Barberis2012_A Model of Casino Gambling

MANAGEMENT SCIENCEVol. 58, No. 1, January 2012, pp. 35–51ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.1110.1435

© 2012 INFORMS

A Model of Casino GamblingNicholas Barberis

School of Management, Yale University, New Haven, Connecticut 06511, [email protected]

We show that prospect theory offers a rich theory of casino gambling, one that captures several featuresof actual gambling behavior. First, we demonstrate that for a wide range of preference parameter values,

a prospect theory agent would be willing to gamble in a casino even if the casino offers only bets with noskewness and with zero or negative expected value. Second, we show that the probability weighting embeddedin prospect theory leads to a plausible time inconsistency: at the moment he enters a casino, the agent plans tofollow one particular gambling strategy; but after he starts playing, he wants to switch to a different strategy.The model therefore predicts heterogeneity in gambling behavior: how a gambler behaves depends on whetherhe is aware of the time inconsistency; and, if he is aware of it, on whether he can commit in advance to hisinitial plan of action.

Key words : gambling; prospect theory; time inconsistency; probability weightingHistory : Received March 3, 2010; accepted March 26, 2011, by Brad Barber, Teck Ho, and Terrance Odean,

special issue editors. Published online in Articles in Advance November 4, 2011.

1. IntroductionCasino gambling is a hugely popular activity. TheAmerican Gaming Association reports that in 2007,55 million people made 376 million trips to casinos inthe United States alone.

If we are to fully understand how people thinkabout risk, we need to make sense of the existence andpopularity of casino gambling. Unfortunately, thereare still very few models of why people go to casi-nos or of how they behave when they get there. Thechallenge is clear. In the field of economics, the stan-dard model of decision making under risk couplesthe expected utility framework with a concave utilityfunction defined over wealth. This model is help-ful for understanding a range of phenomena. It can-not, however, explain casino gambling: an agent witha concave utility function will always turn down awealth bet with a negative expected value.

Although casino gambling is not consistent withthe standard economic model of risk attitudes,researchers have made some progress in understand-ing it better. One approach is to introduce noncon-cave segments into the utility function (Friedman andSavage 1948). A second approach argues that peoplederive a separate component of utility from gambling.This utility may be only indirectly related to the betsthemselves—for example, it may stem from the socialpleasure of going to a casino with friends—or it maybe directly related to the bets, in that the gamblerenjoys the feeling of suspense as he waits for the betsto play out (Conlisk 1993). A third approach suggeststhat gamblers are simply unaware that the odds theyare facing are unfavorable.

In this paper, we present a new model of casinogambling based on Tversky and Kahneman’s (1992)

cumulative prospect theory. Cumulative prospecttheory, a prominent theory of decision makingunder risk, is a modified version of Kahneman andTversky’s (1979) prospect theory. It posits that peopleevaluate risk using a value function that is definedover gains and losses, that is concave over gains andconvex over losses, and that is kinked at the origin,so that people are more sensitive to losses than togains, a feature known as loss aversion. It also statesthat people engage in “probability weighting”: thatthey use transformed rather than objective probabili-ties, where the transformed probabilities are obtainedfrom objective probabilities by applying a weightingfunction. The main effect of the weighting function isto overweight the tails of the distribution it is appliedto. The overweighting of tails does not represent abias in beliefs; rather, it is a way of capturing thecommon preference for a lottery-like, or positivelyskewed, payoff.1

We choose prospect theory as the basis for a possi-ble explanation of casino gambling because we wouldlike to understand gambling in a framework that alsoexplains other evidence on risk attitudes. Prospecttheory can explain a wide range of experimental evi-dence on attitudes to risk—indeed, it was designedto—and it can also shed light on much field evidenceon risk taking: for example, it can address a numberof facts about risk premia in asset markets (Benartziand Thaler 1995, Barberis and Huang 2008). By offer-ing a prospect theory model of casino gambling, our

1 Although our model is based on the cumulative prospect theory ofTversky and Kahneman (1992) rather than on the original prospecttheory of Kahneman and Tversky (1979), we will sometimes referto cumulative prospect theory as “prospect theory” for short.

35

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Barberis: A Model of Casino Gambling36 Management Science 58(1), pp. 35–51, © 2012 INFORMS

paper therefore suggests that gambling is not an iso-lated phenomenon requiring its own unique explana-tion, but that it may instead be one of a family of factsthat can be understood using a single model of riskattitudes.

The idea that prospect theory might explain casinogambling is initially surprising. Through the over-weighting of the tails of distributions, prospect the-ory can easily explain why people buy lottery tickets.However, many casino games offer gambles that,aside from their low expected values, are also muchless skewed than a lottery ticket. It is therefore farfrom clear that probability weighting can explain whythese gambles are so popular. Indeed, given thatprospect theory agents are much more sensitive tolosses than to gains, one would think that they wouldfind these gambles very unappealing. Initially, then,prospect theory does not seem to be a promising start-ing point for a model of casino gambling.

In this paper, we show that, in fact, prospect theorycan offer a rich theory of casino gambling, one thatcaptures several features of actual gambling behavior.First, we demonstrate that for a wide range of prefer-ence parameter values, a prospect theory agent wouldbe willing to gamble in a casino, even if the casinooffers only bets with no skewness and with zeroor negative expected value. Second, we show thatprospect theory—in particular, its probability weight-ing feature—predicts a plausible time inconsistency: atthe moment he enters a casino, a prospect theoryagent plans to follow one particular gambling strat-egy; but after he starts playing, he wants to switch to adifferent strategy. How he behaves therefore dependson whether he is aware of the time inconsistency; and,if he is aware of it, on whether he is able to commitin advance to his initial plan of action.

What is the intuition for why, in spite of his lossaversion, a prospect theory agent might still be will-ing to enter a casino? Consider a casino that offersonly zero expected value bets—specifically, 50:50 betsto win or lose some fixed amount $h—and supposethat the agent makes decisions by maximizing thecumulative prospect theory value of his accumulatedwinnings or losses at the moment he leaves the casino.We show that if the agent enters the casino, his pre-ferred plan is usually to keep gambling if he is win-ning but to stop gambling and leave the casino if hestarts accumulating losses. An important property ofthis plan is that even though the casino offers only50:50 bets, the distribution of the agent’s perceivedoverall casino winnings becomes positively skewed:by stopping once he starts accumulating losses, theagent limits his downside; by continuing to gamblewhen he is winning, he retains substantial upside.

At this point, the probability weighting feature ofprospect theory plays an important role. Under prob-ability weighting, the agent overweights the tails of

probability distributions. With sufficient probabilityweighting, then, the agent may like the positivelyskewed distribution generated by his planned gam-bling strategy. We show that for a wide range of pref-erence parameter values, the probability weightingeffect indeed outweighs the loss aversion effect andthe agent is willing to enter the casino. In other words,although the prospect theory agent would alwaysturn down the basic 50:50 bet if it were offered inisolation, he is nonetheless willing to enter the casinobecause, through a specific choice of exit strategy,he gives his overall casino experience a positivelyskewed distribution, one which, with sufficient prob-ability weighting, he finds attractive.

Prospect theory offers more than just an explana-tion of why people go to casinos; it also predictsa time inconsistency. The inconsistency is a conse-quence of probability weighting: it arises because,as time passes, the probabilities of final outcomeschange, which, in turn, means that the degree towhich the agent underweights or overweights theseoutcomes also changes. For example, when he entersthe casino, the agent knows that the probability ofwinning five bets in a row, and hence of accumulat-ing a total of $5h, is very low, namely, 1/32. Underprobability weighting, a low probability outcome likethis is overweighted. If the agent actually wins thefirst four bets, however, the probability of winning thefifth bet, and hence of accumulating $5h, is now 1/2.Under probability weighting, a moderate probabilityoutcome like this is underweighted.

The fact that some final outcomes are initiallyoverweighted but subsequently underweighted, orvice versa, means that the agent’s preferences overgambling strategies change over time. We notedabove that at the moment he enters a casino, theagent’s preferred plan is usually to keep gamblingif he is winning but to stop gambling if he startsaccumulating losses. We show, however, that once hestarts playing, he wants to do the opposite: to keepgambling if he is losing and to stop if he accumulatesa significant gain.

As a result of this time inconsistency, our modelpredicts heterogeneity in gambling behavior. How agambler behaves depends on whether he is aware ofthe time inconsistency. A gambler who is aware ofthe time inconsistency has an incentive to try to com-mit to his initial plan of action. For gamblers who areaware of the time inconsistency, then, their behaviorfurther depends on whether they are indeed able tofind a commitment device.

To study these distinctions, we consider three typesof agents. The first type is “naive”: he is unaware ofthe time inconsistency. This agent typically plans tokeep gambling if he is winning and to stop if he startsaccumulating losses. After he starts playing, however,

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Barberis: A Model of Casino GamblingManagement Science 58(1), pp. 35–51, © 2012 INFORMS 37

he deviates from this plan and instead gambles aslong as possible when he is losing and stops if heaccumulates a significant gain.

The second type of agent is “sophisticated”—he isaware of the time inconsistency—but is unable to finda way of committing to his initial plan. He there-fore knows that if he enters the casino, he will keepgambling if he is losing and will stop if he makessome gains. This will give his overall casino expe-rience a negatively skewed distribution. Because heoverweights the tails of probability distributions, healmost always finds this unattractive and thereforerefuses to enter the casino in the first place.

The third type of agent is also sophisticated but isable to find a way of committing to his initial plan.Just like the naive agent, this agent typically plans, onentering the casino, to keep gambling if he is winningand to stop if he starts accumulating losses. Unlike thenaive agent, however, he is able, through the use ofa commitment device, to stick to this plan. For exam-ple, he may bring only a small amount of cash to thecasino while also leaving his ATM card at home; thisguarantees that he will indeed leave the casino if hestarts accumulating losses.

In summary, under the view proposed in this paper,the popularity of casinos is driven by two aspectsof our psychological makeup: first, by the tendencyto overweight the tails of distributions, which makeseven the small chance of a large win seem very allur-ing; and second, by what we could call “naivete,”namely, the failure to recognize that after starting togamble, we may deviate from our initial plan of action.

Our model is a complement to existing theories ofgambling, not a replacement. For example, we suspectthat the concept of “utility of gambling” plays at leastas large a role in casinos as does prospect theory. Atthe same time, we think that prospect theory can addsignificantly to our understanding of casino gambling.As noted above, one attractive feature of the prospecttheory approach is that it not only explains why peoplego to casinos, but also offers a rich description of whatthey do once they get there. In particular, it explains anumber of features of casino gambling that have notemerged from earlier models: the tendency to gamblelonger than planned when losing, the strategy of leav-ing one’s ATM card at home, and casinos’ practice ofissuing vouchers for free food and accommodation topeople who are winning.2

In recent years, there has been a surge of interestin the time inconsistency that stems from hyperbolicdiscounting. In this paper, we study a different time

2 It would be interesting to incorporate an explicit utility of gam-bling into the model we present below. The only reason we do notdo so is because we want to understand the predictions of prospecttheory, taken alone.

inconsistency, one generated by probability weight-ing. The prior literature offers very little guidance onhow best to analyze this particular inconsistency. Wepresent an approach that we think is simple and nat-ural; but other approaches are certainly possible.

Our focus is on the “demand” side of casino gam-bling: we posit a casino structure and study a prospecttheory agent’s reaction to it. In §4.2, we briefly discussan analysis of the “supply” side—of what kinds ofgambles we should expect to see offered in an econ-omy with prospect theory agents. However, we defera full analysis of the supply side to future research.3

2. Cumulative Prospect TheoryIn this section, we review the elements of cumula-tive prospect theory. Readers who are already familiarwith this theory may prefer to go directly to §3.

Consider the gamble

4x−m1 p−m3 0 0 0 3 x−11 p−13x01 p03x11 p13 0 0 0 3 xn1 pn51 (1)

to be read as “gain x−m with probability p−m1x−m+1with probability p−m+1, and so on, independent ofother risks,” where xi < xj for i < j , x0 = 0, and∑n

i=−m pi = 1. In the expected utility framework, anagent with utility function U4 · 5 evaluates this gambleby computing

n∑

i=−m

piU4W + xi51 (2)

where W is his current wealth. A cumulative prospecttheory agent, by contrast, assigns the gamble thevalue

n∑

i=−m

�iv4xi51 (3)

where4

�i =

w4pi + · · · + pn5−w4pi+1 + · · · + pn5

for 0 ≤ i ≤ n1

w4p−m + · · · + pi5−w4p−m + · · · + pi−15

for −m≤ i < 01

(4)

and where v4 · 5 and w4 · 5 are known as the valuefunction and the probability weighting function,

3 A sizeable literature in medical science studies “pathological gam-bling,” a disorder which affects about 1% of gamblers. Our paper isnot aimed at understanding such extreme gambling behavior, butrather the behavior of the vast majority of gamblers whose activityis not considered medically problematic.4 When i = n and i = −m, Equation (4) reduces to �n = w4pn5 and�−m =w4p−m5, respectively.

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Figure 1 Prospect Theory Value Function and Probability Weighting Function

–2 –1 0 1 2–4

–2

0

2

4

x

v(x

)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

P

w(P

)

Notes. The left panel plots the value function proposed by Tversky and Kahneman (1992) as part of their cumulative prospect theory, namely, v4x5= x�

for x ≥ 0 and v4x5 = −�4−x5� for x < 01 for � = 005 and � = 2050 The right panel plots the probability weighting function they propose, namely, w4P 5 =

P �/4P � + 41 − P 5�51/�1 for three different values of �. The dashed line corresponds to �= 0041 the solid line to �= 00651 and the dotted line to �= 10

respectively. Tversky and Kahneman (1992) proposethe functional forms

v4x5=

{

x�

−�4−x5�

for x ≥ 01

for x < 01(5)

and

w4P5=P �

4P � + 41 − P5�51/�1 (6)

where �, � ∈ 40115 and � > 1. The left panel in Fig-ure 1 plots the value function in (5) for � = 005 and�= 205. The right panel in the figure plots the weight-ing function in (6) for � = 004 (the dashed line), for�= 0065 (the solid line), and for � = 1, which corre-sponds to no probability weighting at all (the dottedline). Note that v405= 0, w405= 0, and w415= 1.

There are four important differences between (2)and (3). First, the carriers of value in cumulativeprospect theory are gains and losses, not final wealthlevels: the argument of v4 · 5 in (3) is xi, not W +

xi. Second, whereas U4 · 5 is typically concave every-where, v4 · 5 is concave only over gains; over losses,it is convex. This captures the experimental findingthat people tend to be risk averse over moderate-probability gains—they prefer a certain gain of $500to 4$1100011/25—but risk seeking over moderate-probability losses, in that they prefer 4−$1100011/25to a certain loss of $500.5 The degree of concavityover gains and of convexity over losses are both gov-erned by the parameter �; a lower value of � meansgreater concavity over gains and greater convexityover losses. Using experimental data, Tversky andKahneman (1992) estimate � = 0088 for their mediansubject.

Third, whereas U4 · 5 is typically differentiableeverywhere, the value function v4 · 5 is kinked at the

5 We abbreviate 4x1 p301 q5 to 4x1 p5.

origin so that the agent is more sensitive to losses—even small losses—than to gains of the same mag-nitude. As noted in the Introduction, this element ofcumulative prospect theory is known as loss aversionand is designed to capture the widespread aversionto bets such as 4$11011/23−$10011/25. The severityof the kink is determined by the parameter �; ahigher value of � implies a greater relative sensitiv-ity to losses. Tversky and Kahneman (1992) estimate�= 2025 for their median subject.

Finally, under cumulative prospect theory, the agentdoes not use objective probabilities when evaluating agamble but rather transformed probabilities obtainedfrom objective probabilities via the weighting func-tion w4 · 5. Equation (4) shows that to obtain the prob-ability weight �i for an outcome xi ≥ 0, we takethe total probability of all outcomes equal to or bet-ter than xi (namely, pi + · · · + pn), the total proba-bility of all outcomes strictly better than xi (namely,pi+1 + · · · + pn), apply the weighting function to each,and compute the difference. To obtain the probabilityweight for an outcome xi < 0, we take the total prob-ability of all outcomes equal to or worse than xi, thetotal probability of all outcomes strictly worse than xi,apply the weighting function to each, and computethe difference.6

The main consequence of the probability weight-ing in (4) and (6) is that the agent overweights thetails of any distribution he faces. In Equations (3)

6 The main difference between cumulative prospect theory and theoriginal prospect theory in Kahneman and Tversky (1979) is thatin the original version, the weighting function w4 · 5 is appliedto the probability density function rather than to the cumulativeprobability distribution. By applying the weighting function to thecumulative distribution, Tversky and Kahneman (1992) ensure thatcumulative prospect theory satisfies the first-order stochastic dom-inance property. The original prospect theory, by contrast, does notsatisfy this property.

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and (4), the most extreme outcomes, x−m and xn, areassigned the probability weights w4p−m5 and w4pn5,respectively. For the functional form in (6) and for� ∈ 40115, w4P5 > P for low, positive P ; the right panelof Figure 1 illustrates this for � = 004 and � = 0065. Ifp−m and pn are small, then, we have w4p−m5 > p−m andw4pn5 > pn, so that the most extreme outcomes—theoutcomes in the tails—are overweighted.

The overweighting of tails in (4) and (6) is designedto capture the simultaneous demand many peoplehave for both lotteries and insurance. For example,subjects typically prefer ($51000100001) to a certain $5,but also prefer a certain loss of $5 to 4−$51000100001).By overweighting the tail probability of 0.001 suffi-ciently, cumulative prospect theory can capture bothof these choices. The degree to which the agentoverweights tails is governed by the parameter �; alower value of � implies more overweighting of tails.Tversky and Kahneman (1992) estimate � = 0065 fortheir median subject. To ensure the monotonicity ofw4 · 5, we require � ∈ 40028115.7

We emphasize that the transformed probabilitiesin (3) and (4) do not represent erroneous beliefs:in Tversky and Kahneman’s (1992) framework, anagent evaluating the lottery-like ($51000100001) gam-ble knows that the probability of receiving the $5,000is exactly 00001. Rather, the transformed probabilitiesare decision weights that capture the experimentalevidence on risk attitudes—for example, the prefer-ence for the lottery over a certain $5.

3. A Model of Casino GamblingIn the United States, the term “gambling” typicallyrefers to one of four things: (i) casino gambling, ofwhich the most popular forms are slot machines andthe card game of blackjack; (ii) the buying of lot-tery tickets; (iii) pari-mutuel betting on horses atracetracks; and (iv) fixed-odds betting through book-makers on sports such as football, baseball, basket-ball, and hockey. The American Gaming Associationestimates the 2007 revenues from the four types ofgambling at $60 billion, $24 billion, $4 billion, and$200 million, respectively.8

Although the four types of gambling listed abovehave some common characteristics, they also differ in

7 To be precise, Tversky and Kahneman (1992) allow the value of �to depend on whether the outcome that is being assigned a prob-ability weight is a gain or a loss. They estimate � = 0061 for gainoutcomes and � = 0069 for loss outcomes. For simplicity, we usethe same value of � for both gain and loss outcomes and take itsmedian estimate to be the average of 0.61 and 0.69, namely, 0.65.8 The $200 million figure corresponds to sports betting through legalbookmakers. It is widely believed that this figure is dwarfed bythe revenues from illegal sports betting. Also excluded from thesefigures are the revenues from online gambling.

some ways. Casino gambling differs from playing thelottery in that casino games offer bets that are typi-cally much less positively skewed than a lottery. Andit differs from racetrack betting and sports betting inthat casino games usually require less skill.

In this paper, we focus on casino gambling, largelybecause, from the perspective of prospect theory, itseems particularly hard to explain. The buying of lot-tery tickets is already captured by prospect theorythrough the overweighting of tail probabilities. How-ever, because casino games offer bets that are muchless positively skewed than lotteries, it is not at allclear that we can use the overweighting of tails toexplain their popularity.

We model a casino in the following way. There areT +1 dates, t = 0111 0 0 0 1 T . At time 0, the casino offersthe agent a 50:50 bet to win or lose a fixed amount $h.If the agent turns the gamble down, the game is over:he is offered no more gambles and we say that hehas declined to enter the casino. If the agent acceptsthe 50:50 bet, we say that he has agreed to enter thecasino. The gamble is then played out and, at time 1,the outcome is announced. At that time, the casinooffers the agent another 50:50 bet to win or lose $h. Ifhe turns it down, the game is over: the agent settleshis account and leaves the casino. If he accepts thegamble, it is played out and, at time 2, the outcomeis announced. The game then continues in the sameway. If, at time t ∈ 601T − 27, the agent agrees to playa 50:50 bet to win or lose $h, then, at time t + 1, heis offered another such bet and must either accept itor decline it. If he declines it, the game is over: hesettles his account and leaves the casino. At time T ,the agent must leave the casino if he has not alreadydone so. We think of the interval from 0 to T as anevening of play.

We assume that there is an exogeneous date, date T ,at which the agent must leave the casino if he hasnot already done so because we think that this makesthe model more realistic: whether because of fatigueor because of work and family commitments, mostpeople cannot stay in a casino indefinitely.9

Of the major casino games, it is blackjack that mostclosely matches the game in our model: for a playerfamiliar with the basic strategy, the odds of winning around of blackjack are close to 0.5. Slot machines offera positively skewed payoff and therefore, at first sight,do not appear to fit the model as neatly. Later, how-ever, we argue that our analysis may be able to shed asmuch light on slot machines as it does on blackjack.

9 In the online supplement to this paper (available at http://faculty.som.yale.edu/nicholasbarberis), we discuss an infinite horizonanalog of the model presented here, one in which there is no exo-geneous time at which the agent must leave the casino. We arguethere that the results for the infinite horizon case are broadly simi-lar to those for the finite horizon case.

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Figure 2 Binomial Tree Representation of a Casino

Notes. Each column of nodes corresponds to a particular moment in time.The various nodes within any given column correspond to the different pos-sible accumulated winnings or losses at that time. If a node has a black color,then the agent does not gamble at that node. At the remaining nodes, hedoes gamble. The arrows pick out two specific nodes that we refer to in themain text.

In the discussion that follows, it will be helpful tothink of the casino as a binomial tree. Figure 2 illus-trates this for T = 5—ignore the arrows, for now. Eachcolumn of nodes in the tree corresponds to a partic-ular time: the left-most node corresponds to time 0and the right-most column to time T . At time 0, then,the agent starts in the left-most node. If he takes thetime 0 bet and wins, he moves one step up and to theright; if he takes the time 0 bet and loses, he movesone step down and to the right, and so on: wheneverthe agent wins a bet, he moves up a step in the tree,and whenever he loses, he moves down a step. Thevarious nodes in any given column therefore repre-sent the different possible accumulated winnings orlosses at that time.

We refer to each node in the tree by a pair of num-bers 4t1 j5. The first number, t, which ranges from 0to T , indicates the time that the node corresponds to.The second number, j , which, for given t, ranges from1 to t + 1, indicates how far down the node is withinthe column of nodes that corresponds to time t: thehighest node in the column corresponds to j = 1 andthe lowest node to j = t+1. The left-most node in thetree is therefore node 40115. The two nodes in the col-umn immediately to the right, starting from the top,are nodes 41115 and 41125, and so on.

Throughout the paper, we use a simple colorscheme to represent the agent’s behavior. If a node iswhite, this means that at that node, the agent agreesto play a 50:50 bet. If the node is black, this meansthat the agent does not play a 50:50 bet at that node,either because he leaves the casino when he arrives at

that node, or because his actions in earlier rounds pre-vent him from even reaching that node. For example,the interpretation of Figure 2 is that the agent agreesto enter the casino at time 0 and then keeps gamblinguntil time T = 5 or until he hits node 43115, whichevercomes first. The fact that node 43115 has a black colorimmediately implies that node 44115 must also have ablack color: a node that can only be reached by pass-ing through a black node must itself be black.

As noted above, the basic gamble offered by thecasino in our model is a 50:50 bet to win or lose$h. We assume that the gain and the loss are equallylikely only because this simplifies the exposition, notbecause it is necessary for our analysis. Indeed, wehave studied the case where, as in actual casinos,the basic gamble has a somewhat negative expectedvalue—for example, where it entails a 0.46 chance ofwinning $h, say, and a 0.54 chance of losing $h—and find that the results are similar to those that wepresent below.

Now that we have described the structure ofthe casino, we are ready to present the behavioralassumption that drives our analysis. Specifically, weassume that at each moment of time, the agent in ourmodel decides what to do by maximizing the cumu-lative prospect theory value of his accumulated winningsor losses at the moment he leaves the casino, where thecumulative prospect theory value of a distribution isgiven by (3)–(6).

In any application of prospect theory, a key stepis to specify the argument of the prospect theoryvalue function v4 · 5, in other words, the “gain” or“loss” that the agent applies the value function to.As noted in the previous paragraph, our assumptionis that at each moment of time, the agent appliesthe value function to his overall winnings at themoment he leaves the casino. In the language of “ref-erence points,” our assumption is that throughoutthe evening of gambling, the agent’s reference pointremains fixed at his initial wealth when he enteredthe casino, so that the argument of the value func-tion is his wealth when he leaves the casino minushis wealth when he entered.

Our modeling choice is motivated by the way peo-ple discuss their casino experiences. If a friend or col-league tells us that he recently went to a casino, wetend to ask him “How much did you win?” not “Howmuch did you win last year in all your casino visits?”or “How much did you win in each of the games youplayed at the casino?” In other words, it is overallwinnings during a single casino visit that seem to bethe focus of attention.

Our behavioral assumption immediately raises animportant issue, one that plays a central role in ouranalysis. This is the fact that cumulative prospecttheory—in particular, its probability weighting fea-ture—generates a time inconsistency: the agent’s plan,

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at time t, as to what he would do if he reached somelater node is not necessarily what he actually doeswhen he reaches that node.

To see the intuition, consider the node indicated byan arrow in the upper part of the tree in Figure 2,namely, node 44115—ignore the specific black or whitenode colorations—and suppose that the per-periodbet size is h = $10. It is possible to check that fromthe perspective of time 0, the agent’s preferred plan, foralmost all preference parameter values, is to gamblein node 44115, should he arrive in that node. The rea-son is that by gambling in node 44115, he gives him-self a chance of leaving the casino in node 45115 withan overall gain of $50. From the perspective of time 0,this gain has low probability, namely, 1/32, but undercumulative prospect theory, low tail probabilities areoverweighted, making node 45115 very appealing tothe agent. In spite of the concavity of the value func-tion v4 · 5 in the region of gains, then, his preferredplan, as of time 0, is almost always to gamble in node44115, should he reach that node.

Although the agent’s preferred plan, as of time 0,is to gamble in node 44115, it is easy to see that ifhe actually arrives in node 44115, he will instead stopgambling, contrary to his initial plan. If he stops gam-bling in node 44115, he leaves the casino with an over-all gain of $40. If he continues gambling, he has a 0.5chance of an overall gain of $50 and a 0.5 chance ofan overall gain of $30. He therefore leaves the casinoin node 44115 if

v4405≥ v4505w(

12

)

+ v4305(

1 −w(

12

))

3 (7)

in words, if the cumulative prospect theory valueof leaving is greater than or equal to the cumula-tive prospect theory value of staying. Condition (7)simplifies to

v4405− v4305≥ 4v4505− v43055w(

12

)

0 (8)

It is straightforward to check that condition (8)holds for all �, � ∈ 40115, so that the agent indeedleaves the casino in node 44115, contrary to his initialplan. What is the intuition? From the perspective oftime 0, node 45115 was unlikely, overweighted, andhence appealing. From the time 4 perspective, how-ever, it is no longer unlikely: once the agent is at node44115, the probability of reaching node 45115 is 0.5.The weighting function w4 · 5 underweights moderateprobabilities like 0.5. This, together with the concav-ity of v4 · 5 in the region of gains, means that fromthe perspective of time 4, node 45115 is no longer asappealing. The agent therefore leaves the casino innode 44115.

There is an analogous and, as we will see later,more important time inconsistency in the bottom partof the tree. For example, for almost all preferenceparameter values, the agent’s preferred plan, from theperspective of time 0, is to stop gambling in node44155—the node indicated by an arrow in the bottompart of the tree in Figure 2—should he arrive in thisnode. However, if he actually arrives in node 44155,he keeps gambling, contrary to his initial plan. Theintuition for this inconsistency parallels the intuitionfor the inconsistency in the upper part of the tree.

Given the time inconsistency, the agent’s behaviordepends on two things. First, it depends on whetherhe is aware of the inconsistency. An agent who isaware of the inconsistency has an incentive to try tocommit to his initial plan of action. For this agent,then, his behavior further depends on whether he isindeed able to commit. To explore these distinctions,we consider three types of agents. Our classificationparallels the one used in the literature on hyperbolicdiscounting.

The first type of agent is naive. An agent of thistype is not aware of the time inconsistency generatedby probability weighting. We analyze his behaviorin §3.1. The second type of agent is sophisticated—he is aware of the time inconsistency—but is unableto find a way of committing to his initial plan. Weanalyze his behavior in §3.2. The third and final typeof agent is also sophisticated—he is also aware of thetime inconsistency—but is able to find a way of com-mitting to his initial plan. We analyze his behaviorin §3.3.10

3.1. Case I: The Naive AgentWe analyze the naive agent’s behavior in two steps.First, we study his time 0 decision as to whether toenter the casino. If we find that, for some preferenceparameter values, he is willing to enter, we then look,for these parameter values, at his behavior after heenters, i.e., at his behavior for t > 0.

3.1.1. The Initial Decision. At time 0, the naiveagent chooses a plan of action. A “plan” is a mappingfrom each node in the binomial tree between t = 0 andt = T − 1 to one of two possible actions: “exit,” whichindicates that the agent plans to leave the casino ifhe arrives at that node; or “continue,” which indi-cates that he plans to keep gambling if he arrives atthat node. We denote the set of all possible plans asS40115, where the subscript indicates that this is the set

10 In his classic discussion of nonexpected utility preferences,Machina (1989) identifies three kinds of agents: �-types, �-types,and �-types. In the context of our casino, the behavior of these threetypes of agents would be identical to the behavior of the naiveagents, the sophisticates who are able to commit, and the sophisti-cates who are unable to commit, respectively.

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of plans that is available in node 40115, the left-mostnode in the tree. The set S40115 grows rapidly in sizeas T increases: even for T = 5, the number of possibleplans is large.11

For each plan s ∈ S40115, there is a random variableG̃s that represents the accumulated winnings or lossesthe agent will experience if he exits the casino at thenodes specified by plan s. For example, if s is the exitstrategy shown in Figure 2, then

G̃s ∼

(

$301732

3$1019

323−$101

1032

3

− $3015

323−$501

132

)

0

With this notation in hand, we can write down theproblem that the naive agent solves at time 0. It is

maxs∈S40115

V 4G̃s51 (9)

where V 4 · 5 computes the cumulative prospect the-ory value of the gamble that is its argument. Supposethat V 4G̃s5 attains its maximum value for plan s∗ ∈

S40115. The naive agent then enters the casino—in otherwords, he plays a gamble at time 0—if and only ifV ∗ ≡ V 4G̃s∗5 > 0.12 We emphasize that the naive agentchooses a plan at time 0 without regard for the pos-sibility that he might stray from the plan in futureperiods. After all, he is naive: he does not realize thathe might later depart from the plan.13

The nonlinear probability weighting embedded inV 4 · 5 makes it very difficult to solve problem (9) ana-lytically; indeed, the problem has no known analyticalsolution for general T . We therefore solve it numeri-cally, focusing on the case of T = 5. We are careful tocheck the robustness of our conclusions by solving (9)for a wide range of preference parameter values.14

11 Because, for each of the T 4T + 15/2 nodes between time 0 andtime T − 1, the agent can either exit or continue, the number ofplans in S40115 is, in principle, equal to 2 to the power of T 4T +15/2.The number of distinct plans is much lower, however. For example,for any T ≥ 3, all plans that assign the action “exit” to node 40115are effectively the same, as are all plans that assign the actionscontinue, exit, and exit to nodes 40115, 41115, and 41125, respectively.12 Because S40115 includes the strategy of not entering the casino atall—this is the strategy that assigns the action exit to node 40115—the value of V ∗ must be at least zero, the cumulative prospecttheory value of not entering. The agent enters the casino if thereis a plan that involves gambling in node 40115 whose cumulativeprospect theory value is strictly greater than zero.13 We only allow the agent to consider path-independent plans ofaction: his planned action at time t depends only on his accumu-lated winnings at that time and not on the path by which he accu-mulated those winnings.14 For one special case, the case of T = 2, a full analytical character-ization of the behavior of all three types of agents is available. Wepresent this characterization in the online supplement.

The time inconsistency generated by probabilityweighting means that we cannot use backward induc-tion to solve problem (9). Instead, we use the fol-lowing procedure. For each plan s ∈ S40115 in turn,we compute the gamble G̃s and calculate its cumu-lative prospect theory value V 4G̃s5. We then look forthe plan s∗ that maximizes V 4G̃s5 and check whetherV ∗ > 0.

We begin our analysis by identifying the rangeof preference parameter values for which the naiveagent enters the casino. We set T = 5, h = $10, andrestrict our attention to preference parameter triples4�1�1�5 for which � ∈ 60117, � ∈ 6003117, and � ∈

61147.15 We focus on values of � that are less than 4 soas not to stray too far from Tversky and Kahneman’s(1992) estimate of this parameter; and, as noted ear-lier, we restrict attention to values of � that exceed0.3 so as to ensure that the weighting function (6)is monotonically increasing. We then discretize eachof the intervals 60117, 6003117, and 61147 into a set of20 equally spaced points and study parameter triples4�1�1�5 where each parameter takes a value that cor-responds to one of the discrete points. In other words,we study the 203 = 81000 parameter triples in theset ã, where

ã ={

4�1�1�52 � ∈ 801000531 0 0 0 1009471191

� ∈ 80031003371 0 0 0 1009631191

� ∈ 81110161 0 0 0 13084149}

0 (10)

The “+” and “∗” signs in Figure 3 mark the pref-erence parameter triples for which the naive agententers the casino, in other words, the triples for whichV ∗ > 0. We explain the significance of each of thetwo signs below—for now, the reader can ignorethe distinction. To make the marked region easierto visualize, we use a color scheme in which dif-ferent colors correspond to different vertical eleva-tions. Specifically, the blue, red, green, cyan, magenta,and yellow colors correspond to parameter triples forwhich �—the parameter on the vertical axis—takes avalue in the intervals [11105), [10512), [21205), [20513),[31305), and [30514], respectively.16 Finally, the smallcircle marks Tversky and Kahneman’s (1992) medianestimates of the preference parameters, namely,

4�1�1�5= 40088100651202550 (11)

15 The behavior of the three types of agents that we consider doesnot depend on the value of h; we set h = $10 only for the sakeof concreteness. We have also studied the case of T = 10 and findthat the results that we obtain parallel those for T = 5. We do notuse T = 10 as our benchmark case, however, because of its muchgreater computational demands.16 In topographical terms, the region marked by the + and ∗ signsin Figure 3 consists of two “hills”—a steep hill in the right part ofthe figure and a gentler hill in the left part—with a valley in thecenter.

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Figure 3 Preference Parameter Range for Which a Naive Prospect Theory Agent Enters a Casino

0

0.2

0.40.6

0.8

1.0

0.30.4

0.50.6

0.70.8

0.9

1.0

1.5

2.0

2.5

3.0

3.5

4.0

��

�

Notes. The + and ∗ signs mark the preference parameter triples 4�1 �1 �5 for which an agent with prospect theory preferences would be willing to enter acasino offering 50:50 bets to win or lose a fixed amount. The agent is naive: he is not aware of the time inconsistency generated by probability weighting. The+ signs mark parameter triples for which the agent’s planned strategy is to leave early if he is winning but to stay longer if he is losing. The ∗ signs markparameter triples for which the agent’s planned strategy is to leave early if he is losing but to stay longer if he is winning. The blue, red, green, cyan, magenta,and yellow colors correspond to parameter triples for which � lies in the intervals [11105), [10512), [21205), [20513), [31305), and [30514], respectively. Thecircle marks Tversky and Kahneman’s (1992) median estimates of the parameters, namely, 4�1 �1 �5 = 4008810065120255. A lower value of � means greaterconcavity (convexity) of the prospect theory value function over gains (losses); a lower � means more overweighting of tail probabilities; and a higher � meansgreater loss aversion.

Figure 3 illustrates our first main result: Eventhough the agent is loss averse and even though thecasino offers only 50:50 bets with zero expected value,there is still a wide range of preference parametervalues for which the agent is willing to enter thecasino. In particular, he is willing to enter for 1,813of the 8,000 parameter triples in the set ã. Note thatfor the median estimates in (11), the agent does notenter the casino. Nonetheless, for parameter valuesthat are not far from those in (11), he is willing toenter.

To understand why, for many parameter values, theagent is willing to enter, we study his optimal exitplan s∗. Consider the case of 4�1�1�5= 40095100511055,a parameter triple for which the agent enters. Theleft panel in Figure 4 shows the optimal exit planin this case. Recall that if a node has a white color,the agent plans to gamble at that node, should hereach it. By contrast, if a node has a black color, heplans not to gamble at that node. The figure showsthat, roughly speaking, the agent’s optimal plan is tokeep gambling until time T or until he starts accumu-lating losses, whichever comes first.

The exit plan in the left panel of Figure 4 helps usunderstand why it is that even though the agent is

loss averse and even though the casino offers onlyzero expected value bets, the agent is still willing toenter. The reason is that through his choice of exitplan, the agent is able to give his perceived overallcasino experience a positively skewed distribution: byexiting once he starts accumulating losses, he limitshis downside; by continuing to gamble when he iswinning, he retains substantial upside. Because theagent overweights the tails of probability distribu-tions, he may like the positively skewed distributionoffered by the overall casino experience. In particular,under probability weighting, the chance, albeit small,of leaving the casino in the top-right node 4T 115 witha large accumulated gain of $Th is very enticing. Insummary, then, while the agent would always turndown the basic 50:50 bet offered by the casino ifthat bet were offered in isolation, he is nonethelessable, through a specific choice of exit strategy, to givehis perceived overall casino experience a positivelyskewed distribution, one which, with sufficient prob-ability weighting, he finds attractive.17

17 A number of authors (see, for example, Benartzi and Thaler1995) have noted that prospect theory may be able to explain why

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Figure 4 A Naive Prospect Theory Agent’s Planned and Actual Gambling Behavior

Gamble

Exit

Actual behaviorPlanned behavior

Notes. The left panel shows the strategy that a prospect theory agent with the preference parameter values 4�1 �1 �5 = 40095100511055 plans to use when heenters a casino offering 50:50 bets to win or lose a fixed amount. The agent is naive: he is not aware of the time inconsistency generated by probabilityweighting. If a node has a black color, then the agent plans not to gamble at that node. At the remaining nodes, he plans to gamble. The right panel shows theagent’s actual behavior. If a node has a black color, then the agent does not gamble at that node. At the remaining nodes, he does gamble.

The left panel in Figure 4 shows the naive agent’soptimal plan when 4�1�1�5 = 40095100511055. Whatdoes the optimal plan look like for the other prefer-ence parameter triples for which he enters the casino?To answer this, we introduce some terminology. Welabel a plan a “gain-exit” plan if, under the plan, theagent’s expected length of time in the casino condi-tional on exiting with a gain is less than his expectedlength of time in the casino conditional on exitingwith a loss. Put simply, a gain-exit plan is one inwhich the agent plans to leave quickly if he is win-ning but to stay longer if he is losing. Similarly, aplan is a “loss-exit” (“neutral-exit”) plan if, under theplan, the agent’s expected length of time in the casinoconditional on exiting with a gain is greater than (thesame as) his expected length of time in the casinoconditional on exiting with a loss. For example, theplan in the left panel of Figure 4 is a loss-exit planbecause, conditional on exiting with a loss, the agentspends only one period in the casino, while condi-tional on exiting with a gain, he spends five periodsin the casino.

The ∗ signs in Figure 3 mark the preference param-eter triples for which the naive agent enters the casino

someone would turn down a single play of a positive expectedvalue bet—a 50:50 bet to win $200 or lose $100, say—but wouldagree to multiple plays of the bet, a pattern of behavior that issometimes observed in practice. This is a different point from theone we are making in this paper. After all, a prospect theory agentwould turn down T plays of the basic bet offered by our casino forany T ≥ 1. The reason the agent enters the casino hinges on the factthat a casino with T potential rounds of gambling is not the sameas T plays of the casino’s basic bet: in the casino, the agent has theoption to leave after each round of gambling.

with a loss-exit plan in mind.18 In particular, for 1,021of the 1,813 parameter triples for which the naiveagent enters, he does so with a loss-exit plan in mind,one that is either identical to that in the left panel ofFigure 4 or else one that differs from it in only a verysmall number of nodes.

Figure 3 shows that the naive agent is more likelyto enter the casino with a loss-exit plan for low val-ues of �, for low values of �, and for high valuesof �. The intuition is straightforward. By adopting aloss-exit plan, the agent gives his perceived overallcasino experience a positively skewed distribution. As� falls, the agent overweights the tails of probabil-ity distributions more heavily. He is therefore morelikely to find the positively skewed distribution gen-erated by the loss-exit plan appealing. As � falls, theagent becomes less loss averse. He is therefore lessscared by the losses he could incur under a loss-exitplan and therefore more willing to enter with sucha plan. Finally, as � increases, the marginal utility ofadditional gains diminishes less rapidly. The agent istherefore more excited about the possibility of a largewin inherent in a loss-exit plan and hence more likelyto enter the casino with a plan of this kind.

For 1,021 of the 1,813 parameter triples for whichthe naive agent enters the casino, then, he does sowith a loss-exit plan in mind. For the remaining792 parameter triples for which he enters, he does sowith a gain-exit plan in mind, one where he plans togamble for longer in the region of losses than in theregion of gains. These parameter triples are indicated

18 In case it is not clear, the ∗ signs are located in the right half ofthe marked region in the figure. By contrast, the left half of themarked region is made up of + signs.

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by the + signs in Figure 3. As the figure shows, theseparameter triples lie quite far from the median esti-mates in (11): most of them correspond to values of �and � that are much lower than the median estimatesor to values of � that are much higher.

Why does the naive agent sometimes enter thecasino with a gain-exit plan? Under a plan of thiskind, the agent’s perceived casino experience has anegatively skewed distribution, one with a moderateprobability of a small gain and a low probability ofa large loss. If � is very low, the large loss is onlyslightly more frightening than a small loss; and if� is very high, the low probability of the large loss isbarely overweighted. As a result, when � is low and� is high, the agent may find the negatively skeweddistribution appealing.

Our analysis shows that the component of prospecttheory most responsible for the agent entering thecasino is the probability weighting function: for themajority of the preference parameter values for whichhe enters, the agent chooses a plan that correspondsto a positively skewed casino experience; and this,in turn, is attractive precisely because of the weight-ing function. Nonetheless, the naive agent may enterthe casino even in the absence of probability weight-ing, in other words, even if � = 1. For example, heenters when 4�1�1�5 = 40051111025. For this parame-ter triple, the agent’s optimal plan is a gain-exit plan,one that gives his perceived casino experience a neg-atively skewed distribution; but because � = 1 and �is so low, the agent finds it appealing.

We noted earlier that of all casino games, it is black-jack that most closely matches the game in our model.However, our model may also explain why anothercasino game, the slot machine, is as popular as it is.In our framework, an agent who enters the casinousually does so because he relishes the positivelyskewed distribution he perceives it to offer. Becauseslot machines already offer a skewed payoff, they maymake it easier for the agent to give his overall casinoexperience a significant amount of positive skewness.It may therefore make sense that they would outstripblackjack in popularity.

Figure 3 shows the range of preference parametervalues for which the naive agent enters the casinowhen T = 5. The range of preference parameter valuesfor which he would enter a casino with T > 5 roundsof gambling is at least as large as the range markedin Figure 3. To see why, note that any plan that canbe implemented in a casino with T = � rounds ofgambling can also be implemented in a casino withT = � + 1 rounds of gambling. If an agent is willing toenter a casino with T = � rounds of gambling, then,he will also be willing to enter a casino with T = � +1rounds of gambling: at the very least, when T = � +1,

he can just adopt the plan that leads him to enterwhen T = � .

Can we say more about what happens for highervalues of T ? For example, Figure 3 shows that whenT = 5, the agent does not enter the casino for Tverskyand Kahneman’s (1992) median estimates of the pref-erence parameters. A natural question is then, Arethere any values of T for which a naive agent with themedian preference parameter values would be willingto enter the casino? The following proposition, whichprovides a sufficient condition for the naive agent tobe willing to enter, allows us to answer this question.The proof of the proposition is in the appendix.

Proposition 1. A naive agent with cumulativeprospect theory preferences and the preference parameters4�1�1�5 is willing to enter a casino offering T ≥ 2 roundsof gambling and a basic bet of 4$h10053−$h10055 if 19

T−6T /27∑

j=1

4T +2−2j5�[

w

(

2−T

(

T −1j−1

))

−w

(

2−T

(

T −1j−2

))]

>�w

(

12

)

0 (12)

To derive condition (12), we take one particularexit strategy which, from our numerical analysis, weknow to be either optimal or close to optimal for awide range of preference parameter values—roughlyspeaking, this is a strategy where the agent keepsgambling if he is winning but stops gambling if hestarts accumulating losses—and compute its cumula-tive prospect theory value explicitly. Condition (12)checks whether this value is positive; if it is, we knowthat the naive agent enters the casino. Although thecondition is hard to interpret, it is useful because itallows us to learn something about the agent’s behav-ior when T is high without solving problem (9), some-thing which, for high values of T , is computationallyvery taxing.20

It is easy to check that for Tversky and Kahneman’s(1992) estimates, namely, 4�1�1�5 = 4008810065120255,the lowest value of T for which condition (12) holds isT = 26. We can therefore state the following corollary.

Corollary 1. If T ≥ 26, a naive agent with cumu-lative prospect theory preferences and the parameter val-ues 4�1�1�5 = 4008810065120255 is willing to enter a

19 In this expression,(

T−1−1

)

is assumed to be equal to 0.20 Although condition (12) is only a sufficient condition, there is asense in which it is an accurate sufficient condition: at least forthe low values of T where it is possible to check, the set of triples4�1�1�5 that satisfy condition (12) is very similar to the set of triplesfor which the naive agent actually enters the casino with a loss-exitplan. If we denote the left-hand side of condition (12) as X, it ispossible to show that w41/25 > �X is also a sufficient condition forentry. For low T , this last condition accurately approximates theset of triples for which the naive agent enters the casino with again-exit plan.

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casino with T rounds of gambling and a basic bet of4$h10053−$h10055.

We noted earlier that we are dividing our analy-sis of the naive agent’s behavior into two parts. Wehave just completed the first part: the analysis of theagent’s time 0 decision as to whether to enter thecasino. We now turn to the second part: the analysisof what the agent does at time t > 0.

3.1.2. Subsequent Behavior. Suppose that at time0, the naive agent decides to enter the casino. In node4t1 j5 at some later time t ≥ 1, he solves

maxs∈S4t1 j5

V 4G̃s50 (13)

Here, S4t1 j5 is the set of plans the agent could followfrom time t onward, where, in a similar way to before,a “plan” is a mapping from each node between time tand time T − 1 to one of two actions: exit, indicatingthat the agent plans to leave the casino if he reachesthat node, or continue, indicating that the agent plansto keep gambling if he reaches that node. As before,G̃s is a random variable that represents the agent’spotential accumulated winnings or losses if he followsplan s, and V 4G̃s5 is its cumulative prospect theoryvalue. For example, if the agent is in node 43115, theplan under which he leaves at time T = 5, but notbefore, corresponds to

G̃s ∼(

$501 143$301 1

23$101 14

)

0

If s∗ ∈ S4t1 j5 is the plan that solves problem (13), theagent gambles in node 4t1 j5 if and only if

V 4G̃s∗5 > v4h4t + 2 − 2j551 (14)

where the right-hand side is the utility of leaving thecasino at this node.21

To see how the naive agent behaves for t ≥ 1, wefirst return to the example from earlier in this sec-tion in which T = 5 and 4�1�1�5= 40095100511055. Forthese parameter values, the naive agent enters thecasino at time 0. The right panel of Figure 4 showswhat he does subsequently, at time t ≥ 1. Recall thatthe left panel in the figure shows the initial plan ofaction he constructs at time 0.

21 Because S4t1 j5 includes the strategy of leaving the casino in node4t1 j5—this is the strategy that assigns the action exit to node 4t1 j5—the value of V 4G̃s∗ 5 must be at least v4h4t + 2 − 2j55. The agentgambles in node 4t1 j5 if there is a plan that involves gamblingin node 4t1 j5 whose cumulative prospect theory value is strictlygreater than v4h4t + 2 − 2j55.

Figure 4 shows that whereas the naive agent’s ini-tial plan was to gamble as long as possible when win-ning but to stop if he started accumulating losses, heactually, roughly speaking, does the opposite: he gam-bles as long as possible when losing and stops oncehe accumulates some gains. Our model therefore cap-tures a common intuition, namely, that people oftengamble more than they planned to in the region oflosses.

As noted earlier, the time inconsistency is entirelydriven by the probability weighting function. As oftime 0, a strategy under which the agent continuesto gamble if he is winning is very attractive: underthis strategy, the agent could take home $50 in node45115. Although this is unlikely, the low probabilityof it happening is overweighted, making node 45115very appealing. If the agent wins the first few bets,however, reaching node 45115 is no longer an unlikelyoutcome, and, as such, is no longer overweighted.This, together with the concavity of the value func-tion in the region of gains, means that the agent stopsgambling after earning some gains, contrary to hisinitial plan.

A similar mechanism is at work in the lower partof the tree. As of time 0, a plan under which the agentcontinues to gamble if he is losing is very unattractive:such a plan exposes him to low probability losses,which, given that he overweights tail probabilities,is very unappealing. If the agent starts losing, how-ever, losses that were initially unlikely and henceoverweighted are no longer unlikely and therefore nolonger overweighted. This, together with the convex-ity of the value function in the region of losses, meansthat the agent continues to gamble if he is losing, con-trary to his initial plan.22

How typical is the behavior in the right panelof Figure 4? Earlier in this section, we describeda numerical analysis of 8,000 preference parametertriples and noted that the naive agent enters thecasino for 1,813 of these 8,000 triples. We find that forall 1,813 of these triples, the agent’s actual behaviorin the casino is described by a gain-exit strategy thatis either exactly equal to the one in the right panelof Figure 4 or else one that differs from it in only avery small number of nodes; indeed, for triples forwhich � > 0, the agent always gambles until T = 5 in

22 The naive agent’s naivete can be interpreted in two ways. Theagent may fail to realize that after he starts gambling, he will betempted to depart from his initial plan. Alternatively, he may rec-ognize that he will be tempted to depart from his initial plan, butmay erroneously think that he will be able to resist the tempta-tion. After many casino visits, the agent may learn his way out ofthe first kind of naivete. It may take him much longer, however,to learn his way out of the second kind. People often continue tobelieve that they will be able to exert self-control in the future evenwhen they have repeatedly failed to do so in the past.

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Figure 5 Preference Parameter Range for Which a Sophisticated Prospect Theory Agent Enters a Casino

00.2

0.40.6

0.81.0

0.30.4

0.50.6

0.70.8

0.9

1.0

1.5

2.0

2.5

3.0

3.5

4.0

��

�

Notes. The + signs mark the preference parameter triples 4�1 �1 �5 for which an agent with prospect theory preferences would be willing to enter a casinooffering 50:50 bets to win or lose a fixed amount. The agent is sophisticated: he is aware of the time inconsistency generated by probability weighting. Theblue, red, green, and cyan colors correspond to parameter triples for which � lies in the intervals [1,1.5), [1.5,2), [2,2.5), and [2.5,4], respectively. The circlemarks Tversky and Kahneman’s (1992) median estimates of the parameters, namely, 4�1 �1 �5= 4008810065120255. A lower value of � means greater concavity(convexity) of the prospect theory value function over gains (losses); a lower � means more overweighting of tail probabilities; and a higher � means greaterloss aversion.

the region of losses. We noted earlier that for 1,021 ofthe 1,813 triples for which the naive agent enters thecasino, his initial plan is a loss-exit plan. In all 1,021of these cases, then, the naive agent’s actual behavioris, roughly speaking, the opposite of what he initiallyplanned.23

3.2. Case II: The Sophisticated Agent,Without Commitment

In §3.1, we considered the case of a naive agent—an agent who is unaware of the time inconsistencygenerated by probability weighting. In §§3.2 and 3.3,we study sophisticated agents, in other words, agentswho are aware of the time inconsistency. A sophisti-cated agent has an incentive to try to commit to histime 0 plan. In this section, we consider the case ofa sophisticated agent who is unable to find a way ofcommitting to his time 0 plan; we label this agent a“no-commitment sophisticate.” In §3.3, we study thecase of a sophisticated agent who is able to commit tohis initial plan.

23 For the remaining 792 triples for which the naive agent entersthe casino, his actual behavior is more similar to his initial plan:both his initial plan and actual behavior are gain-exit strategies inwhich he gambles longer in the region of losses than in the regionof gains.

To decide on a course of action, the no-commitmentsophisticate uses backward induction, working left-ward from the right-most column of the binomial tree.If he has not yet left the casino at time T , he must exitat that time. Knowing this, he is able to determinewhat he will do at time T −1. This, in turn, allows himto determine what he will do at time T −2, and so on.

Mathematically, the no-commitment sophisticategambles in node 4t1 j5, where t ∈ 601T − 17, if andonly if

V 4G̃t1 j5 > v4h4t + 2 − 2j550 (15)

The term v4h4t + 2 − 2j55 is the utility of leaving thecasino in node 4t1 j5. The term V 4G̃t1 j5 is the valueof continuing to gamble: specifically, it is the cumula-tive prospect theory value of the random variable G̃t1 j

that represents the accumulated winnings or lossesthe agent will exit the casino with if he gambles innode 4t1 j5. The random variable G̃t1 j is determined bythe exit strategy computed in earlier steps of the back-ward iteration. Note that, precisely because he com-putes his course of action using backward induction,the no-commitment sophisticate is time consistent.

We study the behavior of the no-commitmentsophisticate for T = 5 and for each of the 8,000 pref-erence parameter triples in the set ã defined in (10).The + signs in Figure 5 mark the triples for which

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the no-commitment sophisticate enters the casino—in other words, the triples for which V 4G̃0115 > 00 Asbefore, we use a color scheme to make the markedregion easier to visualize. The blue, red, green, andcyan colors correspond to parameter triples for which� lies in the intervals [11105), [10512), [21205), and[20514], respectively.

The figure shows that the agent enters the casino foronly a narrow range of parameter triples: specifically,for just 753 parameter triples, all of which lie far fromthe small circle that marks Tversky and Kahneman’s(1992) estimates in (11). The intuition is straightfor-ward. The agent realizes that if he does enter thecasino, he will gamble longer in the region of lossesthan in the region of gains. This will give his over-all casino experience a negatively skewed distribution.Because he overweights the tails of distributions, heusually finds this unattractive and therefore refuses toenter. Reasoning of this type may in part explain whythe majority of Americans do not gamble in casinos.

For the 753 parameter triples for which the no-commitment sophisticate enters the casino, he followsa gain-exit strategy. When � and � are sufficientlylow and � is sufficiently high, the negatively skewedcasino experience generated by this strategy is actu-ally appealing.

3.3. Case III: The Sophisticated Agent,With Commitment

In this section, we study the behavior of a sophisti-cated agent who is able to commit to his initial plan.We call this agent a “commitment-aided sophisticate.”

We proceed in the following way. We assume that attime 0, the agent can find a way of committing to anyexit strategy s ∈ S40115. Once we identify the strategythat he would choose, we then discuss how he mightactually commit to this strategy in practice.

At time 0, then, the commitment-aided sophisticatesolves

maxs∈S40115

V 4G̃s50 (16)

In particular, because he can commit to any exit strat-egy, we do not need to restrict the set of strategieshe considers. He searches across all elements of S40115until he finds the strategy s∗ with the highest cumula-tive prospect theory value V ∗ = V 4G̃s∗5. He enters thecasino if and only if V ∗ > 0.

The problem in (16) is identical to the problemsolved by the naive agent at time 0. The two types ofagents therefore enter the casino for exactly the samerange of preference parameter values. For T = 5, forexample, the commitment-aided sophisticate entersthe casino for the 1,813 parameter triples marked bythe + and ∗ signs in Figure 3. Moreover, for any givenparameter triple, the commitment-aided sophisticateand the naive agent enter the casino with exactly thesame strategy in mind. For example, for the 1,021

parameter triples indicated by ∗ signs in Figure 3, thecommitment-aided sophisticate enters the casino witha loss-exit plan in mind, as does the naive agent.

The naive agent and the commitment-aided sophis-ticate solve the same problem at time 0 because theyboth think that they will be able to maintain any planthey select at that time. The two types of agents differ,however, in what they do after they enter the casino.Because he has a commitment device at his disposal,the commitment-aided sophisticate is able to stick tohis initial plan. The naive agent, on the other hand,deviates from his initial plan.

For the 1,021 parameter triples indicated by ∗ signsin Figure 3, then, the commitment-aided sophisticatewould like to commit to a loss-exit strategy. The ques-tion now is, How does he commit to such a strategy?For example, in the lower part of the binomial tree,how does he manage to stop gambling when he islosing even though he is tempted to continue? Andin the upper part of the tree, how does he manage tocontinue gambling when he is winning even thoughhe is tempted to stop?24

In the lower part of the tree, one simple commit-ment strategy is for the agent to go to the casino withonly a small amount of cash and to leave his ATMcard at home. If he starts accumulating losses, he issorely tempted to continue gambling, but, because hehas run out of cash, he has no option but to go home.It is a prediction of our model that some gamblers willuse a strategy of this kind. Anecdotal reports suggestthat this is a common gambling strategy.

It is harder to think of a common strategy that gam-blers use to solve the commitment problem in theupper part of the tree. In a way, this is not surprising.An interesting prediction of our model—a predictionthat we have found to hold all the more strongly forhigher values of T —is that the time inconsistency ismore severe in the lower part of the tree than in theupper part. Comparing the two panels in Figure 4,we see that, in the lower part of the tree, the timeinconsistency, and hence the commitment problem, issevere: the agent wants to gamble in every node inthe region of losses even though his initial plan wasto gamble in none of them. In the upper part of thetree, however, the time inconsistency, and hence thecommitment problem, is less acute: the agent’s initialplan conflicts with his subsequent preferences in onlya few nodes. It therefore makes sense that the com-mitment strategies gamblers use in practice seem to

24 We are focusing on the parameter triples marked by ∗ signs inFigure 3 because they are closer to Tversky and Kahneman’s (1992)median parameter estimates than are the triples marked by + signs.Note also that for the latter set of triples, the commitment prob-lem is less severe because the agent’s initial preferences are morealigned with his subsequent preferences: both preferences favorgain-exit strategies.

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be aimed primarily at the time inconsistency in thelower part of the tree.

Although it is hard to think of ways in which gam-blers commit to their initial plan in the upper partof the tree, note that here, casinos have an incentiveto help. In general, casinos offer bets with negativeexpected values; it is therefore in their interest thatgamblers stay on site as long as possible. From thecasinos’ perspective, it is alarming that gamblers aretempted to leave earlier than planned when they arewinning. This may explain the practice among somecasinos of offering vouchers for free food and lodgingto people who are winning. In our framework, casi-nos do this in order to encourage gamblers who arethinking of leaving with their gains, to stay longer.25

4. Further Remarks4.1. Average LossesOur analysis shows that the set of casino gam-blers consists primarily of two distinct types: naiveagents and commitment-aided sophisticates. Which ofthese two types loses more money in the casino, onaverage?

In the context of the model of §3, the answer isstraightforward. Because the basic bet offered by thecasino has an expected value of zero, average win-nings are zero for both types. However, if, as in actualcasinos, the basic bet has a negative expected value,then an agent’s average winnings are the (negative)expected value of the basic bet multiplied by the aver-age number of rounds the agent gambles. To deter-mine whether the naive agent loses more, on average,than the commitment-aided sophisticate, we there-fore need to check whether he gambles for longer, onaverage.

We have studied the gambling behavior of the twotypes of agents when the basic bet offered by thecasino has a negative expected value, for example,when it takes the form 4$h100463−$h100545. We findthat for most preference parameter values (althoughnot all), the naive agent stays in the casino longer, onaverage, than does the sophisticated agent. In mostcases, then, his average losses are larger.

25 When the commitment-aided sophisticate chooses a plan to com-mit to at time 0, he does not put any weight on the preferencesof his future “selves.” For example, when 4�1�1�5= 40095100511055,he commits to the plan in the left panel of Figure 4 even thoughhe knows that, in node 41125, his time 1 self will want to con-tinue gambling. One interpretation of this is that the time 0 selfdisapproves of his future preferences. This, in turn, can be justifiedby arguing that, in wanting to deviate from the time 0 plan, thetime 1 self is committing an error of “consequentialism”: of ignor-ing risk that was previously borne but not realized (Machina 1989).Nonetheless, it may be interesting to study a model in which thetime 0 self puts at least some weight on future selves’ preferenceswhen choosing a plan to commit to.

4.2. Competition from LotteriesA natural question raised by our analysis is, How cancasinos survive competition from lottery providers?After all, the one-shot gambles offered by lotteryproviders may be a more convenient source of thepositive skewness that the casino goers in our frame-work are seeking.

In the online supplement, we study a simple equi-librium model in which there is competitive provisionof both one-shot lotteries and casinos and where bothlottery providers and casinos incur a cost per con-sumer served. Interestingly, we find that both typesof firms manage to break even. In equilibrium, lot-tery providers attract the no-commitment sophisti-cates. These agents prefer lotteries to casinos becausethey know that in a casino, they would face a nega-tively skewed, and hence unattractive, distribution ofaccumulated gains and losses.

Casinos compete with lottery providers by offer-ing slightly better, albeit unfair, odds. This attracts thecommitment-aided sophisticates and the naive agents,both of whom think that, through a particular choiceof exit strategy, they can construct an overall casinoexperience—in other words, a distribution of accumu-lated gains and losses—whose prospect theory valueexceeds the prospect theory value offered by one-shot lotteries. The commitment-aided sophisticatesare indeed able to construct such a casino experience.Because casinos incur a cost per consumer served,they lose money on these agents. They make theselosses up, however, on the naive agents, because, asnoted in §4.1, these agents gamble longer, on average,than they were planning to. In short, then, casinosare able to compete with lottery providers becausetheir dynamic structure allows them to exploit naiveagents’ time inconsistency.

The equilibrium model also sheds light on a relatedquestion, namely, whether casinos would want toexplicitly offer, in the form of a one-shot gamble,the overall casino experience that their customersare trying to construct dynamically. According tothe model, casinos would not want to offer such aone-shot gamble. If they did, naive agents, believ-ing themselves to be indifferent between the one-shotand dynamic gambles, might switch to the one-shotgamble, thereby effectively converting themselvesfrom naive agents to commitment-aided sophisticates.Casinos would then lose money, however, becauseit is precisely naive agents’ time inconsistency thatallows them to break even.

4.3. Predictions and Other EvidenceOur model makes a number of novel predictions—predictions that, we hope, will eventually be tested.Perhaps the clearest prediction is that gamblers’planned behavior will differ from their actual behav-ior in systematic ways. Specifically, if we survey people

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when they first enter a casino as to what they plan todo and then look at what they actually do, we shouldfind that, on average, they exit sooner than plannedin the region of gains and later than planned in theregion of losses. Moreover, if gamblers who are moresophisticated in the real-world sense of the word—in terms of education or income, say—are also moresophisticated in terms of recognizing their potentialtime inconsistency, we should see a larger differencebetween planned and actual behavior among the lesssophisticated.

Some experimental evidence already available inthe literature gives us hope that these predictionswill be confirmed in the field. Barkan and Busemeyer(1999) and Andrade and Iyer (2009) offer subjects asequence of 50:50 bets in a laboratory setting; butbefore playing the gambles, subjects are asked howthey plan to gamble in each round. Both studies findthat, consistent with our model, subjects systemati-cally gamble more than planned after an early loss;and in Barkan and Busemeyer’s study, they also gam-ble less than planned after an early gain.26

5. ConclusionIn this paper, we present a new model of casino gam-bling, one that is rooted in the probability weight-ing component of cumulative prospect theory. Inrecent years, probability weighting has been linkedto a wide range of economic phenomena. For exam-ple, Barberis and Huang (2008) suggest that it isresponsible for several empirical patterns in financialmarkets, including the low long-term average returnon IPO stocks and the apparent overpricing of out-of-the-money options. Taken together with this priorresearch, then, our paper suggests that casino gam-bling is not an isolated phenomenon requiring its ownunique explanation, but rather that it is one of a fam-ily of empirical facts, all of which are driven by thesame underlying mechanism: probability weighting.

AcknowledgmentsThe author is grateful to Isaiah Andrews and Ji Shen forexcellent research assistance. For very helpful comments, the

26 If we reinterpret the binomial tree in §3 as representing not acasino, but rather the evolution of a stock price over time, thenthe results in that section also suggest predictions about how acumulative prospect theory investor would trade a stock over time.We have studied this question in a framework similar to that of §3and find that the model indeed delivers a range of novel predic-tions. For example, a naive investor, who is unaware of the timeinconsistency generated by probability weighting, may exhibit a“disposition effect” in his trading (see Odean 1998) even though heplanned to exhibit the opposite of the disposition effect. This par-ticular contrast between planned and actual behavior has not beennoted in the prior work on the trading of prospect theory investorsbecause this literature has ignored the dynamic effects of probabil-ity weighting.

author is grateful to Ulf Axelson, Daniel Benjamin, StefanoDellaVigna, Yaxin Duan, Keith Gamble, Daniel Gottlieb,Jonathan Ingersoll, Bjorn Johnson, Kristof Madarasz, TedO’Donoghue, Matthew Rabin, Georg Weizsacker, severalreferees, and seminar participants at Caltech, ColumbiaUniversity, Cornell University, DePaul University, HarvardUniversity, the London School of Economics, PrincetonUniversity, Stanford University, the Stockholm School ofEconomics, the University of California at Berkeley, the Uni-versity of Chicago, the Wharton School, Yale University, andthe National Bureau of Economic Research.

Appendix. Proof of Proposition 1Through extensive numerical analysis, we find that whenthe naive agent enters the casino, he often chooses the fol-lowing strategy or one similar to it: he exits (i) if he loses inthe first round; (ii) if, after the first round, his accumulatedwinnings ever drop to zero; and (iii) at time T , if he hasnot already left by that point. Condition (12) simply checkswhether the cumulative prospect theory value of this spe-cific exit strategy is positive. If it is, we know that the agententers the casino.

If the agent exits because he loses in the first round, then,because the payoff of −$h is the only negative payoff hecan receive under the above exit strategy, its contributionto the cumulative prospect theory value of the strategy is−�h�w40055. If he exits because, at some point after the firstround, his accumulated winnings drop to zero, this con-tributes nothing to the cumulative prospect theory valueof the exit strategy, precisely because the payoff is zero.All that remains, then, is to compute the component of thecumulative prospect theory value of the exit strategy thatstems from the agent exiting at date T .

Under the above exit strategy, there are T − 6T /27 dateT nodes with positive payoffs at which the agent mightexit, namely, nodes 4T 1 j5, where j = 11 0 0 0 1 T − 6T /27. Thepayoff in node 4T 1 j5 is 4T + 2 − 2j5h. We need to computethe probability that the agent exits in node 4T 1 j5, in otherwords, the probability that he moves from the initial node40115 to node 4T 1 j5 without losing in the first round andwithout his accumulated winnings dropping to zero at anypoint after that. With the help of the reflection principle (seeFeller 1968), we compute this probability to be

2−T

[(

T − 1j − 1

)

−

(

T − 1j − 2

)]

0

The probability weight associated with node 4T 1 j5 istherefore

w

(

2−T

(

T − 1j − 1

))

−w

(

2−T

(

T − 1j − 2

))

0

In summary then, the exit strategy we described abovehas positive cumulative prospect theory value—and hencethe naive agent is willing to enter the casino—if

T−6T /27∑

j=1

44T + 2 − 2j5h5�[

w

(

2−T

(

T − 1j − 1

))

−w

(

2−T

(

T − 1j − 2

))]

−�h�w40055 > 00

This is condition (12).

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By appropriately modifying the above argument, it isstraightforward to check that if the basic bet offered by thecasino is 4$h1p3−$h11 − p5 for some p ∈ 40115 rather than4$h10053−$h10055, a sufficient condition for the naive agentto enter isT−6T /27∑

j=1

4T + 2 − 2j5�(

w

(

j∑

l=1

pl

)

−w

(j−1∑

l=1

pl

))

−�w41 − p5 > 01

where

pl = pT−l+141 − p5l−1[(

T − 1l− 1

)

−

(

T − 1l− 2

)]

0

ReferencesAndrade, E., G. Iyer. 2009. Planned and actual betting in sequential

gambles. J. Marketing Res. 46(3) 372–383.Barberis, N., M. Huang. 2008. Stocks as lotteries: The implications

of probability weighting for security prices. Amer. Econom. Rev.98(5) 2066–2100.

Barkan, R., J. R. Busemeyer. 1999. Changing plans: Dynamic incon-sistency and the effect of experience on the reference point.Psychon. Bull. Rev. 6(4) 547–554.

Benartzi, S., R. Thaler. 1995. Myopic loss aversion and the equitypremium puzzle. Quart. J. Econom. 110(3) 75–92.

Conlisk, J. 1993. The utility of gambling. J. Risk Uncertainty 6(3)255–275.

Feller, W. 1968. An Introduction to Probability Theory and Its Applica-tions. Wiley, New York.

Friedman, M., L. Savage. 1948. The utility analysis of choicesinvolving risk. J. Political Econom. 56(4) 279–304.

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

Machina, M. 1989. Dynamic consistency and non-expected utilitymodels of choice under uncertainty. J. Econom. Literature 27(4)1622–1668.

Odean, T. 1998. Are investors reluctant to realize their losses?J. Finance 53(5) 1775–1798.

Tversky, A., D. Kahneman. 1992. Advances in prospect theory:Cumulative representation of uncertainty. J. Risk Uncertainty5(4) 297–323.

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Barberis2012_A Model of Casino Gambling

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