Deal or no Deal: Estimating utility functions with the use of simulation techniques

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Deal or no Deal: Estimating utility functions with the use of

simulation techniques

Floris Zoutman

University of Groningen

Faculty of Economics and Business Administration

Thesis submitted for the Degree of Master of Science in

the University of Groningen

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Abstract

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List of Figures

3.1 Mean error in predicting the percentage offer over the rounds for the Netherlands . . . 16 3.2 Mean error in predicting the percentage offer over the rounds for the US . . . 16 3.3 Estimated and actual net worth cumulative distribution in the Netherlands in euros . . . 23 3.4 Estimated and actual net worth cumulative distribution in the US in US dollars . . . 23

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List of Tables

1.1 A game of Deal or No Deal. . . 3

3.1 Summary statistics of the data set . . . 15 3.2 Parameter estimates of the standard model for the Netherlands and the US . . . 17 3.3 Parameter estimates of the standard model without weighting for the Netherlands and

the US . . . 18 3.4 Parameter estimates using the bootstrap for the Netherlands and the US . . . 20 3.5 Parameter estimates using the bootstrap without weighting for the Netherlands and the US 20 3.6 Parameter estimates using the MSL for the Netherlands and the US . . . 21 3.7 Expected value and standard deviation in wealth belonging to the estimated parameters

µW and σW . . . 21

3.8 Parameter estimates using the MSL without weighting for the Netherlands and the US . . 22

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Chapter 1

Introduction

Utility functions are a main input to almost every economic model of individual behavior. Also, many predictions are highly dependent on the shape of utility functions. As a result it is essential for economists to get empirical evidence on utility functions. For a long time the only data on utility functions came from experiments. Experimental economists let participants choose between a certain amount of money and a risky alternative in one way or another, see for example Allais [1953], Ellsberg [1961], Binswanger [1980], Binswanger [1981] and Kachelmeier and Shehata [1992]. In these studies utility functions were estimated using either classical expected utility functions by Von Neumann and Morgenstern [1944] or prospect theory Kahneman and Tversky [1979]. The problem with these studies is that monetary incen-tives for the participants are either small or non-existent.

A solution to this problem is the game show Deal or No Deal which is aired in many developed coun-tries. For this study I use data from the Dutch and the American version of Deal or No Deal. This dataset was also used by Post et al. [2008]. The Dutch dataset consists of 51 contestants and contains 265 observations. The American data set consists of 47 contestants and contains 266 observations. In this game show contestants have to make several choices between a certain amount of money and a risky alternative. Prizes range from €0.01 and €5,000,000 in the Dutch version and between $.01 and $1,000,000 in the US. A choice could literally change a person’s life and as a result the contestant has a very high incentive to maximize his utility. Research has been done by, among others, Post et al. [2008], Baltussen et al. [2007] and De Roos and Sarafidis [2006]. This paper seeks to increase the accuracy of the results obtained by Post et al. [2008] by using computer-intensive econometric techniques.

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Chapter 1: Introduction 2

Before explaining my enhancement on the estimations obtained by Post et al. [2008] it is necessary for the reader to get an idea of the rules of the game. For the Dutch version of the game, in each show there are 26 contestants, each holding one case. The contestants are people participating in the national lottery which sponsors the game show. They are invited to apply for a seat and tickets are randomly distributed to contestants. The contestants know the distribution of the cases but do not know the value in each case. Through quiz questions, one contestant is chosen to play the game. In the first round the player needs to eliminate six cases. Once, the cases are eliminated they are opened and the player learns which amount was inside the eliminated cases. After he has eliminated six cases, he gets an offer from the ’bank’. The bank offer depends on the distribution of cases still in the game. That is, if you eliminate cases with high amount, the bank offer will be less than if you eliminate cases with low amounts. The contestant can either accept the offer, ’Deal’, or play on, ’No Deal’. Before a decision needs to be made, each of the contestants can consult a friend or relative. If he plays on he will eliminate four more cases before receiving a new bank offer. If he continues, he has to eliminate another three cases, in round four he has to eliminate two cases and after round five he eliminates one case each round. The number of prizes left in the game for each round is thus: 20, 15, 11, 8, 6, 5, 4, 3 and 2. If he declines each of the offers he will receive the amount which is within his own case in round 10. Therefore, a contestant gets to make a maximum of nine decisions between a certain amount and a risky alternative. The distribution of the amounts is obviously known to the player and hence, he should be able to make rational decisions based on his utility function. The rules in the US are rather similar. The main difference is that the US contestants are selected by the producer.

A good example of how the game is played can be seen in table 1. It exhibits the game played by the unlucky American contestant Brett aired on 22-3-2006. In the table, cases not marked with an X were eliminated. For example, in round 1 Brett eliminated the cases which contained $.01, $50, $400, $750, $10,000, and $400,000. The mean value in the not eliminated cases was $150,361 and the bank offered him $ 19,000 or about 12.6% of the expected value. He did not accept the bankoffer, as can be seen in the row Deal or No Deal1. Brett was very unlucky in each round, but especially in round 8. In round 8 he eliminated the case containing $300,000 and his expected value dropped from $75,054 to $72. One should notice, that Brett received a very generous bank offer in round 7, in the sense that it was higher than the expected value. However, he refused the offer. Therefore, part of his bad luck is explained by his risk-loving behavior in that particular round. Finally, in round 9 he accepted a bank-offer of only $ 8.

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3

Table 1.1: The game by contestant Brett aired on 22-3-2006

Round 1 2 3 4 5 6 7 8 9 Deal or No Deal 0 0 0 0 0 0 0 0 1 Bankoffer $19,000 $28,000 $34,000 $55,000 $34,000 $52,000 $76,000 $50 $8 $0.01 $1 X $5 X X X X X X X X X $10 X X X X X X X X X $25 X $50 $75 X X X $100 X X X X $200 X X X X X X X X $300 X X X X X X $400 $500 X $750 $1,000 X X $5,000 X X X X X $10,000 $25,000 X X $50,000 X X $75,000 X X X $100,000 X X X $200,000 X $300,000 X X X X X X X $400,000 $500,000 X X X X $750,000 X X $1,000,000 X Mean value $150,361 $120,446 $89,154 $100,702 $50,919 $60,103 $75,054 $72 $8

Post et al. [2008] have estimated utility functions on the basis of revealed preference (see Samuelson [1938]). They hypothesized that contestants maximize their utility by comparing the stopping value (sv) to the continuation value (cv) in each round. The cv is the expected utility of playing another round. They approximate this by taking the expected utility of the upcoming bank offer. The sv is the utility derived from the bank offer. If the cv is larger than sv the utility maximizing decision is No Deal, otherwise it is Deal.

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Chapter 1: Introduction 4

the bank to be completely predictable. Bank behavior is thus partly random. Hence, the model only partially eliminates the second type of uncertainty. Post et al. [2008] does not consider this small second type of uncertainty when they perform their analysis on risk attitudes of the contestants. It is however possible to bring back the second type of uncertainty with the use of simulation techniques. The basic idea is that the expectation of utility towards the second type of uncertainty can be approximated by a Monte Carlo integral, provided that the sample distribution of randomness in bank behavior converges to the asymptotic distribution of randomness in bank behavior. In other words, it can be approximated with the bootstrap. From Jensen’s inequality it can be shown that by excluding the second type of uncertainty, one consistently overestimates the continuation value. Hence, Post et al. [2008] have con-sistently overestimated risk aversion within the sample. With the bootstrap, I expected to get better estimates of the utility function. It turns out that the new estimates show significantly smaller levels of risk aversion portraying the relevance of including both types of uncertainty when making estimations of datasets like Deal or No Deal. Important to note is that my estimates give upper bounds on risk aversion. That is, I make use of the assumption that contestants have perfect knowledge of the bank’s model. If they do not, type 2 uncertainty is higher and risk aversion is even lower.

Another serious issue within the estimates of Post et al. [2008] is the initial level of wealth. Because, within classical utility theory, people are assumed to maximize expected utility of final wealth, initial wealth is an important parameter. Unfortunately, the level of wealth for contestants in Deal or No Deal is not known. Therefore, Post et al. [2008] treat it as a parameter to be estimated. It is not possible to estimate the wealth level for each contestant individually. The number of parameters would become too large which would make the problem unsolvable with current computing technology. Also, estimation would become very inefficient if so many parameters need to be estimated. Post et al. [2008] assume that it is fixed over all contestants in the sample. However, this assumption is quite unrealistic, because it is obvious that people playing the game do not all have the same level of wealth. This could lead to a serious bias in the estimation result. I use the Maximum Simulated Likelihood (MSL) to intro-duce a random effect structure on the level of wealth. The MSL is best explained in Mariano et al. [2000]. The results with respect to this estimation are more difficult to interpret. In general the esti-mated variance in wealth seems too small. Also, the estimation of mean wealth fluctuates wildly over the different countries estimates which does not seem realistic when looking at the real distribution of wealth.

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5

make estimation feasible in a reasonable amount of time. Also, if the simulation is large enough to give accurate results in theory, matrices become far too large for current computers. It is not an option to estimate the model many times for a smaller sample, because as is pointed out by Mariano et al. [2000], the bias of the MSL is severe for small samples.

One should also consider whether the observations needs to be weighted. Post et al. [2008] use the parameter 1/δir which is a function of the other parameters and the data. It is contestant and round

specific. According to the authors it is meant to give more weight to easy decision than difficult deci-sions. They do so by measuring the standard deviation of utility in the next round, δir. If the standard

deviation is high they claim that the decision is difficult, because it is more difficult for a contestant to asses his continuation value. The fit improves by introducing these weights, but there are also some downsides to using δ. Suppose δir = 0 to machine precision for some contestant-round pair, for some

parameter values of the estimation problem. Then those particular parameter values cannot be evalu-ated, because the weights are undefined. Hence, many potential solutions are thrown out. I estimated each of the models with and without weighting and found that each solution to the estimation problem without weighting could not be considered with weighting due to at least one of the observations receiv-ing undefined weight. The decision to weight or not to weight also has large effects on the parameter estimates. As such, it is not particularly clear whether weighting is indeed attractive.

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Chapter 2

The Model

2.1 The continuation value and the stopping value

Contestants maximize utility of final wealth by making discrete choices in which they either accept the bank offer, or continue playing. As such, this is a classical case of revealed preference. In case the bank offer is accepted people the final wealth is the initial wealth plus the bank offer. The stopping value is:

sv(xir, Wi) = u(B(xir, ir) + Wi) (2.1)

In which xir denotes the cases still in the game in round r of contestant i, u is the utility function, B

is the bid function of the bank which is a function of xir and Wi is initial wealth of contestant i. ir

measures randomness in bank behavior.

For a rational contestant the continuation value should be the expected utility of all upcoming bank offers in the upcoming rounds. However, following Post et al. [2008], I assume that the continuation value is simply the expected utility of the next bank offer. The rational behind this assumption is that the difference between the two is very small. To find some information on the next bank offer it is necessary to understand the bankoffer function. I use the following to model bank behavior:

B(xir, ir) = bir(ir) ¯xir

That is the bank offer is a contestant-round specific constant bir(ir), multiplied by the average of the

amounts in cases still in the game ¯xir. Therefore, the bankoffer in the next round, B(xi,r+1, i,r+1)

is both dependent on bank behavior , incorporated in the value of bir, and which cases you eliminate

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7 2.1 The continuation value and the stopping value

between the current and the next round. From the data something more can be deduced on bank behavior: it seems the bank uses the following model:

bi,r+1= bir+ (1 − bir)ρ10−r+ ir (2.2)

In which ρ is a parameter to be estimated and iris an error term with 0 mean and finite variance. Since

in general bir is between zero and one equation 2 can be explained in the following way. The percentage

offer made by the bank increases over the rounds. The reason is that the producers of the game show do not want people to quit the game early in order to keep the game show interesting for viewers. Hence, the bank offer in the first rounds must be too low for people to accept the offer. Further, the offer in round 10 is just the value within the final case. As such, bi,10 = 1. The model is closed because the

contestant always receives the offer in the first round. With this offer and the model he can predict the non-random part of bank behavior in the upcoming rounds.

Before one can find the continuation value, it is also necessary to look into the distribution of the cases. Suppose that in round r there are nrcases and in round r+1 there are nr+1 cases. Then there are:

   nr nr+1   = 1/pr

Different possible combinations of cases in round r+1. prthus represents the probability of each different

combination.

Before the continuation value can be calculated I make one simplifying assumption: the distribution of is independent of which cases you eliminate. Now the continuation value is:

cv(xir, Wi) =

Z X

y∈X_xir

u(B(y, i,r+1) + Wi)prgi,r+1()d (2.3)

In which gi,r+1(.) represents the pdf of the errors which in general could be both round and individual

specific. The continuation value is thus the expected utility of the next bank offer in which contestants take account of both the randomness in which cases they eliminate and the randomness in bank behavior. Before one can proceed with the estimation of the utility function, a functional form of the utility function is necessary. Following Post et al. [2008] I use a very general utility function:

u(z) =1 − exp(−αz

1−β₎

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Chapter 2: The Model 8

In which z denotes final wealth. If β = 0 the utility function exhibits constant absolute risk aversion, and the limit as α goes to zero has constant relative risk aversion. Therefore, it covers two of the main utility function used in economics as well utility functions with increasing and decreasing risk aversion coefficients.

2.2 An ARUM model

The choice of Deal or No Deal can be modelled using an additive random utility model (ARUM) as is explained in Cameron and Trivedi [2005]. It is assumed that the utility of No Deal is given by:

Vir0 = cv(xir, Wi) + ζir0

In which ζ0 is a random error. The utility of Deal can be found in a similar way:

V_ir1 = sv(xir, Wi) + ζir1

The probability of a Deal decision can thus be found by:

P r(DealorN odealir= 1|xir, Wi) = P r(Vir1 > V 0 ir) = P r(ζ 0 ir− ζ 1 ir< svir− cvir) (2.4)

In which DealorN odealir is a dummy variable with value 0 for the decision No Deal and 1 for the

decision Deal. It is important to note that this model requires a scale normalization, because if V1> V0

then aV1> aV0. If it is assumed that:

ζ_ir0 − ζ1

ir∼ N (0, σ 2 ir)

In which N denotes the normal distribution and σ2

ir its variance. 2.4 can be simplified to the following:

P r(DealorN odealir= 1|xir, Wi) = Φ

svir− cvir

σir

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9 2.3 The noise parameter σ

In which Φ denotes the cumulative density function of the standard normal distribution. The probability of a No Deal decision is now easy to find:

P r(DealorN odealir= 0|xir, Wi) = Φ

cvir− svir

σir

(2.6)

Since this is a probit regression model the probability mass function of DealorNodeal is the following:

f (DealornN odealir|xir, Wi) = Φ svir− cvir σir DealorN oDealir Φ cvir− svir σir 1−DealorN oDealir

And the log-likelihood of the sample can be denoted by:

L = N X i=1 9 X r=r0

log(f (DealorN odealir|xir, W ))

in which N denotes the number of contestants and r0the round at which estimation starts.

2.3 The noise parameter σ

To identify the model some structure on σir is assumed. An obvious way to do so is by setting:

σir= σ

Here σ has a double function. First, it is a measure of the noise in the sample. Second, it is used as a scaling parameter of utility. This would lead to the conclusion that it is impossible to compare the noise between the American and the Dutch sample and the different estimation methods, because the scaling of utility might be different between the two samples.

Post et al. [2008] solved this issue by introducing the following variable:

δir) = v u u t Z X y∈X_xir

(u(B(y, i,r+1) + Wi) − cv(xir))2prfi,r+1()d

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this parameter an alternative structure can be imposed on σir:

σir= δirσ

The advantage of this specification is that σ can now be interpreted as a noise parameter. Also, it does not reduce the degrees of freedom because δ is a function of the other parameters and the data only. I will estimate the model under both specifications.

2.4 Randomness in bank behavior: The bootstrap

The gameshow Deal or No Deal is much more interesting if the bank is not (completely) predictable. This is the reason why 2.2 included an error term. An obvious way to approximate the integral in 2.3 is by setting the error term to its expected value, as was done by Post et al. [2008]. To put it differently, instead of finding expected utility for each possible combination of prizes and each possible percentage offer, one finds expected utility for each possible combination of prizes and the expected percentage offer. If the utility function is concave, this approximation should suffer from the Jensen’s inequality:

Z X

y∈X_xir

u(B(y, i,r+1) + Wi)prfi,r+1()d ≤

X

y∈X_xir

u(B(y, 0) + Wi)pr (2.7)

As such, this approach over estimates the continuation value. If the continuation value is overestimated then risk aversion is also overestimated. A priori, it is not clear how important this overestimation is, but one could approximate the integral in 2.3 by using the bootstrap. That is, although there is no information on the asymptotic distribution of ir information can be obtained on its empirical

distribution. This is done in the following way:

er= br+1− br− (1 − br) ˆρ9−r (2.8)

Here er is a column vector for each round. Its size is determined by the number of contestants still

playing in round r + 1. br+1 is a vector with the percentage offer for each of the contestants in round

r + 1, bris a vector with the percentage offer for each of the contestants in round r still playing in round

r + 1, and ˆρ is the estimated value of ρ. ρ is estimated by non-linear least squares on 2.2. It should be noted here that the difference between the predicted value of bi,r+1 and the true value of bi,r+1can

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11 2.5 Wealth: the Maximum Simulated Likelihood

empirical distribution of ir in each round. A smart contestant playing in round r can now approximate

its continuation value in the following way. First, take S random draws of er. Then, find bi,r+1 for each

sample in S, by using equation 2.2. Find utility for each value of bi,r+1and each possible combination of

prizes in the next round. Finally, add the utility up and divide by the possible combinations and S. The reason why I sample from each round individually is to correct for possible heteroskedaticity. Obviously, the accuracy of the approximation of the integral, increases with the size of S. In equation form:

cvbs(xir) = 1/S S X s=1 X y∈X_xir u(B(y, es_i,r+1) + Wi) (2.9)

The bootstrap can thus be used to approximate randomness in bank behavior. But if randomness in bank behavior effects the continuation value, the mean of all possible levels of utility in the next round, then it also effects the standard deviation of all possible levels of utility in the next round, δ. If the bootstrap is used δ can be calculated by:

δ(xir) = v u u t1/S S X s=1 X y∈X_xir (u(B(y, es i,r+1) + Wi) − cvbs(xir))2pr

The effect of using the bootstrap in weighting the observations is easily explained. If in a particular round the bank behavior is more random, than the choice for contestants becomes more difficult. As a result, δ will give observations from that particular round less weight.

2.5 Wealth: the Maximum Simulated Likelihood

Deal or No Deal contestants are not required to give any information on their initial level of wealth. Therefore, the parameter Wi needs to be estimated. The model can be identified if for each contestant

a level of wealth is estimated. However, this would mean that the log-likelihood needs to be estimated over N parameters of wealth as well as the parameters of risk aversion and the noise parameter. In terms of computing time, this is close to impossible. Also, the estimation becomes highly inefficient. Hence, some structure has to be imposed on Wi. A simple solution which is used by Post et al. [2008] is to

assume Wi= W . The advantage of this structure is its simplicity. The disadvantage is that it is highly

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effect on Wi. I assume that the distribution over the sample is given by:

Wi∼ LN (µW, σ2W)

In which, LN denotes the log normal distribution and µW and σW are the parameters of the log-normal

distribution. The log-normal distribution is chosen because in developed countries wealth is positive for almost everybody and the distribution is skewed to the right. These are both properties of the log-normal distribution. Notice that only two parameters on wealth need to be estimated. As such, the expected log-likelihood can be written as:

E(L) = Z N X i=1 9 X r=1 log f (xir, Wi)LNd(W |µW, σW)dW (2.10)

In which LNd(W |µW, σW) denotes the density function of the log-normal distribution with parameters

µW and σW. In Mariano et al. [2000] it is explained how the Maximum Simulated Likelihood(MSL) can

approximate for this integral. Draw NM times from a log-normal distribution with parameters µW and

σW to create M column-vectors of size N with different values of wealth, in which M denotes the number

of simulations. Calculate the log-likelihood for each vector. Take the mean of these log-likelihoods and maximize this mean log-likelihood, with respect to all parameters. The results are consistent in M, but biased for a finite M. However, the bias is much larger if no random effect on W is incorporated. More information on convergence can be found in Mariano et al. [2000]. In mathematics the integral in 2.10 can be Monte-Carlo approximated by:

M SL = 1/M T X m=1 N X i=1 9 X r=1 log f (xir, Wit) (2.11)

2.6 Estimation Procedure

Three basic models need to be estimated. The first model should replicate the results found by Post et al. [2008], the second takes into account randomness in bank behavior by using the bootstrap and the third takes into account randomness in initial wealth with the use of the MSL. The models are estimated by maximizing the log-likelihood. The following three equations, describe the optimization problem for each model: max α,β,W,σ N X i=1 9 X r=r0

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13 2.6 Estimation Procedure max α,β,W,σ N X i=1 9 X r=r0

log(fbs(DealorN odealir|xir, W )) (2.13)

max α,β,σ,µW,σW 1/M M X m=1 N X i=1 9 X r=r0

log(fM SL(DealorN odealir|xir, Wm)) (2.14)

In 2.13 the subscript bs refers to the fact that the bootstrap has been used to incorporate randomness in bank behavior in the continuation value. In 2.14 The subscript MSL refers to the use of the MSL. The superscript m refers to the simulated sample. Because I use data from 2 countries and because I estimate each model with and without the δ parameter, there are a total of 12 estimation results. There are four important constraints on the parameters. First, none of the δxir can be zero if the models are estimated with δ. Second, none of the possible future final wealth levels can be negative. Therefore, I have set the constraint that W ≥ 0. Further, if β > 1 the utility function becomes decreasing in final wealth and finally the parameter σ has to be greater than zero, because it is a standard deviation. I have maximized each of the log-likelihoods with the aid of a Nelder-Mead optimizer, see Nelder and Mead [1965]1, penalizing the function whenever any of the parameters fall outside of their constraints. In principle it would have been possible to derive the gradient and Hessian after which one could use a Newton-method to find the parameters. However, the log-likelihood contains some local maxima as well as some discontinuities and a Nelder-Mead optimizer is much more resistant to these issues. It is important to note that most of these discontinuities are related to the parameter δ. There are several utility functions for which δ is zero to machine precision2_{. Obviously, this issue does not exist when the}

models are estimated without δ.

A final issue is the starting values. The choice for starting values is non-trivial. None of the parameters have a clear upper and lower bound. My first choice for starting values was always the results found by Post et al. [2008]. After convergence to a minimum, I went on to try other starting values randomly. In general I found that the optimizer often converges to local maxima. However, given the time spent in trying different starting values I am quite confident that my results represent global maxima of the log-likelihood functions. The next chapter will describe the data and give estimation results.

1_{Fminsearch in Matlab}

2_{For example: if β is very close to one all outcomes of the utility function are very close to each other and as a}

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Chapter 3

Data and estimation results

3.1 Data

I use the Dutch and US data set also used in Post et al. [2008]. The Dutch show (called ’Miljoenenjacht’ which translates into the hunt for millions) first aired on December 22, 2002. The final show within the dataset was aired on January 1, 2007. The Dutch show has prizes ranging from €0.01-€5,000,000. The data on the Dutch show has 51 contestants playing an average of 5.2 rounds. In total this gives me 265 observations. However, I skip the observations in the first round following Post et al. [2008]. This should not be very important, because the bids in the first rounds are very low. Therefore, nobody makes a deal in the first round and these observations do not contain a lot of information on the utility function of the contestants. The Dutch data set thus consists of 214 observations from round 2-9. The average prize contestants have taken home was about € 122,544.

The US version of Deal or No Deal aired first on December 19, 2005 and the final episode in the data sets was aired in early June 2006. In the data set there were 6 contestants who played with higher top prizes. I have excluded these contestants from the data in the analysis to keep consistency. Excluding these 6 contestants the US data set includes 47 contestants playing an average of about 7.6 rounds. This gives a total of 355 observations. Once again dropping the first round I am left with 308 observations from round 2-9. The average prize taken home by contestants in the US was $131,478. Table 3.1 gives some summary statistics of the data. For a more detailed description of the data set I refer to Post et al. [2008].

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15 3.2 Bank Behavior

Table 3.1: Summary statistics of the data set The Netherlands US

Minimum prize in the game € 0.01 $0.01

Maximum prize in the game € 5,000,000 $1,000,000 Average prize in the game € 404,692 $131,478

Minimum prize taken home € 10 $5

Maximum prize taken home € 1,495,000,000 $464,000 Average prize taken home € 227,265 $116,538

Average number of rounds played 5.20 6.71

# Observations excluding round 1 214 355

3.2 Bank Behavior

As has been established in chapter 2 bank behavior is characterized by equation 2.2. The value of ρ was estimated with the use of non-linear least squared. It was found that ˆρN L = .832 and ˆρU S = .777, in

which the subscript NL is used for the Netherlands and the subscript US is used for The United States. The explanatory value of the model is really high. In fact for the percentage offer in the Netherlands it was found that R2_{N L}= .95 and for the US R2_{U S}= .97 . When one compares the predicted to the actual bankoffers in monetary terms one finds that R2

N L= .97 and R 2

U S= .97. Hence, the fit seems to be very

strong. Figure 3.1 and 3.2 show the mean error per round on the percentage offer for the Netherlands and the US. As can be seen from the figure, the mean error in the Netherlands seems to be somewhat increasing over the rounds which indicates heteroskedaticity. This gives some evidence that the model on bank behavior over predicts bankoffers in the early rounds and under predicts bankoffers at the end of the game. The error structure in the US seems to be more random.

The initial bankoffer after the first 6 cases are eliminated is about 11% in the US and 5.7% in the Netherlands. The conclusion can be that on average US contestants get a better offer in the beginning of the game but the rate of increase, ρ, is smaller in the US than in the Netherlands. In fact, on average Dutch contestants get a higher percentage offer after round 4.

It is important to note that the model does not predict any bid above the expected value in the cases. However, in the Netherlands 11 of the bids were above the expected value. Of these offers only 4 were accepted. In the US 20 offers were above the expected value and only 10 were accepted.

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Chapter 3: Data and estimation results 16

Figure 3.1: Mean error in predicting the percentage offer over the rounds for the Netherlands

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17 3.3 Replication Results

3.3 Replication Results

I was unsuccessful in replicating the results derived by Post et al. [2008]. That is, after carefully following all the steps described in their article and calculating the likelihood at their parameter values, the log-likelihood value was extremely close. The difference could easily be due to a rounding error. However, if I use a Nelder-Mead optimization program to find the maximum likelihood with their parameter estimates as initial value, I find very different parameters. It can be concluded that at least in my configuration1 their results did not maximize the log-likelihood. For the US this could partly be explained by the fact that I do not use the data of the 6 contestants who played the game with different prizes. I can only think of one explanation for the difference in the Dutch estimates which is that I might have a different computer configuration and that this configuration is important. The estimation results can be found

Table 3.2: Parameter estimates of the standard model for the Netherlands and the US

Parameters NL p-value US p-value

α 11.51 0.00 1.08 · 10−6 0.00 β 0.95 0.00 -0.13 0.00 W € 86,367.21 0.00 $ 22.52 0.84 σ 0.42 0.00 0.26 0.00 Log-Likelihood -78.47 -79.76 ARA 2.34 · 10−5 −5.56 · 10−3

in table 3.2. The last row labelled ARA reports the Arrow-Pratt (see Pratt [1964] and Arrow [1971]) measure of absolute risk aversion at initial wealth. It can be calculated as follows:

ARA(W ) = −u

00_{(W )}

u0_{(W )} = αW

−β_{(1 − β) + βW}−1 _(3.1)

It is very difficult to calculate standard errors for this likelihood function. Therefore, I have calculated p-values with the likelihood ratio statistic. That is, fix the parameter for which the p-value needs to be calculated to its value under the null hypothesis. Find the maximum likelihood under this restriction. Compare the restricted to the maximum likelihood and use a likelihood ratio test to find the p-value. The null hypothesis is 0 for each parameter except σ for which the null-hypothesis is that its value is one. As can be seen, all parameters are significant for the Dutch version of Deal or No Deal. Also, the value of absolute risk aversion seems to be in line with other results. For example, Post et al. [2008] find absolute risk aversion of 6.17 · 10−5 for the Dutch version and 1.40 · 10−5 for the US version of Deal or No Deal and De Roos and Sarafidis [2006] estimate risk aversion to be 7 · 10−6 for their static model

1_{I use Matlab R2008a. The smallest difference between two numbers it can recognize is given by eps = 2.2204 · 10}−16_.

The tolerance on convergence was set to T olX = 1 · 10−6_{except when I use the Bootstrap where the tolerance was set at}

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and 5 · 10−5 for their dynamic model using the Australian data of Deal or No Deal2_{. The estimated}

wealth parameter should be interpreted in the following way: on average people play as if they have an initial level of wealth equal to W. In the Netherlands the mean net worth was measured to be € 108,000 per household in 2000, while the median net worth was measured to be € 25,000 3_{. If one takes into}

account the fact that the very rich most probably do not play Deal or No Deal, one would expect the estimated W to be in between the median and the mean wealth and indeed it is. For the US version of Deal or No Deal results are very different. The estimated parameters indicate that US contestants are risk-loving. Although, this result was unexpected it is not as strange as one would imagine. First, within the sample 10 offers were rejected even though they were above the expected value in the cases left in the game. But if equation 2.2 predicts the upcoming percentage offer then these contestants will predict a lower percentage bank-offer next period. By rejecting these offers, contestants act extremely risk-loving, because the expected value of the prizes in the next round will probably be lower than the offer, since the offer is above the expected value now and they also expect the percentage offer in the next round to be worse than in the previous. As such, they have to be extremely lucky to get a better bid in the next round. Second, within the sample 6 contestants play the game until the end, which means the game does not establish a lower bound on risk aversion for them. Therefore, the dataset contains quite a few observations which indicate risk-loving behavior. Hence, the best fit is found when β is negative. The parameter W is insignificant for the Americans. These estimation results indicate that Americans do not take into account their initial wealth when they take decisions in Deal or No Deal.

From the table it can also be concluded that the noise in the American sample is lower than it is in the Dutch sample. This indicates that Americans are better able to asses their continuation and stopping value. I have also estimated utility function without weighting by δ. The results can be found in table

Table 3.3: Parameter estimates of the standard model without weighting for the Netherlands and the US

α -28.44 0.00 1.97 · 10−6 0.00 β 1.00 0.00 1.00 0.00 W €50,237.92 0.00 $67,354.48 0.00 σ 0.49 0.00 4.97 · 10−6 0.00 Log-Likelihood -94.36 -106.224 ARA 1.99 · 10−5 1.48 · 10−5

3.3. From the log-likelihood it can be seen that the fit is worse than it is with δ. This should be expected

2_{In their static model a contestant maximizes utility by choosing either the utility of the bankoffer or the expected}

utility of all cases left in the game, in their dynamic model a contestant maximizes utility by choosing either the bankoffer or the expected utility of all upcoming future bids

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19 3.4 Estimation results using the bootstrap

for the following reason. If δ is used to weight the observations, than the parameters are chosen such that outliers get little weight. As a result, the fit will generally be better when the model is estimated with δ. However, the solution which gives the best fit would have been unacceptable, had the model been estimated with δ. The reason is that β is very close to 1. For this solution at least for one combination of i and r, δ(xir) would have been zero to machine precision. It can thus be concluded that although

δ improves the fit, it also neglects solutions which are potentially optimal. For the Netherlands it can be seen that the parameter estimate of α is extremely different. However, absolute risk aversion at the initial level of wealth is almost unchanged between the two estimations. Also, the wealth parameter is lower than it was in the estimation with δ, but it is still in between the median and the mean net worth in the Netherlands. For the US results have changed dramatically. Americans are now estimated to be risk-averse instead of risk-loving. This is probably a combination of the following two reasons. First, it is obvious that this result would not have been allowed when the δ parameter is used, because β is too close to 1. Second, it could be that the choices which indicate risk aversion were on average more difficult than the choices which indicate risk-loving behavior. As a result more weight would have been given to decisions which indicate risk-loving behavior. But if no weighting takes place, both types of decisions receive equal weight and as a result American contestant are estimated to be risk-averse. Also, the size of the absolute risk aversion is in line with previous studies. For the US it is nice to see that the wealth parameter is now in between the median net worth, $ 46,506, and the mean net worth, $164,635, in 2000. 4 _{Without δ the main role of σ is that it functions as a weighting parameter. Its value can}

now be interpretted in the following way: if the continuation value in terms of utility is about 2σ above the stopping value, the probability of No Deal is 98%5_{. Therefore, it explains more about the scale in}

which utility operates for the particular choices of α and β than it explains on noise. The next section will discuss the results when randomness on bank behavior is taken into account by using the bootstrap.

3.4 Estimation results using the bootstrap

Table 3.4 shows the result of estimating equation 2.13. The sample size I used was 100. It would have been nice to use a larger sample size. However, I ran into memory problems when I increased the sample size. Also, an increase in the sample size from 50 to 100 did not yield very different estimates. From the Dutch results, it can be concluded that Dutch contestants are estimated to be much less risk-averse when it is taken into account that bank behavior is partly random, as was expected from the Jensen

4_{Source is the US Census Bureau.}

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Table 3.4: Parameter estimates using the bootstrap for the Netherlands and the US

α 5.22 0.00 4.84 · 10−14 0.53 β 0.89 0.00 -1.48 0.00 W € 236,946.41 0.00 $ 0.19 0.66 σ 0.60 0.00 0.85 0.02 Log-Likelihood -88.74 -138.8190193 ARA 1.27 · 10−5 _-7.70

inequality in equation 2.7. Further, the wealth parameter went outside of the expected range between median and mean net worth indicating that Dutch contestant play as if their initial wealth is much higher than it actually is. Also, the noise as measured by parameter σ went up because of the extra source of uncertainty. It is important to notice here that a comparatively small extra source of uncertainty can lead to very different results in the estimation of utility functions. Even though, the bank model can predict the upcoming bank offer with an R2higher than .95, the unexplained part of bank behavior plays an important role. A result which is remarkably similar was found by Barberis [2000]. In his article he studied the effect of predictability of stock returns on optimal portfolio weights over the time horizon. Among other things, he found that parameter uncertainty could have huge effects on optimal portfolio weights. He introduced parameter uncertainty through a Bayesian approach. However, in a course on financial econometrics it was shown to me that the same results could be achieved much easier by means of the bootstrap6_{. Although, the problem is completely different, both the approach and the results are}

remarkably alike, namely that small sources of uncertainty can lead to very different quantitive results when you maximize a function such as a utility function or a log-likelihood function.

For the US, risk aversion also went down a lot or rather, the estimation of risk-seeking behavior went up. The parameter α is insignificant in this estimation. Initial wealth is also still insignificant and the noise is now higher in the US estimates than in the Dutch estimates, in this specification.

The results of estimating utility functions with the bootstrap and without weighting can be found

Table 3.5: Parameter estimates using the bootstrap without weighting for the Netherlands and the US

α -28.00 0.00 1.97 · 10−6 0.00 β 1.00 0.00 1.00 0.00 W € 82,004.17 0.00 $ 67,374.65 0.00 σ 0.21 0.00 5.22 · 10−6 0.00 Log-Likelihood -109.15 -158.76 ARA 1.22 · 10−5 1.48 · 10−5

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21 3.5 Parameter estimates using the MSL

in table 3.5. The parameters α and β are not very different from the estimates without the bootstrap. Absolute risk aversion is estimated to be smaller. From equation 3.1 it can be seen that absolute risk aversion is strictly decreasing in W if β is very close to 1 as it is here and therefore, the decrease in ARA is almost entirely due to an increase in W. The US results are almost completely unaffected by the use of the bootstrap. Estimated wealth went up and absolute risk aversion went down but the result is extremely small. In the next section the results of adding a random effect on W with the MSL are discussed.

3.5 Parameter estimates using the MSL

Table 3.6: Parameter estimates using the MSL for the Netherlands and the US

α 19.75 0.00 8.89 · 10−7 0.01 β 0.96 0.00 -0.14 0.00 σ 0.42 0.00 0.27 0.00 µW 11.33 0.00 3.16 0.84 σW 0.20 0.42 0.03 1.00 Log-Likelihood -78.15 -79.75 ARA 2.43 · 10−5 −6.03 · 10−3

Table 3.7: Expected value and standard deviation in wealth belonging to the estimated parameters µW

and σW EV STD NL € 84,881.91 €16,897.41 NL-no δ € 4,651.66 € 1,427.49 US $ 23.56 $ 0.82 US-no δ $ 67,400.77 $ 907.36

Table 3.6 shows the results of estimating equation 2.14. The sample size is again set to 100. For convenience table 3.7 gives the expected value and the standard deviation of W if W follows a lognormal distribution with estimated parameters µW and σW. It should be noted first that the fit has improved

slightly over the standard model. This should be obvious, because the special case where σW = 0 is

exactly equal to the standard model. Also, absolute risk aversion calculated by estimated α, β and expected W has increased although the increase is quite small. The parameter σW is insignificant,

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Table 3.8: Parameter estimates using the MSL without weighting for the Netherlands and the US

α -28.49 0.00 2.57 · 10−6 0.00 β 1.00 0.00 1.00 0.00 σ 0.49 0.00 7.86 · 10−6 0.00 µW 8.40 0.00 11.12 0.00 σW 0.30 0.00 0.01 1.00 Log-Likelihood -87.74 -106.22 ARA 2.15 · 10−4 _{1.48 · 10}−5

The parameter σW is now significantly different from 0. Also, expected wealth is a lot lower than it was

without a random effect on W.

Figure 3.3 and 3.4 depict the estimated and actual net worth cumulative distribution for respectively the Netherlands and the US7. As can be seen from the figures, the variance in the estimated distribution is far too small for any specification. This could indicate that the people playing Deal or No Deal are a very specific sample of the population. A more reasonable explanation might be that people more or less play as if they have the same level of wealth, even when this is not the case. The next chapter concludes.

7_{Source on actual net worth distribution dates from 2000. For the Dutch data the source is the Dutch Central Bureau}

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23 3.5 Parameter estimates using the MSL

Figure 3.3: Estimated and actual net worth cumulative distribution in the Netherlands in euros: The solid line is the actual data, the dashed line is the estimate with δ and the dotted line is the estimate without δ

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Chapter 4

Conclusion

It can be seen from this study that the proper specification is extremely important when one wants to estimate utility functions using this type of data. It seems very straightforward to weight people’s choices by the difficulty of the choice. That is, it makes perfect sense that a difficult choice yields less information on the utility function than an easy choice, because people will probably act less rational if the choice is difficult. If such a weighting decision would also work as a scaling parameter of utility, so much the better. However, making a measure of this difficulty is much more difficult as we have seen with the parameter δ. This research gives evidence of the fact that weighting by δ might not be a proper choice. The reason is that all of the estimation results without the parameter δ cannot even be evaluated with the parameter δ. Therefore, many results with potentially strong explanatory value are thrown out by this particular type of weighting. A particular interesting case with respect to this issue, is the US data of Deal or No Deal. With weighting, American contestants are estimated to be risk-loving, but without weighting they are estimated to be risk-averse.

Another issue is randomness in bank behavior. If randomness in bank behavior is taken into account then choices by contestants contain more risk than considered in previous studies. As such, risk aversion was also overestimated. This thesis establishes an econometric method to take this type of risk aversion into account. From this study it can also be concluded that this overestimation is severe for contestants of Deal or No Deal, especially when the observations are weighted. However, if the observations are not weighted the effect is much smaller for the Dutch case and close to non-existent for the American data.

In expected utility theory people maximize utility of their final wealth level. As such, one would expect

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25

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Bibliography

M. Allais. Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole americaine. Econometrica, 21(4):503–546, 1953.

K. Arrow. The Theory of Risk Aversion. American Elsevier, 1971.

G. Baltussen, T. Post, and J. van den Assem. Risky choice and the relative size of stakes. http://ssrn.com/abstract=989242, 2007.

N. Barberis. Investing for the long run when returns are predictable. Journal of Finance, 55(1):225–264, 2000.

H. Binswanger. Attitudes toward risk: Experimental measurement in rural india. American Journal of Agricultural Economics, 62(3):395407, 1980.

H. Binswanger. Attitudes toward risk: Theoretical implications of an experiment in rural india. Economic Journal, 91(364):867890, 1981.

A. Cameron and P. Trivedi. Microeconometrics: Methods and Application. Cambridge University Press, 2005.

N. De Roos and Y. Sarafidis. Decision making under risk in deal or no deal. http://ssrn.com/abstract=881129, 2006.

D. Ellsberg. Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics, 75(4):643669, 1961.

S. Kachelmeier and M. Shehata. Examining risk preferences under high monetary incentives: Exper-imental evidence from the peoples republic of china. American Economic Review, 82(5):11201141, 1992.

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27 BIBLIOGRAPHY

D. Kahneman and A. Tversky. Prospect theory: An analysis of decision under risk. Econometrica, 47 (2):263291, 1979.

R. Mariano, T. Schuermann, and M. Weeks. Simulation-based Inference in Econometrics: Methods and Applications. Cambridge University Press, 2000.

J. Nelder and R. Mead. A simplex method for function minimization. The Computer Journal, 7(4): 308313, 1965.

T. Post, van den Assem, J., G. Baltussen, and R. Thaler. Deal or no deal? decision making under risk in a large-payoff game show. American Economic Review, 98(1):38–71, 2008.

J. Pratt. Risk aversion in the small and in the large. Econometrica, 32(1-2):122–136, 1964. P. Samuelson. A note on the pure theory of consumer’s behavior. Economics, 5:61–71, 1938.

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Appendix A

Appendix

A.1 resampleD

The appendix contains all the programs necessary to optimize the data. All programs are ran in Matlab The first program performs the bootstrap.

function c=f(m,n,p)

%resampleD(m,n,p) selects m elements from 1:n. For p==1 the selection is %equivalent to random selection of m elements with replacement from the set %{1,2,...,n}.

%resampleD(m,n) with only 2 arguments is the same as resampleD(m,n,1).

%For p in the interval (0,1) (zero is not allowed) ’resampleD’ generates the input required for %the stationary bootstrap. It picks from 1:n randomly blocks of consecutive numbers

%of random size and pieces them together until a string of m numbers is obtained. The %random size of a block,i.e.the number of consecutive elements from 1:n, is

%drawn from the geometric distribution: the probability that the block size %equals j is p*(1-p)^(j-1); so j is the number of times one needs to throw a

%(biased) coin until a head comes up for the first time(p is the probability of obtaining %a head). The average size of a block equals 1/p. Remark: if the combined choice of

%random position and random blocksize would lead us outside of the range 1:n we extend the %sequence by an appropiate number of repetitions of 1:n.

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29 A.1 resampleD

%’resampleD’ can be used in bootstrap analyses. If A is a data matrix containg n

%rows, where each row collects measurements of an item or variables at a point in time,

%A(resampleD(m,n,p),:) represents a bootstrap sample of size m from the empirical distribution. %If the rows are seen as independent draws from a distribution p==1 is appropriate.

%If the rows of A represent points in time,so A collects multivariate timeseries data,

%where independence can be an unwarranted assumption, one could use the full resampleD(m,n,p).

%The choice of p is nontrivial. There is some software that can help you make a more %sophisticated choice.The choice of p does not appear to very critical though. %A major advantage should not be overlooked: one does not need to specify

%a multivariate timeseries model like a VARMA-model;the basic requirement for the %stationary bootstrap is the avalilability of a realization of a stationary %timeseries.

%Acknowledgements:

%The stationary bootstrap is due to D.N.Politis and J.P.Romano.

%See:’The stationary bootstrap’,The American Statistical Association,89,pp %1303-1313,1994.

%The book by S.M.Ross contains a wealth of algorithms for generating random %variables, among them the algorithm for the geometric distribution we use below. %See: ’A course in simulation’,Macmillan Publ.Company,1990, or a more

%recent edition.

%See also D.N.Politis and H.White on ’Automatic block-length selection for

%the dependent bootstrap’,working paper,2003. They refer to the website of Andrew Patton %where you can get the relevant Matlab code,for free. Go to:

%http//fmg.lse.ac.uk/~patton/code.html

%First version:1999-01-07,Theo Dijkstra %This version:2004-26-11,Theo Dijkstra

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Chapter A: Appendix 30 if nargin==2|p==1, c=zeros(1,m); for i=1:m, c(i)=ceil(n*rand); end else total=0; c=[]; while total<m, I=ceil(n*rand); L=floor(log(rand)/log(1-p))+1; L=min(L,m-total); H=kron(ones(1,ceil(L/n)+1),1:n); d=H(I:I+L-1); c=[c,d]; total=total+L; end end

A.2 utility

The upcoming function is the utility function

function u=utility(Y,alpha,beta)

% This program will use the Post et al utility function on data X with % parameters alpha and beta. Input Y is initial wealth+the prize to

% be won. Input can be scalar, vector, matrix or multi-dimensional arrays. if abs(alpha)<eps

u=(Y).^(1-beta)/(1-beta); else

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31 A.3 keep

A.3 keep

Keep removes all variabeles except the ones mentioned function keep(varargin);

%KEEP keeps the caller workspace variables of your choice and clear the rest. % Its usage is just like "clear" but only for variables.

%

% Xiaoning (David) Yang xyang@lanl.gov 1998

% Revision based on comments from Michael McPartland, % michael@gaitalf.mgh.harvard.edu, 1999

% Keep all

if isempty(varargin) return end

% See what are in caller workspace wh = evalin(’caller’,’who’);

% Check workspace variables if isempty(wh)

error(’ There is nothing to keep!’) end

% Construct a string containing workspace variables delimited by ":" variable = [];

for i = 1:length(wh)

variable = [variable,’:’,wh{i}]; end

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Chapter A: Appendix 32

% Extract desired variables from string flag = 0;

for i = 1:length(varargin)

I = findstr(variable,[’:’,varargin{i},’:’]); if isempty(I)

disp([’ ’,varargin{i}, ’ does not exist!’]) flag = 1;

elseif I == 1

variable = variable(1+length(varargin{i})+1:length(variable)); elseif I+length(varargin{i})+1 == length(variable)

variable = variable(1:I); else

variable = [variable(1:I),variable(I+length(varargin{i})+2:length(variable))]; end

end

% No delete if some input variables do not exist if flag == 1

disp(’ No variables are deleted!’) return

end

% Convert string back to cell and delete the rest I = findstr(variable,’:’);

if length(I) ~= 1

for i = 1:length(I)-1 if i ~= length(I)-1

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33 A.4 Optimizer else del(i) = {variable(I(i)+1:length(variable)-1)}; end end evalin(’caller’,[’clear ’,del{:}]) end

A.4 Optimizer

This program prepares the data for the standard model and initiates the maximization of the loglikelihood function mentioned in equation \ref{eqn:estpost}. % This script will optimize and load data

% load NL1 load US global B

global X DEALORNODEAL BANKOFFER % theta0=[5,60000]; % [xa,fvala,exitflag]=fminsearch(@myopica,theta0) % theta0=[.01,xa(1),xa(2)]; % options=optimset(’maxfunevals’,100000,’maxiter’,100000); % [x,fval,exitflag]=fminsearch(@myopic,theta0,options) % roundnumber=NL1a(2:length(NL1a),8);

% Now to find the expected next round bank offer for each contestant:(NOTE % THIS LOOP IS COUNTRY SPECIFIC)

x=zeros(1,length(opencases)); b=zeros(1,length(opencases)); for i=1:length(opencases)

% the amound in the open cases

amountcases=amountvector.*opencases(i,:); % Remove the removed cases

amountcases=nonzeros(amountcases)’;

% Obviously the expected value in the current round is x(i)=mean(amountcases);

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b(i)=bankoffer(i)/x(i); end

% Now average b over the individuals

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35 A.4 Optimizer

end end

% The averages for each round are:

B=[mean(b_1);mean(b_2);mean(b_3);mean(b_4);mean(b_5);mean(b_6);mean(b_7);mean(b_8);mean(b_9)]; % Now to find the bankofferfunction I need the paramter rho

% [rho,delta_square,exitflag]=fminsearch(@bankmodel,.815); %rho=.832 for NL and .777 for the US studies

% rho=.832; rho=.777;

% Now predict the b’s

b_2predic=b_1+(1-b_1)*rho^8; b_3predic=b_2+(1-b_2)*rho^7; b_4predic=b_3+(1-b_3)*rho^6; b_5predic=b_4+(1-b_4)*rho^5; b_6predic=b_5+(1-b_5)*rho^4; b_7predic=b_6+(1-b_6)*rho^3; b_8predic=b_7+(1-b_7)*rho^2; b_9predic=b_8+(1-b_8)*rho^1;

% Now try to match predictions with outcomes where possible (The matching % part takes quite some lines of code

% roundscross=NL1a(2:243,6); % Nobody plays less than 3 rounds b_2predicm=b_2predic; b_3predicm=b_3predic; X=diff(crossection); tmp=[1;X]; tmp=tmp.*roundscross; tmp=nonzeros(tmp);

% Now tmp is a N*1 vector with the number of rounds played by each of the % N contestants. This is used to make vectors of predictions for people % that have actually played with the prediction

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Chapter A: Appendix 36 b_4predicm=nonzeros(tmp2.*b_4predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>4; b_5predicm=nonzeros(tmp2.*b_5predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>5; b_6predicm=nonzeros(tmp2.*b_6predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>6; b_7predicm=nonzeros(tmp2.*b_7predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>7; b_8predicm=nonzeros(tmp2.*b_8predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>8; tmp=nonzeros(tmp2.*tmp); b_9predicm=nonzeros(tmp2.*b_9predic);

% And now it is easy to make an error for each round e_2=b_2-b_2predicm; e_3=b_3-b_3predicm; e_4=b_4-b_4predicm; e_5=b_5-b_5predicm; e_6=b_6-b_6predicm; e_7=b_7-b_7predicm; e_8=b_8-b_8predicm; e_9=b_9-b_9predicm; emean=[mean(e_2),mean(e_3),mean(e_4),mean(e_5),mean(e_6),mean(e_7),mean(e_8),mean(e_9)]; estd=[std(e_2),std(e_3),std(e_4),std(e_5),std(e_6),std(e_7),std(e_8),std(e_9)];

% Obviously the R_squared is an interesting statistic

% ESS=e_2’*e_2+e_3’*e_3+e_4’*e_4+e_5’*e_5+e_6’*e_6+e_7’*e_7+e_8’*e_8+e_9’*e_9; ESS=e_3’*e_3+e_4’*e_4+e_5’*e_5+e_6’*e_6+e_7’*e_7+e_8’*e_8+e_9’*e_9;

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37 A.4 Optimizer

TSS=b_3’*b_3+b_4’*b_4+b_5’*b_5+b_6’*b_6+b_7’*b_7+b_8’*b_8+b_9’*b_9; R_squared=1-ESS/TSS;

% It is now necassary to predict the upfollowing expected value within the % cases x_3=[]; x_4=[]; x_5=[]; x_6=[]; x_7=[]; x_8=[]; x_9=[]; x_10=[]; for i=1:length(opencases)

% The number of open cases in the current period n=sum(opencases(i,:));

% The number of open cases in the upfollowing period if n==20 n_1=15; elseif n==15 n_1=11; elseif n==11 n_1=8; elseif n==8 n_1=6; else n_1=n-1; end

% Then the probability of each case in the next period given the number % of open cases in the current period is:

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amountcases=nonzeros(amountcases)’;

% Now find all the possible combinations of the cases combinations=nchoosek(amountcases,n_1);

% The expected value of the next round cases is then:

% Some more explanations on the loop if the round ==1 for a certain individual, Matlab adds a % vector of about size 15000(the number of different combinations) of

% the mean of what is inside the cases in each combination to the already existing x_2 matrix. if roundnumber(i)==2 x_3=[x_3 mean(combinations,2)]; elseif roundnumber(i)==3 x_4=[x_4 mean(combinations,2)]; elseif roundnumber(i)==4 x_5=[x_5 mean(combinations,2)]; elseif roundnumber(i)==5 x_6=[x_6 mean(combinations,2)]; elseif roundnumber(i)==6 x_7=[x_7 mean(combinations,2)]; elseif roundnumber(i)==7 x_8=[x_8 mean(combinations,2)]; elseif roundnumber(i)==8 x_9=[x_9 mean(combinations,2)]; elseif roundnumber(i)==9 x_10=[x_10 mean(combinations,2)]; end end

% For purposes of keeping the contestants on the row dimensions it is nice % to inverse the x matrices

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39 A.4 Optimizer

x_8=x_8’; x_9=x_9’; x_10=x_10’;

% Now the expected bids in the next round, given which cases are eliminated

% are: Due to stupidity I also use x for this). Unfortenatly this loop takes a long time, but luckily it only has to run once: for i=1:size(x_3,2) x_3(:,i)=b_3predic.*x_3(:,i); end for i=1:size(x_4,2) x_4(:,i)=b_4predic.*x_4(:,i); end for i=1:size(x_5,2) x_5(:,i)=b_5predic.*x_5(:,i); end for i=1:size(x_6,2) x_6(:,i)=b_6predic.*x_6(:,i); end for i=1:size(x_7,2) x_7(:,i)=b_7predic.*x_7(:,i); end for i=1:size(x_8,2) x_8(:,i)=b_8predic.*x_8(:,i); end for i=1:size(x_9,2) x_9(:,i)=b_9predic.*x_9(:,i); end

% The next thing is to match the deal or no deal dummy to the row dimension % of the X’s. While I am at it, let’s do the same for the bankoffer

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41 A.4 Optimizer dealornodeal_8=[dealornodeal_8;dealornodeal(i)]; bankoffer_8=[bankoffer_8;bankoffer(i)]; elseif roundnumber(i)==9 dealornodeal_9=[dealornodeal_9;dealornodeal(i)]; bankoffer_9=[bankoffer_9;bankoffer(i)]; end end

% Now I will put the data in 3-dimensional arrays. The row dimension is the % number of contestants, the column dimension is the rounds and the third % dimensio are the different combinations.

% For the x matrices, first set make equal dimensions in which the missing % values are replaced by NaN.

x_4=[[x_4;NaN(size(x_3,1)-size(x_4,1),size(x_4,2))] NaN(size(x_3,1),size(x_3,2)-size(x_4,2))]; x_5=[[x_5;NaN(size(x_3,1)-size(x_5,1),size(x_5,2))] NaN(size(x_3,1),size(x_3,2)-size(x_5,2))]; x_6=[[x_6;NaN(size(x_3,1)-size(x_6,1),size(x_6,2))] NaN(size(x_3,1),size(x_3,2)-size(x_6,2))]; x_7=[[x_7;NaN(size(x_3,1)-size(x_7,1),size(x_7,2))] NaN(size(x_3,1),size(x_3,2)-size(x_7,2))]; x_8=[[x_8;NaN(size(x_3,1)-size(x_8,1),size(x_8,2))] NaN(size(x_3,1),size(x_3,2)-size(x_8,2))]; x_9=[[x_9;NaN(size(x_3,1)-size(x_9,1),size(x_9,2))] NaN(size(x_3,1),size(x_3,2)-size(x_9,2))]; x_10=[[x_10;NaN(size(x_3,1)-size(x_10,1),size(x_10,2))] NaN(size(x_3,1),size(x_3,2)-size(x_10,2))]; % Now that they are the same dimensions, they can be put in a 3-dimensional

% array.

X=cat(3,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10); X=permute(X,[1 3 2]);

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Chapter A: Appendix 42 dealornodeal_4=[dealornodeal_4;NaN(length(dealornodeal_2)-length(dealornodeal_4),1)]; dealornodeal_5=[dealornodeal_5;NaN(length(dealornodeal_2)-length(dealornodeal_5),1)]; dealornodeal_6=[dealornodeal_6;NaN(length(dealornodeal_2)-length(dealornodeal_6),1)]; dealornodeal_7=[dealornodeal_7;NaN(length(dealornodeal_2)-length(dealornodeal_7),1)]; dealornodeal_8=[dealornodeal_8;NaN(length(dealornodeal_2)-length(dealornodeal_8),1)]; dealornodeal_9=[dealornodeal_9;NaN(length(dealornodeal_2)-length(dealornodeal_9),1)]; % And combine the rows in a matrix

BANKOFFER=[bankoffer_2 bankoffer_3 bankoffer_4 bankoffer_5 bankoffer_6 bankoffer_7 bankoffer_8 bankoffer_9];

DEALORNODEAL=[dealornodeal_2 dealornodeal_3 dealornodeal_4 dealornodeal_5 dealornodeal_6 dealornodeal_7 dealornodeal_8 dealornodeal_9]; keep BANKOFFER DEALORNODEAL X

% Initial estimates US theta=[1E-6,.999,67442,.42]; % Initial estimates NL % theta=[11.16,.95,86367,.42]; options=optimset(’Display’,’Iter’,’MaxFunEvals’,1400,’MaxIter’,500); [theta,fval,exitflag,output]=fminsearch(@nloglikelihood,theta,options)

A.5 nloglikelihood

This function calculates the negative loglikelihood for given parameter values in the standard model.

function [nll]=nloglikelihood(theta) % theta=[.424,.791,75203,.428];

% This function calculates the negative loglikelihood of utility function % with parameters theta on the data of contestants playing deal or no deal. % The likelihood is how likely it is that people have made the decisions % they made given the utility function

global X DEALORNODEAL BANKOFFER alpha=theta(1);

beta=theta(2); W=theta(3); sigma=theta(4);

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43 A.5 nloglikelihood % NO could be complex if W<0. if sigma<0 nll=50003; else if W<0 nll=50001; else

% Calculating the stopping value is piece of cake. SV=utility(BANKOFFER+W,alpha,beta);

% The utility of the bid in the next round is: U=utility(X+W,alpha,beta);

% The continuation value is the expected utility of the bid in the next round: CV=nanmean(U,3);

% The delta parameter is the standarddeviation in utility for the different % decisions. For some reason the standard deviation does not have the % option to work on a chosen dimenion. As a result I take the

% standarddevtiation on the transpose. % DELTA=nanstd(U,0,3);

% If one wants to estimate without DELTA DELTA=ones(size(DEALORNODEAL));

% The if-else loop is necessary to avoid divide by zero. Important to note % that the divide by zero does not lead to optimal values of the parameters % unless NO is always positive, which will never be the case.

if sum(sum(DELTA==0))>0 nll=50000;

else

% I find the normally distributed variable temp=DEALORNODEAL==1;

temp=temp*-1;

temp=temp+(DEALORNODEAL==0); NO=CV-SV;

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% Now divide by the weighted standard deviation to standardize the variable S=NO./(sigma*DELTA);

P=normcdf(S);

% The If-else loop is necesarry to avoid the log of zero. Analytically LL % will be -Inf if log of zero.

if sum(sum(P==0))>0 nll=50002; else

% Now find the loglikelihood of all the values LL=nansum(nansum(log(P)));

% It is more practical to minimize a function so I find the negative % loglikelihood: nll=-LL; end end end end

A.6 Optimizerbootstrap

This program will set up the data and initiate optimization of equation 2.13

% This script will optimize and load data % load NL1

load US global B

global X BANKOFFER DEALORNODEAL x_3 x_4

% Now to find the expected next round bank offer for each contestant for i=1:length(opencases)

% the amound in the open cases

amountcases=amountvector.*opencases(i,:); % Remove the removed cases

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45 A.6 Optimizerbootstrap

% Obviously the expected value in the current round is x(i)=mean(amountcases);

% The value of b is: b(i)=bankoffer(i)/x(i); end

% Now average b over the individuals

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Chapter A: Appendix 46 b_8=[b_8;b(i)]; elseif roundnumber(i)==9 b_9=[b_9;b(i)]; end end

% The averages for each round are:

B=[mean(b_1);mean(b_2);mean(b_3);mean(b_4);mean(b_5);mean(b_6);mean(b_7);mean(b_8);mean(b_9)]; % Now to find the bankofferfunction I need the paramter rho

% [rho,delta_square,exitflag]=fminsearch(@bankmodel,.815); rho=.777;

% rho=.832;

% Now predict the b’s

b_2predic=b_1+(1-b_1)*rho^8; b_3predic=b_2+(1-b_2)*rho^7; b_4predic=b_3+(1-b_3)*rho^6; b_5predic=b_4+(1-b_4)*rho^5; b_6predic=b_5+(1-b_5)*rho^4; b_7predic=b_6+(1-b_6)*rho^3; b_8predic=b_7+(1-b_7)*rho^2; b_9predic=b_8+(1-b_8)*rho^1;

% Now try to match predictions with outcomes where possible (The matching % part takes quite some lines of code

% roundscross=NL1a(2:243,6); % Nobody plays less than 3 rounds b_2predicm=b_2predic; b_3predicm=b_3predic; X=diff(crossection); tmp=[1;X]; tmp=tmp.*roundscross; tmp=nonzeros(tmp);

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47 A.6 Optimizerbootstrap

% that have actually played with the prediction tmp2=tmp>3; b_4predicm=nonzeros(tmp2.*b_4predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>4; b_5predicm=nonzeros(tmp2.*b_5predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>5; b_6predicm=nonzeros(tmp2.*b_6predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>6; b_7predicm=nonzeros(tmp2.*b_7predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>7; b_8predicm=nonzeros(tmp2.*b_8predic); tmp=nonzeros(tmp2.*tmp); tmp2=tmp>8; tmp=nonzeros(tmp2.*tmp); b_9predicm=nonzeros(tmp2.*b_9predic);

% And now it is easy to make an error for each round e_2=b_2-b_2predicm; e_3=b_3-b_3predicm; e_4=b_4-b_4predicm; e_5=b_5-b_5predicm; e_6=b_6-b_6predicm; e_7=b_7-b_7predicm; e_8=b_8-b_8predicm; e_9=b_9-b_9predicm; emean=[mean(e_2),mean(e_3),mean(e_4),mean(e_5),mean(e_6),mean(e_7),mean(e_8),mean(e_9)]; estd=[std(e_2),std(e_3),std(e_4),std(e_5),std(e_6),std(e_7),std(e_8),std(e_9)];

% Obviously the R_squared is an interesting statistic

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TSS=b_2’*b_2+b_3’*b_3+b_4’*b_4+b_5’*b_5+b_6’*b_6+b_7’*b_7+b_8’*b_8+b_9’*b_9; R_squared=1-ESS/TSS;

% Bootstrap the errors to size m. m=100;

% Data is in the following form: e_3 gives m different possibilities of % the set of errors for all contestants getting their second bid, etc. % Obviously for people still active in round 9, the value of b in the next % round is known and non-random.

e_3bootstrap=e_3(resampleD(length(b_2)*m,length(e_3))); e_4bootstrap=e_4(resampleD(length(b_3)*m,length(e_4))); e_5bootstrap=e_5(resampleD(length(b_4)*m,length(e_5))); e_6bootstrap=e_6(resampleD(length(b_5)*m,length(e_6))); e_7bootstrap=e_7(resampleD(length(b_6)*m,length(e_7))); e_8bootstrap=e_8(resampleD(length(b_7)*m,length(e_8))); e_9bootstrap=e_9(resampleD(length(b_8)*m,length(e_9)));

% With the errors one can create a sample of different possible values of % the errors. b_3bootstrap=repmat(b_3predic,m,1)+e_3bootstrap; b_4bootstrap=repmat(b_4predic,m,1)+e_4bootstrap; b_5bootstrap=repmat(b_5predic,m,1)+e_5bootstrap; b_6bootstrap=repmat(b_6predic,m,1)+e_6bootstrap; b_7bootstrap=repmat(b_7predic,m,1)+e_7bootstrap; b_8bootstrap=repmat(b_8predic,m,1)+e_8bootstrap; b_9bootstrap=repmat(b_9predic,m,1)+e_9bootstrap;

Deal or no Deal: Estimating utility functions with the use of simulation techniques