Heuristics as a Proxy for Contestant Risk Aversion in Deal or No Deal

Transcription

1 Heuristics as a Proxy for Contestant Risk Aversion in Deal or No Deal Joy Hua Northwestern University Mathematical Methods in the Social Sciences Senior Thesis 2014 Advisor: Marciano Siniscalchi

2 Abstract The game show Deal or No Deal has risen in popularity as a natural experiment for measuring risk aversion. This paper explores various heuristic measures that contestants may reference throughout the show as a proxy for risk aversion and whether the data shows evidence of the use of such heuristics. Since the model of the entire game is quite complex and requires extensive calculations to solve for the optimal strategy, it is likely that the average contestant relies on more simplistic measures when making the decision. We define three heuristic measures and perform multiple probit regressions to analyze the effect of each measure on the probability of accepting an offer. The results of the regressions are consistent with our assumptions about how the riskiness of the decision varies with each measure and how that affects contestant decisions. We also look at each contestant individually to see which heuristics they adhere to when they take the Deal. We find that the three heuristics are good proxies for risk aversion for roughly two-thirds of the contestants. However, we also find that the measures give contradicting information on risk aversion for the other third of contestants. 2

3 Table of Contents Acknowledgements Introduction Deal or No Deal: Description of the Game Literature Review A Simplified Theoretical Model Description The Decision Tree Solving for the Optimal Strategy Heuristic Measures Safety Net Comparing the Bank Offer to Remaining Prize Values Best vs. Worst Case Scenario Data and Methodology Results Testing the Safety Net Testing p_higher Testing meanspread Conclusion Appendix Table 1: Description of Variables Table 2: Heuristic Measures by ID number for Deal= Table 3: Definition of Risk Tolerance Groups Works Cited

4 Acknowledgements I would first like to thank my advisor, Professor Marciano Siniscalchi for his support throughout the entire process, from figuring out what specific topic to pursue to suggesting ways to enhance the paper as my work progressed. His advice has been tremendously helpful as the direction of my analysis and corresponding results have evolved throughout the year. I would also like to thank Professor Jeffrey Ely who gave me the initial idea to work with Deal or No Deal. Lastly I would like to thank Professor Joseph P. Ferrie for his guidance throughout each quarter, and Professor William Rogerson and Sarah M. Ferrer for all they do for the MMSS program that has played such a significant part in shaping my time at Northwestern. 4

5 1. Introduction As interest increases in the study of risk attitudes, Deal or No Deal has become a popular natural experiment for studies to use in measuring risk aversion because contestants face highstakes lotteries, in a setting untainted by strategic considerations or selection issues relating to skill (De Roos and Sarafidis 2010). Numerous papers have attempted to calculate measures of risk aversion using data from versions of the television show from various countries. The process of solving the optimal strategy for contestants to follow in the game is extremely complicated. We will demonstrate this fact by solving a simplified version of the model that will already require extensive calculations. Because of this, it is highly unlikely that the average contestant on the show is sophisticated enough to optimally solve their decision problem. Rather, it is more realistic that contestants use more descriptive rules of thumb to help make their decision in each round. In this paper, we will explore various heuristic measures that contestants may reference throughout the show as a proxy for risk aversion and whether the data shows evidence of the use of such heuristics. In the remaining sections of this paper, I will first describe how the U.S. version of the show is set up and played. I will then provide a literature review of existing studies done using Deal or No Deal data from various countries. Next, I will show the decision tree and solution process for a simplified version of the game with only four cases in play. I will then describe the three heuristic measures we analyze and discuss the data used in this paper. We will see if each heuristic measure plays a significant role in predicting contestants decisions as a group as well as explore how closely contestants adhere to each one on an individual level. Finally I will discuss the results and corresponding conclusions we can draw about whether or not these heuristics are a good proxy for risk aversion. 5

6 2. Deal or No Deal: Description of the Game The U.S. version of the game consists of 26 briefcases, each containing a monetary value ranging from $0.01 to $1,000,000. Before each game, the values are randomly distributed across the 26 cases by a third party. Neither the contestant nor the banker knows which values are in which cases. At the beginning of the game, the contestant chooses one of the 26 numbered cases to be his or her case. In order to win the monetary value contained in that case, he or she must play through the entire game, rejecting all offers made by the banker. In each round of the game, the contestant opens a pre-determined number of cases out of the remaining cases in play. The number of cases that the contestant must open in each round before receiving another offer is as follows: 6, 5, 4, 3, 2, 1, 1, 1, 1. Whatever monetary value is revealed in those cases is not in the case that the contestant originally chose, so the probability distribution of the value of the contestant s case adjusts with each opened case. At the end of each round, the banker presents the contestant with a monetary offer. He or she must then choose Deal or No Deal. If the contestant chooses Deal, the game is over, and he or she wins the monetary offer. If he or she chooses No Deal, the game continues to the next round. A maximum of nine rounds can be played in the U.S. version of the game. If a contestant makes it to round 9 and chooses No Deal, he or she has the option to swap their original case for the other case left in play. Once that decision is made, the value in the contestant s final chosen case is revealed, and he or she walks away with that amount. After a contestant takes a Deal, the host will ask him or her to continue opening cases and play out the remainder of the game. The banker also continues to give a hypothetical offer corresponding to each remaining round of the game as each additional value is revealed. This gives the contestant and the audience a chance to see what could have happened and whether or not the contestant made a good deal. 6

7 3. Literature Review In recent years, Deal or No Deal has become a very popular natural experiment for measuring risk aversion. At the most basic level, comparing the bank offer to the expected value of the remaining prizes and noting whether the contestant chooses Deal or No Deal will give us a measure of the contestant s risk attitude. In order to model each round of the game as purely a decision problem with no information exchanged, we must be sure that the banker does not strategically choose the value of the offer. Ritcey and Ranjan (2010) explore and test various ways to model the banker s offer. They define a fair offer as one equal to the average of the dollar values in the remaining briefcases in the game. From the data, they see that although never fair, the generosity of the banker in terms of making an offer to the contestant increases as the game progresses. Many of the other existing papers agree and assume that the offer is predictable across all games. The offer is a fraction of the expected value of remaining prizes, with the fraction starting low in the early stages and increasing as the game continues (Blavatskyy and Pogrebna 2008). Blavatskyy and Pogrebna also find that bank offers do not depend on the probability of receiving a large prize as both groups within their study one group with a 20% chance of having a large value case and one group with an 80% chance received qualitatively similar monetary offers. With this assumption in place, the contestant s decision of Deal or No Deal simply depends on the bank offer, the remaining values in play, and his or her risk attitude. Many papers have used data from the television show to calculate the degree of risk aversion exhibited by contestants. These studies estimate coefficients of constant relative risk aversion (CRRA) and constant absolute risk aversion (CARA). The utility function used for CRRA is ( ) ( ), where W is initial wealth, and the utility function used for CARA is 7

8 ( ) ( ) (Deck et al. 2008). Both De Roos and Sarafidis (2010) and Deck et al. (2008) estimate these coefficients using the expected utility framework. Both studies also complete the analysis under different assumptions of what the contestants take into consideration when making their decisions. First they consider how far forward into the game the contestants are looking when making their decision. A contestant is dynamic if he or she considers the effect of the current decision through all rounds of the game until the end and myopic if he or she only looks one round ahead (Deck et al. 2008). De Roos and Sarafidis (2010) find that absolute risk aversion estimates are greater by a factor of 10 in the dynamic model using CARA preferences and a factor of 5 using CRRA preferences. We see the importance of this assumption as a myopic view makes contestants seem less risk averse. An additional distinction that Deck et al. (2008) make in their analysis is whether a contestant is sophisticated or naïve. A naïve contestant expects future offers to be the expected value of the active briefcases, whereas a sophisticated contestant believes that future offers are based on both the expected value and variance of active briefcases. Many of the studies find considerable heterogeneity in risk aversion across contestants. This finding is in line with our expectations that contestants are diverse and have different risk preferences. However, De Roos and Sarafidis (2010) also find that there is a similar level of heterogeneity in the decisions made by the same contestant over time. This suggests the importance of having a stochastic element in the model that can take into account changing levels of risk aversion across different rounds for a single contestant. Beyond just calculating measures of risk aversion, other studies have explored how risk aversion changes under different circumstances. Brooks et al. (2009) find that the degree of risk 8

9 aversion generally increases with stakes. However, they also note that even at very high stakes, there is still considerable heterogeneity, which is consistent with the heterogeneity findings of De Roos and Sarafidis (2010) and Deck et al. (2008). Blavatskyy and Pogrebna (2008) analyze contestants risk attitudes when they face a high or low chance of earning a large prize. They do so by looking only at the round with five cases left in play and grouping contestants into two groups: one with 20% chance of having a large prize in their case and another with 80% chance of having a large prize in their case. They then compare rejection rates of the bank offer across the two groups to see if risk attitudes are affected by the likelihood of large gains. Blavatskyy and Pogrebna find that about the same proportion of contestants rejected the offer in both groups, suggesting that risk attitudes are identical whether facing a high or low probability of a large gain. Many studies have also looked for evidence of framing effects in Deal or No Deal contestant behavior. Prospect theory and the rank-dependent utility model both challenge the expected utility hypothesis used to calculate risk aversion coefficients in previous papers. In the rank-dependent utility model, the weight placed on an outcome depends on its preference ranking in addition to the objective probability (De Roos and Sarafidis 2010). Their analysis suggests evidence of optimism, where contestants are overweighing favorable outcomes. The RDU model also provides better explanatory power over the expected utility model by the same factor that EU outperforms risk neutrality (De Roos and Sarafidis 2010). Post et al. (2008) find that contestants decisions can be explained by outcomes experienced earlier in the game, suggesting the importance of path and reference-dependent considerations. They categorize contestants into three groups winners, losers, and neutral and found that a relatively lower percentage of winners and losers took the deal than those in the neutral group, indicating that they generally become less risk averse after initial expectations have been shattered or surpassed. The unlucky contestants that open all the large cases early in 9

10 the game may experience a break-even effect, in which they are willing to gamble in order to get back to some perceived reference point (Post et al. 2008). In contrast, a contestant that is extremely lucky and eliminates all low-value cases early on in the game may experience a house-money effect, defined as an increased willingness to game when someone thinks she is playing with someone else s money (Post et al. 2008). Thus, outcomes from previous rounds will play a large role in framing the current decision problem a contestant faces in any given round. Another distinction that can be made when framing the contestant s decision problem is whether or not he or she is buying or selling the lottery. The endowment effect suggests that people place higher value on things they own, so for a given object, they would have a higher selling price than buying price. Brooks et al. (2009) and De Roos and Sarafidis (2010) both take advantage of the Chance and Supercase rounds in the Australian version of the show to test for an endowment effect for lotteries. The Chance and Supercase options can be introduced in an extra round at the end of the game at the producer s discretion. They typically arise after contestants have continued to hypothetically open cases after taking a deal and arrive at the stage where only 2 cases are left in play. For the Chance option, contestants are given the option to trade in their accepted offer for a 50/50 gamble between the two remaining cases. Naturally to make the extra round interesting, the Chance round is only offered when there is a large difference between these two case values (Brooks et al. 2009). For the Supercase option, contestants have the option to trade in their accepted offer for whatever is in the Supercase. The value inside can be one of 8 values: $0.50, $100, $1000, $2000, $5000, $10,000, $20,000, or $30,000. Because contestants must forgo, or sell, their already accepted offer in order to play the Chance or Supercase round, prospect theory suggests that the accepted offer becomes the 10

11 new reference point, and contestants now face the possibility of losing money (Brooks et al. 2009). Out of 20 offered Chance rounds and 24 offered Supercase rounds, only seven and eight contestants, respectively, chose to play the extra round. De Roos and Sarafidis (2010) find that although contestants seem to be more risk averse in these special rounds, the effect is only significant when assuming a static choice model. Thus they conclude that the average contestant does not display an endowment effect. On the other hand, Brooks et al. (2009) add dummy variables for Chance and Supercase rounds into their probit regression and find that both variables are statistically significant at the 1% level. They believe that the high level of risk aversion during the Chance and Supercase rounds is consistent with the existence of framing effects. While the endowment effect may not be seen in the U.S. version of the game, we may still want to consider the possible existence of other types of framing effects. 11

12 4. A Simplified Theoretical Model 4.1 Description To demonstrate that solving for the optimal strategy for the full game with 26 cases would be very complicated, we will show the methodology for solving a simplified version of the game. In this version, there are only four cases, with values v 1, v 2, v 3, v 4, and two rounds of play, in which the contestant opens one case in each round. We assume a bank offer function, B i (v j,,v k, X), for when the case containing v i is opened, where v j,,v k are the case values still in play and X represents all other variables that affect the offer value. We also assume that contestants are expected utility maximizers with a CRRA utility function, ( ), where x is their payoff, or amount won, and θ is their degree of relative risk aversion. To solve for the contestant s optimal strategy, we can build the decision tree, calculate the expected utility of each decision, and then use backwards induction to determine the optimal strategy. 4.2 The Decision Tree If the contestant opens the case containing v 4 in the first round, the decision tree would look as follows: 12

13 where the navy boxes represent decisions, blue boxes represent chance outcomes, and teal boxes represent payoffs. Technically in the final round of the real game after contestants have rejected the final offer, they have the choice to exchange the case they originally chose for the other remaining case. However, this does not change the probability distribution of the final two prize values, so it is negligible in the decision tree and solution. In order to model the entire decision problem, we need to include one more chance node at the beginning of the tree, with three more branches representing the cases in which v 1, v 2, and v 3 are opened in the first round. Each of the branches would look exactly like the decision tree above except with v 4 and B 4 switched with v i and B i in the branch where v i is opened first. 4.3 Solving for the Optimal Strategy The solution for the realization where v 4 is opened in the first round is the choice that results in { } where: ( ) ( ) ( ) ( ) ( ) { ( ) ( ) ( )} ( ) { ( ) ( ) ( )} ( ) { ( ) ( ) ( )} The optimal strategy for the entire game would be: If v 1 is opened first, the choice that results in { } If v 2 is opened first, the choice that results in { } If v 3 is opened first, the choice that results in { } If v 4 is opened first, the choice that results in { } where EU Di and EU NDi are calculated as above with v i and B i switched with v 4 and B 4 for i =1,2,3. 13

14 As we can see the solution to the simplified model with only four cases is already very complicated and requires extensive calculations. For each additional case value or round of play, we would have to add another set of nodes and branches for each potential realization and solve the corresponding expected utility maximization problem. With 26 cases, the solution to the entire game would be far more complicated. For this reason, we would expect contestants to make their decisions in another way using more simplistic measures. 14

15 5. Heuristic Measures 5.1 Safety Net Having a safety net is defined as having more large cases left in play than the number of cases to open in the next round. The size of the safety net is defined as the difference between the number of large cases remaining in play and the number of cases the contestant must open in the next round, should he or she choose No Deal in the current round. In the U.S. version, a large case value is defined as all values greater than or equal to $100,000. The term safety net is often used by the host of the show, so it very likely could be a measure that contestants consider when making their decision. Not having a safety net makes No Deal a riskier choice. We would expect more risk averse contestants to take the Deal when they still have a safety net. 5.2 Comparing the Bank Offer to Remaining Prize Values While the theoretical solution and expected utility maximizing assumption inherently take into account the offer in relation to the remaining prize values, contestants are most likely not sophisticated enough to take into account all these intricacies. What contestants can easily see, however, is where the bank offer lies in comparison to the remaining values on the board. Especially in later rounds where there are fewer number of cases left, contestants can determine the fraction of values that are higher or lower than the current bank offer, and thus the probability of winning a higher or lower prize value at the end. The producers will also often show % that contestant s case carries more (less) than $ at the bottom of the television screen. The lower the probability of having a higher case value than the current offer, the riskier a No Deal choice would be. 5.3 Best vs. Worst Case Scenario It is also likely that towards the later rounds of the game, contestants begin to set their sights on the highest possible offer they can receive instead of the actual prize value in their 15

16 chosen case, assuming a myopic view of the game. Opening a low value case in the next round would result in a higher offer whereas opening a high value case could cause the offer to plummet. Thus, contestants may consider the difference in their best and worst case scenario when making their decision. One way to represent this would be to calculate the expected value of remaining prizes for the scenarios in which the lowest and highest remaining cases are opened and take the difference. The larger this difference is, the riskier a No Deal choice would be. 16

17 6. Data and Methodology I obtained the data set that Post et al. (2008) used in their paper, which is available online. This data set consists of observations from versions of Deal or No Deal from the Netherlands, the United States, and Germany. The U.S. data set, which I focus on in this paper, consists of observations for 47 contestants that played a combined total of 355 rounds of the game. The data consists of several descriptive variables, such as gender, age, and education. It also contains information pertinent to the game, such as the remaining case values in each round, the bank offer in each round, the amount won, and whether the contestant chose Deal or No Deal. One of the weaknesses of this data is that due to the nature of the game, we only have one observation per contestant where Deal=1. Thus by design, there are five to eight times the number of observations for Deal=0 than Deal=1. By including all 355 observations in our analysis, our results may be biased due to the greater proportion of Deal=0. Thus, it may be beneficial to restrict our analysis to the later rounds where we actually see contestants choosing Deal, which would be round 5 for the U.S. data set. Table I: Summary of Deal=1 Observations by Round Round Number of contestants Percentage of contestants Play to end To test whether our three heuristic measures play a part in contestants decisions as a whole, we will run various probit regressions with Deal being the binary dependent variable. We reference the probit regressions from Brooks et al. (2009) for other independent variables to 17

18 include in our regression. Their regressions contained a variable for household income, which is information not found in our data set. Instead, we have a binary variable representing high or low education. Since income is typically related to education, we decide to include our HighEducation dummy variable in place of their HIncome variable. Our baseline probit regression results are seen below in Table II. Table II: Baseline Probit Regression for ROUND 5 Deal Coef. Std. Err. z P> z [95% Conf. Interval] BankOffer 3.26E E E E-06 Number of obs 167 Ratio LR chi2(6) Male Prob > chi Age Pseudo R agesq Log likelihood HighEducation _cons While we see statistical significance for the same variables, the magnitudes of our coefficients are much smaller. Since we use data from a different version of the show, we would expect to see slightly different regression results. In the following section, we will add each of the three heuristic measures to this baseline regression to test for significance. 18

19 7. Results 7.1 Testing the Safety Net Before we run the probit regression with the safety net, we take a look at the safety net dummy variable and the percentage of Deals taken when contestants have and do not have a safety net. We can run a chi-squared test to see if the proportion of people who take the Deal when they have a safety net is significantly different than when there is no safety net. Table III: Deal or No Deal Breakdown by Existence of Safety Net for All 9 Rounds Number (percentage) of No Deal Number (percentage) of Deal No Safety Net 191 (88.43%) 25 (11.57%) Yes Safety Net 123 (88.49%) 16 (11.51%) Pearson chi2(1) = Pr = We see that it is about 11.5% in both cases, and the p-value of the chi-square test is very high. This brings us back to the point made previously where due to the nature of the game and the data, each contestant can only take the Deal once. Hence, only one out of roughly every nine data points, or about 11% of the observations, can have Deal=1. We also noted before that no one takes a Deal until at least round 5. Taking both of these into account, we now look at the percentage of Deals taken in each case, restricting it to only data points from round 5 or later. Table IV: Deal or No Deal Breakdown by Existence of Safety Net for ROUND 5 Number (Percentage) of No Deal Number (Percentage) of Deal No Safety Net 72 (74.23%) 25 (25.77%) Yes Safety Net 54 (77.14%) 16 (22.86%) Pearson chi2(1) = Pr =

20 By restricting our analysis to round 5 and later, we begin to see a difference in decisions between when there is or is not a safety net. The p-value of the chi-squared test improves by around 30%, however the result is still not statistically significant. Now we return to our baseline probit regression and add the size of the safety net as another independent variable. We complete the regression both for the entire data set and restricted to rounds 5 and later. Again, we see that the p-value for the safety net variable largely improves when we restrict the analysis to later rounds. The negative coefficient suggests that having a larger safety net will decrease the probability of taking a Deal, which is what we would expect as a general result. Table V: Probit Regression with Safety Net for All Rounds Deal Coef. Std. Err. z P> z [95% Conf. Interval] BankOffer 5.50E E E E-06 Number of obs 355 Ratio LR chi2(6) Male Prob > chi Age Pseudo R agesq Log likelihood HighEducation net _cons Table VI: Probit Regression with Safety Net for ROUND 5 Deal Coef. Std. Err. z P> z [95% Conf. Interval] BankOffer 6.08E E E Number of obs 167 Ratio LR chi2(6) Male Prob > chi Age Pseudo R agesq Log likelihood HighEducation net _cons

21 7.2 Testing p_higher Next we run the probit regression again with p_higher included in place of net. This time we only include observations for round 5 and later. We get a p-value of 0.08, which gives us statistical significance at the 10% level, but not at the 5% or 1% level which would be ideal. We once again see a negative coefficient, so a contestant with more prize values higher than the current offer left in play will be less likely to take the Deal. This is also consistent with our assumptions about this heuristic. Table VII: Probit Regression with p_higher for ROUND 5 Deal Coef. Std. Err. z P> z [95% Conf. Interval] BankOffer 3.93E E E E-06 Number of obs 167 Ratio LR chi2(6) Male Prob > chi Age Pseudo R agesq Log likelihood HighEducation p_higher _cons

22 7.3 Testing meanspread We run the probit regression one final time with the variable meanspread added to the baseline model. Once again we only include observations from round 5 and later. We get a p- value of 0, so the variable is statistically significant, although the magnitude of the coefficient is pretty small. However, the positive coefficient means that the greater the difference between contestants best and worst case scenario, the more likely they are to take the Deal, which is also consistent with our expectations. Table VIII: Probit Regression with meanspread for ROUND 5 Deal Coef. Std. Err. z P> z [95% Conf. Interval] BankOffer E E-06 Number of obs 167 Ratio LR chi2(6) Male Prob > chi Age Pseudo R agesq Log likelihood HighEducation meanspread E E _cons Given our assumptions about how the riskiness of the decision varies with each heuristic, the data shows that each of the measures affects contestant behavior at the overall group level in the direction that is consistent with our expectations of risk preferences. 22

23 8. Conclusion This paper explores the use of heuristics when contestants on the game show Deal or No Deal make their decisions and whether or not it can serve as a good proxy for risk aversion. By using data from the U.S. version of the show, we can see how well contestants decisions are predicted by these measures as well as which contestants individually adhere to each one. Descriptive measures have the potential to be good proxies for risk aversion because they tend to be easier concepts for contestants to grasp during the game. It is unlikely that the average contestant is sophisticated enough to understand the process and actually determine their optimal strategy based on backwards induction of the remaining cases in the moment. In general, we found that contestants were less likely to take the Deal when they have a safety net or when the probability of their case containing a higher value than the offer is greater. They were also more likely to take the Deal when there is a bigger difference between their best and worst case scenarios. These results are consistent with our assumptions about how the riskiness of the decision varies with each of the three heuristic measures, and how that affects the decision of contestants with some level of risk aversion. However, to get a better look at risk aversion at the individual level, we may want to look at which individual contestants decisions are consistent with each heuristic measure. In order to observe and compare the actions of each contestant, we create a table listing each contestant by ID number and the values of the heuristic variables for the observation in which each contestant takes the Deal (see Table 2 in the appendix). The table also allows us to see whether the three heuristics consistently measure risk aversion for a certain contestant. We exclude the individuals that never take a Deal and play until the very end because they always made the riskier choice of No Deal, so they do not exhibit risk aversion. We use the safety net dummy variable in the table because it is easier to group contestants. We consider contestants 23

24 who take the Deal when they still have a safety net to be more risk averse. We include p_higher and assume that a higher value when they take the Deal indicates a more risk averse individual. To make it easier to group contestants, we will distinguish between those with p_higher < 0.5 and p_higher 0.5. It is harder to compare meanspread across contestants because it largely depends on the magnitude of the case values left in play, which varies quite a bit among the contestants. Thus, we would like to create a new variable that is easier to compare. We define a new dummy variable rule3 where { and highestspread is the maximum meanspread value for each contestant. If rule3=1, then the contestant was facing the riskiest choice so far when he or she took the Deal. If rule3=0, then the contestant made a riskier No Deal choice previously. Therefore, we would consider contestants with rule3=1 to be more risk averse. Next we categorize the contestants based on which group they fall in for each of the three heuristic measures. At first glance, we see that there are roughly three types of contestants: those in the more risk averse group for all three, those in the less risk averse group for all three, and those that fall in different groups across the three measures. For the contestants with dsafety_net=1 and rule3=1, we see that there is a fairly equal split between those with p_higher < 0.5 and p_higher 0.5. Since p_higher depends more on the number of cases left in play than the other two measures and is the only non-binary variable, a contestant with dsafety_net=1, rule3=1, and p_higher < 0.5 is likely still relatively more risk averse. Thus, we will create a separate group for these individuals. The definition of each of the four risk tolerance groups is found in Table 3 of the appendix. The composition of these four groups is shown in Table IX. 24

25 Table IX: Breakdown of Risk Tolerance Groups Risk Tolerance Group: Number % We see that there is considerable heterogeneity in risk aversion across all the contestants. This is consistent with much of the existing literature, which finds high levels of heterogeneity in risk aversion among contestants. For roughly two-thirds of the contestants, the three heuristics are good proxies for risk aversion, consistently categorizing them by relative risk aversion levels. However for the other third of the contestants, those in group 2, the heuristic proxies give contradicting information on risk aversion. In order to extend our analysis, we would want to test the accuracy of our regressions on predicting contestant choices in another data set. We were unable to test it with our given data set because there were too few US observations to split into two groups. We could potentially test the accuracy of the regressions on a data set for another country. However, there may be crosscountry differences in risk attitudes based on contestant characteristics such as age and gender. The versions of the game show in other countries also contain different prize values, so the difference in stakes may also affect the accuracy of our probit model. Ideally, we would want to return to our theoretical model, explicitly define the bank offer function, assign numerical values to each of the variables, and observe each of the heuristic measures for the choice given by the optimal strategy. However, with only four cases and two rounds of play, both p_higher and meanspread have fairly trivial values. p_higher will always be 0.5 in round 2 because there are only two cases left, and it would only make sense for the offer to lie between the two remaining values. meanspread is only interesting in round 1 because it just 25

26 becomes the difference between the two remaining case values in round 2. Given only two rounds, rule3 is determined more by a contestant s luck in round 2 rather than risk preferences. Additionally, the concept of the safety net largely depends on the magnitude of the values assigned to v 1 through v 4. Even if we define v 3 and v 4 as large cases the safety net would only be non-zero for the realizations where v 1 or v 2 is opened first. Thus for our simplified model, these heuristic measures are not very informative. As the game becomes more complex with each additional case or round, these measures become more interesting, which supports their relevance in analyzing contestant behavior for the full 26 case, 9 round game. Given that the heuristic measures categorize about a third of the contestants into inconsistent risk aversion groups, these heuristics are most likely too simple to fully capture contestants decision making. Heterogeneity in the way contestants think and what heuristic measures they might notice or come up with while they are playing the game may also explain why no single heuristic can fully represent contestant behavior. However, given the complicated process through which we solve for the optimal strategy, it is likely that contestants are basing their decisions on more simplified notions, such as the three heuristic measures we analyzed in this paper. Thus it is valuable to study Deal or No Deal and the decisions the contestants make through a more qualitative approach. 26

27 Appendix Table 1: Description of Variables Variable Deal Description Dummy variable for whether or not a contestant takes the Deal { BankOffer Bank offer given in the current round Ratio Ratio of the bank offer to the expected value of remaining cases Male Dummy variable for whether or not a contestant is male HighEducation Dummy variable for whether the contestant has a high or low level of education Age Age of contestant agesq = (Age) 2 Large Case Any case in the US version that is greater than or equal to $100,000 (there are 7 total to begin) nlarge_cases The number of large cases left in play cases_to_open The number of cases a contestant must open in the next round, given they choose No Deal net The size of the safety net = nlarge_cases cases_to_open dsafety_net Dummy variable for the existence of a safety net p_higher The probability that the contestant's case contains a higher value than the current offer casex Values in the N remaining cases in ascending order, where case1 is lowest value and casen is highest value meanx Expected value in the next round if casex is opened, where mean1 represents best case scenario (lowest case is opened) and meann represents worst case scenario (highest case is opened) meanspread The difference between a contestant's best and worst case scenarios = mean1 meann 27

28 Table 2: Heuristic Measures by ID number for Deal=1 ID dsafety_net p_higher rule3 RT group *Contestants who never take a Deal are omitted from the table (IDs: 7, 9, 10, 14, 17, 44) 28

29 Table 3: Definition of Risk Tolerance Groups Risk Tolerance Group 1 Conditions dsafety_net = 0 rule3 = 0 2 dsafety_net rule3 3 4 dsafety_net = 1 rule3 = 1 p_higher < 0.5 dsafety_net = 1 rule3 = 1 p_higher

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