HETEROGENEITY IN RISK PREFERENCES: EVIDENCE FROM A REAL-WORLD BETTING MARKET∗ Angie Andrikogiannopoulou Filippos Papakonstantinou King’s College London King’s College London Abstract We develop a structural model of behavior that accounts for rich heterogeneity in individuals’ risk pref- erences, and we estimate it using a panel dataset of individual activity in a sports wagering market. The number and variety of lottery choices we observe enables us to distinguish the key features of prospect theory — value-function curvature, loss aversion, and probability weighting. Our Bayesian hierarchical mixture model enables us to estimate the proportion of distinct utility ‘types’ characterized by the different prospect theory features, and thus to evaluate their relative prevalence in the population. We find that utility curvature alone does not explain the choices we observe and that, while loss aversion is important, probability weighting is the most prevalent behavioral feature of risk attitudes: Two-thirds of individuals exhibit loss aversion, but all exhibit probability weighting. Keywords: Risk Preferences, Prospect Theory, Loss Aversion, Probability Weighting, Discrete Choice, Mixture Model, Hierarchical Model, Bayesian Estimation ∗For helpful comments, we thank participants at the 2014 European Economic Association Annual Congress and the 2018 Center for Economic Analysis of Research Workshop, and seminar participants at Cambridge University, Einaudi Institute for Economics and Finance, Imperial College London Business School, McGill University, Oxford University, Princeton University, the Stockholm School of Economics, the University of Amsterdam, the University of Geneva, and the Wharton School. Send correspondence to Filippos Papakonstantinou, King’s Business School, Bush House, 30 Aldwych, London WC2B 4BG, UK; telephone: +44 (0) 20 7848 3770. E-mail: [email protected]. Since Kahneman and Tversky (1979) proposed prospect theory, a plethora of experimental and field studies have shown that individual behavior deviates from the predictions of expected utility theory and exhibits patterns consistent with loss aversion and probability weighting. But even though prospect theory has become the most widely accepted behavioral theory of choice, the limited availability of rich data, particularly in the field, and modeling limitations have made it difficult to distinguish between the different prospect theory features and assess their relative importance and prevalence in the population. Hence, an important question remains unresolved: which of prospect theory’s features are most useful in explaining behavior? In this paper, we aim to address this question in the field by combining (1) a panel dataset that tracks individuals’ choices in a sports wagering market with (2) a hierarchical structural model that allows for an extensive representation of individual heterogeneity within and across utility types characterized by the different prospect theory features, and (3) a Bayesian algorithm designed to estimate this model. The sports wagering market is an advantageous natural laboratory for risk preference estimation. First, it features choices whose outcomes (i) are determined exogenously by match outcomes, and (ii) are associated with probabilities that are predicted quite accurately by the odds quoted in the market. These features allow for a lottery representation of choices and hence facilitate preference estimation. Second, this market features a variety of lotteries consisting of a wide range of probabilities and prizes, including gains and losses as well as small and large stakes, which enables us to identify and comprehensively study all features of prospect theory. Third, in this setting, individuals typically make over time a large number of choices, which enables us to estimate preference heterogeneity across individuals with improved accuracy. To estimate preference heterogeneity across individuals, extant studies have followed two main ap- proaches. Most commonly, each individual’s preferences are estimated in isolation; but this may lead to inaccurate estimates, especially when only a few choices per individual are observed. As a result, some recent studies have combined individual information with information from the population distribution, which is estimated under specific distributional assumptions (e.g., discrete or normal). This approach allows for information sharing across individuals, but in a restricted manner. For example, it is inappropriate to assume a normal population distribution if individuals cluster into distinct sub-populations. In this paper, we propose a hierarchical mixture model that generalizes previous methodologies. First, we model the population as a flexible mixture of four sub-populations: one type that exhibits neither probability weighting nor loss aversion, two types that each turn on one of these features, and an unrestricted type that exhibits both features. This approach introduces the proportions of each type as model parameters, which are jointly estimated 1 with the entire distribution of preference parameters, and allows us to assess the relative importance of the prospect-theory features. Second, we allow for preference heterogeneity within each sub-population. Thus, we forgo the restrictive assumption that there is a representative agent for each type, which would not only understate the heterogeneity but would also bias the preference estimates (and the type allocations). Overall, our methodology enables us to share information across individuals in a flexible manner dictated by the data and improves inference at both the individual and population level, as what we learn about each individual’s preferences and type allocations feeds into the shape of the population distribution, and vice versa. To estimate our model, we design and implement an appropriate Bayesian Markov chain Monte Carlo algorithm.1 Our model of behavior is based on the cumulative prospect theory (CPT) of Tversky and Kahneman (1992), according to which individuals use a value function defined over gains and losses relative to a reference point, have different sensitivity to losses and gains, and systematically distort event probabilities by weighting the gain/loss cumulative distribution. Our average preference parameter estimates are within the range of those reported in the experimental literature (see Wakker, 2010): the value function curvature is moderate, loss aversion is mild, and individuals significantly overweight extreme positive and negative outcomes. Furthermore, we find that utility curvature alone does not adequately explain individual choices, and that the distinctive features of prospect theory — loss aversion and probability weighting — offer a significant improvement in explaining observed behavior. But, importantly, our study reveals that these two features are not equally prevalent. We find that individuals can essentially be partitioned into two groups: about two-thirds exhibit both behavioral features, and one-third exhibit probability weighting but not loss aversion. These findings indicate that prospect theory should not be viewed as a monolithic theory with a set of features — loss aversion and probability weighting — that all individuals exhibit, but rather reality is more nuanced. This more nuanced view could facilitate and guide the use of prospect theory in applied models. For example, our finding that probability weighting plays a more important role than loss aversion could be useful in finance, where loss aversion has sometimes led to counterfactual predictions for individual investor behavior and for the aggregate stock market (see Barberis, 2013 for a review). Indeed, in an application we show that our preference estimates make realistic predictions when applied to a standard portfolio choice problem, as they yield optimal portfolios similar to those observed in surveys on household portfolio allocations. We conduct a number of robustness checks, necessitated by the fact that we study a field setting hence we 1This estimation solves for an equilibrium density — the joint of the model parameters — which is roughly a stochastic analogue of a fixed point. Our estimation uses a Gibbs sampler, which is the stochastic analogue of an algorithm that solves for a fixed point. 2 do not have the benefit of experimental control. Regarding individuals’ beliefs, our baseline is the standard assumption in the literature that they are rational which, in our setting, can be approximated quite well by the probabilities implied by the market prices. In a sensitivity analysis, we show that our results are robust to al- ternative approximations of the rational beliefs, as well as to small but significant systematic deviations of the subjective from the rational beliefs. We also show that observed bets are unlikely to be driven by (real or per- ceived) superior information or affinity toward specific teams (e.g., home-area teams), since almost all individ- uals place wagers on a large variety of leagues and teams rather than any specific one, and virtually no individ- uals generate significantly positive returns from betting. Regarding the choice sets individuals face, our base- line assumption is that individuals consider all lotteries available in the sportsbook, except for those that have rarely been chosen by any individual in our sample. In a robustness check, we consider an alternative choice set containing only lotteries that respect a tight, but reasonable, budget constraint of a few hundred euros. The