State-Dependent Risk Preferences: Evidence From Online Sports Gambling

Transcription

1 State-Dependent Risk Preferences: Evidence From Online Sports Gambling Angie Andrikogiannopoulou HEC Geneva Filippos Papakonstantinou Imperial College London April 2011 Abstract We use a unique panel dataset of consumer betting activity in an online sportsbook to examine whether individual risk preferences are stable over time or state-dependent. Our econometric model accounts for individual heterogeneity in all utility parameters within a dynamic version of the cumulative prospect theory of Tversky and Kahneman (1992). Our findings suggest that for the majority of bettors the null hypothesis of no state-dependence is rejected. In particular, different forces seems to govern individuals dynamic choice behavior. First, the reference point that separates gains from losses does not update completely to reflect prior bet outcomes but rather remains sticky to some previous level, resulting in prior gains/losses being integrated with current payoffs. In addition to this, the loss aversion parameter significantly increases (decreases) after prior gains (losses), while for a substantial portion of bettors the curvature of the value function is also significantly positively related to past betting performance. Substantial heterogeneity is present in all utility parameters. Methodologically, we use a discrete choice framework and estimate a multinomial mixed logit model using Bayesian econometrics techniques. These findings have potential implications for models in macroeconomics and finance which propose that agents risk preferences are time-varying. Keywords: Risk Preferences, Prospect Theory, State Dependence, Bayesian Estimation, Discrete Choice JEL Classification: D11, D12, D81, G00 HEC, University of Geneva and Swiss Finance Institute, 102 Bd Carl Vogt, CH-1211 Geneva 4, Switzerland, Angeliki.Andrikogiannopoulou@unige.ch. Imperial College Business School, Tanaka Building, South Kensington Campus, London SW7 2AZ, UK, fpapakon@imperial.ac.uk,

2 1 Introduction Individual risk preferences play a central role in many areas of economics and finance. A growing number of studies use experimental or field data to measure individual attitudes towards risk in static choice situations. However, there is very little evidence on how people think about sequences of gains and losses in dynamic settings. This is particularly important in view of a large theoretical literature in finance which proposes that agents risk preferences are state-dependent in order to explain well-known asset-pricing irregularities, such as the equity premium and the volatility puzzles. 1 In this study, we are able to address this issue using a unique panel dataset of real-life individual choices from an online sports betting market. Sports betting has become one of the most common gambling activities in which the average person engages. Its popularity has increased dramatically in recent years with hundreds of bookmakers offering bet opportunities on numerous sporting events. Many people argue that it shares a lot of similarities with stock investing since, in their essence, both activities involve the risk of some capital on the uncertain outcome of future events. Hence, both financial decisions reflect in large part the risk profiles of the decision-makers. Many stock market traders show similar traits with casino gamblers; they are governed by the thrill of chasing higher returns and as a result, after winning a trade, they are trying to force another winning trade, while after losing a trade, they are seeking to quickly recoup their losses on the next trade. 2 But also in terms of their operation, sports betting markets and typical financial markets have a lot of features in common: they are populated with a large number of 1 In the traditional consumption-based framework, state-dependence is usually introduced through habit-formation preferences (Constantinides (1990), Campbell and Cochrane (1999)) and consumption commitments (Szeidl and Chetty (2010)). In the behavioral finance literature, Barberis et al. (2001) introduce time-variation by assuming that investors are loss averse over fluctuations in the value of their financial wealth and the level of loss aversion depends on their previous investing performance. 2 Anecdotal evidence suggests that a large percentage of Wall Street traders engage in sports betting activities while several sports betting syndicates are populated with stock market investors. 1

3 participants with varying degrees of sophistication; information about sports teams and events is publicly available as it is information about companies and stock prices; the sports bookmaker is the analog of the market maker in financial markets while sports handicappers play the role of financial analysts. Using micro-data on asset allocation decisions to make inferences about the dynamics of investors risk preferences within a structural econometric model has been hindered by the fact that it is difficult to know individuals investment horizon as well as their expectations regarding future asset returns. On the contrary, sports bets have an exogenous predefined moment of terminal payoff when the sporting event takes place while the bookmaker s odds provide a natural starting point for bettors beliefs about their probabilities of winning. For these reasons, sports betting markets provide an idealized laboratory setting for a structural elicitation of the risk preferences of individuals who seem to share a similar mentality with some of the participants in financial markets. The cumulative prospect theory (CPT) developed by Tversky and Kahneman (1992) offers a natural framework for explaining how bettors evaluate casino gambles. A central feature of this theory is that people systematically distort the probabilities of events by applying probability weighting functions on objective probabilities. Furthermore, gains and losses are evaluated relative to a reference point which can be affected by the framing of outcomes. In this study, we extend the original CPT framework, which applies only to static choice situations, to allow for possible statedependence in bettors behavior, which could arise, for example, if bettors decisions are affected by the outcomes of their prior bets or by aggregate macroeconomic conditions. We thus allow the value function that people use to evaluate gambles to depend on past betting performance defined as the cumulative payoff of all bets placed over a reasonable time frame in the past. 3 In particular, we first allow the reference point 3 We have left other types of state-dependence such as the effect of business cycle fluctuations, outstanding bets, etc. for future research. 2

4 relative to which prospects are evaluated to not update completely to incorporate previous bet outcomes, resulting in prior outcomes being integrated with current payoffs. In this case, after incurring a gain, subsequent losses will be edited by people as a reduction in this gain rather than as a loss, and similarly, after incurring a loss, subsequent gains that are not large enough to make up for the initial loss will be edited by people as losses rather than as gains. In addition to this, we also allow the loss aversion and the utility curvature parameters to be a linear function of our measure of past betting performance. Our findings suggest that for the majority of bettors the null hypothesis of no statedependence is rejected. Regarding the framing of outcomes, bettors in our sample can be clearly classified into two discrete types: 80% of them have no memory and they immediately incorporate previous gains and losses into the reference point they use to evaluate the outcomes of subsequent gambles, while 20% of them do have memory and therefore their reference point remains sticky to some previous level. We also find that for the majority of bettors (82%) the loss aversion parameter significantly increases (decreases) after prior gains (losses), while for a substantial portion of bettors (46%) the curvature of the value function is also significantly positively related to past betting performance. The total effect on risk-taking is determined by the relative strength of these forces at the individual bettor level. The stickiness of the reference point forces bettors to behave in a more risk averse (risk-seeking) manner than before when confronted with gambles that involve losses that are greater (smaller) than the prior gains. The effect of previous outcomes on the loss aversion and the utility curvature parameters makes bettors effectively more risk averse (risk-seeking) after gains (losses). The idea that individual preferences are state-dependent has first been documented in some experimental studies (e.g., Thaler and Johnson (1990)) and quasiexperimental studies such as game shows (e.g., Post et al. (2008)), where subjects 3

5 are observed to increase (decrease) risk after prior gains (losses), a behavior known as the house money effect. In financial markets, there is evidence of the disposition effect, i.e. investors sell stocks that have risen in value since purchase and hold on to stocks that have fallen in value since purchase (e.g. Odean (1998)). One possible explanation of this behavior is that investors become more risk averse after gains and more risk-loving after losses. Finally, the evidence from data on households asset allocations is mixed: some studies suggest that wealth shocks do not significantly affect the share of wealth that households allocate to risky assets, indicating that risk preferences are not time varying (e.g. Brunnermeier and Nagel (2008)) while others find a significant positive elasticity of the risky share with respect to financial wealth, suggesting that investors exhibit decreasing relative risk aversion (Calvet et al. (2009)). The main difference between this study and the existing finance literature is that we take a more structural rather than reduced form approach to the examination of the dynamics of risk preferences. The advantage of this approach is that the estimated preference coefficients could possibly help us predict individual behavior under any lottery choice situation. One common caveat in the empirical analysis of real-life financial decisions is that the same choices could be explained with several combinations of risk preferences and subjective beliefs with respect to the probabilities of events (e.g. the probability of an accident in insurance choices, the probability distribution of portfolio returns in asset allocation decisions, etc.). Thus, the observation that people s choices are affected by previous outcomes might be either due to the fact that people have time-varying risk preferences or due to the fact that they have time-varying beliefs regarding the various outcomes probabilities. In the context of prospect theory, the first explanation suggests that experience affects the value function that individuals use to evaluate subsequent gambles, while the second explanation suggests that experience triggers a change in the subjective probability that people attach to the outcomes of subsequent 4

6 gambles. 4 This caveat makes it inherently impossible to identify both risk preferences and beliefs without imposing any restrictions on one of the two. In this study, we focus on the risk preference explanation of the dynamics of risk preferences imposing a reasonable constraint on people s beliefs, i.e. that bettors always evaluate the outcomes of events using the probabilities implied by the bookmaker s odds and thus prior outcomes have no effect on bettors subsequent beliefs. 5 Our approach is further supported by the experimental evidence of Barkan and Busemeyer (1999) and Barkan and Busemeyer (2003) who find that subjects behavior when presented with sequential gambles is better explained by the changing reference point model rather than the changing subjective probability model. As an extension, we also consider a fuller econometric specification in which we introduce individualand match-specific errors in the probabilities people use to evaluate outcomes, where these errors can capture possible deviations from the quoted probabilities due to biases in beliefs (e.g. representativeness bias, optimism, etc.), skill/information, or simply probability assessment mistakes. Methodologically, we analyze bettors choices in a multinomial discrete choice framework. We first represent the day lottery chosen by each bettor on each bet day by combining all possible payoffs of all bets placed by the bettor over the period of one day and computing their respective probabilities. Second, we specify the set of alternative lotteries from which the observed day lotteries are selected. This choice set consists of a number of reasonably selected representative lotteries plus a safe lottery which represents the option not to play on a given day. Finally, we set up a multinomial mixed logit model that takes into account both the lotteries 4 For example, according to the gambler s fallacy people expect outcomes in short random sequences to exhibit systematic reversals, e.g. when flipping a fair coin, people believe that after a short streak of heads, the probability of getting tails is increased, while the hot hand bias predicts the opposite, i.e. that patterns in small samples are expected to persist (Croson and Sundali (2005), Camerer (1989), Brown and Sauer (1993)). 5 Depending on how the bookmaker sets the odds, these probabilities might already reflect possible biases in the beliefs of the marginal bettor and/or the correct probabilities of winning. 5

7 chosen conditional on play and the observed frequency of playing for each individual. We estimate the final model with Bayesian econometrics by applying Markov Chain Monte Carlo (MCMC) techniques. The remainder of the paper is organized as follows. Section 2 discusses the relation to the literature. Section 3 describes the data and the analysis thereof. Section 4 lays out the econometric specification employed and Section 4.3 describes its estimation. Section 5 presents the results of our model and Section 6 discusses the caveats and an extension of our baseline econometric specification. Section 7 concludes. 2 Relation to Literature This study is related to two different strands of literature. The first strand focuses on the elicitation of risk preference parameters based on structural econometric models. The vast majority of evidence in this literature comes from laboratory experiments (Hey and Orme (1994), Holt and Laury (2002), Tversky and Kahneman (1992), Choi et al. (2007)), hypothetical survey questions (Donkers et al. (2001)) and data on television game show contestants (Gertner (1993), Metrick (1995), Beetsma and Schotman (2001), Bombardini and Trebbi (2007), Post et al. (2008)). A few attempts have recently been made to estimate risk preferences from regular market participants in such settings as insurance markets (Cicchetti and Dubin (1994), Cohen and Einav (2007), Barseghyan et al. (2010)), labor supply decisions (Chetty (2006)), personto-person lending portfolio choices (Paravisini et al. (2010)). In the sports betting context, several empirical papers (Golec and Tamarkin (1998), Jullien and Salanie (2000), Kopriva (2009)) analyze aggregate price data, mainly from racetrack betting, to infer the preferences of a representative bettor. However, most of the aforementioned studies that utilize micro datasets usually observe very few data points per subject and are therefore not able to address if and 6

8 how individual risk preferences evolve over time. Hence their analysis only applies to static choice situations. A first step towards empirically investigating the dynamics of risk preferences is undertaken by some experimental studies (e.g. Thaler and Johnson (1990)), where subjects are presented with a few sequential gambles, and by some of the game show studies (e.g. Post et al. (2008), Gertner (1993)), where the behavior of the participants is observed over multiple game rounds (usually no more than 10 rounds). In this study, we are using a relatively large panel dataset with more than 35 observations for the average individual (and up to 270 for some individuals) which allows us to address these questions with much greater accuracy than it was possible in the past. On top of that, we observe actual choices of real market participants who risk losing their own money; neither subjects in experiments nor contestants in game shows ever experience actual losses, which casts doubt on whether they truthfully reveal their risk preferences. 6 In addition, experiments (but not game shows) often involve very small stakes and are therefore not very informative about preferences with respect to choices that could have a big effect on one s wealth. The findings of this literature suggest that risk preferences do not remain constant over time but they are in large part affected by previous outcomes. Thaler and Johnson (1990) find that when people are faced with sequential gambles, they are more willing to increase risk after having experienced prior gains. This behavior is labeled the house money effect, reflecting the idea that after a prior gain, losses are coded as reductions in this gain, as if losing the house s money is less painful than losing one s own cash. Gertner (1993) and Post et al. (2008) obtain similar results when studying the behavior of television game show contestants. 6 Game show contestants may experience paper losses during the multiple game rounds; however, they will never have to pay money out of their own pockets. This is less than ideal in the context of behavioral theories, where the framing of outcomes matters, and in this case all possible outcomes are evaluated as gains relative to the status quo. In addition to that, a selection bias is introduced in these studies due to the fact that losses are observed only for people that have previously experienced paper gains. 7

9 The second strand of related literature employs regression analysis to examine the dynamics of investors portfolios in financial markets (e.g. Brunnermeier and Nagel (2008), Calvet et al. (2009), Chiappori and Paiella (2011)). These studies utilize household-level data on income and asset allocation to investigate how the share of wealth that households allocate to risky assets responds to changes in their wealth. 7 The evidence is mixed: while some of the studies suggest that wealth shocks do not significantly affect the risky share, indicating that risk preferences are not time varying (e.g. Brunnermeier and Nagel (2008), Chiappori and Paiella (2011)), other studies find a significant positive elasticity of the risky share with respect to financial wealth suggesting that investors exhibit decreasing relative risk aversion (Calvet et al. (2009)). An important advantage of this literature is that the population under study consists of investors in regular financial markets and so the reported findings could give us direct insights about the formation of asset prices in these markets and potentially help us understand various puzzling phenomena observed at the aggregate level. However, without knowing investors subjective beliefs about asset returns as well as their investing horizon, it is impossible to construct the exact lotteries that each investor takes through his investing decisions, i.e. the subjective probability distribution of the returns of his portfolio. This constraint has made the estimation of a specific structural model of preferences particularly troublesome and thus this literature has hitherto focused solely on reduced form specifications which test whether fluctuations in wealth generate time-varying risk aversion. One complication of this approach, however, is that it becomes necessary to properly account for several other mechanisms that could affect the observed relation between the risky share and wealth (e.g. borrowing constraints, portfolio insurance, etc.). 7 Other datasets that have been used to study the trading behavior of individual investors over time include discount brokerage accounts (e.g. Odean (1998)) as well as 401(k) accounts (e.g. Choi and Madrian (2004)). 8

10 The current study makes a first attempt to bridge the gap between these two strands of literature. On the one hand, our sample of online sports bettors might not be representative of the population of stock market participants and so our findings could not be directly extrapolated to describe individual behavior in financial markets (see Section 6 below). On the other hand, however, two features of sports betting decisions allow us to take a more structural approach towards the investigation of individual risk preferences: i) sports bets have an observable payoff at the conclusion of the relevant sporting event, and ii) the win probabilities implied by the quoted odds are a good approximation for people s subjective probabilities regarding these terminal payoffs. Therefore, contrary to the stock market, in our setting it is possible to construct a reasonably good approximation to the distribution of the lottery that people choose when placing their bets. At the same time, the fact that sports betting markets share significant similarities with financial markets, gives us some confidence that our setting provides an idealized laboratory setting for the empirical analysis of the risk preferences of typical financial markets participants. We therefore view the current study as complimentary to both strands of literature. 3 Data The sports betting data used in this study were obtained from a large online sports bookmaker whose clientele comes primarily from Europe. 8 The dataset includes detailed individual-level information about all bets placed on a variety of sporting events by 100 randomly selected customers of this bookmaker. The sample period extends from October 2005 to May In particular, for each bettor and each bet that was placed since the bettor s initial registration in the sportsbook we observe the 8 The dataset used in this study is the same as that used in Chapter 1 of the dissertation, so we refer the reader to Section 3 of that chapter for a more complete discussion. In this section, we summarize the most important features of the data. 9

11 following characteristics: 1. Bet Characteristics: i) the date on which the bet was placed; ii) the exact event description on which the bet was placed (e.g. final outcome of match m between teams A and B that will take place on mm/dd/yy); iii) the outcome of the event chosen (e.g. home team win/draw/away team win, over/under 2.5 goals scored, etc.); iv) the bet amount; v) the bet type (e.g. single, double, treble, etc.); vi) the odds associated with the outcome selected and vii) the result and final payoff of the bet. A wide variety of sports and events have been selected but the majority of bets have been placed on the final outcomes of soccer matches. Most bets are placed at most one day before the sporting event takes place. The odds associated with the various outcomes are quoted by the bookmaker several days before the event and represent the payoff of a unit wager on the respective outcome (e.g. odds 3/1 imply that the bettor will earn AC3 for every AC1 staked). Bettors usually place a number of multiple bets simultaneously, i.e. they first combine a number of events under a specific type of combination bet (e.g. in a double bet two independent events are selected and the bettor earns a positive payoff if both of his selections win) and then place a number of combination bets simultaneously. Figure 1 provides an example of the betting slip that bettors usually submit at the end of a play session in the online sportsbook and that summarizes all information regarding all bets placed during this visit. Figure 2 shows how the odds are calculated by the bookmaker for the combination bets included in the betting slip. 2. Bettor Characteristics: i) gender; ii) date of birth; iii) country of residence and iv) zip code of area of residence. We match the gender, age and zip code information with census data from each bettor s country of residence and create proxies for each bettor s i) expected annual labor income; ii) expected educational level and iii) expected family situation. The typical bettor in our sample 10

12 is male, around 35 years old with an average annual labor income across zip codes of around AC19,000. Table 1 provides summary statistics for the characteristics of bettors and their selected bets. We now briefly explain how we use the bet information we observe, to represent the risky gambles selected by each individual in our sample. As noted above, bettors submit a number of combination bets per play session so we would like to calculate the probability distribution (lottery) of all possible prize winnings of all bets placed during a given play session. This is achieved in three steps. First, we create a lottery representation for all single bets selected by each bettor. Single bets have two possible prizes: the bettor either loses his stake or wins an amount equal to the amount he bet multiplied by the bookmaker s odds. The probability attached to each of these prizes can be inferred from the odds associated with the selected outcome and the commission (profit) that the bookmaker earns in the specific event. Second, we create a lottery representation for all combination bets by computing all possible outcomes and associated probabilities based on the definition of the specific type of combination bet. For instance, in a double, treble, etc. bet, there are two possible prizes and the probability of winning is equal to the product of the probabilities of each separate event involved. In the final step, we create a lottery representation of the play session, where this is defined as all bets placed by an individual over a period of one day. 9 We thus represent the day lottery by constructing all possible combinations of payoffs of all the bets chosen on the same day, and calculating their corresponding probabilities. 10 Following these manipulations, the final sample consists of 3,755 day lotteries chosen by the 100 bettors over a period of around 3.5 years. The average 9 The choice of the day window is supported by the fact that the majority of the bets in our sample are placed one day before the actual event date. 10 We identify days involving bets on related outcomes and properly deal with the majority of them. We drop days involving related bets for which we did not know the correlation among the win probabilities of the related selections and could thus not construct the day lottery. 11

13 number of day lotteries per year is 21 which implies a bet frequency of around twice per month. Finally, the average across individuals cumulative losses earned over a period of one month is AC2.4 with a standard deviation of Table 2 reports summary statistics of the day lotteries chosen by the bettors in our sample. A typical day lottery contains 2 prizes, has negative expected value ( 2.2), high standard deviation (97.56) and positive skewness (2.10). The bet amounts range from AC0.01 to AC1,768 with the median bet amount being equal to AC10. The maximum prize of the day lotteries ranges from AC0.24 to AC12,000 with the median maximum prize equal to AC62. It is important to note that there is a multitude of day lotteries available in the sportsbook that have been chosen by the bettors in our sample: by combining a wide variety of odds, bet types and bet amounts, bettors can select anything from a nearly safe lottery that returns a tiny payoff with probability 0.99 to a highly skewed lottery that returns a very high payoff with probability Interestingly, we note that only around 10% of the chosen lotteries have very high positive skewness (over 10) and about the same proportion of lotteries are negatively skewed; and that only 4.5% of the chosen lotteries are close to 50:50 bets to win or lose some fixed amount. But as emphasized in Chapter 1 and as can clearly be seen in Table 2, the lotteries considered in this study are much more varied than those faced by individuals in existing studies, where subjects typically choose among few lotteries, while the lotteries they do choose from invariably have a very small number of outcomes (usually two to four) and the stakes are usually limited to less than $50. 4 The Empirical Model In the empirical analysis, we employ a discrete choice framework to analyze bettors choices from a set of mutually exclusive alternatives. Therefore, an essential element of our modeling approach is the specification of the set of alternatives among which 12

14 choice is exercised. In our setting, the true set of alternatives should ideally be the set of all day lotteries considered by each bettor on each bet day. However, it is difficult to determine what constitutes a feasible alternative for any particular individual on any bet day. Therefore, the true choice set is a latent construct to us since nothing is observed about it except for the chosen alternative. 11 Moreover, given the large number of day lotteries that can be constructed from the bets available at any time in the sportsbook, the number of elements in the true choice set is immense. We therefore follow the literature in discrete choice analysis and try to reasonably approximate this universal choice set: specifically, we choose a set of representative lotteries from the set of all day lotteries chosen at least once by any bettor in our sample. In particular, we first reduce the number of alternatives in the choice set by clustering the 3,755 observed day lotteries in 100 clusters, using a hierarchical agglomeration algorithm. Then, we choose the most representative lottery from each cluster, i.e., the lottery that has the smallest mean Wasserstein distance to the other members of the cluster. Finally, we assume that the choice set faced by a bettor on a given day includes the lottery chosen on that day and the 100 lotteries that are most representative of all day lottery choices observed in our sample, augmented by the safe lottery that represents the option not to place any bets on that day. After we specify the choice set, we set up a multinomial discrete choice model of bettors choices and then employ a Markov Chain Monte Carlo approach to estimate it, similar to that used in Chapter 1 of this dissertation. In what follows, we briefly describe the most important general elements of the econometric model, and then we focus on the elements that are specific to this study. 11 For existing studies facing a similar situation of a latent choice set, see, e.g., the study of consumers residential choice by Weisbrod et al. (1980), the study of travelers choice of destination by Daly (1982), and the study of household choice of telephone service options by Train et al. (1987). 13

15 4.1 Random Utility Model We assume a random utility model with random coefficients (mixed logit), where the utility U njt that individual n obtains from choosing lottery j on day t is divided into a deterministic component V njt and a random component ε njt as: U njt = V njt + ε njt where V njt = H (ρ n, P njt ) is a utility function of the probability distribution over the set of monetary prizes of lottery j, P njt, and a vector of individual taste parameters, ρ n, to be estimated from the data. The mixed logit specification assumes that: i) the error terms ε njt are i.i.d. Gumbel-distributed with location parameter η n and scale parameter µ n and ii) the utility parameters ρ n vary over individuals in the population with density f (ρ n θ), which we assume to be the normal density. Under these assumptions, the mixed logit probability L (y njt ) that lottery j is chosen by bettor n on day t out of the set of alternative lotteries C is given by: where ˆ L (y njt θ) = L (y njt ρ n ) f (ρ n θ) dρ n (1) ρ n L (y njt ρ n ) = t e knv njt (2) {e knv nit } i C is the standard logit choice probability with k n being the relative precision of the error. Apart from the lotteries chosen conditional on play, we also observe the frequency with which individuals place their bets. Clearly, both the play frequency and the actual lottery choices are informative of bettors risk preferences. The play frequency is informative of i) the frequency with which bettors have the opportunity to play, which depends on exogenous factors, e.g. the amount of free time available to them and ii) 14

16 the frequency with which bettors accept this opportunity which depends on their risk preferences. To take into account the confounding effect of play frequency, we assume that bettors decision-making process can be partitioned into three sequential stages: First, with probability p nt and 1 p nt respectively, bettor n gets or does not get the opportunity to play on day t. This opportunity to bet or not is determined by an individual s demographic characteristics (i.e., gender, age, family situation, etc.) and time covariates (i.e., day of the week, etc.). Second, an individual with the opportunity to bet, decides whether or not to do so: if he accepts it, he chooses a lottery from the set of alternatives; if he rejects it, he essentially chooses the safe lottery. Letting y n = (y n1,..., y ntn ) be the sequence of daily lottery choices we observe for individual n and x n = (x n1,..., x ntn ) be a set of dummy variables indicating whether we observed play on a given day or not for individual n, the conditional probability of observing data {x n, y n } is given by L (x n, y n ρ nt, p nt ) = T n t=1 {x nt p nt L (y nt ρ nt ) + (1 x nt ) [p nt L (y nt = 0 ρ nt ) + (1 p nt )]}, (3) where probabilities L (y nt = 0 ρ nt ) and L (y nt ρ nt ) are the standard logit choice probabilities as given by Equation 2. The probability p nt is treated as a random parameter to be estimated from the data together with the vector of utility parameters ρ n. According to Equation 3, if we observe no bet on a given day, i.e. x nt = 0, this could be either because the bettor did not have the opportunity to bet on that day (which happens with probability 1 p nt ) or that the bettor had the opportunity to bet on that day but rejected it and chose a safe lottery (which happens with probability p nt L (y nt = 0 ρ nt )). Equation 3 is the likelihood function (not logged) of the observed lottery choices and the observed frequency of playing. 15

17 4.2 Dynamic Choice Cumulative prospect theory (CPT) proposed by Tversky and Kahneman (1992) provides a natural framework for the analysis of gambling choices. Many authors have pointed out the suitability of this theory for explaining why people exhibit a preference for positively-skewed gambles, e.g. purchase of lottery tickets. Besides being able to explain the popularity of bets on longshot outcomes, Barberis (2010) shows that prospect theory can also explain the popularity of 50:50 casino bets and that it captures many of the observed features of gamblers behavior. CPT was reviewed in detail in Section 4 of Chapter 1. We repeat here that according to CPT, an agent evaluates lottery (p m, m m ;... ; p 0, m 0 ;... ; p n, m n ) where m m... m 0 = 0... m n by assigning it the value: π i u (m i ), i where w + (p i p n ) w + (p i p n ) for 0 i n π i = w (p m p i ) w (p m p i 1 ) for m i < 0 (4) where w + ( ) and w ( ) are the probability weighting functions for gains and losses, respectively, and u ( ) is the value function. The value function and probability weighting function proposed by Tversky and Kahneman (1992) take the following functional forms: ((W + m) RP ) a for W + m RP u (m) = λ (RP (W + m)) a for W + m < RP (5) and w (P ) = w + (P ) = P γ (P γ + (1 P ) γ ) 1 γ (6) 16

18 where RP = W is the reference point that separates losses from gains, a (0, 1] measures the curvature of the value function, λ 1 is the coefficient of loss aversion, and γ [0.28, 1] measures the curvature of the probability weighting function. 12 To reduce the number of free parameters, in our estimation we restrict the curvature of the probability weighting function to be equal in the domain of gains and losses. The original version of CPT presented above pertains to static choice situations motivated by experimental evidence from elementary, one-shot gambles. Since there is not much evidence on how people behave when faced with sequential gambles, it is not clear how CPT could be extended to apply in a dynamic setting. Tversky and Kahneman (1992) themselves propose that there are two different ways in which prospect theory could be used to describe individuals behavior in dynamic choice situations: The first way is to assume that people have no memory or equivalently that previous gains and losses are immediately incorporated into the reference point that people use to evaluate the outcomes of successive gambles. The second way is to assume that people do have memory and therefore the reference point relative to which prospects are evaluated does not update completely to incorporate previous bet outcomes but rather remains sticky to some previous level resulting in prior outcomes being integrated with current payoffs. In this case, after incurring a gain, subsequent losses are edited by people as a reduction in this gain rather than as a loss, and similarly, after incurring a loss, subsequent gains that are not large enough to make up for the initial loss are edited by people as losses rather than as gains. Formally, we express this idea by using a variant of the cumulative prospect theory specification described above. Since the framing of outcomes affects the reference point relative to which gains and losses are evaluated, it seems reasonable to allow 12 The main characteristics of this theory are that: i) Utility is defined over gains and losses relative to a reference point; ii) The value function u is concave (convex) in the domain of gains (losses), iii) The value function has a kink at the origin, capturing loss aversion; iv) The framing of outcomes can affect the reference point; and v) People evaluate gambles using transformed probabilities obtained by applying a probability weighting function on objective probabilities. 17

19 this reference point to depend on individuals prior betting performance. First, we create a measure of previous gains/losses experienced by bettor n on day t, denoted by CumP rofit nt, as the cumulative payoff of all the bets previously placed by bettor n that were settled from one month before the current bet day up to one day before the current bet day, i.e., from t 30 up to t 1. Then, we allow the reference point RP nt that bettor n uses to evaluate the payoffs of the lottery of day t to be a convex combination of bettors wealth with and without the payoffs of previous bets, i.e. RP nt = δ n W nt + (1 δ n ) (W nt CumP rofit nt ) where W nt is the wealth of bettor n in the beginning of day t including all gains/losses incurred during the last month up to day t, and δ n [0, 1] is an individual-level parameter to be estimated, that measures the stickiness of the reference point for bettor n. When δ n = 1, the reference point is equal to the current wealth level, i.e. it is instantaneously updated to incorporate the outcomes of all previous bets as the version of CPT with no memory suggests. When δ n = 0, the reference point remains sticky to some previous wealth level, i.e. it does not update completely to incorporate previous bet outcomes as the version of CPT with memory suggests. Apart from the dependence of the reference point on previous betting performance, 13 The monthly window for the previous performance variable is arbitrarily chosen guided by our intuition and given the observed frequency of playing. The specific time frame is not crucial since we are primarily interested in testing the null hypothesis of no path-dependence in people s behavior under a reasonable definition of previous betting performance rather than searching for the definition that better fits the observed choices. Having said that, it would be interesting to examine the robustness of our results under alternative definitions in the future. 14 Note that this variable is just a proxy of bettor n s cumulative gains on day t. First, it is dayspecific rather than bet-specific creating a problem if some bets placed on day t were placed after some previous bets placed on the same day have already paid off. Since we only observe the dates on which bets were placed but not their exact times, this problem cannot be resolved. Second, a bet is considered settled when all uncertainty about it has been revealed, i.e., we know the outcome of all the matches it involves. However, the earliest time at which the payoff of a bet is known is not necessarily the conclusion of the last game involved in it, i.e., the bettor might already know his payoff if he knows the outcome of some, not all, of the games that he has combined in the same bet. For example, in the case of a double, if someone loses the bet on the first match, he knows he has lost the bet no matter what happens in the second match or in the case of a doubles on 3 games, he knows he has won at least some amount of money if he wins 2 of the matches. 18

20 the richness of our dataset allows us to consider an additional, previously unexplored, channel through which prior outcomes could affect bettors subsequent risk-taking. In particular, we consider an extension of standard CPT in which the loss aversion parameter λ and the curvature of the value function α of bettor n on day t are allowed to be a linear function of earlier bet outcomes, i.e. λ nt = λ n0 + λ n1 CumP rofit nt α nt = α n0 + α n1 CumP rofit nt where λ n0, α n0 are individual-specific intercepts and λ n1,α n1 measure the sensitivity of λ and α to previous betting performance. In addition to the dependence on previous bet outcomes, in future research we are planning to explore more types of possible time dependence in bettors behavior. For instance, it would be interesting to allow utility parameters to vary with the GDP of bettors country of origin representing the effect on individuals preferences of aggregate macroeconomic conditions. Furthermore, possible time dependence above and beyond the effect of economic growth and business cycle fluctuations could be captured by a set of time dummies. Finally, we can introduce state dependence in terms of outstanding bets (i.e. bets that have not paid off before the current day lottery is chosen), for example by allowing risk preference parameters to be a linear function of the characteristics of these bets (e.g. mean, variance, skewness). 4.3 Estimation To estimate the posterior density for the latent parameter vector β n = ( ρ n p n ), we employ a hierarchical Bayes model and estimate it using Markov Chain Monte Carlo (MCMC) methods. In particular, we seek the posterior distribution for the utility parameter vector ρ = (ρ 1,..., ρ N ) and the exogenous probability of having 19

21 the opportunity to bet p = (p 1,..., p N ). The vector of individual-specific parameters associated with the CPT specification that includes path-dependence is ρ n = (k n, α n0, α n1, λ n0, λ n1, γ n, δ n ) where k n is the scale of the utility, α n0 and λ n0 are the utility curvature and loss aversion respectively at zero level of previous bet gains/losses, α n1 and λ n1 are the sensitivity to previous bet gains/losses of the utility curvature and loss aversion respectively, γ n measures the curvature of the probability weighting function and δ n measures the degree to which the reference point that separates gains from losses is affected by previous bet outcomes. Selected parameters are allowed to vary with bettors demographic characteristics and variables that capture possible state dependence on bettors risk preferences, such as previous betting performance. Therefore, the elements of β n = ( ρ n p n ) can be expressed as: β n = β 0 + Z nβ 1 + G nt β 2, where Z n is a vector of individual specific attributes and G nt is a vector of individual and time varying covariates. We restrict the effect of the demographics to be fixed, i.e., the same across individuals, and allow the intercept β 0 and the sensitivity to state dependent variables β 2 to vary randomly in the population. Hence, the parameter vector β to be estimated from the data contains a deterministic set of parameters, denoted by ˆβ and a random set of parameters, denoted by β n. We now briefly describe the steps involved in the Bayesian estimation of the multinomial mixed logit. First, we need to specify the prior distributions for the parameters to be estimated. We adopt a hierarchical normal prior for the random parameter vector β n, i.e., we assume that β n b, W N (b, W ), where the population-level parameters b and W follow a multivariate normal-inverse 20

22 Wishart unconditional prior distribution, i.e. b W N (b, W/k) and W 1 W ( v, V 1). 15 Second, we derive the posterior distribution of β n, b, W, and ˆβ, conditional on the ( { } ) N observed data {X, Y }. The joint posterior distribution K βn, b, W, ˆβ X, Y n=1 will be proportional to the product of the likelihood function and the prior densities. Finally, we use simulation procedures to calculate the mean of this posterior distribution, i.e. we first take draws from K βn, b, W, ˆβ X, ( ) Y and then average these draws. ( Since K βn, b, W, ˆβ X, ) Y does not belong to a known family of distributions, we use the Gibbs Sampler algorithm (Gelfand and Smith (1990)) to take draws from it. A detailed description of the Gibbs sampler is provided in the Appendix of Chapter 1 of this dissertation. 5 Estimation Results In this section we present the estimation results of the CPT specification in the presence of state-dependence in individual risk preferences. 5.1 Preliminary Regression Analysis Before proceeding with the structural estimation of the parameters of the CPT specification with state-dependence described in Section 4.2, it would seem natural to start our investigation of the effect of previous betting performance on subsequent risk-taking by conducting some preliminary regression analysis. The results of this analysis, although by no means conclusive, could give us a first indication on the presence of state-dependence in the risk-taking behavior of the typical bettor in our sample. We therefore estimate two models, one for the decision to play in the sportsbook or not, and one for the characteristics of the lotteries chosen given that the bettor 15 The prior hyper-parameters v,v,b and k have been chosen as weak as possible so as to let the data determine the posteriors. 21

23 has decided to play. We shall henceforth refer to the former as the participation model and to the latter as the lottery-choice model. We are interested in how both the participation decision and the lotteries chosen are affected by bettors previous betting performance measured by their cumulative gains over the period of one month (CumP rofit nt ). The participation model is given by: Bet nt = X nt β + ε nt, where Bet nt is an indicator variable for whether bettor n has placed at least one bet on day t or not and X nt is a vector of observable covariates including the variable of interest CumP rofit nt as well as individual fixed effects. The lottery-choice model is given by: Lottery nt = X nt α + v nt, where Lottery nt is the characteristic of the lottery chosen by bettor n on day t, e.g. mean, variance, skewness, etc., and X nt is defined as above. It is important to note here that the variable Lottery nt is observable not only conditional on the bettor having decided to bet in the sportsbook on a given day, but also for all other days of an individual s betting history, where a safe lottery has been selected on days on which we observe no play. Therefore, we do not have to worry about the classic selection bias problem often encountered in this type of analysis and we can estimate the two models independently. Table 3 presents the results of the pooled probit regression (top panel) and the pooled OLS regression (bottom panel) of the participation model and the lotterychoice model respectively. We observe that previous betting performance significantly affects both the participation decision and the characteristics of the chosen lotteries. In the participation model, we find that the variable CumP rofit nt has a significant positive coefficient implying that bettors are more likely continue to play after prior 22

24 gains than after prior losses. In the lottery choice model, we observe that previous performance significantly affects the expected value, the variance and the stake of the gambles that bettors subsequently choose. In particular, the results suggest that bettors who have experienced previous gains (losses) subsequently increase (decrease) their stakes and select lotteries that have lower (higher) variance and expected value. Finally, it should be kept in mind that this preliminary analysis does not properly account for the confounding effect of play frequency, which affects the magnitudes of the parameter estimates in the participation model. Having said that, the results of this section should be interpreted only as a first indication that the null hypothesis of no state-dependence in the risk preferences of the average bettor is rejected. In the following section, we structurally derive individual-level estimates for the various CPT preference parameters using the full econometric model which properly controls for the observed play frequency, and explore the implications of these parameters on the dynamics of risk preferences and how these depend on the gambles that bettors are subsequently confronted with. 5.2 Results Next, we turn our attention to the estimation of the full econometric model described in Section 4.2 of the CPT specification that allows for state-dependence. Figure 3 presents the posterior distribution of parameter δ which measures the stickiness of the reference point that bettors use to separate gains from losses. We observe that for the majority of bettors in our sample (around 80%) this parameter is estimated in the vicinity of 0 implying that the reference point does not update completely to reflect previous bet outcomes but rather remains sticky to some previous level as the version of CPT with memory suggests. On the other hand, less than 20% of the bettors seem to not integrate the outcomes of successive gambles and therefore completely update their reference point as the version of CPT with no 23