set

log4biascond5= + cond 1 (B2)

cond =+ cond 1 (B3)

andcond =

+ condcond

0 (B4)

If the unspecified utility function in Equation (3) is replacedwith a power function ucond4x5= xcond , then Equation (B1) isjust a rearrangement of Equation (3), and, as above, fittingEquations (3) and (B1) is equivalent.

As above, we deal with the repeated observations of eachparticipant by fitting the model as a mixed effects logis-tic regression. The fixed effects part of the model is givenby Equation (B1). The random effects part of the modelincluded random intercepts and full random slopes for eachparticipant. It is the estimates of 0 and 1 and the differ-ence between them that gives the critical test of the differ-ence in utility function curvature between conditions.

For Experiments 2A and 2B, we fit

log[


]

= + cond+log(

y

x

)

+ condlog(

y

x

)

+ log(

q

p

)

+ condlog(

q

p

)

1 (B5)

where wcond4p5= pcond , and bias, , and are given by Equa-tions (B2)–(B4).

For Experiments 3A and 3B, we fit

log[


]

= + cond+log(

t

u

)

+ condlog(

t

u

)

+ log(

y

x

)

+ condlog(

y

x

)

1 (B6)

where ucond4x5= xcond , and bias, , and are given by Equa-tions (B2)–(B4). For Experiment 4, we fit

log[


]

= + cond+log(

y

x

)

+ condlog(

y

x

)

+ log(

t

u

)

+ condlog(

t

u

)

1 (B7)

where wcond4t5 = 1/tcond , and bias, , and are given byEquations (B2)–(B4).

For each experiment, Figure B.1 shows the estimates ofbias, , and for each condition together with bootstrapped95% confidence intervals. The left column shows the param-eter estimates with a solid confidence interval for the posi-tive condition and a dashed confidence interval for the othercondition (uniform or negative, depending on the experi-ment). There are panels for each experiment. The value 1is marked with a horizontal dotted line. For bias, 1 indi-cates no bias; for , 1 indicates probability matching; andfor , 1 indicates a linear utility function, linear proba-bility weighting function, or hyperbolic delay discounting.The right column shows a confidence interval for the dif-ference in the parameter estimates. The top row plots biasand bias differences. That the red confidence intervals donot include zero (the dotted line) for bias difference esti-mates for Experiments 1B, 1C, 2B, and 4 indicates significantdifferences in bias in these studies. The middle row shows and differences. Of most interest is the bottom row,which shows and differences. The 95% confidence inter-vals for differences are all above zero, which indicatesthat in every experiment there is a significant differencein the curvature of the utility, weighting, or discountingfunction.

Raw data and full R source code are available from theauthors.

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Figure B.1 Estimates and 95% Confidence Intervals for Parameter Estimates (Left) and the Difference Between Parameters Between Conditions(Right)

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ReferencesAinslie G (1975) Specious reward: A behavioural theory of impul-

siveness and impulse control. Psychol. Bull. 82(5):463–496.Allais M (1953) Le comportement de l’homme rationel devant

le risque: Critique des postulats et axioms de l’ecole ameri-caine [Rational man’s behavior in face of risk: Critique of theAmerican school’s postulates and axioms]. Econometrica 21(4):503–546.

Beauchamp JP, Benjamin DJ, Chabris CF, Laibson DI (2012) Howmalleable are risk preferences and loss aversion? Work-ing paper, Harvard University, Cambridge, MA. http://economics.cornell.edu/dbenjamin/HowMalleableareRiskPreferencesandLossAversion_3-10-2012-Combined.pdf.

Benartzi S, Thaler RH (2001) Naive diversification strategies indefined contribution saving plans. Amer. Econom. Rev. 91(1):79–98.

Benjamin DJ, Brown SA, Shapiro JM (2013) Who is “behavioral”?Cognitive ability and anomalous preferences. J. Eur. Econom.Assoc. 11(6):1231–1255.

Birnbaum MH (1992) Violations of the monotonicity and contextualeffects in choice-based certainty equivalents. Psychol. Sci. 3(5):310–314.

Birnbaum MH (2008) New paradoxes of risky decision making.Psychol. Rev. 115(2):453–501.

Birnbaum MH, Chavez A (1997) Tests of theories of decision mak-ing: Violations of branch independence and distribution inde-pendence. Organ. Behav. Human Decision Process. 71(2):161–194.

Boyce CJ (2009) Subjective well-being: An intersection betweeneconomics and psychology. Ph.D. thesis, Warwick University,Coventry, UK.

Brown GDA, Gardner J, Oswald AJ, Qian J (2008) Does wage rankaffect employees’ well-being? Indust. Relat. 47(3):355–389.

Burks SV, Carpenter JP, Goette L, Rustichini A (2009) Cognitiveskills affect economic preferences, strategic behavior, and jobattachment. Proc. Natl. Acad. Sci. USA 106(19):7745–7750.

Busemeyer JR, Townsend JT (1993) Decision field theory:A dynamic-cognitive approach to decision making in an uncer-tain environment. Psychol. Rev. 100(3):432–459.

Camerer CF (1995) Individual decision making. Kagel J, Roth AE,eds. Handbook of Experimental Economics (Princeton UniversityPress, Princeton, NJ), 587–703.

Dohmen T, Falk A, Huffman D, Sunde U (2010) Are risk aversionand impatience related to cognitive ability? Amer. Econom. Rev.100(3):1238–1260.

Edwards W (1962) Subjective probabilities inferred from decisions.Psychol. Rev. 69(2):109–135.

Gonzalez R, Wu G (1999) On the shape of the probability weightingfunction. Cognitive Psychol. 38(1):129–166.

Gul F, Pesendorfer W (2005) The case for mindless economics.Levine’s working paper archive, Princeton Univer-sity, Princeton, NJ. http://www.dklevine.com/archive/refs4784828000000000581.pdf.

Haggag K, Paci G (2013) Default tips. Working paper, Columbia Uni-versity, New York. http://hdl.handle.net/10022/AC:P:19386.

Hinson JM, Jameson TL, Whitney P (2003) Impulsive decision mak-ing and working memory. J. Experiment. Psychol.: Learn. 29(2):298–306.

Kahneman D, Tversky A (1979) Prospect theory: An analysis ofdecision under risk. Econometrica 47(2):263–291.

Kahneman D, Knetsch JL, Thaler RH (1991) Anomalies: The endow-ment effect, loss aversion, and status quo bias. J. Econom. Per-spect. 5(1):193–206.

Kornienko T (2011) A cognitive basis for context-dependent utility.Working paper, University of Edinburgh, Edinburgh, UK. doi:10.1.1.296.7140.

Kőszegi B, Rabin M (2006) A model of reference-dependent prefer-ences. Quart. J. Econom. 121(4):1133–1165.

Kőszegi B, Rabin M (2007) Reference-dependent risk attitudes.Amer. Econom. Rev. 97(4):1047–1073.

Loewenstein G, Prelec D (1992) Anomalies in intertemporal choice:Evidence and an interpretation. Quart. J. Econom. 107(2):573–597.

Loomes G (2010) Modeling choice and valuation in decision exper-iments. Psychol. Rev. 117(3):902–924.

Loomes G, Sugden R (1982) Regret theory: An alternative theory ofrational choice under uncertainty. Econom. J. 92(368):805–824.

Luce RD (1959) Individual Choice Behavior (John Wiley & Sons, NewYork).

Luce RD (2000) Utility of Gains and Losses2 Measurement-Theoreticaland Experimental Approaches (Erlbaum, Mahwah, NJ).

Maccheroni F, Marinacci M, Rustichini A (2009a) Pride and diver-sity in social economics. Working paper, University of Bocconi,Milan, Italy. http://www.igier.unibocconi.it/files/pride_and_Diversity.pdf.

Maccheroni F, Marinacci M, Rustichini A (2009b) Social decisiontheory: Choosing within and between groups. Working paper,University of Bocconi, Milan, Italy. http://ideas.repec.org/p/cca/wpaper/71.html.

Matthews WJ (2012) How much do incidental values affect thejudgment of time? Psychol. Sci. 23(11):1432–1434.

Matthews WJ (2013) Personal communication with Neil Stewart,June 20–21.

Mazur JE (1987) An adjusting procedure for studying delayed rein-forcement. Commons ML, Mazur JE, Nevin JA, Rachlin H, eds.Quantitative Analyses of Behavior2 The Effect of Delay and of Inter-vening Events on Reinforcement Value, Vol. 5 (Erlbaum, Hillsdale,NJ), 55–73.

Mazar N, Kőszegi B, Ariely D (2013) True context-dependent pref-erences? The causes of market-dependent valuations. J. Behav.Decision Making, ePub ahead of print July 23, doi: 10.1002/bdm.1794.

Mullett TL (2012) Personal communication with Neil Stewart,December 17–18.

Mullett TL, Tunney RJ (2013) Value representations by rank orderin a distributed network of varying context dependency. BrainCognition 82(1):76–83.

Padoa-Schioppa C, Assad JA (2008) The representation of economicvalue in the orbitofrontal cortex is invariant for changes ofmenu. Nat. Neurosci. 11(1):95–102.

Parducci A (1965) Category judgment: A range-frequency model.Psychol. Rev. 72(6):407–418.

Parducci A (1995) Happiness, Pleasure and Judgment2 The ContextualTheory and Its Applications (Erlbaum, Mahwah, NJ).

Peetz J, Wilson AE (2013) The post-birthday world: Consequencesof temporal landmarks for temporal self-appraisal and motiva-tion. J. Personality Soc. Psychol. 104(2):249–266.

Quiggin J (1993) Generalized Expected Utility Theory2 The Rank-Dependent Model (Kluwer Academic Publishers, Norwell, MA).

Rachlin H, Raineri A, Cross D (1991) Subjective probability anddelay. J. Experiment. Anal. Behav. 55(2):233–244.

Samuelson PA (1937) A note on measurement of utility. Rev.Econom. Stud. 4(2):155–161.

Savage LJ (1954) The Foundations of Statistics (John Wiley & Sons,New York).

Schoemaker PJH (1982) The expected utility model: Its vari-ants, purposes, evidence, and limitations. J. Econom. Literature20(2):529–563.

Scholten M, Read D (2006) Discounting by intervals: A gener-alized model of intertemporal choice. Management Sci. 52(9):1424–1436.

Scholten M, Read D (2010) The psychology of intertemporal trade-offs. Psychol. Rev. 117(3):925–944.

Seymour B, McClure SM (2008) Anchors, scales and the rela-tive coding of value in the brain. Curr. Opin. Neurobiol. 18(2):173–178.

Dow

nloa

ded

from

info

rms.

org

by [

86.1

69.1

1.79

] on

19

Mar

ch 2

014,

at 0

4:24

. Fo

r pe

rson

al u

se o

nly,

all

righ

ts r

eser

ved.


Shamosh NA, Gray JR (2008) Delay discounting and intelligence:A meta-analysis. Intelligence 36(4):289–305.

Shepard RN (1957) Stimulus and response generalization:A stochastic model relating generalization to distance in psy-chological space. Psychometrica 22(4):325–345.

Starmer C (2000) Developments in non-expected utility theory: Thehunt for a descriptive theory of choice under risk. J. Econom.Lit. 38(2):332–382.

Stewart N (2009) Decision by sampling: The role of the deci-sion environment in risky choice. Quart. J. Experiment. Psychol.62(6):1041–1062.

Stewart N, Simpson K (2008) A decision-by-sampling account ofdecision under risk. Chater N, Oaksford M, eds. The Prob-abilistic Mind: Prospects for Bayesian Cognitive Science (OxfordUniversity Press, Oxford, UK), 261–276.

Stewart N, Chater N, Brown GDA (2006) Decision by sampling.Cognitive Psychol. 5

on the origin of utility, weighting, and discounting functions ......stewart, reimers, and harris:...

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