Testing the House Money & Break Even Effects on Blackjack Players in Las Vegas

Transcription

1 Testing the House Money & Break Even Effects on Blackjack Players in Las Vegas May 12, 2008 Zhihao Zhang Department of Economics Stanford University Stanford, CA Advisor: Professor Kenneth Shotts ABSTRACT Building on prospect theory, Thaler and Johnson (1990) observe two effects that govern decision-making under risk. The first effect is that in the presence of prior gains, individuals exhibit increased risk-seeking behaviour. They call this the house money effect. The second effect is that in the presence of prior losses, individuals favour scenarios that offer them the possibility of returning to their original level of wealth. They call this the break even effect. Their findings, however, were based on hypothetical surveys and small-scale laboratory experiments that lack the realism of decision-making in the real world. In this paper, I determine whether the house money and break even effects apply in a real-world context by observing blackjack players in various Las Vegas casinos and analysing the amount they bet when faced with prior gains and losses. Based on my observations, I find that both a house money effect and break even effect exist. The break even effect, however, is less economically significant and this can be attributed to limitations in my data. Keywords: risky decision, gamble, multi-stage lottery, decision making under uncertainty Acknowledgements I would like to thank my thesis advisor Professor Kenneth Shotts for his invaluable assistance and guidance. Our discussions were among the highlights of my academic career. Special thanks must also go to Professor Geoffrey Rothwell for getting me and keeping me on the right track through the Economic Honors Program, and his continued advice throughout the course of my thesis. I am extremely grateful to Stanford Undergraduate Research Programs (URP) for their grant which helped fund part of my research. Finally, I must thank Professor Nick Bloom, Dr. Hilton Obenzinger, and Jonathan Meer for their generous help.

2 Zhihao Zhang 1 1. Introduction Economists have traditionally used expected utility theory to explain how individuals make decisions under risk. Findings from experimental studies, however, have been inconsistent with expected utility theory. This has resulted in alternative theories that take into account behavioural tendencies. One such theory is prospect theory. Thaler and Johnson (1990) build on prospect theory by studying how individuals make decisions based on prior gains and losses. Using hypothetical surveys and small-scale laboratory experiments, they find that in the presence of prior gains, individuals exhibit increased risk-seeking behaviour. They call this the house money effect. They also find that in the presence of prior losses, individuals favour scenarios that offer them the possibility of returning to their original level of wealth. They call this the break even effect. A casino is the perfect place to study these effects since individuals are constantly taking risks in situations where they have experienced prior gains and losses. I choose to study blackjack because it is a popular game and one where it is easy to monitor a player s bets. In blackjack, players receive two cards after making their bet. They can take as many additional cards as they wish as long as they do not exceed a points total of 21, in which case they immediately lose. After the player acts, the dealer must follow a fixed set of rules in deciding whether to take additional cards. The dealer loses if his points total exceeds 21. The aim of the game is to have a higher points total than the dealer without exceeding 21. Players receive a bonus payoff of 1.5 times their original bet if they are dealt a blackjack first two cards are an Ace and a 10 value card (Ten, Jack, Queen, or King). In this paper, 71 randomly chosen blackjack players in various Las Vegas casinos were observed. The amount they bet and the outcome of each hand they played was recorded from the time they sat down at a table and bought in for chips till they left the table. With this data, I analyse the amount individuals bet when experiencing prior gains and losses of

3 Zhihao Zhang 2 different magnitudes, and test for house money and break even effects. If the house money effect were to hold, individuals would increase the amount of risk they are taking by betting proportionally more as they win more. If the break even effect were to hold, individuals would increase their chances of returning to their original level of chips by betting proportionally more as they lose more. In the sections that follow, I begin with a literature review of casino related studies and contrasting theories that model decision-making under risk. This is followed by the methodology section where I evaluate the use of casino observations against traditional laboratory experiments, describe my models and hypotheses, and state my assumptions. Finally, I provide my findings, run robustness checks, and in the conclusion, discuss the implications and limitations of these findings.

4 Zhihao Zhang 3 2. Literature Review 2.1 Decision-making under risk Economists have developed several models to explain how individuals make decisions under risk. Expected utility theory was formulated by Bernoulli (1738). He proposed that when thinking about how individuals value a risky proposition, instead of looking at its expected monetary value, its expected utility should be considered: Expected Utility = p(s)u(s) for all states p(s) = probability of state U(S) = utility of state where utility is a function of an individual s final wealth Bernoulli also introduced the idea of a concave utility-of-wealth function with diminishing marginal utility by noting that there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount. (Bernoulli 1954, p. 24) This established the concept of risk aversion in expected utility theory that would be built on by Arrow (1971) and Pratt (1964), but challenged by Rabin (2000). Bernoulli s (1738) model, however, was simply descriptive. It was von Neumann and Morgenstern (1944) who proved the rationality of expected utility maximisation through a series of axioms that Savage (1954) later expanded upon. Friedman and Savage (1948) extended expected utility theory using a gambling-type problem by explaining why the purchase of insurance (risk-averse behaviour) and lottery tickets (risk-seeking behaviour) by the same individual was not contradictory. They proposed that when individuals buy lottery tickets, they are willing to accept a small loss because they place a high value on the chance of achieving a major increase in personal wealth. To model this behaviour, they proposed a utility-of-wealth function with a concave segment below a certain level of wealth followed by a convex segment, and another concave segment at higher levels of wealth:

5 Zhihao Zhang 4 Figure 1 Utility Wealth Markowitz (1952, p. 151) Buying lottery tickets is thus rational since increases in wealth that are large enough to elevate a consumer into a new socio-economic class yield increasing marginal utility. Expected utility theory is not without critics. Allais (1953) introduced the idea that psychological values play a large role in decision-making under risk and produced experimental findings that were inconsistent with predictions of expected utility theory in what is known as the Allais Paradox. This spurred different models to describe decisionmaking under risk such as Chew s (1983) weighted utility model and Quiggins (1982) rankdependent expected utility model. While Arrow (1971) proved that individuals who maximise expected utility are arbitrarily close to risk neutral when stakes are arbitrarily small, Rabin (2000) took this one step further by claiming that risk aversion does not hold for moderate stakes gambles either. He proved that turning down a moderate stakes gamble implies that the marginal utility of money is diminishing so quickly that it would result in the rejection of a bet with preposterously large winnings when the same diminishing rate of marginal utility is applied. Instead, Rabin and Thaler (2001) explained modes-scale risk aversion using the psychological effects of loss aversion and narrow framing. Loss aversion is the tendency to feel the pain of a loss more acutely than the pleasure of an equal-sized gain (Kahneman and

6 Zhihao Zhang 5 Tversky, 1979). Narrow framing is the observation by Kahneman and Lovallo (1993) that individuals tend to isolate risky choices instead of thinking about them in a broader context. This behaviour stems from mental accounting, which is the nature by which individuals keep track of and evaluate their financial transactions (Thaler, 1990). Kahneman and Tversky (1979) conducted experiments which found that individuals exhibit risk aversion when given choices that involve sure gains and are risk-seeking when given choices that involve sure losses. They also found that individuals behave inconsistently depending on how a choice is framed and attributed this to the tendency to discard common components shared by all potential prospects. Both these behavioural tendencies run contrary to expected utility theory. In response, they developed an alternate model called prospect theory, which assigns value to gains and losses rather than final assets, and uses decision weights instead of probabilities. Prospect theory describes two phases in the decision-making process: an initial editing phase and a final evaluation phase. In the editing phase, several cognitive operations are used to reformulate and simplify outcomes and their associated probabilities. One such operation codes final outcomes as gains and losses with respect to a reference point. In the evaluation phase, the decision maker evaluates each of the edited prospects by calculating its expected value based on a value function and decision weights, and chooses the prospect with the highest value. Decision weights reflect the individual s perceived probabilities (not the actual probabilities) of the outcomes and are generally lower than actual probabilities, except for low probability outcomes which tend to be over-weighted. The value function reflects the subjective value of an outcome (measured in terms of relative gains and losses) and is concave for gains, convex for losses and incorporates loss aversion by being steeper for losses than gains:

7 Zhihao Zhang 6 Figure 2 Kahneman and Tversky (1979, p. 279) Thaler and Johnson (1990) built on prospect theory by noting Kahneman and Tversky s admission that a person who has not made peace with his losses is likely to accept gambles that would be unacceptable to him otherwise. (Kahneman and Tversky 1979, p. 286) They focused on the impact of prior outcomes and proposed new editing rules that modelled how individuals frame multi-stage lotteries. One such rule is the quasi-hedonic editing rule which proposes that individuals faced with a two stage lottery become less risk-averse after a gain in the first stage because a potential second stage loss is perceived as being offset by the first stage gain. When there is a loss in the first stage, however, individuals become more risk-averse because they internalise the first stage loss immediately. The exception is when the second stage lottery offers the opportunity to break even, in which case possible cancellation effects result in risk-seeking behaviour. To test these editing rules, Thaler and Johnson (1990) conducted surveys where individuals were given hypothetical situations of decision-making in the presence of prior gains and losses. They also designed multi-stage real money experiments where participants were faced with the prospect of winning and losing money. Their results supported a house money effect, where in the presence of prior gains, increased risk-seeking behaviour is

8 Zhihao Zhang 7 observed, and a break even effect, where in the presence of prior losses, individuals favour scenarios which offer them the possibility of returning to a break even level. To get compliant participants, however, they had to reduce the stakes and minimise the chances that participants faced losing gambles. Building on their findings, the house money effect was tested in various settings. Keasey and Moon (1996) conducted a hypothetical survey regarding capital expenditure decisions, Clark (2002) designed a small stakes public good experiment, and Weber and Zuchel (2005) conducted surveys based on portfolio choice and betting games. Findings were mixed. Just like Thaler and Johnson s (1990) experiments, they were laboratory based and failed to capture actual decision makers in a real-world context. I believe that better tests can be conducted in casinos where individuals are constantly making decisions under risk for non-trivial sums of money. 2.2 Casino related studies Given the prevalence of gambling-type problems in economics, it is somewhat surprising that very few studies have been conducted in casinos with actual casino games. In an experiment conducted by Lichtenstein and Slovic (1973) in the Four Queen s Casino in Las Vegas, they replicated a previously run laboratory experiment that did not involve an actual casino game. Furthermore, when participants were asked to buy in for 250 chips of a common denomination, of the 53 participants observed, 32 chose 5c chips, 18 chose 10c chips and 3 chose 25c chips. No participants opted for $1 or $5 chips the smallest chip domination in casinos. The study clearly did not capture decision makers in a real-world context. Most studies involving casino games have not been conducted in casinos. Dixon and Schreiber (2002) used computer simulated video poker to test how winning and losing impact the speed of decision-making, speed of play and perceptions of performance. Weatherly,

9 Zhihao Zhang 8 Sauter, and King (2004) used computer simulated slot machines to test the big win hypothesis 1. In both computer simulated experiments, the stakes were minimal. In the video poker simulation, even though a $50 certificate was awarded to the participant with the highest score, no actual money was at stake. In the slot machine simulation, participants started with 100 credits worth $0.10 each for a total value of $10, and the big win jackpot was a mere 16 credits ($1.60). While these results provide a good indicator of gambling behaviour, replicating the true gambling experience is clearly not possible if the stakes are insignificant or nonexistent. For blackjack, previous work has focused on surveys and observations of playing behaviour. In an attempt to study whether individuals behave rationally in a non-laboratory or classroom setting with no experimental intervention, Keren and Wagenaar (1985) administered surveys to blackjack players and observed their play in a Dutch casino. They compared their actions to basic strategy 2 and found that gamblers do not play optimally if their sole objective is to maximise expected value. They concluded that blackjack players decision-making processes contain a rational element based on intuitive statistics and a nonrational element based on misperceptions and incorrect beliefs. With this in mind, Keren, Wagenaar and Pleit-Kuiper (1984) conducted surveys on 77 experienced blackjack players and used principal component analysis to analyse their behaviour. They found that even though subjects generally agree that a high expected return is important, they greatly diverge in risk attitudes and beliefs about how luck and skill influence the outcome. These divergences were thought to explain why gamblers differ from one another and do not always play optimally. In one of the more ambitious projects to date, Bennis (2004) documented the playing strategies and beliefs of blackjack players, and examined how the decision-making process in 1 Big wins that occur early in the gambling process create mistaken expectations of winning and encourage individuals to continue gambling even after suffering losses. 2 An optimal blackjack strategy that maximises the player s expected return in every possible scenario.

10 Zhihao Zhang 9 gambling is influenced by factors such as experience, beliefs, and the socio-cultural context of gambling decisions. Commenting that prior research conducted in a real-world environment was lacking, Bennis argued that individuals behave differently in the real world as opposed to laboratory or classroom settings where novice decision makers engaged in artificial or inconsequential tasks (Bennis, 2004, p. 8). After spending more than one and a half years as a blackjack dealer and player, and conducting approximately two hundred interviews with gambling specialists and gamblers, he found that the socio-cultural context is crucial to understanding gamblers decision-making processes. After reviewing the literature, I find that much of the theory that incorporates psychological tendencies in explaining how individuals behave under risk has primarily been tested through experimental surveys and laboratory experiments. I propose that a casino would be a better place to test these theories because it captures actual decision makers in a real-world context. By observing blackjack players in Las Vegas casinos and using their behaviour to test Thaler and Johnson s (1990) house money effect and break even effect, I hope to contribute towards the literature of decision-making under risk that has developed from prospect theory.

11 Zhihao Zhang Methodology Since blackjack requires players to make a bet before seeing their cards, and increasing or decreasing the amount bet can be viewed as varying the level of risk taken, I was able to test the house money and break even effects in a real-world context by unobtrusively observing 71 blackjack players and determining whether the amount they bet is dependent on prior gains and losses. Additional details about my data collection process can be found in Appendix I. Before delving into my model and stating my assumptions, I will first evaluate the strengths and weaknesses of casino observations against standard laboratory experiments. 3.1 Evaluating Casino Observations against Laboratory Experiments There are several advantages to observing individuals in casinos as opposed to conducting laboratory experiments. Firstly, casinos capture individuals in a habitual environment. Unlike a laboratory, where the set up is contrived and participants may consciously or subconsciously attempt to decipher the intent behind the experiment and give the right answer, the individuals in my study did not know that they were being observed. This guarantees that their behaviour models real life as closely as possible. Furthermore, a problem with laboratory experiments is that the participant pool is usually limited by factors such as geographical location and level of remuneration. In gambling-type experiments, this can result in novices individuals with little experience and knowledge of gambling being put in unfamiliar situations and asked to make quick determinations of how they would behave. This can result in inconsistent behaviour due to inexperience. Casinos, on the other hand, capture individuals from various backgrounds who usually have some gambling experience. They are likely to have a good understanding of the game they are playing, leaving less ambiguity about whether they actually comprehend the gamble being offered.

12 Zhihao Zhang 11 Unlike laboratory experiments where getting compliant participants usually involves limitations on the level of stakes, as witnessed in studies conducted by Thaler and Johnson (1990), Lichtenstein and Slovic (1973), Dixon and Schreiber (2002) and Weatherly, Sauter, and King (2004), casinos allow decision makers to be studied in a real-world context with significant stakes on the line. While there is the option of conducting studies in less wealthy subject populations such as third world countries, this might be less effective in dealing with gambling-type problems where the ideal stakes are on one hand non-trivial, but on the other hand not overly large. It is a difficult medium to achieve and ideal stakes clearly differs from person to person. The advantage of casinos is that they offer varying levels of stakes. In Las Vegas, blackjack stakes range from $3 minimums in smaller casinos to $25,000 maximums in high-end casinos, and in between there are many $5, $10, $15, $25, $100 and $200 minimum bet tables. One would imagine that there is some self selection and individuals typically choose to play at a table where the stakes are sufficiently non-trivial in that they give them some excitement, but are not excessively high and over their heads. At the same time, there are disadvantages to observing individuals in casinos. Firstly, there is a much greater chance that their decision-making could be affected by external factors which are unobservable. Casinos offer complimentary drinks to all gamblers and several of the individuals whom I observed consumed alcoholic beverages while at the table. It is impossible to know how much alcohol they had consumed prior to arriving and to what extent the alcohol was affecting their behaviour. This problem would be absent in a laboratory setting. In casinos, betting behaviour can also be influenced by other people at the table such as the dealer, a spouse, or other gamblers. A controlled laboratory setting has the ability to remove all these factors, or add them if the experimenter so chooses. A disadvantage of casinos specific to my study is that it is likely that the individuals observed had gambled recently and won or lost an indeterminate amount of money at another

13 Zhihao Zhang 12 table or casino. I was unable to collect any data on these prior wins or losses which could be affecting their behaviour. In a laboratory setting, it is much less likely that participants would have experienced recent gambling wins or losses and even if they had, they probably would not take them into consideration when participating in the experiment. One final advantage of a laboratory is that it is possible to control the cards that the player receives and to a certain extent, the outcome of the hands 3. In my study, I had substantially more observations of individuals who were winning since those who lost all their chips would leave the table and my observations of them ceased. With greater control over outcomes in a laboratory, it is possible to better control the number of hands that each individual plays. Different situations can also be created to see how people react to them. 3.2 Basic Variables Variable betsize netgain Description Amount bet (betting units) Net money won prior to player placing bet (betting units) up 1 if netgain > 0 down 1 if netgain < 0 middle late 1 if 31 st -60 th hand played 1 if > 60 th hand played Note: I define a betting unit as the mode of all bets that an individual makes. 4 For example, if an individual with a betting unit of $25 bets $100 and has won $400, betsize = 4 and netgain = 16. The reason for measuring true betsize and true netgain in betting units is that a prior gain of $200 to a person betting $10 a hand should be treated differently than a prior gain of $200 to a person betting $200 a hand. Normalising betsize prevents a few large bettors from driving any of the effects. 3 In blackjack, it is not possible to completely control the outcome of hands since the outcome is dependent on player decisions. This could be solved in a laboratory by using a pre-programmed computer simulation. 4 Using the mode bet might seem arbitrary, and so other definitions of a betting unit, such as median bet and 10 th percentile bet, are tested in the robustness section.

14 Zhihao Zhang Simple Model Model 1 betsize = α + β 1 * up * netgain + β 2 * down * netgain + ε In this model, β 1 captures the increase in amount bet (in betting units) when prior gains increase by one betting unit. This represents the house money effect. β 2 captures the increase in amount bet (in betting units) when prior losses increase by one betting unit. This represents the break even effect. Based on the house money and break even effects, I expect β 1 and β 2 to be positive since this implies that people are betting proportionally more as they win more and proportionally more as they lose more. I also expect β 1 and β 2 to be less than 1. If β 1 is greater than 1, this implies that individuals want to risk more than their prior gains. If β 2 is greater than 1, this implies that they want to risk more than required to break even. It would run contrary to the house money and break even effects. Furthermore, when an individual is winning, the maximum amount they can bet is equal to the sum of how much they have won and how much they initially started with. When losing, the amount they can bet is limited by the amount of money they have left and other bankroll constraints. Therefore, I expect β 1 to be greater than β 2. Even though normalising bets controls for bettors who are playing at different stakes, there could be other factors which affect the betting patterns of individuals as alluded to by Keren, Wagenaar and Pleit-Kuiper (1984) and Bennis (2004). To account for this, I also run the regression while controlling for individual fixed effects. With a suitable sample size and range of observations per individual, controlling for individual fixed effects should not affect the signs or significance of my previous findings.

15 Zhihao Zhang Advanced Models Model 2 A possible omitted variable in my simple model is the amount of time an individual spends at the table. I thus add two dummy variables, middle and late, as indicators for the number of hands played: betsize = α + β 1 * up * netgain + β 2 * down * netgain + β 3 * middle + β 4 * late + ε β 1 and β 2 capture the house money effect and break even effect respectively. β 3 captures the increase in amount bet when individuals are playing their 31 st - 60 th hand compared to their 1 st 30 th hand. β 4 captures the increase in amount bet when individuals are playing any hand greater than their 60 th hand compared to their 1 st 30 th hand. I expect β 3 and β 4 to be positive, since there are several possible explanations why individuals bet more the longer they are at a table. They might be consuming free cocktails and find themselves under the influence of alcohol. They might be growing increasingly desensitised to chip values and need to bet more to get the same amount of excitement as they previously did. It is also possible that, after sitting down for a while and becoming more comfortable at the table, they are influenced by other gamblers and feel a need to loosen up and bet more. Given this logic, I would expect β 4 to be greater than β 3. In addition, since it would seem that individuals are more likely to have won or lost larger amounts the longer they are at a table, i.e. middle and late are positively correlated with up* netgain and down* netgain, the inclusion of middle and late will result in β 1 and β 2 being smaller compared to Model 1. Model 3 To test if an increase in amount bet the longer a person is at the table depends on whether the individual is winning or losing money or is independent of prior outcomes, I interact the late variable with gains and losses:

16 Zhihao Zhang 15 betsize = α + β 1 * up * netgain + β 2 * down * netgain + β 3 * late + β 4 * up * netgain * late + β 5 * down * netgain * late + ε Again, β 1 and β 2 capture the house money effect and break even effect respectively. β 3 captures the increase in amount bet when individuals have played more than 60 hands and netgain is 0. β 4 captures the increase in amount bet when individuals have played more than 60 hands and netgain is positive. β 5 captures the increase in amount bet when individuals have played more than 60 hands and netgain is negative. In this model, if individuals are betting more solely because they are at a table for a long time, then only β 3 will be positive (out of β 3, β 4 and β 5 ). I do not expect this to be the case. Instead, I predict that β 4 will be positive, since individuals who are winning after more than 60 hands are more likely to have won a large amount of money, be desensitised to chip values, and have the chips for bigger bets. If they are losing after 60 hands, while they could be more desperate to get back to a break even level, they are more likely to have fewer chips to bet and so β 5 will be smaller than β 4, but still positive. Using the same logic as in Model 2, β 1 and β 2 should be smaller compared to Model Assumptions I make the following assumptions: 1. Each time a person buys in for chips at a new table, they mentally account for it as a new session, and isolate wins and losses of that session from any prior outcomes. This assumption is grounded in Kahneman and Lovallo s (1993) narrow framing where individuals tend to evaluate gambles in isolation. In this case, the session, which starts when they first join a table and buy in for chips and ends when they leave the table, is mentally accounted for as a new session. They do not take into account any previous wins or losses that might have occurred at other tables or in other casinos.

17 Zhihao Zhang My values for betsize are continuous. At first glance, betsize might seem to be discrete, since the amount that individuals can bet is determined by the denomination of chips in their possession, and casinos only have a set variety of chip denominations. The individuals I observed mainly had $1, $5, $25 and $100 chips, with $1 chips typically only coming into their possession when they bet an amount that ended in 5 ($15, $25, $45 etc.), and were paid 1.5 times their bet after receiving a blackjack. 5 However, when normalising bets by dividing an individual s true betsize by their betting unit, I was able to obtain reasonably continuous values for betsize. 3. The relationship between betsize and netgain is linear and an OLS model suitably approximates the house money and break even effects even though I am dealing with censored data (betsize is always positive). 4. The individuals I observed were not card counters. 6 Card counters do not take into account prior gains or losses when betting. Instead, they vary their bets based on the composition of the deck. This would introduce random shocks into my data. I tried to eliminate the presence of card counters in my observations by counting cards myself and discarding a player s observations if I believed they were counting cards. This happened on two occasions. 5. The individuals I observed had some idea of how much they were winning or losing. For individuals who arranged their chips in neat stacks and counted them periodically, I could be reasonably sure that they knew how much they had won or lost. For others, it was harder to ascertain, but I believe that gamblers generally keep track of how they are faring. 5 A player gets dealt a blackjack approximately once every 21 hands. 6 In blackjack, the casino has an edge in the long run due to the rules of the game. In any given situation, however, the casino s edge is based on the composition of the deck that has yet to be dealt. Card counters take advantage of this by varying their bets based on the composition of the deck systematically increasing their bets when there are more 10 value cards and Aces left in the deck.

18 Zhihao Zhang Descriptive Statistics Figure 3: Scatter plot of betsize against netgain with fitted lowess line betsize netgain Lowess defined using bandwidth 0.8 A preliminary look at my raw data in the form of a scatter plot of betsize against netgain with a fitted lowess line shows a positive correlation between betsize and netgain when netgain is positive. This indicates that individuals are betting more in the presence of prior gains and supports the house money effect. When netgain is negative, the negative correlation is slight and individuals do not seem to be betting more as they lose more to increase their chances of returning to their break even point. While there is some evidence of a break even effect, it is less conclusive.

19 Zhihao Zhang 18 Table 1: Basic Variables Summary Statistics Variable Description Obs. Mean Standard Deviation Min. Max. betsize Amount bet (betting units 7 ) netgain Net money won prior to placing bet (betting units) up 1 if netgain > down 1 if netgain < middle 1 if 31 st 60 th hand played late 1 if > 60 th hand played Note: 59% of the observations are of individuals who were winning (mean of up = 0.59). While this may seem counterintuitive since the casino has an edge in blackjack and one would think that I should have more observations of people who are losing, I observe more winners because individuals who were losing had shorter sessions. They often left the table when they ran out of money, needed a break or wanted to change up their luck. 7 Mode of bets that individual made.

20 Zhihao Zhang 19 Table 2: Interaction Terms Summary Statistics (Conditional) Variable Description Obs. Mean Standard Deviation Min. Max. up * netgain down * netgain up * netgain * late down * netgain * late Amount ahead (given netgain > 0) Amount behind (given netgain < 0) Amount ahead given > 60 th hand Amount behind given > 60 th hand Summary statistics of the interaction terms conditional on the interaction terms being positive are shown in Table 2. The summary statistic of ( up * netgain ) indicates that there are 2157 observations when a player is winning, with a mean win of 7.6 betting units. The maximum win is 51.5 betting units. The summary statistic of ( down * netgain ) indicates that are 1314 observations when a player is losing, with a mean loss of 4.5 betting units. The maximum loss is 52.8 betting units. The summary statistic of ( up * netgain * late ) indicates that there are 484 observations when a player is winning after playing more than 60 hands, with a mean win of 12.4 betting units. The summary statistic of ( down * netgain * late ) indicates that there are 281 observations when a player is losing after playing more than 60 hands, with a mean loss of 7.3 betting units.

21 Zhihao Zhang 20 A further breakdown of the 2157 observations where ( up * netgain > 0 ) Table 3: Distribution of netgain when ( up * netgain ) > 0 netgain Observations Percentage 55% 22% 8% 4% 4% 3% 1% 1% 1% A further breakdown of the 1314 observations where ( down * netgain > 0 ) Table 4: Distribution of netgain when ( down * netgain ) > 0 netgain Observations Percentage 75% 15% 5% 2% 0.2% 0.2% 0.1% 0.5% 1% Even though the magnitude of the largest win and loss is approximately equal, Tables 3 and 4 indicate that there is a much wider range of observations when individuals are winning. When winning, 55% of wins are less than or equal to 5 betting units and 10% are greater than 20 betting units. In comparison, 75% of losses are less than or equal to 5 betting units and only 2% are greater than 20 betting units. A further analysis of the data shows that one individual accounts for 8% of total losing observations and is responsible for all 23 observations of losses greater than 26.5 betting units. In comparison, seven individuals are responsible for the 126 observations of gains greater than 26.5 betting units. This suggests that one individual may be driving the break even effect. I will acknowledge this in the robustness checks section.

22 Zhihao Zhang Findings OLS regressions on the three models yield the following results: 5.1 Model 1 Results betsize = α + β 1 * up * netgain + β 2 * down * netgain + ε As shown in Table 6 column 1, both β 1 and β 2 are positive and statistically significant. This supports the house money and break even effects. The magnitudes of the coefficients suggest that while the house money effect (β 1 = 0.08) is economically significant, the break even effect (β 2 = 0.02) is less so. These coefficients can be interpreted as follows: A person with a betting unit of $10 increases the amount they bet by $8 when the amount they are winning increases by $100, but only increases it by $2 when the amount they are losing increases by $100. After controlling for individual fixed effects, β 1 is still positive and significant but β 2 is no longer statistically significant (Table 6 column 2). A possible explanation for the break even effect losing statistical significance is that among individuals who experience losses, there are few observations of individuals who have a large range of net losses. Most individuals hover around small net losses. In addition, one individual accounts for all 23 losses greater than 26.5 betting units. 5.2 Model 2 Results betsize = α + β 1 * up * netgain + β 2 * down * netgain + β 3 * middle + β 4 * late + ε As hypothesised, β 1 and β 2 are smaller than in Model 1 (Table 6 column 3). The reduction, however, is slight and β 1 and β 2 are still positive and significant, which is in line with the house money and break even effects. When individual fixed effects are controlled for (Table 6 column 4), β 4 is significant and large (0.305). This supports the notion that the longer a person is at a table (as defined by a session that lasts for more than 60 hands), the

23 Zhihao Zhang 22 more they bet. This is not so for β 3 which might suggest that the 31 st 60 th hand is not a good measure of a milestone length at the table. 5.3 Model 3 Results betsize = α + β 1 * up * netgain + β 2 * down * netgain + β 3 * late + β 4 * up * netgain * late + β 5 * down * netgain * late + ε Without controlling for individual fixed effects, β 3 and β 5 are not significant (Table 6 column 5). This suggests that the number of hands played does not affect the amount bet by someone who is even or is losing money. β 4 is positive and significant (0.031). Combined with the findings on β 3 and β 5, this suggests that when winning, individuals who play more than 60 hands increase their amount bet by of a betting unit for a one betting unit increase in winning, compared to if they play 60 or fewer hands. When summed with β 1, this suggests that when winning, individuals who play more than 60 hands increase their amount bet by of a betting unit for a one betting unit increase in winning. While the estimator for β 2 is very similar to Models 1 and 2, β 1 is smaller in magnitude but still positive and significant. This indicates that the house money effect still applies to individuals who are playing 60 or fewer hands. Controlling for individual fixed effects yields similar results (Table 6 column 6).

24 Zhihao Zhang Robustness Checks As mentioned in the descriptive statistics section, one individual accounts for 8% of total losing observations and is responsible for all 23 observations of losses greater than 26.5 betting units. He appears to be an outlier. To determine whether he is driving the break even effect, I drop all 113 observations of his play and re-run the regressions in Models 1, 2 and 3 (Table 7). β 2 remains positive and significant, and actually increases in magnitude, lending support to the break even effect. At first glance, there appears to be selection bias because there are significantly more observations when people are winning. This should not substantially affect estimates of the house money and break even effects because my models essentially estimates the two effects separately by looking at them conditional on an individual winning or losing. But if there is a fundamental difference between people who are winning and people who are losing, the house money effect that I find would only apply to winners and the break even effect to losers. One such difference could be skill level. Blackjack, however, is a game of chance and card counters aside, the difference in the expected returns of a great player and an average player is marginal. Over a single session, the main factor driving outcomes is not skill, but cards received an aspect that is completely random. There is very little fundamental difference between individuals who are winning and losing. Bias in this form is not a problem. There could, however, be a fundamental difference between individuals who play fewer hands and individuals who play more hands. Given that the casino has an edge in blackjack, I am likely to have fewer observations of individuals who bet larger amounts when winning and losing, since they are more likely to lose all their chips and leave the table. It is also possible that risk-seeking individuals buy in for smaller amounts (in terms of betting units), and are thus more likely to lose all their chips. Following this same line of thought, I

25 Zhihao Zhang 24 will have more observations of individuals who bet the same amount regardless of how much they are winning or losing, and conservative people who buy in for larger amounts (in terms of betting units). This results in a form of selection bias that could lead to the underestimation of the house money and break even effects, since individuals who exhibit this behaviour have fewer observations and are under-represented in my data compared to individuals who bet conservatively and play numerous hands. To account for the fact that certain types of individuals play more hands than others, a regression is run where observations are weighted by the number of hands an individual plays. As shown in Table 8 column 1, β 1 and β 2 remain identical in sign and significance, and similar in magnitude to my previous regressions. Another regression is run where the data is restricted such that only the first 30 hands are included (i.e. middle = 0 and late = 0). If an individual plays less than 30 hands, all their hands are included. By focusing only on the first 30 hands, this gives equal weighting to all individuals while allowing enough hands to gain some variation in netgain. For this restricted regression, β 1 and β 2 are and respectively without controlling for individual fixed effects and β 1 is when controlling for fixed effects (Table 8 columns 2 and 3). The estimators are statistically and economically significant, and lend strong support for the house money effect and break even effect. To deal with the arbitrary use of 30 and 60 hands as cut-off points for the middle and late variables, I replace the middle and late variables with a more general proxy for time spent at the table. I choose two variables ln(hands) and (hands) 0.5 where hands is how many hands an individual has played when making their bet. The reason for using such functional forms is the belief that the relationship between betsize and hands is a concave function. This would seem reasonable since there is an upper limit to how much someone can bet no matter how many hands they have played. The results are almost identical to the regressions using the middle and late variables (Table 9).

26 Zhihao Zhang 25 Another concern is that the outcome of the previous hand could affect betsize if gamblers are superstitious or follow betting systems that encourage people to press their bets when winning. My estimates for the house money and break even effects will only be biased, however, if the outcome of the previous hand is correlated with netgain. The outcome of the previous hand should be largely uncorrelated with netgain but to account for this, I create a dummy variable called win_previous_hand and include it in the original regressions. win_previous_hand takes on a value of 1 if the previous hand resulted in a win and a value of 0 if it resulted in a tie or push or was the first hand played. As shown in Table 10, β 1 and β 2 are similar to the original regressions. The coefficients of win_previous_hand in the various regressions indicate that winning the previous hand results in individuals increasing the amount they bet by around 0.25 betting units. It is both statistically and economically significant and supports the notion that gamblers tend to press their bets when they have won the previous hand. Lastly, the definition of a betting unit could affect the magnitudes of the house money and break even effects. This is because the definition of a betting unit scales netgain and betsize for different individuals in a different manner. I chose mode bet in an attempt to reflect the psychology of gamblers who might view increased or decreased risk-taking relative to their most common bet. This might not necessarily be true. In Tables 11 to 20, all the regressions are re-run using alternate definitions of a betting unit. In Tables 11 to 15, a betting unit is defined as the 10 th percentile bet and in Tables 16 to 20, a betting unit is defined as the median bet. The results are largely similar in sign and significance to the previous regressions although there is some variation in magnitude of estimators because of scaling.

27 Zhihao Zhang Conclusion Many economists have attempted to model how individuals behave under risk. Building on prospect theory, Thaler and Johnson (1990) propose that in the presence of a prior gain, individuals exhibit increased risk-seeking behaviour (house money effect), and in the presence of a prior loss, favour gambles that offer the possibility of returning to their break even level (break even effect). I test these two effects by observing 71 people play blackjack in various Las Vegas casinos. Using an OLS model with estimators that capture the house money and break even effects, I determine whether individuals bet proportionally more when winning more (house money effect), and proportionally more when losing more (break even effect). My findings indicate that the house money effect is statistically and economically significant. The break even effect is less conclusive due to limitations in my data. Economic significance is only achieved when observations are restricted to reduce selection bias, which occurs because individuals who try to get back to a break even level by betting larger amounts are more likely to lose all their chips and have fewer observations than individuals who continue to bet small amounts. Furthermore, when individual fixed effects are controlled for, the break even effect loses statistical significance because of the lack of observations of multiple individuals over a range of losses. Two other findings of note are that individuals bet significantly more after winning the previous hand, and when winning, bet larger amounts the more hands they have played. There could be a variety of explanations for these findings but without interviewing players or collaborating with casinos, it would be difficult to pinpoint these explanations since extensive table conditions data is necessary. I will now discuss the broader implications of my findings. While the wealth levels of the individuals I observed were not known, it would be fair to assume that any bets they made, and gains or losses they experienced, were small fractions of their total wealth. Expected

28 Zhihao Zhang 27 utility theory would predict risk neutrality and that the amount bet is largely independent of prior gains or losses. My findings, however, support prospect theory and the idea that utility is a function of net gains and losses relative to a reference point rather than final assets. Gambling experts often point to poor money management as the main reason why people suffer gambling losses. They claim that people start betting more when they are ahead and eventually lose all their winnings. When they are behind, they chase their losses by betting even more. It appears that this perceived lack of money management can be traced back to the house money and break even effects rationalisable behaviour governing decision-making under risk. From the casino s perspective, the house money and break even effects are highly beneficial. Casinos have an edge in all casino games and their expected win is simply a function of the amount bet. The larger the amount bet, the more the casino is expected to win. Since the house money and break even effects propose that players bet proportionally more when they are both winning more and losing more, it is advantageous to the casino. In addition, given that people who are winning bet more the longer they are at the table, casinos should try to keep winning gamblers at the table for as long as possible something I am confident they already know. Finally, I will highlight the limitations of my study. One limitation is that I did not have a wide range of observations from individuals who were losing money and one individual accounted for all 23 observations of losses greater than 26.5 betting units. Given my method of casino observations, it might be difficult to gather data that would support the break even effect at an individual level because if it were to hold, one would expect to see individuals with large losses betting large amounts with two likely outcomes. The first is that they lose and leave the table because they are out of money. The second is that they win and gets back to a break even level. In both cases, the number of observations where the player

29 Zhihao Zhang 28 has a net loss is limited, and unless an individual has deep enough pockets to sustain a wide range of losses and chooses to stay at the same table, there would be insufficient observations. Another limitation of my data collection is that I did not account for tipping. I simply looked at how much an individual bet and the outcome of the hand, and calculated how much they had won or lost prior to making each bet. In blackjack, players occasionally tip the dealer and cocktail waitresses. This affects the amount of money they have in front of them, which in turn affects how much they bet since a player is typically not keeping track of wins ands losses, net of tips. While total tips rarely added up to more than 1 betting unit, they could have been recorded for greater accuracy. Lastly, without interviewing players or cooperating with casinos, I was unable to gather any data on an individual s wins or losses prior to sitting down at the table. While narrow framing is assumed, it is hard to determine the extent to which it holds. It would be interesting to see how players bets change as they move from table to table. Do they factor in previous wins or losses into their bets at a new table in the same casino or do they believe that this table change makes it an entirely new gamble? Much of the theory governing decision-making under risk has not been tested in a real-world context. Observing individuals in a habitual environment offers many advantages over laboratory experiments in capturing decision-making behaviour. Given that lotteries and gambles form the basis of many experimental findings, casinos seem a logical place to investigate decision-making in risky situations. If tapped properly, they can provide a great avenue for further research.