WHO REALLY WANTS TO BE A MILLIONAIRE? ESTIMATES ...

WHO REALLY WANTS TO BE A MILLIONAIRE? ESTIMATES OFRISK AVERSION FROM GAMESHOW DATA

ROGER HARTLEYa, GAUTHIER LANOTb* AND IAN WALKERc

a Department of Economics, School of Social Sciences, TheUniversity ofManchester,OxfordRoad,ManchesterM13 9PL,UnitedKingdomb Department of Economics, Umeå University, SE-901 87 Umeå, Sweden

c Department of Economics, The Management School, Lancaster University, Bailrigg, Lancaster LA1 4YX, United Kingdom

SUMMARYThis paper estimates the degree of risk aversion from one of the most popular TV gameshows ever. The format ofthe show is straightforward; it involves no strategic decision making; we have a large number of observations; andthe prizes are cash, which is paid immediately and covers a large range: from £100 up to £1 million. We providenon-parametric estimates of the utility function and then we test some parametric restrictions. We find that,although the restriction to CRRA utility is statistically rejected, a log function approximates the utility functionquite well over a large range of potential winnings. Copyright © 2013 John Wiley & Sons, Ltd.

Received 4 June 2012; Revised 3 May 2013

Supporting information may be found in the online version of this article.

1. INTRODUCTION

Despite many misgivings, expected utility theory (EUT) remains an important framework for thinkingabout decision making under uncertainty in many contexts. Empirical challenges to EUT have typi-cally compared some non-EU model against a specific parametrization of a model based on EUT.Within the EUT model, we focus our analysis instead on the ‘shape of the utility function’ that explainsthe behaviour of contestants in a UK TV gameshow. For this particular game, the range of prizes islarge: starting at £100 and potentially reaching £1,000,000, depending on the quality of the contestant.Unlike other games used in the literature, WWTBAM is a game of skill. The importance of skill makesour estimation problem more complex. Thus we estimate a structural model where individuals makechoices on the basis of backward induction and we rely on EUT as the maintained hypothesis whentesting for specific functional forms. We begin by estimating (scaled) utility values associated witheach prize level in the game. We then provide minimum distance estimates of the parameters of com-mon functional forms for the utility function based on our estimates of a flexible specification and testthe validity of these parametric restrictions.The validity of the EUT as an adequate description of behaviour has been tested. For example, Hey

and Orme (1994), in a seminal study that provides statistical tests of EUT against several alternativesusing laboratory data, conclude that ‘EUT emerges fairly intact’. More recently, an innovative paper byHarrison and Rutström (2009) estimates a mixture model of specific parametrizations of EU and pros-pect theory (PT), due to Kahneman and Tversky (1979), which yields estimates that suggest that the‘probability of the EUT and PT specifications indicate that each is equally likely’. In common withmost of the literature, both studies focus on testing EUT together with a utility function of the constantabsolute risk aversion (CARA) or constant relative risk aversion (CRRA) form.

* Correspondence to: Gauthier Lanot, Department of Economics, Umeå University, 901 87 Umeå, Sweden.E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 29: 861–879 (2014)Published online 18 September 2013 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/jae.2353

The shape of the utility function is important for the modelling of choice under risk. Rabin’s (2000)calibration result shows that expected utility theory (EUT) with a concave utility function (over wealthoutcomes) cannot explain observed aversion to small gambles whatever the wealth level, without im-plying an implausibly high degree of aversion to risk when facing large gambles. The concavity of theutility function is crucial to Rabin’s juxtaposition of large and small gambles. While Rabin’s argumentmay not be the final death blow to EUT (see Rubinstein, 2001; Watt, 2002; Palacios-Huerta andSerrano, 2006), it illustrates the role that the assumption of concavity of the utility function plays.Friedman and Savage (1948) suggest, for example, that marginal utility need not be everywhere con-cave, and Hartley and Farrell (2002) extend this argument to an intertemporal setting and show thatdifferent borrowing and lending rates can account for persistent gambling. Indeed, in more complexenvironments individuals may behave differently with respect to small-stake gambles than they dowhen facing large stake gambles for reasons that do not enter into the Rabin argument. For example,Chetty and Szeidl (2007) show that individuals may remain committed to their current levels of con-sumption of some goods and services because of adjustment costs for some elements of consumption.This effectively raises their risk aversion over small and low-skew gambles, relative to risk aversionover large gambles, or small highly skewed gambles where extreme outcomes would justify payingthe cost to adjust all components of consumption. Similar ideas can be found in Rosen (1997), whereindivisibilities in consumption can generate risk-loving behaviour over a range of wealth values.

We find that, although the restrictions to CRRA or CARA are statistically rejected, a ‘simple’ utilitymodel is consistent with our non-parametric estimates over a large range of the value of prizes.Remarkably, over that very large range (£2 k–500 k), this simple model with a coefficient of riskaversion close to 1 provides a good fit. Above and below this level we have too few informativeobservations to provide reliable non-parametric estimates.

The paper is structured as follows. In section 2 we briefly outline the existing evidence that relies ongameshow data. Section 3 explains the operation of the game used here in more detail. In section 4 weprovide a simplified structural model of the game, which we then extend to account for heterogeneousbeliefs and for the availability of lifelines. In Section 5 we present the econometric details and thelikelihood. In Section 6 we give some summary details of our data and explain how we estimate riskaversion using these data. In Section 7 we present some results and consider possible shortcomingsof the work. In Section 8 we draw together some conclusions.

2. EXISTING EVIDENCE

Our own analysis elicits preferences from responses to offers of very specific gambles in a gameshowcontext. Since our contestants face a sequence of lotteries which change across rounds of the game, wecan obtain an estimate for each lottery faced. A related literature elicits preferences from responses tohypothetical (and sometimes real) lotteries. Many papers use relatively low-value lotteries on relativelysmall samples of selected participants (often students). For example, Harrison and Rutström (2009)estimate their finite mixture model (of EU and PT) using data generated by 158 student subjects facinglotteries with payouts of up to just $15. In contrast Harrison et al. (2007) use data from a randomsample of 253 Danish individuals drawn from the general population who chose between a range oflarger lotteries, with expected values up to 3850 DKK (approx. US $670) . They estimate an EUCRRA model and find a coefficient of 0.78.

Levitt and List (2007) have questioned the reliability of data generated in laboratory experimentsand some research has cast light on the contrast between field and lab experiments. For example, Holtand Laury (2002) compare large and small gambles within the field comparing real and hypotheticalstakes that are typical of lab studies with larger stakes. The authors estimate the extent to which pref-erences differ across the real and hypothetical lotteries. Their analysis features prizes that range up toseveral hundreds of dollars and find that the results are similar only for small gambles. Holt and Laury

R. HARTLEY, G. LANOT AND I. WALKER862

Copyright © 2013 John Wiley & Sons, Ltd. J. Appl. Econ. 29: 861–879 (2014)DOI: 10.1002/jae

(2002) estimate an expo-power expected utility function that allows for non-constant relative risk aver-sion (RRA), using real payoffs.Rabin and Thaler (2001) and Rabin (2000) suggest that risk aversion increasing with stakes is a nec-

essary condition for EUT to provide a unified account of attitudes towards risk. Consistent withRabin’s calibration theorem, Holt and Laury (2002) find small degrees of RRA (around 0.3) at lowprize levels and higher (around 0.9) at high prize levels which, together, fit the data well. However,even the largest payouts (up to $350) considered in Holt and Laury (2002) are small compared to typ-ical gameshows.An extensive review by Harrison and List (2004) concludes that field and lab-based experiments

complement each other. The existing recent literature that uses field (as opposed to lab-based) exper-iments is dominated by data that have been generated by a game of chance called Deal or No Deal(DOND), or similar in other languages. DOND is a game of chance in which contestants eliminate ran-domly a number of cash prizes and, at regular intervals, contestants are given the option to trade againstcash the option of winning one of the remaining cash prizes.1 Post et al. (2008) consider US, Germanand Dutch DOND data and they also replicate the game in the classroom and generate data from stu-dent participants. Their estimates of the degree of relative risk aversion, maintaining the concavity ofthe utility function, lie in the range 1–2. However, they find it hard to reconcile the data with simpleEUT and note that the behaviour of individuals in similar positions is coloured by the path by whichthey got to that position—they refer to this as path dependence.2 Deck et al. (2008) use the MexicanDOND game, estimate a structural model and report CRRA estimates that are approximately 1.Bombardini and Trebbi (2012) use DOND data (from Italy) to estimate a structural dynamic stochasticmodel using maximum likelihood. They estimate the average degree of RRA to be close to 1. de Roosand Sarafidis (2010) use Australian DOND data to estimate structural CARA and CRRA expected util-ity functions and adopt both random effects and random coefficient specifications of the stochasticchoice model and find estimates of relative risk aversion around 2.5 (evaluated at the average incomelevel). They also test a rank dependent utility (RDU; see Quiggin, 1982) characterization of preferencesagainst EUT and find that RDU significantly outperforms EUT—they find that contestants are system-atically ‘optimistic’. Of course, DOND is a game of chance that is broadcast for its entertainment value.It seems very likely that players are selected on to the show for the way in which their behaviour islikely to contribute to viewing figures and this might well be consistent with contestants being optimis-tic. Optimism has the added advantage, to its producers, that the costs of running the show are likely tobe less than would otherwise be the case. Indeed, at least in the UK game, the compere, the audienceand fellow show participants all seem to encourage risk taking by the current contestant.Andersen et al. (2008) reviews the evidence from analyses of gameshow data, including the papers

which use data from DOND. The estimate that Andersen et al. (2008) provides for the degree of riskaversion in a CRRA specification, using the US DOND gameshow data, is around 0.20 (compared to0.45 from their laboratory data). They generalize this to allow for non-constant risk aversion and find anear-zero risk aversion coefficient at prizes below $10,000, rising to 0.75 at $1 million. When they testagainst CRRA they easily reject in favour of the non-constant RRA specification.Much of this literature estimates EUT models; most of the estimates of the degree of relative risk

aversion are between 0.3 and 3.0, with several studies producing estimates of the coefficient of RRAthat are close to 1; and some findings reject CRRA in favour of a specification that implies low

1 Conte et al. (2012) provide estimates of attitudes to risk under alternative modelling hypotheses of the banker’s function, and itadds to the discussion of functional form. Mulino et al. (2009) also consider the Australian DOND game but choose not to es-timate a structural model, while Mulino et al. (2006) provide structural estimates that conclude that players are risk loving butassumes that players are myopic. Blavatskyy and Pogrebna (2008) show that DOND players have identical risk attitudesirrespective of whether lotteries yield positive outcomes with low or high probability.2 Baltussen et al. (2008) also use DOND data from a variety of countries. They do not estimate a specific structural model butnote that behaviour is quite different across games with different prize structures but similarly responsive to relative stakes.

WHO REALLY WANTS TO BE A MILLIONAIRE? 863


(orno) risk aversion at low stakes and higher RRA when evaluated at high stakes. Among the studieswhich maintain EUT, most suggest that CRRA (and CARA) is too strong an assumption and preferexpo-power specification, which allows for a variable degree of RRA and nests both special cases(see Holt and Laury, 2002; Saha, 1993).

Wilcox (2011) and Blavatskyy and Pogrebna (2010) point out that observed choices under risk re-quire a theory of how alternatives are valued to be combined with a theory of how the choices betweenalternatives are made in the light of their valuations. Wilcox (2011) refers to the evidence from manyexperimental studies to argue that choices under risk are highly stochastic. He shows, using the Heyand Orme experimental data, how the model of stochastic choice adopted by the researcher mattersfor evaluating EUT against RDU. Similarly, Blavatskyy and Pogrebna (2010) use DOND data to ex-tend the analysis of Hey and Orme (1994). They emphasize that whether EUT is dominated by someother model of preferences is not independent of the assumption made about how players make choiceson the basis of their preference calculations. Their DOND results suggest that EUT is the preferred ex-planation if the decision rule used by players to make choices in the game is sufficiently complex—forexample, a random utility model of stochastic choice. Across all models that they consider they findthat the (equal) best explanation of the data is provided by EUT using an expo-power utility function(equal with regret theory; see Loomes and Sugden, 1982).

3. THE WWTBAM GAMESHOW

Our data come from what has probably been the most popular TV gameshow of all time:Who wants tobe a millionaire? (hereafter WWTBAM). This particular game has a number of design features thatmake it well suited to our task: the format is straightforward; in stark contrast to DOND the compereof WWTBAM takes great care to neither encourage nor discourage contestants; and the prizes are paidimmediately in cash and cover a large range: up to £1 million in the UK case. In this gameshow thecontestant is faced with a sequence of 15 multiple-choice questions. At each stage she can guess theanswer to the current question and might double her current winnings but at the risk of losing and leav-ing the game with a question-specific amount, or she can quit, opt not to answer the current question,and leave the game with her winnings to date. At each stage contestants are reminded that their win-nings so far belong to them—to risk, or walk away with. The prizes start at a modest level but reachvery high levels. The widespread of possible outcomes makes WWTBAM a considerable challengefor EUT, and for simple parametric forms in particular. The sequential nature of the game gives riseto an important complication: in all but the last stage of the game, answering a question correctly givesan option to hear the next question and this itself has a value, over and above the value of the additionto wealth associated with the question. This option value depends on the stage of the game, the contes-tant’s view about the difficulty of subsequent questions, and the degree of risk aversion.

While skill complicates our analysis, it allows for the possibility that an individual can make ex post‘mistakes’ in her decisions because she may find that subsequent questions are more or less difficultthan she anticipated. In our model it is the variability in ‘skills’ which explains the different trajectoriesthrough the game of otherwise identical individuals. Thus our model has an explicit role for ‘noise’ andso does not rely on the preference specification alone to fit the data. One feature of the game that addsconsiderably to complexity is the existence of ‘lifelines’. These are opportunities for the player to im-prove her chances of providing correct answers. The decision to play a given lifeline can be viewed asan ‘investment’ decision which involves managing the trade-off between playing the lifeline and in-creasing the chances of success in the current round, and not playing the lifeline now, so increasingthe likelihood of success in some future round. This investment will change the individual’s perceptionof the difficulty of the current question. Understanding the timing and sequence of use of the lifelinesprovides further information which should help reject a model based on EUT. We model these lifelinesin simple but intuitive and flexible ways.



The game features a sequence of 15 ‘multiple-choice’ questions with associated prizes that, in theUK, start at £100 (approximately US $150, €120) and (approximately) double with each correctly an-swered question, so that the final question results in overall winnings of £1 million. The contestant hasthe choice of quitting with her accumulated winnings or gambling by choosing between the four pos-sible answers given after being asked each question. If the chosen answer is correct the contestant dou-bles her existing winnings (and is then asked the next question). If the chosen answer is incorrect shegets some ‘fallback’ level of winnings and leaves the game. The design is such that the difficulty ofquestions rises, on average, across the sequence of rounds, and the fallback level of winnings also rises(in just two steps). At least in the UK, questions are chosen randomly by computer from a sequence ofquestion banks and there is no attempt to manipulate the fortunes of individual contestants during play.To complicate our analysis, contestants are endowed with three ‘lifelines’ which are use-once oppor-

tunities that they may use to improve their odds. The three lifelines are: ‘fifty-fifty’ (50:50 hereafter),which removes two of the three incorrect answers; ‘ask-the-audience’ (ATA), which provides ahistogram showing audience member opinions of the correct answer; and ‘phone-a-friend’ (PAF),which allows the contestant to call a friend for advice.Contestants are not randomly selected on to the show. The details of how this is done varies across

countries but in the UK aspiring contestants ring a premium rate phone number, where calls cost ap-proximately £1, and are asked a question of medium difficulty. If correct, their name is entered intoa draw to appear in the studio. Ten names are drawn for each show (plus two reserves). Aspiring con-testants can improve their odds of appearing by ringing many times, so having many entries in thedraw. Once at the recording studio, aspiring contestants compete with each other to provide the fastestcorrect answer to a single question and the winner is selected to enter the main game that we analysehere.During play the compère is very careful, at least in the UK game, to ensure that contestants are sure

they want to commit themselves at every stage—contestants have to utter the trigger phrase ‘finalanswer’ to indicate commitment. At each of the two fallback stages, the compère hands a cheque tothe contestant for that level of winnings and ensures that the contestant understands that this moneyis now theirs and cannot be subsequently lost. The UK compère makes every effort to ensure thatcontestants behave rationally—he strongly discourages quitting at the question corresponding to thefallback levels (where there is no downside risk), and quitting with unused lifelines. He is also carefulto ensure that contestants understand the magnitude of the risks that they face in each round of thegame. Neither compère nor the game format seems to encourage cautious or hesitant contestants to takeundue risks. The contrast between WWTBAM and DOND could not be sharper.

4. A SIMPLE MODEL

4.1. Dynamic Aspects of the Game

The model of play we present here accounts for the dynamic structure of the complete game. We focusinitially on a simplified version of the game in which contestants are risk neutral and hence areexpected income maximizers. In Section 4.2 we generalize to consider risk aversion in a very straight-forward way that allows us to retain the structure that we have here. Modelling lifelines requires thatwe are precise about how we specify the beliefs that contestants hold concerning the difficulty of ques-tions—which we do in section 4.3. We postpone consideration of the ‘lifelines’ to Section 4.4 and, forthe moment, assume that questions are selected by independent random draws from a pool of questionsof identical difficulty.Let p denote the probability that the contestant (of some given ability) is able to correctly answer a

question, where p is a realization of the random variable P with cumulative distribution function F.Rounds of the game are denoted by the number of questions remaining, i.e. n=N, …, 1. Let an



bethe accumulated winnings after the contestant has successfully completed N� n questions and there

are n questions remaining. The prizes (in £1000 units) are given by the sequence anf g16n¼1¼1000; 500; 250; 125; 64; 32; 16; 8; 4; 2; 1; 0:5; 0:3; 0:2; 0:1; 0f g. Similarly, let bn be the value of the fall-back level of winnings, i.e. the winnings that can be kept in the event of an incorrect answer, which is

given by bnf g15n¼1¼ 32; 32; 32; 32; 32; 1; 1; 1; 1; 1; 0; 0; 0; 0; 0f g.At the start of the game, the value of playing the game is given byV15(p) =max{a16, p( f14� b15) + b15}

where f14 =E[V14(P)] is the value of continuing optimally and, at this stage, a16 = b15 = 0. This is the firststage of a recursion, such that when there are n questions to go and the question asked can be answeredsuccessfully with probability p, the value of the game is Vn( p) =max{an + 1, p( fn� 1� bn) + bn}, wherefn� 1 = E[Vn� 1(P)] and we set f0 =a1. Note that the decision to quit or not is made after the question hasbeen asked. At any round of the game, there exists a critical value of p, pn ¼ anþ1 � bnð Þ= f n�1 � bnð Þfor an income maximizer, such that if p≤pn the individual quits the game and therefore Vn(p) = an+1.Otherwise p > pn and the individual offers an answer to the question and the value of the gameis Vn( p) = p( fn� 1� bn) + bn. Note that the immediate value of answering correctly is an + 1 andp( fn� 1� bn) measures the sum of the option value of continuing to play the game, p( fn� 1� an), if thecontestant were to answer correctly, and the expected value of giving a correct answer over and abovethe assured gains, p(an� bn).

These dynamic programming equations lead to the following relationship for {fn},

f n�1 � f n ¼ f n�1 � bnð ÞZ 1

pn

F pð Þdp . To obtain the likelihood we need to evaluate the probability

of winning. The probability of continuing to participate by choosing an answer to the nth question, but priorto seeing the next question, is Pr ’Play’½ � ¼ 1� F pnð Þ≡F pnð Þ. The probability of giving a correct answer,

having decided to answer, is given by Pr½’Win’ ’Play’� ¼Z 1

pn

pdF pð Þ=1� F pnð Þ≡G pnð Þ=F pnð Þ�� , where

the probability of answering correctly is simply Pr ’Win’½ � ¼ G pnð Þ . Thus the likelihood of acontestant reaching round k and then quitting (i.e. refusing to give an answer to question k) is

L k; 0ð Þ ¼ 1� F pkð Þ� �∏15

n¼kþ1G pnð Þ. The probability of a contestant reaching round k and then giving

an incorrect answer is L k; 1ð Þ ¼ F pkð Þ � G pkð Þ� �∏15

n¼kþ1G pnð Þ. Finally, the probability of a contestant

reaching the final round, 1, and then winning (£1 million) is L 1; :ð Þ ¼ ∏15

n¼1G pnð Þ.

4.2. Risk Aversion

The extension to allow for risk aversion, as opposed to neutrality, is very simply accommodated byallowing prizes to be measured in utility terms, i.e. we consider a′n = u(an), n = 1… 16, and b′n = u(bn), n= 1… 15, for some increasing utility function u(x) with u(0) = 0.

4.3. Questions, Answers and Beliefs

It seems reasonable to assume that the contestant chooses the answer with the highest subjectiveprobability of being correct. Hence, once the distribution of that probability is defined, it becomesstraightforward to describe the probability distribution of the maximum belief and, more generally,of the order statistics.

A question, and a list of possible answers, is drawn uniformly (at each stage of the game) from apool of questions (assuming just one pool for the moment). The question and its possible answers



are presented to the contestant, who is endowed with a draw from the belief distribution concerning thelikelihood of each answer being correct. The formation of beliefs for all contestants is assumed to fol-low this process in an identical and independent manner. Hence, given a particular question, two oth-erwise identical individuals can hold distinct beliefs concerning the likelihood of each answer beingcorrect.Formally, suppose that X is an k-dimensional random vector, with k≤ 4 since there are at most four

possible choices, on the simplex Δk ¼ x : xi≥0 ∀i ¼ 1…:k;Xk

i¼1xi ¼ 1

n o. We assume that x

has the probability density function ψk(x) and we require it to exhibit the following symmetry property:if x; x∈Δk, such that x is obtained from x by a permutation of its components, thenψk xð Þ ¼ ψk xð Þ. Thisproperty reflects the fact that the ordering of the potential answers does not contain any information initself. This property is weak in general. In order to develop the model we construct a specific family ofdistributions of beliefs which satisfies the symmetry condition and which will prove convenient duringthe estimation process. We start with a probability density function (pdf) ϕ on [0,1] that is symmetric,

i.e. such that ϕ(x) =ϕ(1� x) for all x in [0,1]. Note that ϕ’s symmetry impliesZ 1

0ϕ xð Þ 1� xð Þdx ¼ 1

2

andZ 1

0ϕ xð Þ 1� xð Þ2dx ¼

Z 1

0ϕ 1� xð Þ xð Þ2dx ¼ μ2 , where μ2 is the second moment of ϕ. If Φ

denotes the distribution function corresponding toϕ, it is straightforward to show thatΦ[1� z] = 1�Φ[z].From ϕ we deduce the belief distribution for two, three, and four possible answers and find:

ψ2 x1; x2ð Þ ¼ 12ϕ x1ð Þ þ ϕ x2ð Þ½ � (1)

ψ3 x1; x2; x3ð Þ ¼ 13

Xi; j;kf g∈P3

ϕ xkð Þϕ xj1� xk

� �(2)

ψ4 x1; x2; x3; x4ð Þ ¼ 112μ2

Xi; j;k;lf g∈P4

ϕ xlð Þϕ xk1� xl

� �ϕ

xj1� xl � xk

� �(3)

wherePk is the set of all permutations of {1,…,k}, andμ2 ¼Z 1

0x2ϕ xð Þdx. In each case the summation

of the set of permutations arises because of the unobserved random (uniform) order in which the

possible answers are presented. Because ϕ is symmetric andXk

i¼1xi ¼ 1, some simplifications are

possible, so that for the cases of interest we have

ψ2 x1; x2ð Þ ¼ ϕ x1ð Þ (4)

ψ3 x1; x2; x3ð Þ ¼ 23

ϕ x2ð Þϕ x21� x1

� �þ ϕ x2ð Þϕ x3

1� x2

� �þ ϕ x3ð Þϕ x1

1� x3

� �� (5)

ψ4 x1; x2; x3; x4ð Þ ¼ 16μ2

Xl;k¼1;…;4k≠l

ϕ xlð Þϕ xk1� xl

� �ϕ

xj1� xl � xk

� �(6)

These simplifications are useful since the number of terms involved is halved. It is straightforward toverify that the integral of ψk over Δk, k = 2, 3, 4 is unity, and ψk satisfies the symmetry property required



above. If ϕ is the density of the uniform distribution between 0 and 1, then ψk is the uniform distributionover Δk, k=2, 3, 4. This specification of the beliefs distribution is, of course, restrictive even among the dis-tributions satisfying the imposed symmetry property (not all symmetric distributions necessarily satisfyequations (4), (5) and (6)). It leads, however, to simple specifications for the distribution of the order statis-tics and for the distribution of the maximum (see the online Technical Appendix, supporting information).

In Figure 1 we show possible density distributions of the belief that an answer is correct. We assume thatϕ is the density function of a symmetric beta distribution, i.e. such that the two parameters of the densityfunction are equal to some positive value α. Note that a uniform distribution of the belief arises whenα =1. In the top row, we plot ϕ, and therefore ψ2, for α = 0.8 (on the left) and α = 1.2 (on the right). Inthe former case, a contestant who draws his belief for the question at hand according to this distribution islikely to draw a large value of her belief that one of the two answers is highly likely: i.e. the density is largefor small (p< 0.1) or large (p> 0.9) values of the belief that the first answer is the correct one. In the sec-ond row, we describe the contour levels for ψ3 over Δ3. The sides of the triangle have length 1; hence anypoint inside the triangle corresponds to a particular value of the beliefs (x1,x2,x3) that either answer is thecorrect one. The lighter a region, the higher is its density. For α = 0.8, the regions of highest density are in thecorner and along the edges of the simplex, while for α = 1.2 the region of highest density is in the middle ofthe simplex. Roughly speaking, if the beliefs attached to the answers of a given question are described by aBeta distribution with parameters (0.8,0.8), contestants believe that they ‘know’ the answer to the question,since they attach a large belief that one of the answers will be correct. In the other case, where the beliefs aredistributed according to a Beta distribution with parameters (1.2,1.2), contestants are more likely to be‘unsure’ about the correct answer, since the likely values of the belief are around (1/3,1/3,1/3).

To illustrate the differences between the belief distributions when four answers are possible wedescribe instead the distribution of the maximum belief in Figure 2. A contestant will be more likelyto be confident about the correct answer for a particular question the more probability mass is given

Figure 1. Density distributions over beliefs



to larger values of the maximum belief. The figure illustrates this clearly, when α is small (consider firstα= 0.1) the distribution of the maximum belief allocate a large positive probability to a maximum be-lief being close to 1 (the distribution function in this case intersect y = 1 from below and appears as-ymptotic to x = 1), whereas for a large value of α (i.e. α= 10) the probability that the maximumbelief is near 1 is clearly small (the distribution function in this case is asymptotic to y = 1). For thevalues of α of 0.8 and 1.2 the differences are less acute; nevertheless the probability that the maximumbelief takes a value near 1 is larger when α = 0.8 than when α = 1.2.

4.4. Lifelines

The specification of the distribution of the beliefs becomes important when we introduce the lifelines. In-deed, we think of each lifeline as a source of (imprecise) information which is combined with the originalbelief to produce a new set of beliefs on which the contestant acts. Contestants are endowed with threedistinct lifelines that can each be played once and only once at any stage in the game. Multiple lifelinescan be used on one single question. Consider first the game with only one lifeline remaining. To accountfor the lifeline, the state space has to be extended.WewriteWn(p;γ) for the expected value of the game to acontestant faced with a question with belief vector p, where γ=0 if the lifeline has been used and γ=1 if itis still available. Whether to use a lifeline or not may depend on all components of p so the value ofWn is afunction of the whole vector of subjective probabilities. However, p=max ipi is a sufficient statistic for pin the contestant’s decision problem with no lifeline left and we will write this value function as Vn(p;0).We assume that the lifeline is a draw of a new belief, say q, given the current belief p.For example, the use of 50:50 reduces two components of the belief vector to 0. For the other

lifelines the audience and/or one (among several) friends will provide some information which isthen combined with the initial belief. Finally, we define kn(p) ≡ Eq|p[Vn(q,0)|p] to be the value ofplaying the lifeline at stage n, where q=max iqi. When no lifeline is left we have the familiar equationVn(p,0) =max{an+1, p( fn� 1(0)� bn) + bn}, where fn(0) =E[Vn(P;0)]. If the lifeline remains the contes-tant chooses the best of three options, i.e. quit, Wn(p,1) =max{an+1, kn(p),p( fn� 1(1)� bn) + bn}, uselifeline, fn� 1(1)≡E[Wn� 1(P,1)] or play and save the lifeline, f0(1) = f0(0) = a1. The treatment of contes-tants with more than one remaining lifeline is a straightforward extension and the details are given in theonline Technical Appendix.

Figure 2. Distribution of maximum belief



5. DATA

Our data are from the videotapes that cover all 515 actual contestants from the 11 series fromSeptember 1998 to June 2003. In addition, there were 1859 participants who were not chosento compete on the show and were merely spectators. A major concern about the findings of thegameshow literature is that the data are generated by selected samples. To investigate this issuea questionnaire was sent to all of the 2374 potential contestants who had ever been invited tothe studio for all broadcast shows. 791 replies were received. Table I shows the means of thesedata for all of the 515 actual competitors from the videotapes and the 243 (47%) of these whoresponded to our questionnaire. The survey respondents who competed are clearly typical ofthe contestants on the tapes, except that there were more likely to be big winners and so respon-sive to a questionnaire about their experience.

The questionnaire provides data comparable to that available from a number of large official randomsamples of the population. In Table I it was immediately obvious that men were heavily overrepre-sented among both contestants and spectators relative to the population at large. Once the populationsurvey datasets are reweighted, in Table II, the observable differences between the questionnaire dataand the official survey data are small and statistically insignificant. Two variables are particularlynoteworthy: the proportion of individuals who report that their household contents are uninsured issimilar to the population value; and the proportion which reports being regular lottery ticket buyersis also quite similar.

Thus our questionnaire dataset does not suggest that those that play, or wish to play, WWTBAM areheavily selected according to observable variables—except gender; and that their behaviour towardsrisk and gambling in everyday life seemed similar to the population at large. The 243 WWTBAMplayers are more likely to be male, are a little younger, and have slightly more years of education thanthe 548 in the sample who were merely WWTBAM spectators in the studio.

The distribution of contestants’ winnings depends on whether the contestant quit or failed toanswer the last question asked. Almost all contestants who survived beyond £125,000 quit rather thanfailed—only one contestant failed at £500,000 and so left with just £32,000 instead of quitting andleaving with £250,000. Only three contestants failed at a sub £1000 question and left with nothing.Three contestants won the £1 million prize. Two-thirds of contestants quit and one-third failed.‘Failures’ left the studio with an average of £17,438 (£15,000 for women and £18,244 for men), while‘quitters’ went away with an average of £72,247 (£68,182 for women and £73,411 for men), where wecategorize those that won the maximum £1 million as quitters. Finally, the use of lifelines is an impor-tant part of observed behaviour that our model attempts to explain. Out of the 515 contestants, 501played ATA, 488 played 50:50, and 484 played PAF. There were no examples of individuals quittingwith unused lifelines.

Table I. Questionnaire contestant sample and population of contestants

Questionnaire sample of contestants Population of contestants on videotapes

Mean SD Mean SD

Male 0.76 0.43 0.77 0.43Winnings £000 61.96 104.1 54.26 105.9% quit at last Q 0.68 0.47 0.67 0.47N 243 515

Note: We categorize three players who won the maximum £1 million as quitters.



6. ECONOMETRIC SPECIFICATION AND ESTIMATION

6.1. Specification of the Belief Distribution

The distribution of the beliefs describes the unobservables in the model. Under the assumptions in Section4.3 the joint densityψ4(�) can be constructed from some symmetric densityϕ. We assume thatϕ(x) comesfrom a symmetric Beta random variable, B(α,α) on [0,1], i.e. ϕ(x) = (Γ (2α)/Γ (α)2)xα� 1 (1� x)α� 1, withα> 0, and Γ (.) is the gamma function. Given the symmetry ofϕ(x) the Beta distribution provides us with aone-parameter specificationwhich is convenient and relatively flexible. Any random variable distributedB(α,α) has an expectation of ½ and a second moment (α+1)/2(2α +1).It is convenient to use draws from the joint distribution of ordered statistics of the belief distribution.

Because of the symmetry assumptions that we impose on ψ4(�), the joint density function of the orderstatistics (i.e. the vector of beliefs in decreasing order) is simply 4!ψ4 epð Þ, where ep is a vector of valuesordered in decreasing order. The only remaining issue is to draw from a multidimensional random var-iable with joint density χ4(�). In the online Technical Appendix Section A.3 we show how this can beachieved using three independent draws from Beta distributions with parameters (α, α + 2), (α, α+ 1)and (α,α), respectively.

6.2. Likelihood

The contribution of an individual history to the likelihood is the product of the probabilities of success,and of the particular pattern of use for the lifelines for that individual history, up to and including thepenultimate question, multiplied by the probability that for her last question she wins a million, loses orquits and the observed use of the lifelines for this last question.In estimation, we fix some parameters to specific values. Because the £64,000 question will always

be attempted there is no information about the utility from observed behaviour at this point, since the£32,000 prize money is guaranteed. Furthermore, the number of failed answers at the £1000 and £2000level is very small (1 and 0 respectively in our data). Therefore, we fix the corresponding value of the

Table II. Questionnaire sample and population data

Population survey dataa WWTBAM players WWTBAM spectators

Mean SD Mean SD Mean SD

Male 0.52 0.40 0.76 0.43 0.66 0.48Age 44.41 10.21 43.14 9.36 47.86 11.67Married 0.80 0.44 0.79 0.41 0.76 0.43Education years 13.88 4.10 13.71 3.99 12.82 3.22Smoker ++ 0.25 0.42 0.22 0.41 0.26 0.44Renter 0.25 0.33 0.144 0.35 0.177 0.38Contents uninsured + 0.09 0.26 0.07 0.27 0.06 0.31House value (£k)b ++ 178.9 157 190.8 127 184.8 188Employed 0.652 0.44 0.638 0.48 0.593 0.49Self-employed 0.155 0.38 0.193 0.40 0.189 0.39Not working 0.194 0.40 0.160 0.37 0.195 0.40Gross earnings (£k pa)c 27.08 23.0 31.17 24.0 28.67 22.7Regular lottery player +++ 0.67 0.40 0.63 0.41 0.65 0.41Observations Various 243 548

aSurvey datasets have been reweighted to reflect the gender mix in the WWTBAM data. Population data come from the 2002Labour Force Survey with the exception of: ++Family Expenditure Survey 2002 data; ++++British Household Panel Study2001 wave; and ++++++Gambling Prevalence Survey 2002.bIf owner occupier.cIf employed. The gross earnings figure for our survey is the average of the mid-points of a number of ranges of income. The LFSfigure was obtained by multiplying the survey week data by 52.



parameters of the belief distribution to be 0.045, which implies that contestants are confident of an-swers for these early rounds. Furthermore, since we have few observations at the £1 million questionto estimate reliably, we fix the parameter of the belief distribution for the £1,000,000 question (at avalue of 1.95), which suggests that for such a question contestants’ beliefs are distributed away fromcertainty. Similarly, we fix the probability that the ‘friend’ knows the answer for the last two questionsat 1/3 (our parameter estimate of the probability of correctly answering the £250,000 question is 0.39).Finally, we fix the value of the utility of the £1000 prize to a value close to 0 but strictly positive. Noneof our results is sensitive to the precise numbers we choose.

We estimate the most general model we are able to: we normalize the utility of the million poundprize to 1 and the utility of no winnings to 0; we then allow the utility function u(x) to be definedby a single parameter for each step of the game: £2000, £4000, £8,000 etc. Therefore eight utility pa-rameters need to be estimated, which are u(an) in our previous notation—we lose a further parameterbecause there is no decision to be made at £64,000. At this stage we make no assumption about theshape of u(an) apart from the presumption that it is increasing.

We further allow the average difficulty of questions to be round specific, which introduces eight fur-ther parameters which are components of θ in our previous notation. Finally, the probability that thefriend, contacted when the PAF lifeline is played, knows the correct answer is also allowed to be roundspecific, which adds a further nine parameters. The 50:50 lifeline does not add further parameters. TheATA lifeline adds two further parameters, which we estimate just from the histograms of the audiencebeliefs. Hence the likelihood depends on 25 distinct parameters.

The issue of dealing with lifelines is complex. At each stage of the game we estimate the value ofplaying each lifeline, and combinations of lifelines, then available. Our approach to modelling 50:50is parsimonious but intuitively compelling. Our approach to ATA relies only on the observed data fromthe audience behaviour. The appropriate modelling of PAF is simple but more debatable—we assumethat a friend is an expert who either knows or doesn’t know the answer: if she doesn’t know then thecontestant does not update his belief; if she does then he believes as if he knows the answer with cer-tainty. The contribution to the likelihood of contestant i’s history, which ends at stage n�i of the game,has the general form

L̂S;i LL k; ið Þ; ll k; ið Þð Þf gn�ik¼1; θ; λ̂ν̂

¼ ∏n�i �1

k¼1Ω̂LL k;ið Þ

ll k;ið Þ;k

( )Ω̂

LL n�i ;ið Þll n�i ;ið Þ;n�i

where LL(k,i) indicates the number and nature of the lifelines available to the contestant i at stage k,and ll(k,i) selects the relevant probability depending on the lifeline used by contestant i at stage k.Ω̂LL k;ið Þ

ll k;ið Þ;k is the simulated probability for the event indexed by ll(k,i), LL(k,i) and k. Section A.4 in theonline Technical Appendix describes how the simulations are performed. θ is the vector of parametersof interest, i.e. the parameters of the utility function at the various prize levels, the values of theparameters of the belief distribution and the values of the probability that the friend knows the correctanswer when the PAF lifeline is used for each round. Finally, (̂λ; ν̂) are the independent estimates of theparameters of the density of the updated belief which results from the use of ATA—see the onlineTechnical Appendix for further details.

7. ESTIMATION AND RESULTS

The model assumes that contestants understand every aspect of the game and, in particular, are awareof the distribution of their own beliefs at every round of the game as well as the information contentand value of the lifelines. Furthermore, the model assumes that contestants are aware of the valueof their lifelines in later rounds involving more valuable but harder questions. Indeed we observe



that lifelines are played throughout the game. Furthermore, players do fail even though they retainunused lifelines.We first estimate a model for the histograms that the lifeline ATA produces. Modelling ATA

requires more than simply applying Bayes’ rule to the current belief draw. In particular, we mustallow the contestant to learn from the information provided by the lifeline, i.e. here theproportions of the audience’s votes in favour of each alternative answer. The difficulty here isto understand why and how a ‘perfectly informed’ rational individual should revise his/her prioron the basis of someone else’s opinion. The route we follow here was proposed by French (1980)in the context of belief updating after hearing an expert opinion. The histograms record thepercentage of the audience that selects each of the four possible answers whenever asked. This

preliminary estimation is used to determine the two parameters ðλ̂ν̂Þ for the particular Dirichletdistribution of the audience’s distribution of opinions described in Section A.2.3 of the onlineTechnical Appendix.To estimate these parameters we use the data we have collected on these histograms and our knowl-

edge of the correct answer. The parameter estimates are presented in Table II and imply that, on aver-

age, the expected histogram of the lifeline ATA is 0.63, 0.12, 0.12, 0.12, i.e. λ̂λ̂þ3ν̂

; ν̂λ̂þ3ν̂

; ν̂λ̂þ3ν̂

; ν̂λ̂þ3ν̂

.

That is, audiences indicate the correct answer with a peak in the histogram describing the distributionof opinions and on average the peak is approximately 63%. For any particular draw, however, themodel allows for arbitrary histograms. The online Technical Appendix, Section A.2.3, formallydescribes this aspect of the model. The parameter estimates allow us to evaluate the quality of theATA lifeline: if λ= ν the audience’s opinion does not provide any information and the contestant’sbeliefs are left unaltered; when the difference λ� ν is positive the contestant changes her belief inthe direction of the audience’s histogram; and if λ� ν is negative the contestant changes her beliefaway from the signal provided by the audience.As an example, consider a completely ignorant contestant, i.e. the initial belief is (¼, ¼, ¼, ¼). As-

sume now that ATA produces the histogram (0.7, 0.1, 0.1, 0.1) then Technical Appendix (sectionA.2.3) shows that the contestant’s revised belief would become q1 = 0.25(0.7)

λ� ν/[0.25(0.7)λ� ν+ 0.75(0.9)λ� ν]≃ 0.998, where we have deduced, from Table III, that λ� ν = 3.84; and correspondingly smallbeliefs for the remaining answers. Our example shows that the contestant can make extreme revisionsto her beliefs even though the signal from the histogram is not particularly definitive. Here the audienceis convincing enough to lead the contestant to near confidence in the first answer being correct.Treating (λ,ν) as constants at their estimated values, we then proceed to estimate the remaining param-eters of the model.While it is true that we are making parametric assumptions about PAF (and ATA, which we can es-

timate independently of contestant behaviour), we are completely agnostic about the properties of u(an)at this stage, apart from it being increasing. The eight estimated utility values (marked as x) and their95% CIs (marked as triangles) are shown in Figure 3 (note the horizontal axis has a log scale). It is clearthat these estimates would not support the imposition of the usual smooth functional forms and this is

Table III. ML estimates of the parameters of the distribution of histograms from ATA

Parameter Estimate SE

λ 4.754 0.210ν 0.914 0.030Number of observations 501Log-likelihood 1526.41

Note: These results are based on 501 observations for which the use of ATA is observed.



confirmed in Table IV, where we provide estimates for the parameters which minimize the distance tothe estimated utility values for some common utility functions. In all cases we fit the scaled utility, i.e.[u(x)� u(0)]/[u(1000)� u(0)], to the estimated utility values. The minimum distance statistics quotedare χ2 and the P-values confirms the rejection of the conventional forms. CARA performs much worsethan CRRA and the power-expo (HPE) form, which nests both CARA (if ρ= 0), and CRRA (whenα = 0).3 We find that HPE does not improve substantially on the fit of CRRA. Note that the estimateof the CRRA coefficient is very precisely estimated and is very close to unity.4 In Figure 3 we also plotthe estimate utility levels for a functional form which is essentially logarithmic but only over the range[2, 500]: which we refer to as log with jump (LWJ). Although this simple functional form does best outof all the utility functions we propose here, we can still reject this as an adequate description of theutility values. Of course the LWJ utility function ignores, by construction, the behaviour of utilityabove £500,000 and below £2000 and therefore it is not informative of the behaviour of the utilityfunction outside this range. In fact, a comparison of the original utility values and of the predictedvalues using the functional form show that the log function is ‘too concave’ even within the range overwhich it is fitted—in particular, it seems too concave for values above £250,000. Furthermore, thisform overestimates the value of utility at £2000 (this pattern is true for most of the other simplerestrictions). We conclude therefore that the usual models of utility, CARA, CRRA, log, or evenHPE, are not satisfactory representations of the behaviour of the utility function over a large rangeof values (from £2000 to £1 million).

In Figure 4 we present the nine estimated probabilities for the PAF lifeline together with their ap-proximate asymptotic standard errors. We would expect PAF to be less likely to be useful (i.e. theprobability that your friend can help) the longer the contestant remains in the game because the ques-tions get harder. Our estimates suggest that a friend remains relatively informative (around a 40%chance of knowing the answer at £64,000 and above). The additional parameters which allow forthe distribution of the initial belief to change with each round of the game are also estimated.

3 When α> 0 the HPE form arises from an increasing and convex transformation of a CRRA utility function and hence charac-terizes more risk-averse preferences than CRRA. When α= 0 and ρ= 1 the HPE utility reduces to u(x) = log(x+ γ).4 We also tested the robustness of this finding for the CRRA/HPE model by distinguishing between male and female players.Female contestants indeed appear to be slightly more risk averse. Results are available on request.

Figure 3. Estimated utility values and common functional forms



Table IV. Restrictions to common utility functions

Parameter estimates (asym. SE) P-value

HPEu xð Þ ¼ exp � α

1�ρ xþ γð Þ1�ρ

sign αð Þα 0.0001 (0.23)ρ 0.9804 (0.23)γ 0.0001 (–)Minimum distance 334.77 0.00CARAu xð Þ ¼ 1� 1

α exp �αxð Þα 0.3564 (0.0037)Minimum distance 1832.21 0.00Logu(x) = ln(x+ 10� 5)Minimum distance 386.02 0.00CRRAu xð Þ ¼ 1

1�ρ xþ γð Þ1�ρ

γ 0.0001 (0.0002)ρ 0.9782 (0.0227)Minimum distance 334.71 0.00Log with jump

u xð Þ ¼κ1 if 1000 ¼ x

log xþ γð Þ if 2≤x < 1000

κ2 if x < 2

8><>:κ1 10.5800 (0.0153)κ2 �7.7800 (0.0165)γ 0.0000 (0.0307)Minimum distance 125.75 0.00

Note: Minimum distance estimates are obtained using a grid search. The (asymptotic) standard errors are then derived using the‘delta method’. For the HPE specification the calculation of the variance–covariance of the parameters leads to impreciseestimates; in this case we show the calculations assuming γ= 0.0001 fixed (the search nevertheless was carried over all the threeparameters). The minimum distance statistics reported are given by the value of the objective at the estimated parameters.

Figure 4. Probability of ‘Friend knows the answer to the question’



To illustrate how the distribution of the beliefs changes as the game progresses we have calculated,in Figure 5, the distribution of the maximum belief when one question, and three, five, eight and 10questions, remain to be played. We are interested in the maximum belief because, in the absence ofany lifeline, the distribution of the maximum belief tells us how confident individuals are about theirability to answer the question correctly. If the distribution of the maximum belief is close to 1 then con-testants are very confident that they know the answer. Note that the questions get more difficult in thesense that the distribution at any stage is stochastically dominated by the distribution at earlier stages.Indeed, with ten questions to go, the corresponding probability density function (i.e. the slope of thedistribution function) increases up to, and is concentrated close to, p= 1. Whereas, with few questionsleft the density function peaks before 1, and is concentrated around p= 0.45 for the last question.

The estimates allow us to predict the probabilities of quitting and failing at each question and com-pare these with the observed distributions. We find, as expected, there are many fails and no quits whenthere are exactly four more questions to come, i.e. when confronted with the £64,000 question, sincethere is no risk at this point. The model predicts the overall share of quits and fails qualitatively well,although we reject the null hypothesis that the model fits the data of quits and fail at the all levels ofsignificance (χ2df¼15 ¼ 170). Our estimates of V also allow us to predict when lifelines will be played.A comparison of the observed and predicted distributions (see Figure A1 in the online TechnicalAppendix) for the use of each lifeline shows good agreement between our model and the data.Although the fit is not perfect (in the PAF case the χ2df¼8 ¼ 18:95, P-value = 0.015, while for ATAP-value = 0.188) we feel the model captures the main features of this aspect of our data well.

Finally, it is appropriate to discuss the robustness of our results and in particular the role ofunobserved heterogeneity that our model ignores. First, we are concerned about the role of wealth,which is omitted from our specification. If the utility function is concave, and for a game likeWWTBAM when the upside is large relative to wealth and the downside is little different from the safeoption of not playing at all, then initial (unobserved) wealth does not contribute much to the descriptionof choices. The argument is as follows. In a simplified setup, assume that the safe option is worth s,assume that the contestant is facing an uncertain lottery where the upside is worth M with probability(1� p) and the downside is worth d< s with probability p. Then, for a utility function u(x), the com-petitor will choose the risky option over the safe option if p(u(d +w0)� u(s+w0)) + (1� p)(u(M +w0)u(s+w0))≥ 0. Note that, using the intermediate value theorem, this condition requires that the

Figure 5. Distribution of the maximum belief



inequality pu′(k1 +w0)(d� s) + (1� p)(u(M)� u(s+w0) + u′(k2)w0)≥ 0 be satisfied, for some con-stants, where k1, d< k1< s, and k2,M< k2<M+w. Clearly the first term on the left-hand side of thisinequality is negative whereas the second term is always positive. For a large value of M, u′(k2) is neg-ligible, and the inequality is satisfied provided the product u′(k1 +w0). (d� s) is not too large. This willhold for individuals who are sufficiently rich such that d� s is small relative to initial wealth, w0. Inparticular, when the downside and the safe option are not too different, the above inequality is satisfiedwhatever the value of the initial wealth. In all other cases, initial wealth (or any other unobserved in-dividual specific characteristic which determines marginal utility) matters since it contributes to themarginal utility attached to the possibility of losing.Hence our estimates fail to consider the possible differences across contestants of the utility weight

attached to losing. Whether accounting for this possibility would change the estimates significantly isdifficult to assess without measuring individual wealth and providing a model of marginal utility whichcaptures the effect of initial wealth. This may be an important caveat to our results. In the case of a log-arithmic utility function u(x) = ln(x+w0), for example, a richer individual is less risk averse than apoorer one. And as a consequence, if we ignore the value of continuation, we may expect less wealthycontestants to quit as soon as the quantity u′(k1 +w0)(d� s) = (d� s)/(k1 +w0) is sufficiently negative.Clearly, for some questions, the difference d� s can be sizeable relative to many contestants’ wealth(for example, in the case of any question above £32,000 and certainly for the questions above£125,000). A similar argument can be made in the case of any CRRA utility function. The literatureoffers few convenient solutions to this problem. Post et al. (2008) seems to be alone in estimatingan average value of wealth in the context of specific parametric utility functions. Bombardini andTrebbi (2012) use annual income as a measure of individual initial wealth and allow the coefficientof relative risk aversion to depend on individual characteristics. They report an average coefficientof relative risk aversion distributed around unity. Their result would provide support for our ownfindings.To assess the potential for significant bias in the measurement of risk aversion, we experimented

with accounting for unobserved heterogeneity in a power-expo parametric utility framework using a(2-point) discrete mixture where the unobserved components acts as a wealth shifter. In this context,we identify the distribution of the unobserved component to the wealth distribution, or at least to thatpart of wealth which determines the preferences. Perhaps reassuringly, our experiments did not suggestthat accounting for unobserved heterogeneity, in this fashion at least, changes our estimate of the co-efficient of relative risk aversion significantly, since the estimate remained close to unity over a largerange of prize values (the details results are available from the authors upon request).

8. CONCLUSIONS

This paper provides new evidence about the degree of individual risk aversion. The analysis of prefer-ences is effectively nonparametric but our testing of parametric forms is firmly embedded in theexpected utility paradigm. Our contribution is to exploit behaviour on the popular gameshow WhoWants to be a Millionaire?, where the outcomes can vary enormously across contestants. The gamepresents a natural way of incorporating stochastic choice because it is a game of skill. The skill elementof the game requires us to make parametric assumptions about the distribution of beliefs over contes-tant knowledge of the answers to each question. This amounts to us making assumptions about the dis-tribution of beliefs and leads us to estimating these distributions jointly with the preference parameters.This is usually not required in a laboratory experiment or in other field experiments where the out-comes are determined only by statistical chance.Perhaps surprisingly, we find that our model is quite effective in explaining behaviour in the game.

Moreover, our headline result is that the degree of RRA is approximately 1 over a large range (but not



over the whole range) of the distribution of prizes. This figure is broadly consistent with the results ofrecent work on other gameshows that feature large stakes.

The literature we review estimates EUT models which impose restrictive forms of concavity, as amaintained hypothesis. Our findings suggest that such restrictions may not be supported by the facts.It is possible (but outside of the scope of this paper) that maintaining the usual functional forms ex-plains part of the difficulty of distinguishing between EUT and its alternatives. We suggest that furtherwork should attempt to use the most flexible model of the utility function (i.e. saturate the utilitymodel) in order to be able to rule out–or in–the form of the interaction between the probabilities andthe utility function which best describe observed behaviour. In particular, our general EUT model fitsthe sequence and timing of the use of lifelines during the game reasonably well, although this aspect ofbehaviour, akin to an investment decision, adds further restrictions on what would be expected from anexpected utility maximizer and therefore should help to rule out a choice model based on EUT. We findthat our EUT model describes these features of the data reasonably well.

ACKNOWLEDGEMENTS

We are grateful to Celador PLC, the gameshow’s creator and the UK production company, for theirhelp with gathering the data, and their permission to use it. We are particularly grateful to Ruth Settle,who provided detailed advice that helped our understanding of the game. The research was supportedby grant R000239740 from the Economic and Social Research Council. We are also indebted to seminarparticipants at Copenhagen, Bristol, Melbourne, Cardiff, Exeter, Kent, Toulouse, the EconometricsSociety World Congress in London, Moscow Higher School of Economics, and Paris II. However,the views expressed herein are those of the authors alone.

REFERENCES

Andersen S, Harrison GW, LauMI, Rutström EE. 2008. Risk aversion in game shows. In Research in ExperimentalEconomics, Vol. 12, Isaac RM, Norton DA (eds). Emerald Group Publishing: Bingley, UK; 361–406.

Baltussen G, Post T, van den Assem MJ. 2008. Risky choice and the relative size of stakes. Available: http://dx.doi.org/10.2139/ssrn.989242 [30 August 2013].

Blavatskyy P, Pogrebna G. 2008. Risk aversion when gains are likely and unlikely: evidence from a naturalexperiment with large stakes. Theory and Decision 64: 395–420.

Blavatskyy P, Pogrebna G. 2010. Models of stochastic choice and decision theories: why both are important foranalyzing decisions. Journal of Applied Econometrics 25(6): 963–986.

Bombardini M, Trebbi F. 2012. Risk aversion and expected utility theory: a field experiment with large and smallstakes. Journal of the European Economic Association 10(6): 1348–1399.

Chetty R, Szeidel A. 2007. Consumption commitments and risk preferences. Quarterly Journal of Economics22(2): 831–877.

Conte A, Moffatt PG, Botti F, Di Cagno DT, D’Ippoliti C. 2012. A test of the rational expectations hypothesisusing data from a natural experiment. Applied Economics 44(35): 4661–4678.

Deck C, Lee J, Reyes J. 2008. Risk attitudes in large stake gambles: evidence from a gameshow. Applied Economics40(1): 41–52.

de Roos N, Sarafidis Y. 2010. Decision making under risk in Deal Or No Deal. Journal of Applied Econometrics25: 987–1027.

French S. 1980. Updating of belief in the light of someone else’s opinion. Journal of the Royal Statistical Society143: 43–48.

Friedman M, Savage LJ. 1948. The utility analysis of choices involving risk. Journal of Political Economy 56(4):279–304.

Harrison GW, List J. 2004. Field Experiments. Journal of Economic Literature 42: 1009–1055.Harrison GW, Rutström EE. 2009. Expected utility and prospect theory: one wedding and a decent funeral.

Experimental Economics 12(2): 133–158.



Harrison GW, Lau MI, Rutström EE. 2007. Estimating risk attitudes in Denmark: a field experiment. ScandinavianJournal of Economics 109: 341–368.

Hartley R, Farrell L. 2002. Can expected utility theory explain gambling? American Economic Review 92(3): 613–624.Hey JD, Orme C. 1994. Investigating generalizations of expected utility theory using experimental data.

Econometrica 62: 1291–1326.Holt CA, Laury SK. 2002. Risk aversion and incentive effects. American Economic Review 92(5): 1644–1655.Kahneman D, Tversky A. 1979. Prospect theory: an analysis of decision under risk. Econometrica 47(2): 263–292.Levitt SD, List J. 2007. What do laboratory experiments measuring social preferences reveal about the real world?

Journal of Economic Perspectives 21: 153–174.Loomes G, Sugden R. 1982. Regret theory: an alternative theory of rational choice under uncertainty. Economic

Journal 92: 805–824.Mulino D, Scheelings R, Brooks R, Faff R. 2006. Is a dollar in the hand worth two in a lottery? Risk aversion and

prospect theory in Deal Or No Deal. Mimeo. Monash University: AustraliaMulino D, Scheelings R, Brooks R, Faff R. 2009. Does risk-aversion vary with decision-frame? An empirical test

using recent gameshow data. Review of Behavioural Finance 1: 44–61.Palacios-Huerta I, Serrano R. 2006. Rejecting small gambles under expected utility. Economics Letters 91(2):

250–259.Post T, Baltussen G, Van den Assem M. 2008. Deal Or No Deal: decision making under risk in a large payoff

gameshow. American Economic Review 98(1): 38–71.Quiggin J. 1982. A theory of anticipated utility. Journal of Economic Behaviour and Organization 3: 323–343.Rabin M. 2000. Risk aversion and expected-utility theory: a calibration theorem. Econometrica 68(5): 1281–1292.Rabin M, Thaler RH. 2001. Anomalies: risk aversion. Journal of Economic Perspectives 15(1): 219–232.Rosen S. 1997. Manufactured Inequality. Journal of Labor Economics 15(2): 189–196.Rubinstein A. 2001. Comments on the risk and time preferences in economics. Mimeo, Tel-Aviv University.Saha A. 1993. Expo-power utility: a ‘flexible’ form for absolute and relative risk aversion. American Journal of

Agricultural Economics 75: 905–913.Watt R. 2002. Defending expected utility theory. Journal of Economic Perspectives 16: 227–229.Wilcox NT. 2011. Stochastically more risk averse: a contextual theory of stochastic discrete choice under risk.

Journal of Econometrics 162(1): 89–104.



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