Explaining the Favourite-Longshot Bias in Tennis: An Endogenous Expectations Approach

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1 Explaining the Favourite-Longshot Bias in Tennis: An Endogenous Expectations Approach Andrew Zeelte University of Amsterdam, Faculty of Economics and Business A Master s Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of: Master of Science in Economics University of Amsterdam Under the Supervision of Professor Arthur Schram * June 2012 ABSTRACT Numerous explanations have been posed in an attempt to explain the robust empirical finding that a favourite-longshot bias exists in a large number of wagering markets. Attempts by economists have spanned a period of over fifty years, yet there still remains no consensus as to why returns to bets on favourites often yield higher expected returns than bets on longshots. This paper uses a large data set and a multitude of methods to test the relationship between bookmaker odds and monetary return in the market for men s professional tennis, and finds the presence of a robust positive longshot-bias. It is then demonstrated that a supply-side explanation involving the manipulation of prices by bookmakers in response to insider trading may provide a partial explanation to this bias. Finally, a unique approach is largely successful in finding that a utility function incorporating expected emotions of bettors, which are endogenously dependent on the expectation of winning, can provide an alternative explanation for the longshot-bias. * I am very grateful to Professor Arthur Schram, for his helpful comments throughout. Any remaining errors are my own responsibility. Longshots are defined as competitors for whom the bookmaker have set relatively high potential returns (betting odds ), as a result of their relatively low perceived probability of winning. Conversely, favourites are defined as competitors with a relatively low potential return, as a result of their relatively high perceived probability of winning.

2 I. INTRODUCTION The favourite-longshot bias (hereon denoted FLB) is a widely proposed empirical regularity which has shown that betting odds on a variety of sports provide biased estimates of winning. The seminal paper by Griffiths (1949) and the majority of those that have followed have argued that given the objective probabilities of winning, longshots are overbet whereas favourites are underbet. The result of this bias is that a betting strategy on favourites often yields significantly higher returns than a betting strategy focusing solely on longshots (Ali 1977, Dowie 1976, Snyder 1978). The argument advocating the presence of a FLB is made more convincing given that similar biases exist within other financial markets. For example, in the stock market a type of negative-longshot bias exists, where a relatively high expected return to assets is associated with relatively high-risk stock investments, yet investors have continued to invest in lower risk government bonds. This has led economists to attempt to explain the behaviour manifested in the so-called equity premium puzzle in these financial markets (Mehra and Prescott, 1985). With widespread evidence discovered in a variety of markets, the apparent presence of the FLB and its underlying causes should therefore motivate studies into the sports market to focus on the following question: Why do individual bettors persist in betting on longshots when expected returns are lower relative to a strategy of betting on favourites? Classical economic theory suggests that individuals should seek to maximise their own expected monetary payoff, yet the findings of the FLB conflict with this statement as bettors appear to consistently make decisions that to do not satisfy this maximisation condition. Attempts to answer the stated research question are therefore important for economists to achieve a better understanding of individual decision making under uncertainty and risk. The importance of this topic is clear to see when one looks at the numerous previous attempts of economists to reconcile the behaviour behind the FLB. Existing theories can be divided into three categories. The first group of theories contain the local risk preference in a representative agent models (Weitzman 1965, Ali 1977, Quandt 1986). Representative agents in these models maximise an expected utility function, which is solely dependent on the monetary payoff and the probability of winning. Unlike the classic utility of money curve these models propose sections of different curvature, and hence differing marginal utilities of money (an idea first 1

3 proposed by Friedman and Savage (1948)) to reconcile both the initial decision to bet and the decision to bet on longshots. The second group of theories originated from the ideas of Thaler and Ziemba (1988), and state that concepts from behavioural economics such as mental accounting and prospect theory can be used to explain the FLB. It could be possible that one of the foundations of prospect theory (Kahneman and Tversky, 1979), the misperception of probabilities, explains individual behaviour consistent with the FLB. The third group of theories provides a completely different supply-side explanation to the FLB. In essence, these models suggest that bookmakers deliberately trim the odds of longshots for particular events when they believe a bettor has insider information on a longshot (Shin 1992). Results emanating from the racetrack market provide evidence that the FLB is therefore mitigated in high-profile handicap races, where insider trading is less likely to be an issue (Vaughan Williams and Paton, 1997). The collective limitations of these papers, which are discussed in more detail in Section III, have left a requirement for different potential explanations of the FLB to be explored which may advance our understanding of the forces that determine market prices (Sauer, 1998:2062) and therefore provide a better explanation of the behaviour behind the FLB. The gap in the literature which this paper seeks to fill can be divided into two parts. Firstly, this paper involves a study into the presence of the FLB in the sports betting market for men s professional tennis. In comparison to other sports, in particular horse racing, relatively few studies have been conducted in this market. In fact only one study into the FLB has been completed in the market for men s professional tennis (Forrest and McHale, 2007). With a significantly larger data set now available the FLB can be tested for with more certainty. Furthermore, the paper by Forrest and McHale (2007) provided a test for the supply-side explanation of the FLB through comparing returns in Grand Slam and non-grand Slam Tournaments, based on the relatively higher profile nature of Grand Slam events. This paper provides an alternative approach of comparing returns in the Early and Late rounds of tournaments, based on the relatively higher profile nature of the latter rounds of a tournament. By comparing the results of these contrasting tests, there is scope for a more informed conclusion on the success of supplyside models. Secondly, as Section III will also discuss in more detail, none of the existing theories provide entirely convincing explanations for the FLB regardless of the specific market 2

4 being studied. One of the main reasons for this is that a number of proposed theories normalise uniform emotional responses to losses across all bets (Snowberg and Wolfers, 2007:729, Jullien and Salanie, 2000:506)). This suggests that bettors obtain the same emotional response from losing a bet independent of whether it was placed on a favourite or an outsider. Using the model of reference-dependent preferences proposed by Koszegi and Rabin (2006), and the psychological evidence presented on emotionbased choice by Mellers, Schwartz and Ritov (1999), there is a strong argument that such an assumption is consistently violated in an individual s actual utility function for losses. A similar argument for winning bets and utility gains will also be made. The idea proposed in this paper attempts to solve the problems of the existing models through introducing an endogenous-reference point based on the expectation of winning, a new approach that current theories have not included in their analyses. The method of studying the FLB in tennis involves treating the wagering market as a simple generic financial market. This approach is supported by Jullien and Salanié, who claim in their study on the FLB that sports markets can be considered as a test bed for alternative theories of behaviour under risk (2000:504). In the UK market, bookmakers quote odds (prices) which represent the potential returns to one-unit bets, making it analogous to a one-period financial market. As Gabriel and Marsden (1990) note, both types of markets involve a large number of investors (bettors) that all have access to widely available information. As well as this, investments (bets) in both types of markets entail decisions under uncertainty and risk, which predictably results in investors having heterogeneous beliefs (Levitt, 2004:223). Once the investment is made, the return on that investment is uncertain. The vital difference is that the wagering market has the advantage of a well-defined termination point: bets are placed on a sporting event; the event takes place; and ex post empirical returns are observed. This makes studying behaviour with relation to financial returns in these markets a feasible target, and led Thaler and Ziemba (1988) to point out that in this respect wagering markets are better suited to tests for market efficiency and rational expectations than the stock market. Therefore, theories that are explored in this particular field may have important applications in other financial contexts such as the stock market, by providing a better understanding of individual behaviour under uncertainty and risk. A number of results emerge from the analysis. Analogous to findings in other markets, significant evidence for the presence of a positive FLB in the wagering market for men s 3

5 professional tennis is found. The subsequent analysis on the Early and Late rounds of tournaments appears to disagree with the findings of Forrest and McHale (2007) to a certain extent, providing some evidence consistent with supply-side theories that there is a difference in the FLB between relatively low and high profile tennis matches. Finally, an expected utility model incorporating expected emotions based on expectations of winning showed success in predicting the direction of behaviour manifested in the FLB. The remainder of this paper is organised as follows: Section II discusses the main characteristics of the tennis market and incorporates the motivation for choosing this particular sport as the focus of the analysis. Section III contains a discussion of the existing literature on the topic, and includes a review of existing theories on the FLB as well as those that influenced the proposed theory in this paper. Before the data analysis, the hypotheses encompassing this paper are introduced in Section IV. The beginning of Section V explains the data to be used in the succeeding analysis, followed by a series of tests for the presence of the FLB in the betting market for men s singles professional tennis in the years , finishing with a comparison to findings in other sports. In Section VI, the supply-side theory proposed by Shin (1992) is subject to statistical testing by comparing the extent of a FLB in Early and Late rounds of tournaments. The ideas considered in Section VII relate to the possibility that a utility theory of reference-dependent endogenous expectations may be successful in explaining the individual behaviour that causes the FLB. Finally, Section VIII provides a summary and discussion of the paper s findings. II. THE TENNIS MARKET Studies on the FLB have historically tended to focus on racetrack betting, though other sports markets such as the English Premier League, the NFL, and the AFL have also been investigated (Cain et al. 2003, Levitt 2004). In this paper, the betting market in men s professional singles tennis matches is chosen as the focus of analysis. Since only one paper in this area has been conducted to date (Forrest and McHale, 2007), this study provides a useful contribution to a largely unexplored market. As shall be discussed in Section III, the solitary existing paper uses a limited data set compared to what is now available and in its attempts to explain the FLB seems only to focus on a classical local risk preference approach, which is not a preferred explanation. 4

6 In addition to the limited work conducted in this particular wagering market, there are many advantageous characteristics of the tennis market that make it a favourable case in which to study the FLB. Firstly, the tennis market allows for analysis to be conducted on betting opportunities across almost the whole probability-odds range (from 0 to 1) within a single market. The data set being used has a significant amount of data for all different types of bets which makes any results regarding prices and returns more reliable. To demonstrate this in direct comparison to studies conducted within other sports, one can look at the paper by Cain et al. (2003) that studied the FLB in a number of different sports including baseball, boxing, cricket, horse racing, greyhound racing, soccer, and snooker. The distribution of prices in the markets that they studied is shown in comparison to the distribution of the data set in this paper in Table 1. The data suggests that the betting market for tennis is suitable to study the relationship between prices and returns because it has the best available distribution of prices. Even when ignoring the substantially large sample size (which itself is a big advantage), the difference between the odds ranges with the lowest and highest proportions of data is the smallest in the tennis market, which is a reflection of the relatively even distribution of the data. Secondly, there is an argument that suggests a sentiment effect in team sports has the potential to distort odds in sports markets (Avery and Chevalier, 1999). However, individual tennis players relatively rarely have a committed following of fans who bet on them in high volumes in comparison to certain team sports. For example, within the UK betting market this could be betting on England to win the World Cup in football, or backing your favourite Premiership side. Within the UK tennis market, the only conceivable sentiment effect would be the support in previous years for British players Tim Henman and Andy Murray during the Wimbledon tournament 1. With less distortion of starting prices due to these factors, the study of return-risk preferences amongst bettors can be conducted with more confidence (Forrest and McHale, 2007). Additionally, in singles tennis matches only two players compete, which provides a clear distinction between who is the favourite and who is the longshot. In contrast, the majority of horse races consist of a significantly higher number of competitors, making the distinction between favourite and outsider less defined. There are also additional potential distorting factors that may wield influence in racetrack betting, such as the 1 However, comparison of subjective probabilities implied by bookmaker odds and the objective probabilities of these players winning during the Wimbledon tournament shows no evidence of a sentiment effect. The objective probabilities are in fact higher than the subjective probabilities, going against the intuition of a sentiment effect. 5

7 Data Set Range of Prices Baseball Boxing Cricket Horses 1978 Greyhounds Soccer Snooker Current Data Set ,777 3, , , ,879 1,995 5, , , , , , ,601 > ,822 Sum 49, ,119 6,000 8,565 1,294 45,352 Proportion of Total Data Set (%) > Min Max Range Table 1: Table showing the distribution of prices in a variety of different sports betting markets. The table is adapted from the paper by Cain et al. (2003). 6

8 owner of the horse and the jockey, whereas tennis is preferably limited to two competitors that compete relatively frequently and whose performance is less dependent on any of these distorting factors. As has been previously mentioned the mitigation of distorting effects is an advantage to this particular data set. Furthermore, one could argue that bettors in the tennis market are likely to be better informed than those in other betting markets; in particular horse racing and greyhound racing. The history of these sports is largely based on the culture of gambling, with many people going to the racetrack for the purpose of social enjoyment, rather than with the sole of aim of making a profit. Indeed Ashton, referring as far back as the early 1600s, notes that in the early days of horse racing in England betting, almost immediately, attended the popularity of the sport (1968:176). With this in mind the extent of noise traders is likely to be higher in these particular markets. It can be argued that such a culture exists to a lesser extent in the tennis market, which implies bets placed in this market are more informed. The net result of this argument is that the pricing mechanism is likely to be more efficient than in some other sports markets. Forrest and McHale (2007) argue that any bias emerging in the tennis market as a result of noise traders is also relatively likely to be arbitraged away as transaction costs (the bookmaker s commission / over-round ) are much lower than in horse racing. Within this data set, the average commission in each match is only 7.3% 2, which is less than half the commission in horse racing (Forrest and McHale, 2007:755). A higher level of market efficiency relative to other sports, and noise that is largely arbitraged away due to low transaction costs, forms an important part of the basis of a reliable market in which to test for the presence of the FLB. The study of tennis matches also appears to eliminate the possibility of a last race of the day effect that some authors have proposed exists in the racetrack betting market (McGlothlin 1956, Ali 1977). This theory suggests that bettors disproportionately bet on longshots in the final race of a meeting in order to try and recoup losses made in previous races, and thus the rate of return on moderate longshots falls in these races (Snowberg and Wolfers, 2010). The presence of this phenomenon is by no means a certainty, but even assuming it does exist, it would clearly not apply to the tennis market for a number of reasons. During the tennis season there are a high frequency of matches, making any particular importance placed on the last match of one day seem unrealistic. In addition it is simply unclear what the last match of the day is in tennis, given that there are often numerous matches and even tournaments taking place at the same time. 2 Standard Deviation =

9 The final advantage of using the tennis market relates to issues of economic modelling in wagering market analyses. In many papers on this topic, this one included, a series of strong assumptions have to be made when carrying out tests on the FLB. For example, in the paper by Jullien and Salanié (2000) on British racetrack betting, three key assumptions on the decision process of bettors are necessary for their analysis: firstly, agents decide whether or not to bet; secondly, they decide how much to bet; thirdly, they decide on which horse to bet. Focusing on the third part of this process, analysis on the FLB requires the restriction of focusing on one win bet per race only. In racetrack betting, this ignores other important types of bets such as each-way betting (betting on a horse to place in one of the top three or four positions), exotic bets (which entail predicting the first two or three positions in a race) and spread betting (where bettors bet on more than one horse per race to spread their risk). Attempts have been made to analyse exotic bets, the paper by Jullien and Salanié (2000) being an example, yet this is not an issue the tennis market. Given there are only two players competing in each match, spread betting, exotic bets and each-way betting are not feasible strategies for a bettor. As a result, the assumption in this paper that bettors place only one win bet per tennis match gives a closer approximation of the true behaviour of bettors in comparison to studies in other markets. The beneficial aspects of studying this market compared to other sports that have been discussed provide the main motivation for choosing tennis as the focus of analysis in this paper. Although a number of strong modelling assumptions are required, these appear to be more reasonable than in some other papers and overall a study in this specific market seems to provide a useful contribution to the topic of the FLB. III. LITERATURE OVERVIEW The FLB was first noted by psychologists Griffith (1949) and McGlothlin (1956). With few exceptions, it has since been found consistently both in pari-mutuel and bookmaker (which is the focus here) markets (Dowie 1976, Henery 1985, Bird and McCrae 1994). However, although the FLB may be the most widely established empirical regularity in racetrack gambling, it certainly is not universal (Asch and Quandt, 1987). Understanding these anomalous results is important to developing a fuller understanding of the causes behind this empirical regularity. Until relatively recently there existed two broad groups of theories that attempted to explain the FLB, which shall now be the focus of discussion. 8

10 Advocates of neoclassical economic theory have argued that it is possible to reconcile observed gambling behaviour and the longshot bias through localised risk-loving preferences. One of the key proponents of such ideas can be found in the paper by Weitzman (1965), whose expected-utility hypothesis rests on three main pillars of assumptions. First, the personal utility of an individual in a risky situation is reliant only upon the probability of winning and the possible monetary reward. Second, in order to reconcile individual behaviour consistent with the FLB the resulting utility of money curve must contain sections of varying curvature depending on different economic levels. Weitzman turns to the work of Friedman and Savage (1948), and then Markowitz (1952) to present a utility of money curve shown in Figure 1, where the origin is not zero income but present or customary income (Weitzman, 1965:19). Utility (U) b a Money (m) Fig 1: Figure showing the money utility curve proposed by Markowitz (1952). Moving along the curve below the origin until point b and similarly to point a above the origin the curve displays increasing marginal utility. Past these points instead yields decreasing marginal utility, with the exact position of these inflection points depending on an individual s attitude toward risk. Such a utility function, Weitzman argues, can be used to account for the apparent contradictory attitudes toward risk as the simultaneous taking of insurance and participation in gambling (1965:19). With the proposed money utility curve, the third main assumption of Weitzman was a group of homogenous bettors with identical beliefs and risk preferences, simplified to the representative agent Mr. Avmart ( average man at the race track ). The derived money utility function of Mr. Avmart was then found to be strikingly similar to the Markowitz curve, suggesting that bettors must demonstrate in at least some range of decisions, a risk-loving attitude. A 9

11 number of related papers followed such as Ali (1977), also on American racetrack data. Ali argued that bettors with a utility function such as the one proposed by Weitzman are not rational (1977:811). By restricting betting opportunities to a single race, and again assuming bettors are sophisticated expected utility maximisers, Ali comes to a similar conclusion that the representative bettor (in this case Mr. B ) must be a risk lover who takes more risk as his capital declines. This conclusion also supported the last race of the day effect that was mentioned in Section II. However, Ali also accepts the same bias can be explained by risk neutral bettors who are not sophisticated and make estimation errors. The paper by Quandt (1986) also demonstrates that the FLB is an equilibrium condition when bettors are risk lovers with mean-variance utility functions, but also accepts that in reality, other motives may well be present and may also explain the observed regularity (1986:206). Golec and Tamarkin (1998) take a different view, claiming that data on racetrack betting is just as consistent with risk averse bettors as it is for risk loving bettors, when one introduces the idea that bettors instead enjoy the high skewness offered by longshot bets. However, there are a number of reasons why these models alone are not sophisticated enough to comprehensively explain the FLB and the individual behaviour under uncertainty and risk that is behind it. As a basis for this statement, one can turn to the following quote that emanated from Samuelson: When I go to the casino I go not alone for the dollar prizes but also for the pleasures of gaming (1952:671). Just because an individual decides to bet (in particular on a longshot), one cannot automatically assume that they are risk-seeking. Linked to this, a fundamental problem with the risk-loving theories is that it is inconceivable that recreational gambling activities like those being studied are completely wealth-orientated. Therefore looking only to theories which attempt to reconcile utility maximising behaviour in models that are limited to including the probability of winning with relation to changes in monetary levels is not sufficient to explain individual behaviour in this context. Such a problem is mentioned by Weitzman (1965), who accepts that his proposed personal utility function involves abstracting from the influence of all other possible variables (1965:19), but claims that in any specified time period these influences are of fixed quality and therefore can be ignored. This is a counterintuitive assumption, given that it is highly unlikely other variables do not consistently affect the decision making process of a bettor. The argument for a more sophisticated set of models is made more convincing given the small stakes involved in recreational betting. Most bets that people make (for 10

12 example, a 10 bet on a player at odds of 3.00) are incapable of generating significant wealth changes of any consequence (Sauer, 1998), implying there must be other relevant variables that should be accounted for. Furthermore, in the beginning of this section it was stated that the anomalies in the FLB should not be ignored, a sentiment which is now important in assessing the strength of these models. In some sports, no significant FLB of has been found (Busche and Hall 1988, Busche 1994, Vaughan Williams and Paton 1998, Swidler and Shaw 1995). In other markets, a negative FLB has been found (Woodland and Woodland, 1994). The basis of the theories proposed by Weitzman and Ali require a homogenous risk attitude amongst bettors, and subsequently assume all bettors have the same utility function (through the use of the representative agent). If this is true, then the FLB should be found explicitly in all sports markets that are studied. The fact that these outlying studies cannot be explained by these models therefore confirms that a more sophisticated approach is required. In response to this criticism, Thaler and Ziemba (1988) suggested that tools from behavioural economics such as mental accounting and prospect theory should be used to try and develop a more psychologically inclusive explanation of the FLB. As early as the seminal paper by Griffith (1949), it was suggested that biases may exist in assessments of the probability attached to different outcomes by bettors. For example, people may underestimate the chances of favourites and overestimate those of longshots, given that it has been shown that people tend to overestimate probabilities of small probability events (Kahneman and Tversky, 1979). Thaler and Ziemba (1988) also suggest an array of other psychological influences on betting behaviour which may contribute to the FLB. It is possible that certain bettors derive utility from holding a bet on a longshot itself (see also Conlisk, 1993). Given that average stakes on events are relatively low, even a small intrinsic utility from gambling (on a longshot in particular) may sometimes be sufficient to persuade an individual to place a bet, regardless of the expected return. Bettors may also find it more fun to pick a longshot to win than a favourite, as the individual can claim bragging rights for picking a more difficult winner. The same paper by Thaler and Ziemba (1988) also proposed that some bettors may choose horses for irrational reasons, such as the name, yet given the discussion on the lack of a sentiment effect in tennis betting in Section II this seems not be a plausible contributing factor to the FLB in this particular market. The conclusion of Thaler and Ziemba (1988) appears to directly favour a psychological approach compared to the risk-loving models, as does that of Snowberg 11

13 and Wolfers (2010), who take an original approach in comparing the two sets of models by using a large dataset on complex bets from the racetrack. They conclude that their results are consistent with the favourite-longshot bias being driven by misperceptions rather than risk-love (2010:744). However, this is not to suggest that probability misperception theories provide a convincing explanation of the FLB. Experiments (Dwyer et. al, 1993) have shown that probability estimates improve markedly as individuals acquire more experience in their environment. The nature of the betting market points towards it being a highly repetitive environment (Sauer, 1998) which implies that systematic and consistent probability misperceptions are not a convincing explanation of the FLB in the long run. A more speculative argument would also suggest that in the tennis market especially, the probability misperception approach is not sufficient. Whilst in some sports markets such as racetrack betting there is likely to be a high turnover of inexperienced (and hence more likely to be uninformed) bettors, in the tennis market one may predict that this turnover is relatively lower. This is because of the culture behind racetrack betting, which as previously discussed contains a high number of leisure bettors and therefore has fewer individuals who bet in a high volume and are therefore more likely to make probability perception errors. With a relatively lower turnover of bettors in the tennis market, this leaves a higher proportion of the same individuals within a highly repetitive environment, and hence any probability misperceptions may be less pronounced. Therefore, whilst introducing psychological aspects to the decision making process is important, existing theories in this field are ultimately insufficient and require further development. A third group of models has also emerged focusing on potential supply-side explanations of the FLB, rather than the preferences of bettors. Such models are predictably only applicable to markets where there is a bookmaker such as in the market being studied in this paper, and cannot be applied to pari-mutuel betting markets. The original advocate of this third strand of models is Shin (1992). In his paper, Shin builds on the analogy that the betting market is a particularly good example of a contingent claims market (1992:426), where the bookmaker is the market maker and the bettors are the traders. Within this market, Shin argues the bookmaker faces an adverse selection problem in which some bettors may be trading on the basis of having superior information (insider trading). Evidence provided by Crafts suggests that British racing offers considerable 12

14 potential for profitable insider trading (1985:303), implying that it is possible bookmakers could hold sufficient beliefs on these activities to alter their optimal pricing strategy. The model consists of two bookmakers competing to set odds on races, and a set of bettors are modelled as being either insiders or outsiders. The model then focuses on incidences of insider trading when a longshot (rather than a favourite) is tipped to win, which is in accordance with intuition. Then if an insider trader is chosen from the population with a high enough probability, which may be seen as a proxy for the bookmaker s belief that insider trading exists in the market, relative to the price of the longshot, then the favourite-longshot bias would seem to be a fairly general feature of models of this kind (Shin, 1992:431). In essence, this paper suggests that bookmakers may deliberately trim the odds of longshots when they believe with a high enough probability a bettor has insider information on a longshot. An example of a successful application of this theory can be found in Vaughan Williams and Paton (1997). Their model once again includes a separation of bettors into informed and uninformed categories. Interestingly, in the utility function for informed traders an extra consumption benefit is included for bets on favourites. This is motivated by the evidence in Griffith (1994), who suggests increased absolute frequency of success increases the motivation of bettors, and Bruce and Johnson (1992), who draw a link between the perceived skill of bettors and self-confidence in comparison to other bettors. The final important aspect of this model is that whilst uninformed bettors place bets of a fixed amount, informed bettors vary their stake according to their expected return up to a predetermined stake limit and subject to the constraint that utility from any bet is non-negative (Vaughan Williams and Paton: 1997:1506). As a result, their model predicts that in the presence of transaction costs the favourite is underbet relative to the objective probability of winning i.e. a positive FLB exists. This prediction is subsequently tested on a racetrack betting data set, in which it is assumed that in highergrade (higher-profile) handicap races there is no insider trading as nearly all information is publicly available and the market is made up solely of informed bettors, which their model predicts will result in no FLB. Their findings confirm this, and show a bias does not exist where the chance of asymmetric information is very low. Additionally, their results also propose that the presence of insiders in lower-profile races can account for the existence of a positive longshot bias at any level of transaction costs (Vaughan Williams and Paton, 1997:1510), suggesting there is a marked difference in the bias between relatively high and low profile events, independent of transaction costs. 13

15 In summarising the three groups of models so far discussed, it seems a necessity that any demand-side approach should embrace psychological influences given the severe limitations of the class of risk-seeking models. Any accurate attempt to model behaviour under uncertainty should not simply ignore the potential effects psychology may have on the outcome. Whilst Weitzman separates the aims of the psychologist as explaining existence and reality on an individual level and of the economist as understanding implications in market behaviour, attitudes towards risk, theory of demand (1985:22), the two should instead be studied in conjunction with one another. It is of course not realistic to model the exact psychology behind every individual bettor, but existing work in the field of behavioural economics on theories such as Prospect Theory (Kahneman and Tversky, 1979) and quasi-hyperbolic discounting (Loewenstein and Prelec, 1992) have shown that individuals often make decisions that consistently differ from the predictions of traditional economic theory, and therefore should be crucial to accurately modelling market behaviour and individual attitudes towards risk. In addition to this, the supply side models also appear to have some influence in explaining the FLB, in particular in markets that have a bookmaker. The testing of this theory forms the basis of Hypothesis 2 in the following section. The existence of these competing theories, all of which claim to have explanatory power, makes attempts to explain the causes of the FLB extremely complicated. This is a view shared by Thaler and Ziemba, who accept that bettor s behaviour seems to depend on numerous factors such as how they have done in earlier races, and which bets will yield the best stories after the fact (1988:172) among other influences. A similar complication also lies in explaining behaviour in other financial markets, which helps to create the interesting challenge of finding a comprehensive explanation of the FLB. In opposition to the aforementioned opinion of Weitzman (1985), the approach laid out in Section VII and whose supporting literature follows, is supportive of the argument that more sophisticated and enriched models of human behaviour are fundamental to understanding the market forces behind the FLB. There are a number of papers which influenced the development of the ideas to be laid out in the third hypothesis in Section IV, which suggests that the expectations of bettors (and the attached expected emotions) may play a role in the decisions of bettors. The key influence originates from the field of behavioural economics, and in particular the effect of emotions on decisions involving risk. The paper by Rick and Loewenstein (2008) 14

16 makes clear both the limitations of the Expected Utility model (von Neumann and Morgenstern 1944) and the importance of emotions in the types of decisions that are studied in this paper. In their paper, emotions are divided into two categories: expected and immediate emotions (Loewenstein & Lerner, 2003). Expected emotions are those that entail an anticipation of how one will feel as a result of a variety of different possible outcomes. On the other hand, immediate emotions are not anticipated, and only experienced at the moment of choice. Proponents of emotion theory suggest that although traditional economics suggests individuals seek to choose the most desirable options and hence maximise their utility function, this does not imply that consequentialist decision makers are devoid of emotion or immune to its influence (Rick & Loewenstein, 2008:138). Expected emotions in particular are perfectly consistent with the traditional consequentialist approach to economics. For example, in the betting market, utility theory does not discount the possibility that the decision of bettors to bet on a favourite or outsider is influenced by the bettor s predicted emotions regarding both potential results. One particular axiom of EU theory challenged in this paper is hence the assumption that utility is strictly defined over realised outcomes. A reference-dependent model of preferences, proposed by Koszegi & Rabin (2006), claims that gains and losses should be defined as relative to expectations, instead of the status-quo theory introduced by Kahneman and Tversky (1979). In essence this means that gain-loss utility is derived from standard consumption utility and the reference point is determined endogenously by the economic environment (Koszegi & Rabin, 2006:1133). More specifically the reference point is an individual s rational expectations held in the recent past about outcomes. The authors claim that this theory may be preferable in a number of situations, as the nature of status-quo dependent reference points may not always be realistically applicable to certain situations. For example, Koszegi & Rabin claim the result of the famous field experiment conducted by List (2003) can be explained by their model. List found that the endowment effect amongst sports memorabilia traders was greater amongst inexperienced traders. If it is the case that more experienced traders come to expect a high probability (expectation) of parting with items they have just acquired (Koszegi & Rabin, 2006:1142), then this may explain why the endowment effect is mitigated amongst these traders: because they expect to sell their items with a higher probability, they are willing to sell at a relatively lower price than inexperienced traders who may have a lower probability of trading. More generally and with relevance to the topic of this paper, a status-quo theory of the reference point seems inappropriate to model the behaviour of a bettor given that he 15

17 should expect to either win the bet or lose it: there is no expectation that the status quo is maintained so it seems unrealistic that this forms the basis of the reference point. This idea is developed more formally in Section VII. From a psychological point of view, the theory of Koszegi & Rabin is heavily influenced by the work of Mellers, Schwartz, Ho and Ritov (1999), who introduced Decision Affect Theory. The basis of this theory is the importance of anticipated emotions in predicting choices and develops on previous work that was conducted on two strands of research: regret theory (Bell, 1982) and disappointment theory (Loomes & Sugden, 1986), both of which focus on counterfactual comparisons. In simplified terms regret theory assumes that people anticipate the regret they may experience in making a decision. Disappointment theory predicts that people anticipate future disappointment if an alternative state of the world is realised. Both theories have been successful in describing violations of EU theory (Loomes, Starmer & Sugden 1989), though the work of Mellers et al. (1999) was the first to test directly the effect of anticipated emotions on choices. As will be seen, the literature encompassing the ideas in the final part of this section provides the basis of the model in this paper and will be relatively successful in providing a realistic alternative explanation of the FLB in market for men s professional tennis. IV. THEORETICAL PREDICTIONS & HYPOTHESES The hypotheses in this paper have been arranged into three distinct sections and are all relevant to the proposed research question. Through investigating these hypotheses, the result should be a better understanding of the nature and background of the FLB in the wagering market for men s professional tennis. The analysis in Section V consists of a series of statistical tests in an attempt to provide robust evidence for the presence of the FLB in the data. The study for the presence of a FLB in the tennis market is also synonymous with a test of market efficiency. Thaler and Ziemba define strong efficiency as all bets should have expected values equal to (1 t) times the amount bet and weak efficiency as no bets should have positive expected values (1988:163) 3. Any violation of efficiency would be especially significant given that wagering markets have a better chance of being efficient because the conditions are those which usually facilitate learning (Thaler and Ziemba, 1988). Despite this, the vast 3 Where t is equal to the level of transaction costs 16

18 majority of papers on this topic (see Section III) have found a significant bias in a wide range of sports, and in particular violations of strong efficiency (although in some occasions weak efficiency is also violated, see Hausch et al and Ziemba and Hausch 1986). The results of the existing literature on the FLB therefore form the basis of the first hypothesis. Hypothesis 1 There is a significantly positive favourite-longshot bias in the market of professional men s tennis. In particular, expected returns from betting in men s professional tennis increase as the bookmaker odds of the player decreases. The second part of the paper is motivated by the work conducted by Forrest and McHale (2007). In their paper, they interestingly assess if there is a difference in the FLB between Grand Slam and non-grand Slam tennis tournaments. The theoretical basis behind this is found in the studies conducted by Shin (1992) and Vaughan Williams and Paton (1997) that have proposed supply-side explanations of the FLB in racetrack betting, which is in contrast to the majority of theories on the topic which focus on the preferences of bettors. They argue that the market maker (bookmaker) faces an adverse selection problem in which a customer (bettor) may be trading on the basis of superior information (Shin, 1992:426). The net result of this problem is that if the bookmaker believes with a high enough probability that there exists a bettor (or group of bettors) with superior information on a longshot, it will trim the price of this horse in line with its optimal pricing strategy. This strategy requires the bookmaker to raise enough revenue from outsiders to pay insiders their winnings (Cain et al., 2000:26), and as a consequence the relative negative manipulation of prices of longshots can reconcile the presence of the FLB in racetrack betting. An inference that can be drawn from this theory which has been previously tested is that there should be a lower probability of insider trading in higher-grade handicap races, in which nearly all information is available publicly (Vaughan Williams and Paton, 1997:1507). In these higher profile races, in which the asymmetric information problem is diminished, there was no FLB independent of the level of transaction costs (Vaughan Williams and Paton, 1997:1510). Forrest and McHale (2007) acknowledged this theory and looked to test it in the wagering market for men s singles tennis matches. They hypothesised the idea that if there is less scope for insider trading in racetrack betting, this could also be applied in 17

19 the tennis market. In particular, matches in Grand Slam tournaments are more high profile than matches in non-grand Slam tournaments. Therefore, the aforementioned supply-side side model would predict the FLB to be mitigated in a Grand Slam subsample. However, their conclusion was that there is no evidence of a role for inside information in accounting for the positive longshot bias (Forrest and McHale, 2007:763) in the tennis market. The second hypothesis in this paper constructs an alternative test for the theories proposed by Shin (1992) and Vaughan Williams and Paton (2007). The tennis market provides a unique opportunity to conduct analyses in different rounds within the same tournament. One can expect that as a tournament progresses, matches become more high-profile for two main reasons. Firstly, matches in the latter point tournaments naturally receive more media coverage and attract more interest from followers of the sport. Secondly, it is shown in Section VI that lower ranked players are predictably often eliminated in earlier rounds, leaving predominantly higher ranked players in the draw for latter rounds. It can be safely argued that bettors are likely to be more informed on the qualities of higher ranked players than lesser known players and that the presence of higher profile players in matches also makes the match itself higher profile. What impact (if any), does this have on the extent of a positive favourite-longshot bias? If supply side models such as those proposed by Shin (1992) have any influence, the FLB may be mitigated in higher-profile events i.e. the matches in the latter rounds of tournaments. This is consistent with asymmetric information amongst bettors becoming less prominent (Vaughan Williams and Paton, 1997). However, significant empirical evidence from a number of papers has argued against this theory. For example, Hurley and McDonough (1995) find experimental evidence that suggests in pari-mutuel markets, costly information (and equivalently asymmetric information) and transaction costs amongst bettors cannot explain the FLB. Although this study was conducted in a pari-mutuel and not fixed odds market as is being studied in this paper, their findings on information asymmetries amongst bettors remains relevant. The empirical study on the National Football League (NFL) by Levitt (2004) came to the conclusion that it cannot be the case that a significant fraction of bettors have better information that the bookmaker (2004:243), and instead the bookmakers actually exploit their information advantage over bettors. Finally, in their study on the tennis market Forrest and McHale (2007) found no evidence for a role for inside information in accounting for the positive longshot bias (2007:763) found in the tennis market. 18

20 The net result of the proposed theory therefore provides the motivation for second hypothesis in this paper, although the prediction it contains is a cautious one given the limited work that has been thus far conducted. Hypothesis 2 The extent of the favourite-longshot bias will be significantly different in the latter rounds of tournaments when compared to early rounds. This is based on the aforementioned testing of the supply-side models, with the null hypothesis in the statistical tests being that there is no significant difference in the FLB between early and late rounds of tournaments. Theories on the FLB thus far have only focused on utility models which are defined over realised outcomes. However, a strong argument can be made that suggests the decisions of bettors are also influenced by other potential outcomes. Such counterfactual thinking can be linked to emotion-based choice, as well as to the beliefs an individual has over potential outcomes. Therefore it can be argued that bettors derive utility both from expected monetary returns and from how outcomes deviate from the expectations of each outcome occurring. The latter can be modelled through the use of a reference point that differs from the status-quo approach of Kahneman and Tversky (1979), which instead suggests individuals base their reference point on the current state of the world. The paper by Koszegi and Rabin hypothesises that the status-quo theory of the reference point is especially unsatisfying when applied to the many economic activities that involve no ownership of physical assets (2006:1143). For example, a person expecting to go to the dentist but eventually does not, would have a positive utility gain (assuming he does not enjoy going to the dentist). This utility gain cannot intuitively be the same as someone who was not expecting to go to the dentist, and who also does not go (and thus has zero utility). With the status-quo approach to the reference point unable to reconcile these types of situations, an alternative theory is required. In this paper, a version of the model proposed by Koszegi and Rabin (2006) is applied in attempt to provide an alternative explanation to the FLB which has not been tested before. The intuition follows a logical path. When gamblers place a bet, their expectation of winning is likely to be related to the odds offered by the bookmaker. If the odds are relatively low (a favourite), a relatively high expectation of winning can be foreseen in the bettor. If the odds are relatively high (an outsider), a relatively low expectation of 19