Risk Aversion in Game Shows
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Risk Aversion in Game Shows by Steffen Andersen, Glenn W. Harrison, Morten Igel Lau and Elisabet E. Rutström † Forthcoming, J.C. Cox and G.W. Harrison (eds.), Risk Aversion in Experiments (Greenwich, CT: JAI Press, Research in Experimental Economics, Volume 12, 2007) Abstract. We review the use of behavior from television game shows to infer risk attitudes. In many cases these shows provide evidence when contestants are making decisions over very large stakes, and in a replicated, structured way. Inferences are generally confounded by the subjective assessment of skill in some games, as well as the dynamic nature of the task in most games. We consider the game shows Card Sharks, Jeopardy!, Lingo, and finally Deal Or No Deal. We provide a detailed case study of the analyses of Deal Or No Deal, since it is one of the cleanest games for inference and has attracted considerable attention. We describe the manner in which the analyses have been undertaken, and propose a general method to overcome the curse of dimensionality that one encounters when estimating risk attitudes in the context of a dynamic, stochastic programming environment. † Centre for Economic and Business Research, Copenhagen Business School, Copenhagen, Denmark (Andersen); Department of Economics, College of Business Administration, University of Central Florida, USA (Harrison and Rutström) and Department of Economics and Finance, Durham Business School, Durham University, United Kingdom (Lau). E-mail: [email protected], [email protected], [email protected], and [email protected]. Harrison and Rutström thank the U.S. National Science Foundation for research support under grants NSF/IIS 9817518, NSF/HSD 0527675 and NSF/SES 0616746. We are grateful toPavlo Blavatskyy, Daniel Mulino, Ganna Pogrebna, Thierry Post and Martijn van den Assem for discussions and comments. Table of Contents 1. Previous Literature ............................................................. -2- A. Card Sharks............................................................ -2- Estimates of Risk Attitudes............................................ -3- EUT Anomalies..................................................... -6- B. Jeopardy!............................................................... -8- C. Lingo ................................................................ -10- 2. Deal Or No Deal.............................................................. -14- 3. Previous Deal Or No Deal Analyses ............................................... -18- A. Bombardini and Trebbi [2005]............................................ -19- B. Mulino, Scheelings, Brooks and Faff [2006].................................. -21- C. Post, van den Assem, Baltussen and Thaler [2006a][2006b] . ................... -23- Interval Responses . ........................ -23- Prior Outcomes.................................................... -25- Stake Effects . ........................ -27- Main Inferences.................................................... -28- Prospect Theory . ........................ -29- Maximum Likelihood Estimation . ......................... -30- D. De Roos and Sarafidis [2006]............................................. -35- E. Other Studies.......................................................... -37- 4. A General Estimation Strategy................................................... -38- A. Basic Intuition......................................................... -38- B. Formal Specification .................................................... -41- C. Estimates ............................................................. -45- 5. Conclusions . .............................................................. -47- References..................................................................... -57- Observed behavior on television game shows constitutes a controlled natural experiment that has been used to estimate risk attitudes. Contestants are presented with well-defined choices where the stakes are real and sizeable, and the tasks are repeated in the same manner from contestant to contestant. We review behavior in these games, with an eye to inferring risk attitudes. We describe the types of assumptions needed to evaluate behavior, and propose a general method for estimating the parameters of structural models of choice behavior for these games. We illustrate with a detailed case study of behavior in the U.S. version of Deal Or No Deal. In section 1 we review the existing literature in this area that is focused on risk attitudes, starting with Gertner [1993] and the Card Sharks program. We then review the analysis of behavior on Jeopardy! by Metrick [1995] and on Lingo by Beetsma and Schotman [2001].1 In section 2 we turn to a detailed case study of the Deal Or No Deal program that has generated an explosion of analyses trying to estimate large-stakes risk aversion. We first explain the basic rules of the game, which is shown with some variations in many countries. In section 3 we review in some detail previous analyses of behavior in Deal Or No Deal versions shown in Australia, Italy, the Netherlands, Belgium, Germany and the United Kingdom. Section 4 proposes a general method for estimating choice models in the stochastic dynamic programming environment that most of these game shows employ. We resolve the “curse of dimensionality” in this setting by using randomization methods. We illustrate the application of the method using U.S. behavior, and estimating a simple structural model of expected utility theory choice behavior. The manner in which our inferences can be extended to other models is also discussed. Finally, in section 5 we identify several weaknesses of game show data, and how they might be addressed. We stress the complementary use of natural experiments, such as game shows, and laboratory experiments. 1 Behavior on Who Wants To Be A Millionaire has been carefully evaluated by Hartley, Lanot and Walker [2005], but this game involves a large number of options and alternatives that necessitate some strong assumptions before one can pin down risk attitudes rigorously. We focus on games in which risk attitudes are relatively easier to identify. -1- 1. Previous Literature A. Card Sharks The game show Card Sharks provided an opportunity for Gertner [1993] to examine dynamic choice under uncertainty involving substantial gains and losses. Two key features of the show allowed him to examine the hypothesis of asset integration: each contestant’s stake accumulates from round to round within a game, and the fact that some contestants come back for repeat plays after winning substantial amounts. The game involves each contestant deciding in a given round whether to bet that the next card drawn from a deck will be higher or lower than some “face card” on display. Figure 1 provides a rough idea of the layout of the “Money Cards” board before any face cards are shown. Figure 2 provides a clearer representation of the board from a computerized laboratory implementation of Card Sharks in Andersen et al. [2006c]. In Figure 2 the contestant-subject has a face card with a 3, and is about to enter their first bet. Cards were drawn without replacement from a standard 52-card deck, with no Jokers and with Aces high. The contestant decides on the direction of the next card, and then on an amount to bet that their choice is correct. If they are correct their stake increments by the amount bet, but if they are incorrect their stake is reduced by the amount bet.2 Every contestant starts off with an initial stake of $200, and bets could be in increments of $50 of the available stake. After three rounds in the first, bottom “row” of cards, the contestant moved to the second, middle “row” and received an additional $200 (or $400 in some versions). If the contestant’s stake had gone to zero in the first row, they went straight to the second row and received the new stake; otherwise, the additional stake was theirs to bet with. The second row included three choices, just as in the first row. After these three choices, and if the subject’s stake had not dropped to zero, they could play the final bet. In this case they had to bet at least one-half of their stake, but otherwise the bets were the same. One feature of the game is that contestants sometimes had the option to switch face cards in the hope of 2 If the new card is the same as the face card, there is no change in the stake. There are minor variations in the versions of the game show. -2- getting one that was easier to win against.3 The show aired in the United States in two major versions. The first, between April 1978 and October 1981, was on NBC and had Jim Perry as the host. The second, between January 1986 and March 1989, was on CBS and had Bob Eubanks as the host.4 The maximum prize was $28,800 on the NBC version and $32,000 on the CBS version, and would be won if the contestant correctly bet the maximum amount in every round. This only occurred once. Using official inflation calculators5 this converts into 2006 dollars between $89,138 and $63,936 for the NBC version, and between $58,920 and $52,077 for the CBS version. These stakes are actually quite modest in relation to contemporary game shows in the United States, such as Deal Or No Deal described below, which typically has a maximal stake of $1,000,000. Of course, maximal stakes can be misleading, since Card Sharks and Deal Or No Deal are both “long shot” lotteries. Average earnings in the CBS version used by Gertner [1993] were $4,677, which converts to between $8,611 and