Who really wants to be a millionaire?
Estimates of risk aversion from gameshow data*
Roger Hartley,
University of Manchester, Manchester M13 9PL, UK
Gauthier Lanot,
Keele University, Keele ST5 5BG, UK
Ian Walker,
Lancaster University, Lancaster LA1 4YX, UK
10 March 2016
Keywords: Risk aversion, gameshow
JEL Class: D810, C930, C230
Abstract
This paper estimates the degree of risk aversion from the behaviour of contestants in one
of the most popular TV gameshows ever. This gameshow has a number of features that
makes it well suited for our analysis: the format is extremely straightforward; it involves
no strategic decision-making; we have a large number of observations; the prizes are cash
and paid immediately; and cover a large range – from £100 up to £1 million. We develop
a complete structural model for the behaviour of the contestant of the game and we
estimate a flexible functional form for the utility function and then test some parametric
restrictions. We find that, although the restriction to the CRRA utility is statistically
rejected, it approximates the utility function quite well over a large range of the values of
potential winnings.
* We are grateful to Celador PLC, the gameshow’s creator and the UK production company, for their help
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 supported by grant
R000239740 from the Economic and Social Research Council. We are also indebted to seminar participants
at the Institute for Fiscal Studies, Her Majesty’s Treasury, the Welsh Assembly, CAM Copenhagen, the
Universities of Bristol, Melbourne, Durham, Toulouse, the Cardiff Business School and Exeter University.
However, the views expressed herein are those of the authors alone.
1.
Introduction
Rabin (2000) suggests that “Expected-utility theory seems to be a useful and adequate
model of risk aversion for many purposes, and it is especially attractive in lieu of an equally
tractable alternative model.” Despite various misgivings, including those of Rabin himself,
expected utility theory (EUT) remains the dominant framework for thinking about decision
making under uncertainty in most contexts. EUT has been challenged in a number of ways
in recent research but formal statistical testing against alternatives is still in short supply. Hey
and Orme (1994) is a seminal study that provides statistical tests of EUT against alternatives
and concludes that EUT fits “no worse than other models”. They conclude that “EUT
emerges fairly intact” and state that the behaviour in their data is best described by “EUT plus
noise”. Similar sentiments are expressed by Harrison and Rutström (2009) who model both
EUT and Prospect Theory (PT, see Kahneman and Tversky, 1979) and claim that “both
models have roughly the same support” in their data derived from a gameshow.
The existing structural dynamic stochastic modelling literature that uses field
experiments is dominated by data that has been generated by a game of chance called Deal or
No Deal (DOND, or similar in other languages). 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 RRA that are close to 1; and some findings reject
CRRA if favour of a specification that implies low (or no) risk aversion at low stakes and
higher RRA when evaluated at high stakes. Whether EUT is the appropriate model of
preferences is finely balanced in these studies – most studies fail to reject EUT. Within EUT
studies, most suggest that CRRA (and CARA) is too strong an assumption and prefer Expopower specification which nests both special cases (see Holt and Laury (2002) and Saha
(1993)).
1
In the light of the apparent robustness and usefulness of EUT, this paper analyses the
degree of risk aversion that is exhibited in the behaviour of contestants in a TV gameshow
which is, in part, a game of skill. This element of skill makes our estimation problem, in the
dynamic stochastic framework, more complex but adds some flexibility. Thus, estimate a
fully structural model where individuals make choices on the basis of backward induction,
and we make EUT a maintained hypothesis in testing for specific functional forms.
Our data comes from what has probably been the most popular TV gameshow of all
time, Who wants to be a millionaire? (hereafter WWTBAM). Notwithstanding that
gameshow data has a number of drawbacks for the purpose of estimating attitudes to risk, this
particular game has a number of design features that make it particularly well-suited to our
task: the format is straightforward; it involves no strategic decision-making; in stark contrast
to DOND the compere of WWTBAM takes great care to neither encourage nor discourage
contestants; and the prizes are paid immediately in cash and cover a large range – up to £1
million.1
In this gameshow the contestant is faced with a sequence of 15 multiple-choice
questions. At each stage: she can guess the answer to the current question and might double
her current winnings but at the risk of losing and leaving the game with a question-specific
amount; or she can quit and leave the game with her winnings to date. At each stage
contestants are reminded that their winnings so far belong to them - to risk, or walk away
with. The prizes start at a modest level but reach very high levels. The wide spread of
possible outcomes makes WWTBAM a considerable challenge for EUT, and for simple
parametric forms such as CRRA in particular. The sequential nature of the game gives rise to
1
We also have supplementary survey information from players that allows us to compare their observable
characteristics to check the representativeness of the contestants. There is a marked gender imbalance in the data
but otherwise the contestants seem broadly representative, especially with regard to risky behaviour. See the
Data Appendix for details.
2
one important complication – in all but the last stage of the game, answering a question
correctly gives an option to hear the next question and this itself has a value, over and above
the value of the addition to wealth associated with the question. This option value depends on
the stage of the game, the contestant’s view about the difficulty of subsequent questions, and
the degree of risk aversion.
The mechanism of the game is well known and simple. It is, however, a game where
skill matters. While this complicates our analysis, it has the advantage that it allows for the
possibility that an individual can make ex post “mistakes” in her decisions because she found
that subsequent questions were more or less difficult than had been anticipated. In our model
it is the variability in “skills”, here the ability to answer a given question, which explains the
different trajectories through the game of otherwise identical individuals. Thus, our model
has an explicit role for “noise” and so does not rely on the preference specification alone to
fit the data. Hence, we model the subjective difficulty of questions and so allow for the
possibility that individuals are, on average, optimistic or not with respect to their skill. Of
course, we need to make a number of assumptions to capture some of these details of the
game but they are relatively weak assumptions made for parametric convenience.
Furthermore, one feature of the game that adds considerably to complexity is the existence of
“lifelines”. These are opportunities for the player to improve her chances of providing correct
answers. The decision to play a given lifeline can be viewed as an “investment” decision
which involves managing the tradeoff between playing the lifeline, and increasing the
chances of success in the current round, and not playing the lifeline, and increasing the
likelihood of success in some future round. This investment will change the individual’s
perception of the difficulty of the current question. Understanding the timing and sequence of
use of the lifelines provides further information which should help reject a model based on
EUT. We model these lifelines in simple but intuitive and flexible ways.
3
We begin by estimating utility values associated with each stage of the game in a nonparametric framework. We then provide Minimum Distance estimates of flexible functional
forms for the utility function based on our non-parametric estimates and so test some
parametric restrictions. We find that, although the restriction to CRRA is statistically
rejected, it broadly fits the data over a large range of the value of prizes. Remarkably, over
that very large range, it seems likely that the value of the coefficient of risk aversion is very
close to 1.
In summary, the contribution of the paper is that we provide a fully structural,
effectively non-parametric, model that describes the choices that contestants make, but we
make parametric assumptions about the nature of the constraints that individuals believe they
face. Earlier research, uses the DOND gameshow, and effectively separates these steps by
making strong assumptions about the behaviour of the game operator: they calculate expected
utility under assumptions about this “banker” and then model choices discretely making
assumptions about the distribution of risk aversion. Here, we estimate risk aversion using
WWTBAM gameshow data where skill plays a role and where individual contestants
exercise some control over the uncertainty they face through the use of lifelines. We find a
degree of consistency between our findings and that based on other gameshows which also
yield a CRRA close to 1 - but only over a range of the data.
The paper is structured as follows. In section 2 we briefly outline the existing
evidence that relies on gameshow data. Section 3 explains the operation of the game in more
detail. In section 4 we provide a simplified structural model of the game which we then
extend to account for heterogeneous beliefs and for the availability of lifelines. In section 5
we present the econometric details and the likelihood. In section 6 we give some summary
details of the UK data and explain how we estimate risk aversion using this data. In section 7
4
we present some results and consider possible shortcomings of the work. In section 8 we
draw together some conclusions.
2. Existing Evidence
There are several distinct strands to the empirical literature. Firstly, considerable
attention has been given to the estimation of Euler equations derived from lifecycle models of
consumption and savings (see Hall (1988) and Attanasio and Weber (1989)) where the
coefficient on the interest rate in a log-linearised model is the elasticity of substitution. If
utility is time separable and exhibits CRRA then this interest rate coefficient is also the
inverse of the degree of relative risk aversion, ρ. The typical result in such analyses, usually
based on macro data, is that consumption and savings are relatively insensitive to interest
rates so the elasticity of intertemporal substitution is small. Thus, the macro-econometric
literature largely suggests that the degree of risk aversion is large. Some of this literature
(notable contributions are Epstein and Zin (1989, 1991)) considers two assets and backs out
risk aversion from the excess returns on equities. Since individual portfolios are typically
highly concentrated in relatively safe assets this work also implies that the degree of risk
aversion is implausibly large. However, this method, which relies on portfolio allocations,
has only been applied to micro-data in a handful of studies. One example is Attanasio, Banks
and Tanner (2002) who provide a very plausible estimate of the degree of relative risk
aversion, ρ, of just 1.44 using a large UK sample survey.2
Another strand to the literature takes an experimental approach where participants are
offered either real or hypothetical gambles. Many papers use low value lotteries on relatively
2
If utility is intertemporally separable then the extent to which utility varies with income is related not just to
consumption and savings, but also to labour supply. This idea has been exploited by Chetty (2006) who derives
estimates of risk aversion from evidence on labour supply elasticities. He shows that ρ, in the atemporally
separable case, depends on the ratio of income and wage elasticities and that typical estimates in the literature
imply a ρ of about 1. Indeed, Chetty (2006) shows that weak separability and a positive uncompensated
elasticity is sufficient to bound ρ to be below 1.25.
5
small samples of selected participants (often students). For example, Andersen, Harrison, Lau
and Rutström (2008) uses data from a sample of 253 individuals who were representative of
the Danish adult population who chose between low value lotteries, and Harrison and
Rutström (2009) which use data generated by 158 student subjects. Harrison and List (2004)
and Levitt and List (2007) have questioned the reliability of data generated in laboratory
experiments and some research has compared field and lab experiments. For example, Holt
and Laury (2002) compares estimates from hypothetical lotteries with the same lotteries
where the prizes are real. The authors estimate the extent to which preferences differ across
the real and hypothetical lotteries and find that they are similar only for small gambles. Their
analysis features prizes that range up to several hundreds of dollars which they feel allows
them to address the critique raised in Rabin and Thaler (2001) and Rabin (2000) - who
suggest that risk aversion increasing with stakes is a necessary condition for EUT to provide
a unified account of attitudes towards risk. Holt and Laury (2002) estimate an Expo-power
expected utility function that allows for non-constant RRA, using real payoffs. Consistent
with Rabin’s calibration theorem, they find small degrees of RRA (around 0.3) at low prize
levels and higher (around 0.9) at high prize levels which, together, fit the data well. However,
even the largest payouts considered in Holt and Laury (2002) are small compared to typical
gameshows.
The present paper belongs to the final strand of the empirical literature that relies on
data generated by gameshow contestants. Fullenkamp et al (2003) uses the Hoosier
Millionaire TV gameshow to analyze decision-making. Unlike earlier gameshows this, like
WWTBAM, involves relatively high stakes. They use a large sample of simple gambling
decisions to estimate risk-aversion parameters. However, with this game the prizes are
annuities and so their value to contestants will depend on time preference. They find that,
assuming a discount rate of 10%, contestants display risk-aversion with the mean ρ ranging
6
from 0.64 to 1.76. Beetsma and Schotman (2001) estimate risk aversion in Lingo, a TV word
game that requires players to make a judgement about their own skill. Their best estimate of
risk aversion in a CRRA model is an implausibly large 6.99.
Andersen et al (2008a) reviews the evidence from analyses of gameshow data,
including a recent spate of papers which uses data from a recent gameshow, Deal or NoDeal. They also provide new estimates of risk aversion in a CRRA specification using US
DOND data, and with data generated by implementing DOND in the laboratory using smaller
prizes. The attractive feature of this game is that it involves a number of potential prizes,
varying from the very small up to very large amounts, which are won only by chance.
Contestants simply make a sequence of decisions about staying in the game as prizes are
eliminated at random. On several occasions during this process the contestant is made
financial offers to quit and so needs to compare the offer made with the value of the
remaining prizes. While there is no skill involved, contestants need to take a view about what
the level of future offers to quit might be if they are to appropriately value the option of a
current offer. While it is not clear what the process is that generates these offers, this can be
estimated along with the risk aversion parameters. The estimate that Andersen et al (2008a)
provides for the degree of risk aversion in a CRRA specification, using the gameshow data, is
around 0.20 (compared to 0.45 from the laboratory data). They generalise this to allow for
non-constant risk aversion and find a near zero risk aversion coefficient at prizes below
$10,000 rising to 0.75 at $1m. When they test against CRRA they easily reject in favour of
the non-constant RRA specification.
DOND differs in format across many countries, and some studies exploit only some
features of the data. Data from these games have been analysed in several recent papers.
Bombardini and Trebbi (2012), which was the first available study to our knowledge to use
DOND data (from Italy) to estimate a structural dynamic stochastic model. They estimate the
7
degree of RRA to be close to 1 in the whole sample, but also estimate a degree of risk
aversion with the low stake observations only that is close to zero. Deck et al (2008) uses the
Mexican DOND game, estimates a structural model, and reports CRRA estimates that are
approximately 1.
Post et al (2008) considers US, German and Dutch DOND data and they also replicate
the game in the classroom and generate data from (Dutch) student participants. Their
estimates of the degree of relative risk aversion, which also uses the expo-power functional
specification, lie in the range 1 to 2. However, they find it hard to reconcile the data with
simple EUT and note that the behaviour of individuals in similar positions is coloured by the
path by which they got to that position - they refer to this as path-dependence.3
de Roos and Sarafidis (2010) use Australian DOND data to estimate structural CARA
and CRRA expected utility functions and adopt both random effects and random coefficients
specifications of the stochastic choice model.4 Their models produce plausible estimates of
risk aversion – their CRRA specification suggests a value of around 2.5 (evaluated at the
average income level). They also test a Rank Dependent Utility (RDU, see Quiggin, 1982)
characterisation of preferences against EUT and find that RDU significantly outperforms
EUT - they find that contestants are systematically “optimistic”. Of course, DOND is a game
of chance that is broadcast for its entertainment value. It seems very likely that players are
selected onto the show for the way in which their behaviour is likely to contribute to viewing
figures and this might well be consistent with contestants being optimistic. Optimism has the
added advantage, to its producers, that the costs of running the show is likely to be less than
3
Baltussen et al (2008) also use DOND data from a variety of countries. They do not estimate a specific
structural model but note that behaviour is quite different across games with different prize structures, but
similarly responsive to relative stakes.
4
Mulino et al (2009) also consider the Australian DOND game but choose not to estimate a structural model,
while Mulino et al (2006) provide structural estimates that conclude that players are risk loving but assume that
players are myopic.
8
would otherwise be the case. Indeed, at least in the UK game, the compere, the audience, and
fellow show participants all seem to encourage risk taking by the current player
Wilcox (2011) and Blavatskyy and Pogrebna (2010) point out that observed choices
under risk require a theory of how alternatives are valued to be combined with a theory of
how the choices between alternatives are made in the light of their valuations. Wilcox (2011)
refers to the evidence from many experimental studies to argue that choices under risk are
highly stochastic. He shows, using the Hey and Orme experimental data, how the model of
stochastic choice adopted by the researcher matters for evaluating EUT against RDU.
Similarly Blavatskyy and Pogrebna (2010), uses DOND data to extend the analysis of Hey
and Orme (1994). They emphasise that whether EUT is dominated by some other model of
preferences is not independent of the assumption made about how players make choices on
the basis of their preference calculations. Their DOND results suggest that EUT is the
preferred explanation if the decision rule used by players to make choices in the game is
sufficiently complex – for example, a random utility model of stochastic choice. Across all
models that they consider they find that 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).
Our own analysis elicits preferences from responses to offers of very specific gambles
in a gameshow context. A similar literature elicits preferences to responses to hypothetical
(and sometimes real) lotteries. For example, Harrison et al (2007) carefully implements such
a method in Denmark using lotteries with real, but modest, payments. Their findings reject
neutrality using a power-expo functional form.
9
3. The WWTBAM gameshow
The game features a sequence of fifteen “multiple-choice” questions with associated
prizes that, in the UK, start at £100 (approximately US$160, €120) and (approximately)
double with each correctly answered question so that the final question results in overall
winnings of £1m. The contestant has the choice of quitting with her accumulated winnings or
gambling by choosing between the four possible answers given after being asked each
question. If the chosen answer is correct the contestant doubles her existing winnings (and is
then asked the next question). If the chosen answer is incorrect she gets some “fallback” level
of winnings and leaves the game. The design is such that the difficulty of questions rise, on
average, across the sequence of rounds, and the fallback level of winnings also rises (in just
two steps).
To complicate our analysis, contestants are endowed with three “lifelines” which are
use-once opportunities to improve their odds – so, when faced with a difficult question,
contestants may use one or more lifelines to improve their odds. The three lifelines are:
“fifty-fifty” (50:50 hereafter), which removes two of the three incorrect answers; “ask-theaudience” (ATA), which provide the contestant with the distribution of opinion in the
audience concerning the correct answer in the form of a histogram; and “phone-a-friend”
(PAF), which allows the contestant to call on a friend for advice. At least in the UK,
questions are chosen randomly by computer from a sequence of question banks and there is
no attempt to manipulate the fortunes of individual contestants during play.
Contestants are not randomly selected onto the show. The details of how this is done
varies across countries but in the UK aspiring contestants ring a premium rate phone number
and are asked a question of medium difficulty. If correct, their name is entered into a draw to
appear in the studio. Ten names are drawn for each show (plus two reserves). Aspiring
10
contestants can improve their odds of appearing by ringing many times so having many
entries in the draw. Once at the recording studio, aspiring contestants compete with each
other to provide the fastest correct answer to a single question and the winner is selected to
enter the main game that we analyse here.
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 “final answer” to indicate commitment. At each of the two fallback stages,
the compère hands a cheque to the contestant for that level of winnings and ensures that the
contestant understands that this money is now theirs and cannot be subsequently lost. The
compère makes every effort to ensure that contestants behave rationally – he strongly
discourages quitting at the question corresponding to the fallback levels (where there is no
downside risk), and quitting with unused lifelines. He is also careful to ensure that contestants
understand the magnitude of the risks that they face in each round of the game. Neither
compère, nor the game format, seem to encourage cautious or hesitant contestants to take
undue 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 focus initially on a simplified version of the game in which contestants are risk
neutral and hence are expected income maximisers. In Section 4.2 we generalise to consider
risk aversion in a very straightforward way that allows us to retain the structure that we have
here. We postpone consideration of the “lifelines” to section 4.4 and, for the moment,
assume that questions are selected by independent random draws from a pool of questions of
identical difficulty. Modelling lifelines requires that we are precise about how we specify the
11
beliefs that contestants hold concerning the difficulty of questions – which we do in section
4.3.
Let p denote the probability that the contestant (of some given ability) is able to
correctly answer a question, where p is a realisation of the random variable P with c.d.f. F
(we provide a model for this distribution in the next section). Rounds of the game are denoted
by the number of questions remaining, i.e. n = N,…,1. Let an be the accumulated winnings
after the contestant has successfully completed N-n questions and there are n questions
remaining. In the televised game N=15 and the prizes (in £1000 units) are given by the
sequence an n 1  1000,500, 250,125, 64,32,16,8, 4, 2,1, 0.5, 0.3, 0.2, 0.1, 0 . Similarly, let bn
16
be the value of the fallback level of winnings, i.e. the winnings that can be kept in the event
of an incorrect answer. In the UK game the sequence of fallback prizes is given by,
bn n1  32,32,32,32,32,1,1,1,1,1, 0, 0, 0, 0, 0 .
15
Now consider the decision problem at the start of the game when the contestant is
faced with the first of 15 questions. The value of playing the game is given by
V15  p   max a16 , p  f14  b15   b15  where f14  E V14  P   is value of continuing optimally
and, at this stage a16  b15  0 . This is the first stage of a recursion, such that when there are n
questions to go and the question asked can be answered successfully with probability p, the
value of the game is Vn  p   max an1 , p  f n1  bn   bn  , where f n 1  E Vn 1  P   and we
set f0=a1. Note that the decision to quit or not to quit is made after the question has been
asked. At any round of the game, there exists a critical value of p, pn   an1  bn 
 fn1  bn 
for an income maximiser, 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 game is Vn  p   p  f n1  bn   bn . Note that the immediate value of answering
12
correctly is an 1 and p  f n1  bn  measures the sum of the option value of continuing to play
the game, p  f n1  an  , if the candidate were to answer correctly, and the expected value of
giving a correct answer over and above the assured gains, p  an  bn  . These dynamic
programming equations lead to the following relationship for  f n  ,
f n 1  f n   f n 1  bn   F  p  dp .
1
pn
To obtain the likelihood we need to evaluate the probability of winning. The
probability of continuing to participate through offering an answer to the nth question, but
prior to seeing the next question, is Pr "Play"  1  F  pn   F  pn  . The probability of
giving a correct answer, having decided to answer, is given by
Pr "Win " | "Play"


1
pn
p dF  p 
1  F  pn 

G  pn 
.
F  pn 
Hence the probability of answering correctly is simply Pr "Win"  G  pn  . Thus, the
likelihood of a contestant reaching round k and then quitting (i.e. refusing to give an answer
15
to question k) is L  k , 0   1  F  pk   G  pn  . The probability of a contestant reaching
n  k 1
round k and then giving an incorrect answer is L  k ,1  F  pk   G  pk   G  pn  .
15
n  k 1
Finally, the probability of a contestant reaching round 1 and then winning (£1m) is
15
L 1,.   G  pn  .
n 1
The empirical model we estimate is an extension of this simple model where: we
allow for risk averse behaviour (section 4.2); we provide a model for the belief structure
(section 4.3); and we provide for the updating of the belief which arises after the use of each
lifeline (section 4.5).
13
4.2
Risk aversion
Section 4.1 outlines the structure of the model but imposes the assumption of risk
neutrality. The extension to allow for risk aversion, rather than neutrality, is very simply
accommodated by allowing prizes to be measured in utility terms, i.e. we consider an  u (an ) ,
n  1...16 ,
and bn  u (bn ) , n  1...15 , for some increasing utility function u  x  with
u  0  0 .
4.3
Questions, Answers and Beliefs
The purpose of this section is to propose a model for the distribution of the beliefs that
an individual holds each time she is confronted with a question and a list of possible answers
(in the real game, four) of which one only is correct. It seems reasonable to assume that the
contestant chooses the answer with the highest subjective probability of being correct. Hence,
once the distribution of that probability is defined it becomes straightforward to describe the
probability distribution of the maximum belief and, more generally, of the order statistics.
The specification of the distribution of the beliefs becomes important when we
introduce the lifelines (in the next subsection). Indeed we think of each lifeline as a source of
(imprecise) information which is combined with the original belief to produce a new set of
beliefs on which the contestant acts.
The process of generating questions (and corresponding answers) can be described as
follows. A question, and a list of possible answers, is drawn uniformly (at each stage of the
game) from a pool of questions (assuming just one pool for the moment). The question and
its possible answers (in a randomized order) are presented to the contestant who is then
endowed with a draw from the belief distribution concerning the likelihood of each answer
being correct. The formation of beliefs for all contestants is assumed to follow this process in
an identical and independent manner. Hence, given a particular question, two otherwise
14
identical individuals can hold distinct beliefs concerning the likelihood of each answer.
Furthermore, all individuals are assumed to be able to evaluate the distribution of possible
beliefs over the population of questions involved at any given stage of the game.
Formally, suppose that X is an k-dimensional random vector, with k≤4 since there are

at most 4 possible choices, on the simplex k  x : xi  0 i  1....k ,

k

x  1 . We
i 1 i
assume that x has the probability density function  k  x  and we require it to exhibit the
following symmetry property: if x, x   k , such that x is obtained from x by a permutation
of its components, then  k  x    k  x  . We construct a family of distributions of beliefs by
starting with a probability density function  on 0,1 that is symmetric, i.e. such that
  x    1  x  for all x in 0,1 . Note that  ’s symmetry implies
   x 1  x 
1
0
2
1
   x 1  x  dx  2
1
0
and
dx    1  x  x  dx  2 where  2 is the second moment of  . If
1
2
0
 denotes the distribution function corresponding to  , it is straightforward to show that
 1  z   1    z  . From  we deduce the belief distribution for two, three and four
possible answers, we find:
1
2
 2  x1 , x2     x1     x2   ,
 3  x1 , x2 , x3  
 x
1
  xk    j

3 i , j ,kP3
 1  xk
 4  x1 , x2 , x3 , x4  
(1)

,

 x
1
  xl    k

122 i , j ,k ,lP4
 1  xl
(2)
xj
 
 
  1  xl  xk

,

(3)
where Pk is the set of all permutations of 1,..., k , and 2   x2  x  dx . In each case the
1
0
summation of the set of permutations arises because of the unobserved random (uniform)
15
order in which the possible answers are presented. Because  is symmetric and

k
x 1,
i 1 i
some simplifications are possible, so that for the cases of interest we have
 2  x1 , x2     x1  ,
(4)
2
3
 x1  
 x2 
 x3 
 ,
    x2   
    x3   
 1  x1 
 1  x2 
 1  x3  
 3  x1 , x2 , x3     x2   
 4  x1 , x2 , x3 , x4  
xj
 x  

1
  xl    k   
, .

62 l ,k 1,...,4
1  xl   1  xl  xk 

k l
(5)
(6)
These simplifications are useful since the number of terms involved is halved. It is
straightforward to verify that the integral of  k over  k , k  2,3, 4 is unity, and  k satisfies
the symmetry property required above. If  is the density of the uniform distribution between
0 and 1, then  k is the uniform distribution over  k , k  2,3, 4 . This specification of the
beliefs distribution is, of course, restrictive even among the distributions satisfying the
imposed symmetry property. It leads, however, to simple specifications for the distribution of
the order statistics and for the distribution of the maximum amongst  x1 ,..., xk  , k  2,3, 4
(see the Technical Appendix).
4.4
Lifelines
Contestants are endowed with three distinct lifelines that can each be played once and
only once at any stage in the game. Multiple lifelines can be used on one single question. Let
us first consider the game with only one remaining lifeline. To account for the lifeline, the
state space has to be extended. We write Wn  p;   for the expected value of the game to a
contestant faced with a question with belief vector p, when   0 if the lifeline has been used
and   1 if it is still available. Whether to use a lifeline or not may depend on all
components of p so the value is a function of the whole vector of subjective probabilities.
16
However, p  max i pi is a sufficient statistic for p in the contestant’s decision problem with
no lifeline left and we will write this value function as Vn  p;0 . In what follows we assume
that the lifeline is a draw of a new belief, say q , given p , the current belief.
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 is then combined with the initial belief p . The new belief is the outcome
of this process and q is then used instead of p in the decision problem. We therefore assume
that the conditional distribution function of q given p is well defined for each lifeline.5
Finally, we define kn  p   E q|p Vn  q, 0  | p  to be the value of playing the lifeline at stage n
where q  max i qi . The values Wn  p,1 and Vn  p,0 are then related according to the
updated dynamic programming equations below. When no lifeline is left we have the familiar


equation Vn  p,0   max an1 , p  f n 1  0   bn   bn , where f n  0   E Vn  P;0   . When the
lifeline remains the contestant will choose the largest of three options, i.e. quit, lifeline, play,
where:


Wn  p,1  max an1 , kn  p  , p  f n 1 1  bn   bn ,
f n 1 1  E Wn 1  P,1  , and f0 1  f0  0  a1 .
The treatment of contestants with more than one remaining lifeline is a straightforward
extension of this approach. Details are given in the Technical Appendix.
5
Note that contestants will never strictly prefer to quit with a lifeline left unused. However, it is still possible
that, for some p, a contestant may be indifferent between quitting and using the lifeline if she would
subsequently choose to quit for any realisation of q contingent on p. For example, a contestant for whom
p   1 4 , 1 4 , 1 4 , 1 4  will have q=½ no matter which two incorrect answers are removed if she were to play
50:50 (which two answers are discarded is decided when the question is constructed). This may still fall short of
the initial value attached to answering the question. A contestant who would reject such a 50:50 gamble would
place no value on the lifeline. Except in these circumstances, the lifeline will be used if an1  kn  p  , otherwise
the contestant will answer and retain the lifeline for future use.
17
5.
Data
Our data has been extracted from videotapes of the broadcast shows, kindly made
available to us by Celador, the production company and inventor of the show. These tapes
cover all shows in the eleven series from its inception in September 1998 to June 2003.6 This
amounted to 515 separate contestants who were Fastest Finger7 winners and so played the
second stage of the game.
A major concern about the findings of the gameshow literature is that the data is
generated by selected samples. To investigate this issue a questionnaire was sent to all of the
2374 potential contestants who had ever been invited to the studio for all UK shows in the
first eleven series of shows broadcast. 791 replies were received. Table 1 shows the means of
the data for the 515 second stage competitors on our videotapes who get to play the game,
and the 243 of these who responded to our questionnaire. The means for the 243 respondents
to our own survey suggest that bigger winners were slightly more likely to respond.
[ INSERT TABLE 1 HERE ]
The Data Appendix shows, by comparing the questionnaire responses to official
population surveys, that the contestants, once balanced for a large gender discrepancy, was
fairly representative on the UK population. While, there may well be unobserved differences
in preferences over risk, it is notable that the similarity between potential contestants, actual
contestants, and the UK population data extended to the contestants making similar decisions
on lottery play and insurance as the UK population. The 243 “Fastest Finger” winners who
go on to become WWTBAM players are more likely to be male, are a little younger, and
have slightly more years of education than the 548 that failed at this first round and end up
simply as spectators.
6
One show was excluded because it involved a legal dispute over an incident of cheating.
7
A knock-out stage of the game where contestants play for the right to play the second round.
18
The distribution of winnings, for the second round contestants, depends on whether
the contestant quit or failed to answer the last question asked. Almost all contestants who
survived beyond £125,000 quit rather than failed – only one contestant failed at £500,000 and
so left with just £32,000 instead of quitting and leaving with £250,000. Only three contestants
failed at a sub £1000 question and left with nothing. Three contestants won the £1m 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 we categorise those that
won the maximum £1m as quitters).
Figure 1 shows the overall distribution (the unshaded blocks) of fails/quits as the
game progresses from the first relevant question (i.e. when only 10 questions are left to play
before winning the million pounds) to the last. Note that there are a disproportionate number
of male contestants so we plot the rates rather than the frequencies for each gender. Men tend
to fail less in earlier stages of the game, while women tend to quit earlier, but the overall
pattern is broadly comparable across genders.
[INSERT FIGURE 1 HERE]
Finally, the use of lifelines is an important part of observed behaviour that our model
attempts to explain. Out of the 515 contestants in our videotapes, 501 played ATA, 488
played 50:50, and 484 played PAF. There were no examples of individuals quitting with
unused lifelines. There was a systematic tendency for lifelines to be played in order. ATA
was played, on average, with 8.5 questions remaining; 50:50 was played with, on average, 7.0
questions left; and PAF was used with just 6.9 questions remaining.
19
6.
Econometric specification and estimation
6.1
Specification of the belief distribution
The distribution of the beliefs is one of the main elements of the model since it
describes the distribution of the unobservables in the model. Under the assumptions we make
below (see section 6.2) the joint density  4 

can be constructed from some symmetric
density  over the unit interval. We assume that   x  is the density of a symmetric Beta
random variable, B  ,   on [0,1], i.e.   x  
  2 
  
2
x 1 1  x 
 1
, with  some positive
parameter, and where  . is the gamma function. Given the symmetry restriction on   x  ,
the Beta distribution provides us with a one parameter specification which is both convenient
and relatively flexible. Any random variable distributed according to B  ,   has
1   1 
expectation ½ and second moment  2  
.
2  2  1 
In what follows, it will prove convenient to use draws from the joint distribution of
ordered statistics of the belief distribution. Because of the symmetry assumptions that we
impose on  4 
 , the joint density function of the order statistics (i.e. the vector of beliefs in
decreasing order) is simply 4! 4  p  , where p is a vector of values ordered in decreasing
order.
Since it is straightforward to rank four numbers in decreasing order, the only
remaining issue is to draw from a multidimensional random variable with joint density
4   .
In Appendix A.3 we show how this can be achieved using three independent draws
from Beta distributions with parameters  ,  2 ,  ,  1 and  ,   , respectively.
20
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 the penultimate question, multiplied by the probability that for
her last question she wins a million, loses or quits 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 specific information about the utility of the
participants reaching this point from their observed behaviour: the £32,000 prize money is
guaranteed and everyone chooses to answer. Furthermore, the number of failed answers at the
£1000 and £2000 level is very small (1 and 0 respectively in our data). Therefore, we fix the
value of the parameters of the belief distribution to be 0.045 which implies that candidates are
very likely to know the answers for these early rounds. Finally, we fix the value of the utility
of the £1,000 prize to a value close to 0 but strictly positive. None of our results are sensitive
to the precise numbers we choose.
We estimate the most general model we can: we normalise the utility of the million
pound prize to 1 and the utility of no winnings to 0, we then allow the utility function u  x 
to be defined by a single parameter for each step of the game - £2000, £4000, £8,000 etc.
Therefore eight utility parameters need to be estimated which are u(an) in our previous
notation – we lose a further parameter because, as stated above, there is no decision to be
made at £64,000. At this stage we make no assumption about the shape 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 11 further parameters which are components of  in our previous notation. Finally,
21
the probability that the friend contacted by the life line phone-a-friend knows the correct
answer is also allowed to be round specific which adds a further 11 parameters. The 50:50
lifeline does not add further parameters. The ATA lifeline adds two further parameters which
we estimate just from the histograms of the audience beliefs. Hence potentially the likelihood
depends on 30 distinct parameters.
The issue of dealing with lifelines is complex. At each stage of the game we estimate
the value of playing each lifeline, and combinations of lifelines, at that stage. Our approach to
modelling 50:50 is parsimonious but intuitively compelling. Our approach to ATA relies only
on the data from the audience behaviour and we assume only that their knowledge about each
question is uncorrelated with the contestant’s beliefs. The appropriate modelling PAF is
simple but more debatable – we assume that a friend is an expert knows or doesn’t and that if
she does then the contestant does not update his belief. The contribution to the likelihood of
contestant i’s history, which ends at stage ni* of the game, has the general form:

L S ,i   LL  k , i  , ll  k , i  


n 1
LL k ,i  
 LL ni* ,i 
  i
; ;  ,    ll  k ,i ,k  ll n* ,i ,n* ,
i  i
k 1
  k 1


ni*
 
*
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
LL k ,i 
contestant i at stage k. ll  k ,i ,k is the simulated probability for the event indexed by ll  k , i  ,
LL  k , i  and k. Appendix section A.4 describes how the simulations are performed.  is the
vector of parameters of interest, i.e. the parameters of the utility function at the various prizes
levels, the values of the parameters of the belief distribution and the values of the probability
that the friend knows the correct answer when the PAF lifeline is used for each round. Finally
22
(  , ) are the independent estimates of the parameters of the density of the updated belief
which results from the use of ATA – see the Technical Appendix for further details.
7.
Estimation and results
The model assumes, very explicitly, that contestants understand every aspect of the
game and, in particular, are aware of the distribution of their beliefs at every round of the
game as well as the information content and value of the lifelines. It is clear from the raw
data that lifelines are not played very quickly as would be the case in a perfectly myopic
model.
The estimation of the parameters of the model requires that we first estimate our
model for the histograms that the lifeline ATA produces. These histograms record the
percentage of the audience that selects each of the four possible answers whenever asked.
 
This preliminary estimation is used to determine the parameters ˆ,ˆ of the distribution
described in Section A.2.3 of the Technical Appendix. We are assuming that the audience is
the same for all contestants.
To estimate the ATA parameters we use the data we have collected on these
histograms and on our knowledge of the correct answer. Appendix Section A.2.3. formally
describes this aspect of the model. These parameter estimates allow us to evaluate the quality
of the ATA lifeline. The parameter estimates are presented in Table 2 where we have ordered
the responses so that the first is correct. These estimates imply that, on average, the expected
 ˆ
ˆ
ˆ
ˆ 
,
,
,
histogram of the lifeline ATA is 0.63, 0.12, 0.12, 0.12; i.e. 
 .
ˆ
ˆ
ˆ
ˆ
   3ˆ   3ˆ   3ˆ   3ˆ 
On average, audiences believe that they the correct answer 63% of the time. But, for any
particular draw, the model allows for arbitrary histograms. The contestant updates her belief
23
on the basis of the specific histogram for that stage. Treating these parameters as constants
we then proceed to estimate the remaining parameters of the model.
[INSERT TABLE 2 HERE]
We take these ATA estimates as given and maximise the likelihood with respect to
the remaining 30 parameters. While it is the case that we are making parametric assumptions
about PAF (and ATA which we can estimate independently of contestant behaviour), we are
completely agnostic about the properties of u(an) at this stage apart from it being increasing.
Using the resulting eight utility value estimates, shown as crosses in Figure 2 (note the
horizontal axis has a log scale), we test the restrictions of our general case to some of the
common utility functions used in economics. We therefore provide, in Table 3, parameters
estimates for these common utility functions which minimise the distance to the estimated
utility values. In all cases we fit the scaled utility, i.e. u  x   u  0   u 1000   u  0   , to
the estimated utility values.
[INSERT TABLE 3 HERE]
We report the parameters estimates for each functional form which are obtained from
a minimum distance estimator. The Minimum Distance statistics quoted are  2 and the P
values confirms the rejection of conventional forms. We find that Constant Absolute Risk
Aversion form performs much worse relative to the Constant Relative Risk Aversion form.
We estimate the Power Expo form, which nests both CARA and CRRA, but find that it does
not improve the fit of the CRRA form substantially, effectively because CARA is so roundly
rejected. Note that the estimated value of the coefficient of relative risk aversion is very
precisely estimated and is very close to unity8. However, from Figure 3, where we plot what
8
We tested the robustness of this finding for the CRRA/HPE model in different ways. We estimated the model
distinguishing between male and female contestants. Female contestants indeed appear to be more risk averse
with a slightly larger estimated value for the risk aversion parameter. Furthermore we attempted to account for
unobserved heterogeneity in the parameter g using a mixture with two mass points. This does indeed improve
24
these forms look like, it is clear that our data does not support the use of the usual smooth
functional forms.
[INSERT FIGURE 3 HERE]
We also report estimates 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 out of all the utility functions we propose here, we still reject this as
an adequate description of the utility values. Of course, the LWJ utility function ignores, by
construction, the behaviour of utility above £500k and below £2k and therefore it is not
informative of the behaviour of the utility function outside this range. In fact, a comparison of
the original utility values and of the predicted values using the functional form show that the
log function is “too concave” even within the range over which it is fitted – in particular, for
values above £250,000. Furthermore, this form overestimates the value of utility at £2,000
(this pattern is true for most of the other simple restrictions). Figure 3 illustrates this finding.
In particular, we note that the utility function may not even be concave over the range of
values of the prizes. We conclude therefore that the usual models of utility, CARA, CRRA,
log, or even HPE, are not satisfactory representations of the behaviour of the utility function
over a large range of values (from £2k to £1000k).
In Figure 4 we present the nine estimated probabilities for the PAF lifeline together
with their approximate asymptotic standard errors. We would expect PAF to be less likely to
be useful (i.e. the probability that your friend can help) the longer the contestant remains in
the game because the questions get harder. Our estimates suggest, however, that a friend
remains relatively informative (around a 40% chance of knowing the answer at £64,000 and
above) even quite late in the game, although the CI around the estimated p increases at later
questions because we have few observations.
the fit of the model but it does not change the conclusion concerning the estimate of the degree of relative risk
aversion. These findings are available upon request from the authors.
25
The additional parameters which allow for the distribution of the initial belief to
change with each round of the game are also estimated. 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 1 question, and 3, 5, 8, and 10 questions, remain to be played. We
are interested in the maximum belief because, in the absence of any lifeline, the distribution
of the maximum belief tells us how confident individuals are about their answers. If the
distribution of the maximum belief is close to 1 then contestants are very confident that you
know the answer. Note that the questions get more difficult in the sense 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 the
distribution function) increases up to, and is concentrated close to, p=1. Whereas, with few
questions left the probability density function peaks before 1, and is concentrated around
p=0.45 for the last question.
[INSERT FIGURE 5 HERE]
Figure 6 shows the value of continuing to play the game, V, implied by our estimates,
at each stage of the game – i.e. as a function of the number of questions remaining (on the
horizontal axis) and the set of lifelines remaining. As we would expect, the value of
continuing to play rises as the number of remaining questions falls, and unused lifelines add
positive value to playing. If you had only one lifeline left then ATA appears to be the most
valuable, while 50:50 and PAF have almost comparable but lower values. 50:50 dominates
early in the game while PAF dominates later on. V0 is when no lifelines left. Having all three
lifelines is always better. If you only have two the worst pair is 50:50 and PAF. Whenever
ever you have ATA left then having PAF dominates towards the end of the game while 50:50
would be better to retain at the beginning of the game.
[INSERT FIGURE 6 HERE]
26
In Figure 7 we show the predicted probabilities of quitting and failing at each
question, computed from the general specification estimated, and compare these with the
observed distributions. Although there is nothing in our specification that imposes this, we
find, as expected, there are many fails and no quits when there are exactly four more
questions to come, i.e. when confronted with the £64,000 question, since there is no risk at
this point. In the bottom panel, we broadly capture the peak in quits immediately before and
after this point. We underpredict fails in the middle part of the game while we predict the
quits quite well. Note that the model predicts the overall share of quits and fails quite well.
[INSERT FIGURE 6 HERE]
Our estimates of V also allow us to predict when lifelines will be played. Figure 7
presents a comparison of the observed and predicted distributions for the use of each lifeline.
Although, again this is not a perfect fit, the agreement between our model and the data is
striking for most lifelines. In fact, it is for the 50:50 lifeline, for which our modelling process
seemed to be the most obvious approach to take, that the estimated model fares worst. In all
cases the model overpredicts the use of this lifeline in the middle part of the game (for
questions £8,000, £16,000 and £32,000 questions for 50:50 and PAF and for the £4,000 and
£8,000 questions for ATA). Although the fit is not perfect we feel the model captures the
main features of the data well (in term of decay as the game progresses and the general
“shape” of the distribution of use).
[INSERT FIGURE 7 HERE]
Finally, it is appropriate to now discuss the robustness of our results and in particular
the role of unobserved 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,
i.e. if the marginal utility decreases as the value of the outcome increases, and for a game like
27
WWTBAM when the upside is large relative to wealth and the downside is little different
from the safe option of not playing at all, then initial (unobserved) wealth does not contribute
much to the description of choices.
In a simplified setup, assume that the safe option is worth s, assume that the candidate
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 competitor
will choose to answer the question (the risky option) over quitting (the safe option) if
p  u  d  w0   u  s  w0    1  p  u  M  w0   u  s  w0    0 .
intermediate
value
theorem,
this
condition
Note
requires
pu  k1  w0  d  s   1  p  u  M   u  s  w0   u  k2  w0   0
that
be
that,
the
satisfied,
using
the
inequality
for
some
constants, k1 , d  k1  s , and k2 , M  k2  M  w . Clearly the first term on the LHS of this
inequality is negative while the second term is always positive. For a large value of M,
u  k2  is negligible, and the inequality is satisfied provided the product u  k1  w0  .  d  s  is
not too large. This will hold for individuals who are sufficiently rich such that d-s is small
relative to initial wealth, w0. In particular, when the downside and the safe option are not too
different the above inequality is satisfied whatever the value of the initial wealth. In all other
cases, initial wealth (or any other unobserved individual specific characteristic which
determines marginal utility) matters since it contributes to the marginal utility attached to the
possibility of losing.
Hence our estimates fail to consider the possible differences across candidates of the
utility weight attached to losing. Whether accounting for this possibility would change the
estimates significantly is difficult to assess without measuring individual objective or
subjective wealth and providing a model of marginal utility which captures the effect of
initial wealth. This may be an important caveat to our results. In the case of a logarithmic
28
utility function u  x   ln  x  w0  , for example, a richer individual is less risk averse than a
poorer one. And as a consequence, if we ignore the value of continuation, we may expect less
wealthy candidates to quit as soon as the quantity u  k1  w0  d  s  
d s
is sufficiently
k1  w0
negative. Clearly, for some questions, the difference d-s can be sizeable relative to many
candidates’ wealth (for example in the case of any question above £32k and certainly for the
questions above £125k). A similar argument can be made in the case of any CRRA utility
function.
The literature offers few convenient solutions to this problem. Post et al (2008) only
estimate an average value of wealth in the context of specific parametric utility functions.
Bombardini and Trebbi (forthcoming) use annual income as a measure of individual initial
wealth and allow the coefficient of relative risk aversion to depend on individual
characteristics. They report an average coefficient of relative risk aversion distributed around
unity. Their result would provide support for our own findings.
Botti et al (2008) adopt a statistical framework which accounts for both idiosyncratic
and individual specific components (in the spirit of a panel probit model) to determine the
coefficient of risk aversion (in either the CARA or the CRRA case) without allowing for the
differences in marginal utility between individuals caused by differences in initial wealth.
They report that a utility function of the CARA type is better supported by the data.
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 points) 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 that part of wealth which determines the preferences.
Perhaps reassuringly, our experiments did not suggest that accounting for unobserved
29
heterogeneity, in this fashion at least, changes our measurement of the coefficient of relative
risk aversion significantly, since the estimate remained close to unity over a large range 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 preferences is effectively nonparametric but our testing of parametric forms is
firmly embedded in the expected utility paradigm. Our contribution is to exploit a “natural
experiment”, based on the popular gameshow Who Wants to be a Millionaire? where the
outcomes can vary enormously across contestants. The game features a wider range of prizes
than does DOND - that has generated much attention from researchers. Moreover, the game
presents a natural way of incorporating stochastic choice because it is a game of skill rather
than chance, as in DOND.
The skill element of the game requires us to make parametric assumptions about the
distribution of beliefs over contestant knowledge of the answers to each question. The
assumptions, in effect, amount to making assumptions about the distribution of prizes and
lead us to estimating these distributions jointly with the preference parameters. This is usually
not required in a laboratory experiment or in field experiments where the outcomes are
determined only by statistical chance.
Perhaps surprisingly, we find that our model is fairly effective in explaining
behaviour in this 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 is consistent with the results of recent work on other gameshows that feature
large stakes.
30
The literature we review estimates EUT models which impose concavity, or
restrictive forms of concavity, as a maintained hypothesis. Our findings suggest that such
hypotheses may not be supported by the facts. It is possible (but outside of the scope of this
paper) that maintaining the usual functional forms explains part of the difficulty of
distinguishing between EUT and its alternatives. We suggest that further work should attempt
to use the most flexible model of the utility function (i.e. saturate the utility model) in order
to be able to rule out, or in, the form of the interaction between the probabilities and the
utility function which best describe observed behaviour. In particular, our general EUT
model fits the sequence and timing of the use of lifelines during game reasonably well,
although this aspect of behaviour, akin to an investment decision, adds further restrictions on
what would be expected from an Expected Utility maximizer and therefore should help to
rule out a choice model based on EUT. We find that our EUT model describes these features
of the data reasonably well.
31
Data Appendix
The questionnaire was designed to identify differences between contestants and the
population as a whole. The questions aimed to provide data that was comparable to that
available from official social surveys of large random samples of the population.
Questionnaire replies were received in 791 cases, a response rate of 33%, where 243 (32%)
of these cases were Fastest Finger winners and so played the second round game. These 243
represent a response rate of 47% of the population of second round contestants. Not
surprisingly, these second round contestants were more likely to respond to the survey
because they were well disposed towards Celador, having had the opportunity to win
considerable amounts of money.
It was immediately obvious that men were heavily overrepresented in both datasets.
Once the population datasets are re-weighted the observable differences between the
questionnaire data and the population survey data tend to be quite small. Two variables are
particularly worthy of note: the proportion of individuals who report that their household’s
contents are not insured is similar to the population value (in fact slightly smaller suggesting
more risk aversion); and the proportion who report being regular lottery ticket purchasers is
also quite similar. Thus, our questionnaire dataset does not suggest that those that play (in the
second round of) WWTBAM are heavily selected according to observable variables – except
gender. Indeed, for those variables which might be expected to reflect risk attitudes we find
no significant differences with our population surveys. However, whether the same can be
said about the videotape information which is the population of WWTBAM contestants
depends on the questionnaire respondents being representative of this underlying population.
Thus, in Table A1, we compare the questionnaire data for the sample of 243
contestants who responded to the questionnaire with the population of 515 actual contestants.
We have no consistent information on the characteristics of contestants in the population
apart from that which is recorded on the videotapes. Thus, Table A1 records only gender and
the outcomes of play. There are no significant differences in gender and although the
outcomes information shows, as might be expected, that the questionnaire respondents were
bigger winners on average, these differences are not significant. Thus, we can have some
confidence that the representativeness of contestants (in the questionnaire data) carries over
to the population data in the videotapes.
32
Table A1
Questionnaire Sample and Population Data
Population survey
data*
WWTBAM
players
WWTBAM
spectators
Mean
Std Dev
Mean
Std Dev
Mean
Std Dev
Male
0.52
0.40
0.76
0.43
0.66
0.48
Age
44.41
10.21
43.14
9.36
47.86
11.67
Married
0.80
0.44
0.79
0.41
0.76
0.43
Education years
13.88
4.10
13.71
3.99
12.82
3.22
Smoker ++
0.25
0.42
0.22
0.41
0.26
0.44
Renter
0.25
0.33
0.144
0.35
0.177
0.38
Contents uninsured +
0.09
0.26
0.07
0.27
0.06
0.31
House value (£k) ** ++
178.9
157
190.8
127
184.8
188
Employed
0.652
0.44
0.638
0.48
0.593
0.49
Self-employed
0.155
0.38
0.193
0.40
0.189
0.39
Not working
0.194
0.40
0.160
0.37
0.195
0.40
Gross earnings (£k pa) ***
27.08
23.0
31.17
24.0
28.67
22.7
Regular lottery player +++
0.67
0.40
0.63
0.41
0.65
0.41
Observations
various
243
548
Notes: * the survey datasets have been re-weighted to reflect the gender mix in the WWTBAM data. Population
data comes from the 2002 Labour Force Survey with the exception of: + from Family Expenditure Survey 2002
data, ++ from British Household Panel Study 2001 wave, and +++ from the Gambling Prevalence Survey 2002.
** if owner occupier. *** if employed. The gross earnings figure for our survey is the average of the mid points
of a number of ranges of income. The LFS figure was obtained by multiplying the survey week data by 52.
33
Technical Appendix
A. 1 Distribution of the maximum belief
The dynamic model outlined involves the distribution, F, of the contestant’s
assessment of her chance of answering the question successfully. Without lifelines, F  Fn is
the distribution of max  x  if x has the probability density function,  n  x  . Indeed max  x 
x n
x n
measures the individual assessment of her likelihood of answering the question correctly
when faced with n alternatives. In this section, we describe formulae for the distribution of
the highest order statistic Fn  z   Pr 

n
i 1
 X i  z , given that
X is distributed with density
function  n . In particular we can show that
if z  1 2
0

F2  z   2  z   1 if 1 2  z  1,
1
otherwise.

1 
and, as a consequence, the density function f 2  z  has support  ,1 where is satisfies
2 
f 2  z   2  z  . The distribution function at higher orders can be obtained from F2
recursively.
F4  z  
Whenever
z   0,1 ,
we
1
z
F3  z   2 F2   y  y  dy ,
1 z
 y
have
and
z
F3   y 2  y  dy , and the relevant density functions, say f3 and f 4 , can be
2 1 z  y 
1

1
shown to exist and to be continuous everywhere inside  0,1 . For example, in the uniform
case where   x   1 for x 0,1 , and 0 elsewhere, we find that
0

3
 4 z  1
F4  z   
3
2
44 z  60 z  24 z  3
3

1  4 1  z 
if 0  z  1 4
4
z
3
 z  12
if
1
if
1
if
1
2
1
3
 z 1
In this latter case it is easy to verify that the density function is continuous and that the
derivatives match at the boundaries of each segment. The distribution functions Fn do
depend on the density  in an important fashion. We interpret  as a description of the
individual’s knowledge. When  is diffuse over 0,1 (e.g. uniform) all points on the simplex
34
 n are equally likely and in some instances the individual will have the belief that she can
answer the question correctly while in some cases the beliefs will be relatively uninformative,
while if  is concentrated around, or in the limit at, ½ the individual is always indecisive.
Finally, when  ’s modes are located around 0 and 1, the individual is always relatively
informed about the correct answer.
A. 2 Lifelines
Extending the model above to allow for the lifelines makes the analysis more difficult
but also enables us to exploit more aspects of the data. We show how, in the first sub-section
below, the model can be modified when only one lifeline is allowed for. In a second subsection we show how the model can be modified for all three lifelines. We then present the
precise assumptions that allow the modelling of each lifeline in particular.
A. 2.1 The complete game
There are three lifelines available which can be played at most once. As above each
lifeline generates a new belief, q, which is used in the decision process. Given the initial
belief p, the new belief is drawn from a separate distribution for each lifeline, say H1  q | p 
for 50:50, H 2  q | p  for ATA and H3  q | p  for PAF. We write γ   1 ,  2 ,  3  for the
“lifeline state” vector where  i  0 if the ith lifeline has been played and 1 otherwise. We use
Wn  p; γ  to denote the optimal expected value of the game at stage n, when the probability
vector of the current question is p and the lifeline state is  1 ,  2 ,  3  . As above, Vn  p  is
used as a shorthand for Wn  p;  0,0,0   and Vn  p  satisfies the recursive dynamic
programming equations set out in section 5.1 above. Below we write the dynamic
programming equations using the notation f n    1 ,  2 ,  3    E Wn  Pn ;   1 ,  2 ,  3    , where
the expectation is with respect to Pn , the distribution of the belief vector p at stage n. When
there are one or more lifeline left, i.e.  1   2   3  1 , the contestant has three options: (i)
quit, (ii) answer the question, (iii) use one of the remaining lifelines. The recursive equation


is Wn  p; γ   max an1 , p  f n 1  γ   bn   bn , kn  p; γ  where kn  p; γ  denotes the maximum
expected value from using a lifeline when the belief is p and the lifeline state vector is γ .


Here, kn  p; γ   max E Wn  Qi ; γ  ei  | p  where I  γ    j : e j  γ using e i to denote the
iI  γ 
35
i-th unit vector in R 3 and Q i is distributed according to H i  q | p  . This formulation does not
preclude an individual from using more than one lifeline on the same question, a behaviour
we observe in some contestants.
A. 2.2 “50:50”
This is the simplest lifeline to model. It provides the contestant with “perfect
information” since two incorrect answers are removed. Ex-ante (i.e. before the lifeline is
played) the contestant believes that the correct answer is i (=1,…,4) with probability pi . The
50:50 lifeline removes two of the incorrect answers, retaining j  i , say, with equal
probability (1/3). By Bayes Theorem, the probability that answers i,j survive this elimination
process is pi 3 . The answers i and j can also be retained if j is correct and i survives
elimination. This occurs with probability p j 3 . Applying Bayes Theorem gives the updated
belief vector q
i , j
, where qk
i , j
equals
pj
pi
if k  i,
if k  j , and 0 otherwise.
pi  p j
pi  p j
Hence H1  q; p  is a discrete distribution with support

q  
i, j
i , j1,2,3,4
such that

H1 qi , j ; p   pi  p j  3 , and 0 elsewhere.
A. 2.3 “Ask the Audience”
Modelling ATA requires more than simply applying Bayes’ rule to the current belief
draw. In particular we must allow the contestant to learn from the information provided by
the lifeline, i.e. here the proportions of the audience’s votes in favour of each alternative
answer. The difficulty here is to understand why and how should a “perfectly informed”
rational individual revise his/her prior on the basis of someone else’s opinion? The route we
follow here was proposed by French (1980) in the context of belief updating after the opinion
of an expert is made available. French suggests that the updated belief that some event A is
realised after some information inf has been revealed should be obtained from the initial
belief, Pr  A , the marginal probability that a given realisation of the information is revealed,
Pr inf  , and the individual’s belief about the likelihood that the information will arise if A
subsequently occurs, Pr[inf | A] according to the following rule, related to Bayes theorem:
Pr  A | inf   Pr inf | A Pr[ A] / Pr[inf] . In this expression Pr[inf | A] is understood as another
component of the individual’s belief - her assessment of the likelihood of the signal given
36
that the relevant event subsequently occurs. Introducing A , A’s alternative event, this is
rewritten as
Pr  A | inf  
Pr inf | A Pr[ A]
Pr inf | A Pr[ A]  Pr inf | A  Pr[ A]
.
In our context we treat asking the audience as an appeal to an expert, and assume that the
events of interest are the four events “answer k is correct”, k=1,2,3,4. We assume that
contestants “learn” some information about the quality of the expert in particular the
distribution of the quantities Pr q   q1 , q2 , q3 , q4  | answer k is correct    k , where qk is the
proportion of votes allocated to the kth alternative. Following French’s proposal, the kth
4

component of the updated belief  given the information q is  k   k pk
j 1
j
p j . Let us
assume for now that each contestant knows the joint distribution of the vector
  1 ,2 ,3 ,4  . In fact the above expression implies that, without loss of generality, we
can normalise the  k to sum to one. Denote I   the density function of  given some
initial belief p. Given p, the density of the updated belief H 2  ; p  can be calculated as
4
 4
 4

H 2  ; p   I   ; p     pk    k pk 1  , with i  ; p    i pi 1

 k 1   k 1
4

k 1
k
pk 1 . The term
4
 4
 4 k 
p

k

    arises because of the change of variable from  to  .
 k 1   k 1 pk 
The quantities Pr q   q1 , q2 , q3 , q4  | answer k is correct    k represent the added
information obtained from using the lifeline and are estimable from the data provided we
assume a form of conditional independence. In particular we require that the contestant’s
choice to ask the audience does not influence the audience’s answer. Furthermore, we assume
that there is no information contained in the position of the correct answer, hence we expect
the following symmetry restrictions to hold :


 Pr q   q   , q   , q , q    | answer 3 is correct 


 Pr q   q   , q   , q   q ,  | answer 4 is correct  ,


Pr q   q1 , q2 , q3 , q4  | answer 1 is correct   Pr q  q 1 , q1 , q  2 , q 3 | answer 2 is correct 


37
 1
 2
1
 3
  1
  2
  3
1
where  1 ,   2  ,   3  ,   1 ,    2  ,    3 
permutations of
and   1 ,    2  ,    3 
are some
 2, 3, 4  . The symmetry restrictions, the conditional independence
assumption, and the uniform random allocation of the correct answer among four alternative
answers allow us to estimate the likelihood of the information given the position of the
correct
answer,
and
therefore
provide
empirical
estimates
for
Pr q   q1 , q2 , q3 , q4  | answer k is correct  . In practice we assume that, given answer k is
correct, information q has a Dirichlet density D  q;  k   ,   , k=1…4, defined over  4 such
that D  q;  k   ,   
  3     4  1   
q  qk , where the symmetry assumption is
3  i
        i 1

imposed through the parameter vector  k  ,     ek     with ek is a vector of zeros
with a 1 in position k. This vector of parameters for the Dirichlet density depends on two free
parameters only,  and  . These two parameters can be estimated (independently from the
other parameters of the model) by maximum likelihood from the observation of the
information obtained from the audience (i.e. the histograms) whenever the lifeline is used,
and the observation of which answer is the correct answer (even when the contestant chooses
an incorrect answer, the compére always reveals the correct one). For completeness note that
 k can be defined in terms of the elements of q as k  qk  
4
 q   . The information
j 1

j
density which the contestant expects is therefore the mixture D  q; p, ,  of the previous
densities D  q;  k   ,   , k=1…4, conditional on a given answer being correct, we have:
4
D  q; p,  ,    pi D  q;  i   ,   
i 1
  3     4  1  4
q   pi qi  
3  i
        i 1
  i 1

 , where the mixing

weights are the initial beliefs pi , i  1...4 .
A. 2.4 “Phone a Friend”
To use this lifeline the contestant determines, ahead of the game, six potential experts
(“friends”) and when she plays the lifeline she chooses one from this list of six. We imagine
that the contestant engages in some diversification when drawing up the list (i.e. the range
and quality of “expert knowledge” of the friends on the list is in some way optimised), and at
the time of playing the lifeline the contestant chooses the expert to call optimally. There is
however little information available to us about this process. As a consequence our model for
38
this particular lifeline is somewhat crude. We assume that the entire process can be modelled
as an appeal to an expert who knows the answer with some probability  , and is ignorant
with the probability 1   . We assume that the expert informs the contestant of his confidence
(contestants invariably ask the friend how confident they feel – although the answer is usually
not quantitative). Hence either the contestant knows the answer and her opinion “swamps”
the contestant’s belief, or the expert is ignorant and conveys no information and the
contestant’s belief is left unchanged. The density of the updated belief is therefore
H3  ; p    1 1,0,0,0  1    1 p .

A. 3

Proposition (factorisation of 4  x1 , x2 , x3 , x4  ):
The joint density:  4  x1 , x2 , x3 , x4  
xi  0
for
all
i,
2
2
 x2  

x3
 
 , such that
 1  x1   1  x1  x2 
  x1   
can
be
factorised
as
4
x
i 1
i
1,
follows:
4  x1 , x2 , x3 , x4   fU  x1  fU |U  x2 ; x1  fU |U ,U  x3 ; x1 , x2  ,
1
fU1  u  ,
fU1  u 
2
fU2 |U1  v; u  ,
fU 2 |U1  v; u   2
3
0u 1 ,
2
are
(conditional)
densities
such
that
with 2   1  x    x  dx ,
1
2
0
1  u  v    v  1

 0v1u 
2
1  u   1  u  
fU3 |U1 ,U 2  w; u , v  
1
fU3 |U1 ,U2  w; u, v  ,
2
1  u   u 


1
2
1
and
1
 w 

.
1
1  u  v  1  u  v  0 w1u v
Proof: It is easy to verify, by simple integration for fU1  u  , fU2 |U1  v; u  , and by construction
for fU3 |U1 ,U2  w; u, v  , all three are well defined densities over the relevant ranges. Moreover
their product is equal to  4 . . This implies that if U1 , U 2 and U 3 are three random variables
each distributed with densities fU1  u  , fU2 |U1  v; u  , and fU3 |U1 ,U2  w; u, v  , then the random
vector P  U1 U1U 2 U1U 2U 3 U1U 2U 3  , with U i  1  U i for all i=1..3, is distributed with
joint density:  4  x1 , x2 , x3 , x4  
2
2
 x2  

x3
 
 . Note that, by construction,
 1  x1   1  x1  x2 
  x1   
P ' e  1 , and P  0 . Since 4  x  and  4  x  share the same joint density for the order
39
statistics, i.e. 4! 4  x  where x has elements in descending order, to sample from 4! 4  x 
we sample first from  4 . and sort the resulting vector in descending order.
A.4
Probabilities and Simulated Likelihood
Here we describe the evaluation of some of the probabilities that lead to the log
likelihood. A complete description of the calculations can be obtained from the authors.
Calculating the probabilities when only one lifeline is available.
When the contestant has used all her lifelines, the events of interest are quitting or
losing (and, for the last question, the event that the contestant wins the million prize). The
probabilities of these events can be calculated directly from the analytical expressions given
in section 5.1 using the formulation for F we derive in section 5.3. When one or more
lifelines are available the calculations are made more complicated because of the information
which is gained when the lifeline is used and which allows the contestant to update her belief.
Hence, given the initial draw of the belief we determine whether this particular draw leads to
the use of the lifeline and, if so, whether the updated belief, or the original belief, if the
lifeline is not played, is informative enough to lead to an answer attempt. Finally, we evaluate
the probability that the answer is correct (under the original or the updated belief). We will
write ijk
k , n  p  as the probability that given p at stage n event k (which is defined precisely
below) is observed, given that the contestant is in the lifeline-state ijk, where i, (respectively j
or k) is one if the first (respectively second or third) lifeline is yet to be played and zero
ijk
otherwise. Let, ijk
k , n be the expected value of k , n  p  over all possible realisations of p, i.e.
ijk ,ij k 
 ijk

ijk
 p  stand for the probability that given p at stage n event
k , n  E   k , n  P   . Finally k , n
k is observed given that the contestant starts the question in the lifeline-state ijk and transit to
lifeline-state i’j’k’. We consider below representative events for each lifeline.
“50:50” is the only lifeline available at stage n.
The contestant uses “50:50”, plays and wins (moves to the next stage or wins the million
prize).
First, define the probability that 50:50 is used, she plays and wins, given a draw
(ordered
in
decreasing
order)
p
from
Pr 50:50  plays  wins | stage n  , p 
the
belief
distribution,
as
≡
40
100
1, n  p   1 k1  p ,0,0,0  p  f

n
1

100,000
1, n
with  jk 

n1 1,0,0   bn   bn 
1,100,000
p  ,
n
where
4
1 3
 p    p j  1  jk  p   fn1  0,0,0   bn   bn  an1  ,
3 j 1 k  j 1
pj
p j  pk
. This last expression is the probability that, given p, the contestant
correctly answers after using the lifeline. Hence, the unconditional probability satisfies
Pr 50:50  plays  wins | stage
  1k1 p,0,0,0 p
4

n
1
 f n1 1,0,0bn bn 
n
  1,100n
100,000
 p  4o  p  dp,
1, n
where  4 is the subset of the 4-simplex where p1  p2  p3  p4  0 . In order to determine
the probabilities we have used the fact that a contestant with a lifeline available will either
use it (and perhaps then quit), or play. It is straightforward to verify that the five expressions
above sum to unity; in particular the sum of the first three expressions is the probability that
the contestant uses the lifeline and this is the complement of the sum of the last two
100
probabilities. Each term of the sum that determines 100
1  p  (and similarly 2  p  and
100
3  p  ) is the product of the probability that a given two of the four options remain after the
lifeline is played,
with probability
p
 pk  / 3 , multiplied by the probability that the
j
remaining alternative with the largest updated belief is correct, with probability
 jk  p  
pj
p j  pk
with p j  pk , multiplied by the indicator that, given the updated belief, the
contestant decides to play.
“Ask the Audience” is the only lifeline left at stage n.
The contestant uses “Ask the Audience”, plays and loses,
010
o
Pr ATA  plays  loses | stage n   010
2, n    2, n  p  4  p  dp,
4
 2
 010,000  p  ,
010
2, n  p   1  k n  p, 0, 0, 0   p1  f n 1  0,1, 0   bn   bn   2, n
010,000
p  
2, n
4
1    q; p   1   q; p   f  0, 0, 0   b   b
1
1
n 1
n
n
 an 1  D  q; p,  ,  dq
where
and
where
  q; p  stands for the revised belief after information vector q is made available and
1  q; p  is the largest element in   q; p  .
41
“Phone a Friend” is the only lifeline left at stage n.
The contestant uses “Phone a Friend” and quits,
Pr PAF  quits | stage n   3,001n   3,001n  p  4o  p  dp,
where
4
3,001n  p   1k 3 p ,0,0,0 p  f

n
1
n1
 0,0,1 bn bn 
3,001,000
 p  and 3,001,000
 p   1    1 p  f
n
n

1
n1
 0,0,0bn bn an1 
.
General Case: all the lifelines are available
When more than one lifeline is available at a given stage, the number of elementary
events of interest increases, since not only can the contestants decide to play one lifeline
among many but the contestant can play more than one lifeline to answer a single question.
Hence while there are only five elementary events of interest when only one given lifeline is
left there are nine such events when two lifelines are available and seventeen when all three
lifelines are available, ignoring the order in which the contestant uses the lifeline and not
counting events with zero probability ex-ante (for example observing an event such as
quitting while the three lifelines are available). In this section we present the relevant
expressions needed to obtain the probabilities of a few selected elementary events. All other
probabilities can be obtained in a similar fashion.
The contestant uses the three lifelines (in any order), plays and loses.
Pr uses all life lines  plays  loses | stage n   111
2, n

1 k 1
4

n
 p ,0,1,1 max p1  f n1 1,1,1 bn   bn , kn2  p ,1,0,1, kn3  p ,1,1,0 
1k 2 p ,1,0,1  max

n


1k 3 p ,1,1,0  max

n

111,011
p 
2, n
p  f
1

n1
1,1,1 bn  bn , kn1  p ,0,1,1, kn3  p ,1,1,0 

p1  f n1 1,1,1 bn   bn , kn1  p ,0,1,1, kn2  p ,1,0,1 

111,011
p 
2, n
111,101
p 
2, n
111,110
 p  4o  p  dp.
2, n


1 3 4
011


2, n  j , k  p  ,  k , j  p  ,0,0  ,
3 j 1 k  j 1
111,101
 p    101
2, n
2, n    q; p   D  q; p,  ,  dq ,
4
and
110
111,110
p   110
2, n
2, n 1,0,0,0   1    2, n  p  . Inspection of these expressions reveals that the
probabilities of events in which more than one lifeline is available, here 111
2, n , can be defined
recursively in terms of the conditional probability of events with one fewer lifeline, given the
101
110
initial belief draw, here 011
2, n  p  , 2, n  p  and 2, n  p  . In turn, each of these conditional
42
probabilities can be calculated from conditional probabilities involving only one lifeline, i.e.
100
010
001
2, n  p  , 2, n  p  and 2, n  p  . This property is a consequence of the recursive definition
of the value function over the lifeline part of the state space (see section 5.4.b). Recall,
however, that the number of events of interest when the three lifelines are available is larger
than when only two or less are available. Hence the definition of 17 probabilities with three
lifeline at stage n, i.e. 111
m, n , m=1…17, will involve the 27 conditional probabilities with two
101
110
lifelines, i.e. 011
m,n  p  , m,n  p  and m,n  p  , m=1…9. In turn each of these conditional
probabilities will depend on the 15 probabilities with one lifeline as defined in the previous
010
001
section, i.e. 100
m,n  p  , m,n  p  and m,n  p  m=1…5.
The three lifelines are available, the contestant uses “50:50” , plays and wins.
Pr 50:50 only  plays  wins | stage n   111
10, n
  1k1 p ,0,1,1  max
4

n


p  f
1
111,011
p 
10, n
with
n 1
1,1,1 bn  bn , kn2  p ,1,0,1, kn3  p ,1,1,0 
111,011
 p  4o  p  dp.
10, n


1 3 4
8,011n  j ,k  p  ,  k , j  p  ,0,0  where 8,011n  p  is the probability


3 j 1 k  j 1
that with ATA and PAF available, for some belief p, the individual plays and wins.
Three lifelines are available, the contestant does not use any, plays and loses.
Pr does not use any of the 3 life lines  plays  loses | stage n  111
17, n
  1 p
4

1
 f n1 1,1,1 bn  bn  maxkn1  p ,0,1,1, kn2  p ,1,0,1, kn3  p ,1,1,0 
1  p1  4o  p  dp.
Simulation and smoothing
The evaluation of the probabilities rst
m,n  p  , n=1..15, m=1..17,
 r , s, t   0,1
3
and
of the conditional expectations knj  p, r, s, t  , n=1..15, j=1..3, and  r , s, t   0,1 requires the
3
use multidimensional integration techniques. If rst
m, n is not defined for some m, and some
r,s,t we assume rst
m,n  0 . Simulation methods (as described in Gouriéroux and Monfort
(1996) and Train (2003)) are well suited and have been applied successfully in similar
context (see the examples discussed in Adda and Cooper, (2003)).
Clearly the specification of the belief lends itself to a simulation based likelihood
methodology since simulations of Beta variates are obtained simply from Gamma variates
43
(see for example Poirier (1995)). In turn, Gamma variates can be obtained directly, using the
inverse of the incomplete Gamma function. Numerically accurate methods to evaluate the
inverse of the incomplete Gamma function are detailed in Didonato and Morris (1996). This
is implemented in Gauss in the procedure gammaii (contained in the file cdfchic.src). The
main advantage of their results is that it allows for simulations that are continuous in the
parameters of the Gamma distributions. Evaluation by simulation of an integral involving the
density of a 4 dimensional Dirichlet random vector, D  q; p, ,  , is obtained directly by the
simulation
of
each
of
its
100
o
100
1, n   1, n  p  4  p  dp   1k1  p ,0,0,0 p  f
4
4
approximated
1,n  S  
100
by

n
1
component.

n1 1,0,0  bn  bn 
For
100,000
p  4o p  dp,
1, n
example
can
be
1 S 100
1 S

p

1k1 p ,0,0,0 p f 1,0,0b b  100,000
 p s ,


1, n  s 
1, n
1,s  n 1
n
n
S s 1
S s 1  n s
where p s is one of S (the number of simulations) independent draws from the distribution of
the order statistics of the belief,  4o . . In fact the accuracy of this simulated probability, and
of all others which involve draws from  4o . , can be improved through antithetic variance
reduction techniques which involve the permutations of the gamma variates used to generate
each individual beta variate (as explained, for example, in Train (2003)).
111
10, n   1 k1 p ,0,1,1  max
4
be

n


p  f
evaluated
1
n1
1,1,1bn bn ,kn2  p ,1,0,1,kn3  p ,1,1,0
using
Moreover,
111,011
 p  4o  p  dp. is a quantity that can
10, n
the
simpler
S
111,011
ˆ 111  S   1  1 1

10,
ps  ,
2
3
10, n
n
S s 1 kn ps ,0,1,1 max p1,s  fn1 1,1,1bn bn ,kn ps ,1,0,1,kn ps ,1,1,0
formula
or
any
improvement of it. Similarly 111,101
 p    101
2, n
2, n    q; p   D  q; p,  ,  dq can be evaluated
4


4
S
ˆ 111,101  p; S   1  p  101   q ; p   , where q s ,i is one of S independent draws
by 
2, n
i
2, n
s ,i

S i 1 s 1 
from D  q;  i   ,   . In practice, we use S=96.
Finally all quantities kn2  p, r , s, t   E 2 |p Wn  , r , s, t  | p  which involve a multi
dimensional integral and the joint density D  q; p, ,  can be obtained in a similar fashion:
1
for example, using kˆn2  p, r , s, t ; S  =  pi  Wn  q s ,i , r , s, t  , where q s ,i is one of S
S i=1 s 1
4
S
independent draws from D  q;  i   ,   . In practice these expression are modified in order to
44
smooth out the discontinuities that are created by the indicator terms. Hence, the indicator
functions 1v1  maxv2 ,v3 ,v4  ,

versions,

1v  maxv ,v  ,

1
2
3

1
1 e
  v2 v1 
e
  v3 v1 
e
  v4 v1 
,
or 1v1 v2  ,
are replaced by their smoothed
1
1 e
  v2 v1 
e
  v3 v1 
, and
1
1 e
  v2 v1 
respectively,
where  is a smoothing constant. In the limit as    the smoothed versions tend to the
indicators.
45
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Table 1
Questionnaire Contestant Sample and Population of Contestants
Questionnaire sample of
contestants
Mean
Std Dev
0.76
0.43
61.96
104.1
0.68
0.47
243
Male
Winnings £,000
% quit last Q
N
Population of contestants on
videotapes
Mean
Std Dev
0.77
0.43
54.26
105.9
0.67
0.47
515
Note: We categorise three players who won the maximum £1m as quitters.
Table 2
ML Estimates of the Parameters of the Distribution of Histograms (ATA)
Parameter
Estimate
Std. error

4.754
0.210
0.914
0.030

Number of observations
Log-Likelihood
501
1526.41
Note: These results are based on 501 observations for which the use of ATA is observed.
50
Table3: Restrictions to common utility functions
Parameter estimates
(asymptotic Standard Errors)
p-value
HPE
 
1  
u  x   exp  
 x     sign  
 1 

α
ρ
γ
Minimum distance
CARA
1
u  x   1  exp   x 

0.0001 ()
0.9804 ()
0.0001 ()
240.20
0.00
0.3564 (0.0037)
1832.21
0.00
386.017
0.00
0.0001 (0.0002)
0.9782 (0.0227)
334.71
0.00
10.5800 (0.0153)
-7.7800 (0.0165)
0.0000 (0.0307)
125.75
0.00

Minimum Distance
Log
u  x   ln  x  105 
Minimum Distance
CRRA
1
1 
u  x 
x  
1 


Minimum Distance
Log with Jump
1 if 1000  x


u  x   log  x    if 2  x  1000

 2 if x  2

1
2

Minimum Distance
Note: Minimum distance estimates are obtained using a grid search. The (asymptotic) standard
errors are then derived using the “delta method”. The Minimum Distance statistics reported are
given by the value of the objective at the estimated parameters.
51
Figure 1
Observed Fails Rates
52
Figure 2
Figure 3
Estimated Utility function
Common Functional Forms
53
Figure 4
Figure 5
Probability “Friend knows the answer to the question”.
Distribution of the Maximum Belief
54
Figure 6
Value of playing the game at stage n, given the lifeline state.
Figure 7
Observed versus Predicted Frequencies of Fails and Quits
Note: The predicted frequencies are shown as shaded bars, observed frequencies are shown as
empty bars
55
Figure 8
Observed versus Predicted Use of Lifelines
Note: Predicted frequencies shown as shaded bars, observed frequencies as empty bars.
56