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

21 winning is predicted. These varying expectations can have a significant impact on how betting wins and losses are felt by bettors. For example, imagine one individual places a bet of 1 on a tennis player at odds of 1.05 (a favourite ), and another places a bet of the same stake on a tennis player at odds of 3 (a longshot ). If both tennis players then lose their respective matches, the monetary loss for both bettors is the same i.e The majority of papers on this topic use this assumption in their models, including Forrest and McHale (2007). However, it is clear that given the favourite bet had a higher expectation of winning, losing this bet yields an additional utility loss compared to the bet on the longshot. This is supported through the findings of Mellers, Schwartz and Ritov, who find evidence that the pleasure of winning and the pain of losing are more intense when outcomes are surprising; that is, responses are more extreme when outcomes are unexpected (1999:336). The decision-affect model proposed in this paper also draws on theories on emotional-based choice such as disappointment theory (Loomes and Sugden, 1986) and regret theory (Bell, 1982). Building on these models, this hypothesis proposes that psychological emotions such as surprise, disappointment and regret may have a significant impact on betting decisions. The empirical regularity that longshots are overbet could be reconciled through the notion that these types of bets involve a lower expectation of winning and hence have the potential to elicit stronger expected positive emotions (and weaker expected negative emotions) than bets on favourites. Hypothesis 3 A model incorporating endogenous expectations and expected emotions can accurately explain behaviour consistent with the favourite-longshot bias in the betting market for men s professional tennis. V. DATA & HYPOTHESIS 1 RESULTS Data for betting odds on men s professional tennis is readily available on the internet in excel format, from The data consists of betting prices, results, rankings and other information for all professional men s singles matches, ranging the seasons. Odds are available from 5 reputable bookmakers. The bookmaker Bet365 is a highly reputable company in the UK market, and has the most numerous data within the set, so was chosen in this paper. In total, the data set originally 20

22 compiled constituted approximately 23,562 matches, after the removal of matches for which the starting prices for both players were not available. Given two players play in each match, this therefore equals approximately 47,000 bets. Throughout the analysis, tennis matches in which players had the same starting price odds were also been excluded from the analysis. This was because in these matches it was not possible to discern between a favourite and a longshot based on the starting price odds. The starting prices reflect the betting trends resulting from bets made with bookmakers, and represent the last odds offered as bookmakers attempt to round their books (Gabriel and Marsden, 1990:887). This is in comparison with pari-mutuel betting, where payoffs are taken from the pool of successful bets after taxes have been deducted. It was also necessary to test whether matches in which one player failed to complete the match (as a result of a retirement/walkover) should be excluded from the analysis. The cause of such a result is often an injury to a player. If a player has injury suspicions before a match this should be reflected in the odds, and thus the market should work efficiently. However, if a player must withdraw through an injury or any other reason which occurs solely within the match itself, this will not be captured by the starting odds for this match. Therefore these matches should be considered for removal from the data set if they significantly bias the results of the analysis. Any bias either towards favourites or outsiders compared to the set of completed matches would also bias any relationship between return and odds, and thus should be controlled for. The result of the relevant tests on this issue, which are discussed further in Appendix 1, found that inclusion of these matches could bias the relationship between return and odds. As a result, following detailed analysis these matches were excluded from the data set for the forthcoming empirical testing. The sample size as a result of these changes remained large, with 22,676 matches, and therefore 45,352 bets. 1. There is a significant positive favourite-longshot bias in the market of professional men s tennis. The research method for the first hypothesis is consistent with numerous previous studies conducted on the FLB and consists of four different methods in order to allow a convincing conclusion to be reached. 21

23 The first method which clearly indicates the presence of a bias in the wagering market for men s singles tennis is a visual representation, see Figure 2 4. To construct this figure, the data was divided into percentiles (at 5% intervals), and the mean return per Great British Pound (GBP/ ) bet was calculated for each percentile. For example, the 50% percentile used all bets with bookmaker odds of 1.85 or greater. Consistent with previous studies, each bet is normalised to be of the same value, in this case 1. This means a winning bet yields a net return of Decimal odds 1, and a losing bet yields a net return of -1. Figure 2 shows that a betting strategy comprising of betting on players with odds of 6.5 or higher yields a return of -45.9%, compared to a rate of return to making all possible bets (bookmaker odds of 1 or greater) of -10.4%. These returns, as well as the observable negative relationship between bookmaker odds and mean percentage return throughout the whole odds range, suggest the presence of a positive FLB in this market. Mean Return per Bet (%) Decimal Odds (Logarithmic Scale) bandwidth =.8 Fig 2: Figure showing the favourite long shot bias: the rate of return on win bets declines as bookmaker odds increases. The sample includes 45,352 bets on men s professional tennis matches from 2003 to Lines reflect Lowess smoothing. 4 A logarithmic scale is used to represents odds in order to better show the relevant range of odds. 22

24 Prior to conducting the next part of the statistical analysis, odds are converted into probability odds through the simple transformation: Probability Odds = 1 Decimal Odds The second method undertaken to indicate the presence of the FLB in the market is analogous to the statistical analysis conducted in Forrest and McHale (2007). The data was divided into probability odds ranges at 0.1 intervals, and the mean return calculated for each range of odds. This analysis forms the basis of Table 2 and shows an almost monotonic relationship between odds and return, with the exception of one anomaly. One can observe that in the range the mean return is , a figure lower than for the range of However, a one-sided t-test shows that the mean return from bets placed in the odds range is not significantly greater than the mean return from bets placed in the range Thus with the exception of one statistically insignificant anomaly, Table 2 also provides robust evidence for the presence of the FLB in men s professional tennis. Odds Range Total Return # of bets Mean Return Total Table 2: Table showing mean returns by odds range, for bets on men s singles tennis from An alternative proof of the FLB using the data in Table 2 is to test for differences in the subjective probability (implied from the bookmaker odds) of a bet being successful and the objective (empirical) probability for each of the odds ranges. A positive FLB would predict that the objective probabilities are consistently lower than the subjective probabilities for bets on outsiders (low probability odds), and the opposite for bets on favourites (high probability odds). In order to calculate this, the average commission on 5 t-statistic =

25 each bet that was calculated in Section II (7.5%) was taken off the probabilities implied by the bookmaker odds, as failing to do so would of course always result in the subjective probability being higher than the objective probabilities. This could be done with confidence across all bets given the extremely low variation in the level of commission across tennis bets (SD = 0.01). The average win probabilities for each range were then calculated. Only by grouping the data was an unbiased estimate of the corresponding average objective probabilities able to be calculated, a factor ignored in the studies by Griffith, McGlothlin and Weitzman (Ali, 1977). The subsequent results allow for a clear graphical representation to be achieved and the prediction of the FLB is confirmed by the evidence provided in Figure 3. 1 Subjective Win Probability Objective Win Probability Subjective = Objective Win Probability Subjective Win Probability Fig 3: Figure showing the favourite-longshot bias: the subjective probabilities of bettors overestimate (underestimate) the objective probabilities of outsiders (favourites). Although the information provided so far indicates the presence of the FLB in this particular wagering market, as Forrest and McHale (2007) state, it does not provide sufficient evidence for a precise relationship between odds and return. Another advantage of the following method is that it avoids the potential for measurement error that exists in grouping bets into artificial odds categories (Busche and Hall, 1988). 24

26 To statistically establish the presence of the FLB in men s professional tennis, a series of OLS regressions were conducted. Given the nature of the data being analysed, if OLS estimation was carried out on the whole data set the resulting statistical inference would be biased. This is because not all returns are independent: within a single match one player wins, and one loses. Hence if the return to one player is positive (if he wins), the return to the other player is necessarily -1. Therefore, regressing return on odds would generate downwardly biased standard errors (Forrest & McHale, 2007:757). To correct for this, one player from each match was randomly selected in order to achieve an independent data set. This still leaves a sample size of approximately 22,676. It should be noted this is approximately three times the size of any other study conducted within the tennis market on this topic. This allowed for the following linear regression model to first be tested: (1) Return = β 0 + β 1 ProbabilityOdds + u i The prediction made in Hypothesis 1 requires two clear results from this linear regression. Firstly, the constant term (β 0 ) should be significantly negative, representing the fact that bettors on average lose money. Secondly, the estimated coefficient on probability odds (β 1 ) should significantly exceed zero. This would prove that betting on favourites (as probability odds approach 1) yields higher returns. This linear regression can equally be viewed as a test of market efficiency. A negative constant term and positive slope coefficient would also suggest a violation of strong efficiency, whilst weak efficiency would hold. The results of the linear regression strongly support the predictions of Hypothesis 1 6. The intercept term is significantly negative as expected 7, and the estimated coefficient on probability odds is significantly greater than zero 8. Specifically, the estimated relationship from the linear regression was: (2) Return = Probability odds 6 All OLS estimation conducted with heteroscedasticity-robust standard errors 7 t-statistic = t-statistic =

27 Therefore even with the imposition of linearity, the presence of a bias cannot be rejected at the usual significance levels. This is consistent with the findings in Forrest and McHale (2007), with the assumption of strong efficiency clearly violated. However, the evidence presented so far suggests that the relationship between return and probability odds is not linear, and may instead be captured more accurately through a non-linear relationship. This argument is made more convincing when one takes into account Figure 4, which graphically represents a comparison of the data with the fitted linear regression line, and a polynomial (power 5) regression line. In order for a better visual representation to be achieved, a similar percentile method to that used in the creation of Figure 2 is used to divide the data and plot the relevant points, except in this case probability odds are used. Mean Return per Bet (%) Probability Odds Mean Return (%) Linear Fitted Values Polynomial (^5) Fitted Values Fig 4: Figure showing a comparison of linear and non-linear regression techniques. The diagram clearly shows the presence of a non-linear relationship between return and probability odds. In addition, the random selection procedure of selecting one bet from each match was once again conducted in order for the observed regression lines to be unbiased. The real data points can then be compared to the estimated regression lines. It is clear to see that 26

28 the relationship between return and odds is non-linear and that instead a higher order polynomial regression equation fits the data more accurately. For example, one could estimate the following equation: (3) Return = β 0 + β 1 PO + β 2 PO 2 + β 3 PO 3 + β 4 PO 4 + β 5 PO 5 + u i Where PO = Probability Odds. This result is strengthened through the OLS estimation results of the above equation: each individual coefficient is significant, as is the F- statistic. 9 In comparison, when a regression equation including the independent variable to the power of 6 is estimated, the t-statistics of the individual coefficients are no longer as significant. This suggests the relationship between return and probability odds is closest to a 5 th -order polynomial regression equation. Comparison of the extent of a FLB in different markets The best available comparison of the FLB in different sports markets can be found in Coleman (2004). The method for comparing results involved estimating an equivalent regression for each of the papers that were included of the form: (4) Expected Return (%) = β 0 β 1 Ln(Dividend) + u i In this paper Dividend is equivalent to the decimal odds. An equivalent regression for this data set was conducted and is compared to the results of Coleman in Table 3: Wagering Market β 0 β 1 Tennis UK Bookmakers Australasia Thoroughbreds (Horses) North America Parimutuel Asia Table 3: Table showing the comparison of the FLB in a variety of sports markets Table 3 shows that on average the FLB in sports markets that have bookmakers is significantly greater than pari-mutuel markets. The results in this paper are largely 9 F-statistic =

29 synonymous with the results of other papers that have also focused on UK and/or bookmaker markets. To conclude this section, a plethora of evidence has been provided which unanimously points toward the presence of a positive FLB in the betting market of men s professional tennis matches. The result is further strengthened when one acknowledges that the extent of the FLB is also consistent with other studies into the UK and bookmaker sports betting markets. The consequence of this is that betting on favourites players with higher probability odds yields significantly higher returns than betting on longshots, though the relationship is not a linear one. Instead, returns change at a faster rate in the lowest probability odds ranges compared to the highest odds ranges. The next logical step is to provide an explanation for this empirical regularity, both for the tennis market and other sports. The next section first looks at a potential supply-side explanation, before an entirely new approach is attempted. VI. HYPOTHESIS 2 RESULTS 2. The extent of the favourite-longshot bias will be significantly different in the latter rounds of tournaments when compared to early rounds. Fundamental to an accurate assessment of this hypothesis was a reliable method to distinguish between Early and Late rounds of men s professional tennis tournaments. Two characteristics of these tournaments meant that this was not a simple task. Firstly, not all tennis tournaments consist of the same number of rounds, with Table 4 displaying a division between 5, 6 and 7 round competitions. This importantly implies that the definition of Early and Late cannot be the same in these different types of tournaments. Rounds in Tournament # of Matches Total Table 4: Table showing the distribution of matches amongst tournament types. Round Robin and Masters Cup matches were excluded from this section. Secondly, consideration also has to be given to the fact that there is a higher volume of matches in the earlier rounds of the tournaments being studied due to the elimination process that takes place. Therefore, any method to divide matches within tournaments 28

30 faced the challenge of providing a sufficient amount of data to allow for accurate statistical testing in a Late rounds subsample. Before describing the chosen method, it is first also important to acknowledge that the rationale behind undertaking the analysis in this section was based on the assumption that matches become more high profile as a tournament progresses, partly due to the fact that higher ranked players are significantly more likely to contest matches in the latter rounds. In order to statistically verify this, the following regression was conducted for each of the three different types of tournaments 10 : (5) Average Ranking = β 0 + β 1 Round + u i The variable Average Ranking was calculated as the average ATP World Ranking of both players within a single match. If the stated assumption is correct, then the coefficient β 1 will be significantly negative. This prediction was confirmed, as Table 5 shows. Type of Tournament B 1 t-stat 5 Rounds Rounds Rounds Table 5: Table showing the relationship between the average rankings of players within each match, and the round of the tournament. Results are given separately for each type of tournament. Despite the limited number of matches for the latter rounds of each type of tournament, the regression results explicitly show that higher ranked, and therefore higher profile players, are significantly more likely to be present in the Late rounds of men s professional tennis tournaments than lower ranked players. With one of the crucial assumptions behind this hypothesis confirmed, the method of separating Early and Late rounds can now be presented. It was previously suggested that any accurate method for separating the Early and Late rounds of tournaments should take into account that the number of rounds varies across tournaments, as well as the fact there is significantly more data available for earlier rounds of tournaments. In addition to this, the evidence provided on the relationship between rankings and rounds implies that any method of dividing the data 10 Matches with incomplete data on rankings were necessarily removed from this part of the analysis. 29

31 should also achieve a significant difference in average ranking between the two subsamples of data. Through separating the data in the way displayed in Table 6, the above criterion are largely achieved. Although the sample size for Early rounds is significantly greater than for Late rounds, this could not be avoided without compromising the difference in average rankings too much, and both samples are still sufficiently large for reliable data analysis to be conducted 11. Type of Tournament Early Rounds Late Rounds 5 Rounds 1,2 3,4,5 6 Rounds 1,2 3,4,5,6 7 Rounds 1,2,3 4,5,6,7 Table 6: Table showing the rounds for each type of tournament that constitute the Early and Late subsamples respectively. This approach is also motivated by the evidence provided in Figure 5, which shows average ranking against the progression of rounds in the three types of tournaments. For tournaments with 5 rounds, it is clear there is a significant drop in average ranking following the second round of the tournament. Average Ranking Round Average Ranking (5 Round Tournaments) Average Ranking (6 Round Tournaments) Average Ranking (7 Round Tournaments) Fig 5: Figure showing the average ranking of players in a match, dependent on the round of the tournament for the three different types of tournament discussed. 11 Sample size of Early rounds is 17,380 matches, sample of size of Late rounds is 5,060 matches. 30

32 For tournaments consisting of 6 and 7 rounds respectively, the curved nature of the lines suggest that larger drops in average rankings can be observed between the earlier rounds of tournaments compared to the latter rounds. One additional feature of Figure 5 should be mentioned. The third round in 5-round tournaments, which is classed as Late, has approximately the same average ranking as the second round in 6 and 7-round tournaments, which are classed as Early rounds. Thought at first glance this seems counterintuitive, within a 5-round tournament the third round is a quarter-final at the midway stage of the competition. In comparison, the second rounds in the other types of tournament lie below the midway stage of a tournament, and are likely to attract significantly less attention within the course of the tournament compared to a quarterfinal, which supports the decision to class these as Early rounds. The motivation behind disproportionately biasing Early rounds towards the beginning of each type of tournament has hence been provided, in comparison to a method of splitting the number of rounds in a tournament equally. The stated goals from separating the data have therefore been achieved through providing a clear distinction between relatively high and low profile matches, whilst keeping the size of the Late rounds subsample sufficiently large. With this in place, the hypothesis can now reliably be tested, to see whether the FLB is mitigated in the latter rounds of tournaments. In order to determine the success of this hypothesis, a series of tests similar to those conducted for the first hypothesis can be constructed for the Early and Late round subsamples. Firstly, the mean returns to bets within different odds ranges were compared between the Late round subsample and Early round subsample. Odds Range Total Return # of bets Mean Return Late Total Table 7: Table showing the mean return to placing a 1 bet for all matches in the Late rounds subsample. 31

33 It can be observed through Table 7 that there appears to be less of a pure monotonic relationship between odds and return in the Late rounds subsample compared to the complete data set. Anomalies exist in the and ranges 12, though the overall pattern of the data still suggests the presence of a positive FLB: a betting strategy on favourites yields better returns in general than a betting strategy on outsiders. In contrast, Table 8 shows that the relationship between odds and return in the early rounds of tournaments is equivalent to that of the data set as a whole. This is to be expected, given the size of the Early rounds sample is significantly larger than the Late rounds subsample, and is hence likely to be comparatively closer to the results in the complete data set. In comparing the two sets of results, an argument can be made to suggest that the FLB is less pronounced in the latter rounds of tournaments. Recalling from previous sections the assumption of strong efficiency in wagering markets, for which the FLB demonstrates a significant violation (as different betting strategies yielded different returns), it is here suggested that the extent of this violation can be used to measure for the degree of bias in the market. The degree of strong efficiency can be measured through simple descriptive statistics of the mean return data in each subsample; in particular the range and standard deviation. It turns out that these statistics are smaller in the Late rounds subsample than for Early rounds, suggesting the level of strong efficiency violation, and therefore FLB in the market, is mitigated 13. Odds Range Total Return # of bets Mean Return Early Total Table 8: Table showing the mean return to place a 1 bet for all matches in the Early rounds subsample. 12 Neither are significantly more negative than the preceding odds range at the 5% level 13 Early: Range = 0.709, SD = ; Late: Range = 0.559, SD =

34 The second statistical test involves a regression analogous to that performed in Equation (1) within each subsample. Under the stated hypothesis, the coefficient on Probability Odds should not be significantly different between the two sub samples, which would support the findings of Forrest and McHale (2007). On the other hand, if the coefficients are significantly different then one can look to the models proposed by Shin (1992) and Vaughan Paton and Williams (1997) as being relevant to the wagering market for men s tennis. The results of the relevant regressions are displayed below 14, and show that in both samples the assumption of strong efficiency is violated, but since there are no strategies which yield a positive return the assumption of weak efficiency still holds. Early Rounds Late Rounds (6) Return = Probability odds (7) Return = Probability odds If one looks at the coefficients on Probability Odds, it would appear that to some extent the FLB has been mitigated in the latter rounds of tournaments. To test for this formally, the same regression was once again performed on the entire data set, but also including a Dummy variable for Early or Late rounds, and an interaction term. A significant interaction term (IT) would indicate that the FLB is significantly mitigated in the latter rounds of men s professional tennis tournaments. Interaction (8) Return = Probability odds Dummy IT The coefficient on the interaction term is significant, but only at the 10% significance level. 15 There is therefore reasonable evidence to suggest that to some extent the FLB is mitigated in higher profile matches, but before it can be stated with confidence that this is consistent with the supply-side theories previously discussed some further work is needed. This work entails analysing any other possible differing characteristics between the two sub-samples that could explain the presented results. 14 All coefficients are significant at the 5% significance level 15 t-statistic =

35 One of these characteristics may be differences in bookmaker odds patterns between the two sub-samples. At initial thought, one may expect that there would be larger differences between the bookmaker odds of the favourite and of the longshot in earlier rounds of tournaments, as higher ranked players often come up against relatively lowly ranked players more often. As a result, the bookmaker odds of the longshot may on average be higher, and the bookmaker odds of the favourite on average lower in the earlier round of tournaments compared to latter rounds. It could be the case that the classical risk-loving theory of the FLB has some influence in the betting market for tennis, and therefore relatively higher odds of the longshot in earlier rounds would disproportionately increase bets on these players in comparison to latter rounds. This could therefore provide an alternative explanation for the apparent difference in the FLB between the two subsamples. Equally, if longshots also on average have lower decimal odds (higher probability odds) in latter rounds, the conclusion of Hypothesis 1 would suggest the FLB may be mitigated in these rounds, especially given the non-linear nature of the relationship between odds and return. However, statistical analysis on the odds characteristics in the two subsamples shows this is not completely the case. Rounds Average Odds of Favourite Average Odds of Average Odds Longshot Disparity Early Late Table 9: Table showing characteristics of bookmaker odds between the Early and Late Rounds subsamples. Table 9 shows that the average odds of favourites is remarkably consistent across Early and Late rounds of tournaments, though there is a decrease in the average odds of the longshot (and hence disparity in odds) in the latter rounds of tournaments. By looking more into the detail of tennis tournaments the overall apparent consistency of bookmaker odds in the earlier and latter rounds of tournaments can be explained. As previously stated, it is true that high ranked players may face lower ranked players more often in the earlier rounds. But it is also the case that some low ranked players will also meet other low ranked players in the early rounds on some occasions, in which cases the odds disparity will be relatively smaller. With some lower ranked players therefore progressing by necessity, they are likely to face a higher ranked player in the subsequent round(s), and hence the odds disparity in some of these matches will be relatively high. These occurrences within tournaments can therefore, at least partly, explain the higher 34

36 than expected consistency in the characteristics of bookmaker odds between Early and Late subsamples. It should be mentioned that the difference in average odds disparity is significant at the 5% level 16. However, the difference is small. Therefore, whilst this particular characteristic of the fixed odds betting market may explain to some extent the apparent mitigation of the FLB in the latter rounds of tournaments, it certainly does not on its own provide a comprehensive explanation. Another possible influencing factor on the regression result in the interaction regression, which cannot be directly tested for with the available data, is that it is likely a much higher amount of money is placed on bets in the latter rounds of tournaments compared to the earlier rounds, though the exact implication of such a trend is uncertain. Firstly, bettors may be more confident in their bets given that they know more about the higher profile players in the latter rounds, and therefore bet higher stakes. This should be combined with the fact that serious bettors (whose aim is to make a profit) are already relatively informed on the qualities of players and should on average be even more informed on the quality of players in latter rounds, given that they are higher profile and are hence more well-known to those who follow tennis closely. This suggests that higher confidence may be justified, and bettors may be better at predicting probabilities in the latter rounds which could be manifested in a mitigated FLB. On the other hand, it is certainly the case that not all bettors are serious bettors with the sole aim of making a profit. Such leisure bettors, who are sometimes referred to as noise traders, are likely to be more prevalent in higher profile matches and hence also increase the total volume of bets placed. Examples of high profile events in other sports support this claim: the Grand National and The Derby in British horse racing, major international championships in football and the Playoffs and subsequent Superbowl in American Football. The high-profile nature of certain sporting events attracts those who are not frequent bettors, and are therefore likely to be more uninformed about the qualities of competitors compared to informed bettors. With a higher proportion of uninformed bettors this is likely to distort prices in the market, potentially in favour of longshots (as leisure bettors more often do not simply base their decisions on making a profit), which could in fact exacerbate the presence of the FLB in the latter rounds of tournaments. The result of these opposing forces makes it difficult to conclude whether the increased volume of betting in the Late round of tournaments is likely to mitigate or strengthen the presence of a positive FLB. 16 t-statistic =

37 However, it is not possible to say that this potential effect or any difference in bookmaker odds patterns that was previously discussed provides enough evidence to discount the likely possibility that the supply-side models have an influence in explaining the FLB in the tennis market. This is made more convincing if one looks back to Table 3 (in Section V), where it can be observed that the FLB tends to be far more pronounced in bookmaker markets than pari-mutuel markets. The main difference between the two is that the bookmaker has influence over the odds that are set in the market, suggesting the actions of the bookmaker are influential in intensifying the FLB in comparison to pari-mutuel markets. It can therefore be strongly concluded that in contrast to the findings of Forrest and McHale (1997), information asymmetry amongst bettors, and the beliefs and subsequent actions of bookmakers in response to such insider trading is likely to be significant in explaining at least part of the positive FLB that exists in the betting market for men s professional tennis matches. VII. HYPOTHESIS 3 RESULTS 3. A model incorporating endogenous expectations and expected emotions can accurately explain behaviour consistent with the favourite-longshot bias in the betting market for men s professional tennis. The detailed discussion in Section III made it apparent that existing theories regarding the FLB do not provide a sufficient explanation. Especially from the point of view of a behavioural economist, the existing explanations of the FLB fail to account for the emotions experienced by bettors. For example, evidence has arisen that shows counterfactual thoughts can induce varied expected emotions such as regret, disappointment and elation, through considering alternative outcomes that may occur (Rick and Loewenstein, 2008). These expected emotions can then have an effect on the decision making process. As Mellers et al. suggest, our imagined feelings of guilt, elation or regret influence our decisions (1999:332) and therefore factors such as disappointment and regret which have not yet been considered may provide part of the explanation of the FLB in sports betting markets. Consequently, the aim of this section is to put forward a model which incorporates expected emotions into a standard Expected Utility function through the use of endogenously determined reference points, in order to achieve an alternative explanation to the FLB. The proposed model has influences from a number of different theories related to emotion-based choice. The Decision Affect Theory of Mellers et al. (1997) models responses which are based on obtained outcomes, relevant comparisons to other 36

38 potential outcomes and the beliefs attached to each of these outcomes. Such an approach is supported by the model proposed by Loomes and Sugden (1986), which encapsulates the emotions of disappointment and elation. In their theory, people seek consistently to maximise expected satisfaction, where that expectation includes the anticipation of possible disappointment and elation (1986:280). Such an approach seems consistent with the previously proposed idea that bettors are rational but also care for more than just monetary returns in their betting decisions. The paper by Koszegi & Rabin (2006) follows a similar path and separates utility into two separable parts, which are central to the model that will be introduced in due course. On the one hand, consumption utility corresponds to the outcome-based utility classically studied in economics (2006:1134). On the other hand, gain-loss utility refers to a reference bundle which endogenously depends on the beliefs attached to each potential outcome. The beliefs that bettors hold are closely related to the emotions they would expect to feel as a result of each potential outcome. This is consistent with the ideas of Mellers et al., who found in their experiment that surprising wins were more pleasurable than expected wins, and surprising losses were more painful than expected losses (1997:333), highlighting the close correlation between beliefs and expected emotions. Such an approach also rests on the notion that individuals are able to accurately and consistently predict future emotions. Such an assumption has been widely approved in previous studies, an example being Loewenstein and Schkade, who state that it is foolish to conclude that people typically mispredict their own feelings (1997:31). Building on the intuition that has so far been discussed, the dynamics of the proposed utility function in the context of the betting market can be expressed as follows. When considering winning, a bettor gains expected utility from the potential monetary return (consumption utility) and an additional expected utility gain that depends on how this result compares to his expectations of winning (gain-loss utility). If the bettor has a low expectation of winning the bet, the expected gain-loss utility is higher than if the bettor has a relatively high expectation of winning. With respect to expected emotions, this is synonymous with the proposal that surprising wins are more pleasurable than expected wins. For an unsuccessful bet, the bettor loses utility from the 1 stake lost (consumption utility) and an additional utility loss that again depends on the comparison of the result with expectations (gain-loss utility). If the bettor has a high expectation of losing this additional utility loss is relatively low, whereas a low expectation of losing induces a high additional utility loss. This is again consistent with the ideas in the decision affect 37

39 model, which suggest that losses are more painful when they are surprising. Therefore, these dynamics can be reconciled through the predictions of the expected emotions theories previously discussed, which will now be formally introduced into a standard EU model. In a standard EU model, the expected utility for a risk-neutral bettor from placing a bet (of value 1) would be: (9) U = q W U W + q L U L (10) U = q w r + (q L 1) (11) U = q w r q L Where q w and q L are the beliefs of winning and losing held by the bettor, q w + q L = 1, U W and U L are the utilities from winning and losing a bet, and r is the return from winning a bet (which itself is equal to the decimal odds). The model used in this paper can be viewed as a modified EU model. Building on the traditional EU model, the model displayed below separates U W and U L into two parts. The original pillars of expected utility theory remain in the utility function, which as discussed previously is denoted as consumption utility, comprising U c,w and U c,l. In addition to this is a reference point that is dependent on bettors expectations of winning and losing. This gain-loss utility is measured in an intuitive way, through the squared difference between the utility gain (loss) from winning (losing) a bet, and the expected utility based on the belief that this outcome would occur, and comprises U g,w and U g,l. The net result of the theory and intuition behind this model is displayed in a modified expected utility function 17 : (12) U = q W (U c,w + U g,w ) + q L (U c,l + U g,l ) (13) U = q w [r + {r (rq w )} 2 ] + q L [ 1 { 1 ( q L )} 2 ] From the model one can see that if the belief of losing (q L ) is a very small number ( 0), then losing comes as a big surprise and U g,l tends towards -1 as disappointment levels are high. In comparison, when q L is very large ( 1), losing comes as no surprises and U g,l tends toward 0. On the other hand, if the belief of winning (q W ) is a very small number ( 0), then winning comes as a big surprise and U g,w tends toward r 2 as the level of elation is relatively high. In comparison, when q W is very large ( 1), winning comes as no surprises and U g,w tends toward One can see that the utility function assumes risk neutral bettors 38

40 The non-linear quadratic function used in this paper is largely consistent with the ideas proposed by Loomes and Sugden (1986), who suggest that gain-loss utility should be convex for all positive values and concave for all negative values (1986:272). Importantly, the dynamics of the specified model largely follow the intuition described above but also achieve some interesting deviations from it, which in hindsight appear to offer an interesting potential insight in to the psychology of a bettor. As Appendix 2 proves, U g,w (U g,l ) q W (q L ) is predictably decreasing (increasing) with q W (q L ) for all values above one-third. However, whenever the relevant belief is small and falls below one-third, these relationships are reversed. This suggests that as the belief of a bettor becomes small, any expected emotions have less of a significant effect on the decision making process given the very small probability attached to the event of experiencing these emotions. The results of the model are now discussed, with the solution (see Appendix 3 for proof) in terms of return displayed below: (14) U = r 2 (q 2 L q 3 L ) + r(1 q L ) + (2q 2 L q 3 L 2q L 1) = 0 In order to determine the empirical success of this model, the set of results were formulated in comparison to the empirical results. This involved applying the data previously used to compare the objective and subjective probabilities in the betting market (see Figure 3). Included in this data were the mean subjective win probabilities for each odds range (P), derived from the mean price (odds) within each range once commission had been deducted. It also included the actual (objective) win probabilities (π) within each price range, and the disparity of these with the subjective probabilities formed the basis of the FLB in Section IV. This data is displayed in Table 10. π W P W +/ = Table 10: Table showing the discrepancy between subjective and objective probabilities in the betting market for men s tennis. 39

41 Table 10 shows the subjective probability of winning on relative longshots is higher than the objective probability, with the reverse being the case for favourites. This therefore implies that longshots are overbet and favourites are underbet. The proposed model should be successful in replicating a similar pattern to that shown in Table 10. Before the model is tested directly, Table 11 also displays the same data, but instead converted into decimal odds form (where Dπ of greater than 2 indicates a longshot). Dπ D P +/- Net Result OVERBET OVERBET OVERBET OVERBET OVERBET = CORRECT UNDERBET UNDERBET UNDERBET UNDERBET Table 11: Table showing the discrepancy between subjective and objective prices in the betting market for men s tennis, and how this predicts that favourites (longshots) will be underbet (overbet). Table 11 allows for a clearer analysis of how the differences in subjective and objective probabilities affect expected returns. In order to test the model, the objective win probabilities (π W ) from Table 10 were entered into the quadratic solution for the proposed model as a proxy for bettor beliefs (see Equation (12)), and the equilibrium return calculated. An equilibrium condition requires bettors to be indifferent between betting and not betting, as well as being indifferent between the two bets on offer. This implies that the expected modified utility from placing a bet should be equal to the normalised utility of not betting, which is assumed to be equal to the betting stake of 1 (see Appendix 3). In order for successful results to be achieved, the calculated equilibrium prices should resemble the relationship in Table 11, with prices lower than the objective prices for longshots, and higher for favourites. The results of the model appear to be successful in matching the same pattern of results that the empirical evidence implies, as Table 12 demonstrates. Through looking at the distribution of positive and negative signs and the net result this is likely to have on betting patterns, the model is very successful in predicting the direction of behaviour manifested in the FLB. Such success is not achieved through the use of the standard EU model. 40

42 π W Model Equil. Price +/- Net Result OVERBET OVERBET OVERBET OVERBET OVERBET UNDERBET UNDERBET UNDERBET UNDERBET UNDERBET Table 12: Table showing the equilibrium prices predicted through the proposed model Subjective Probability Objective Probability Subjective = Objective Win Probability Model Win Probability Subjective Win Probability Fig 6: Figure showing the discrepancies between subjective and objective probabilities in the tennis market compared to the predictions of the proposed model. Naturally the results of any model on this topic cannot solely explain with perfect accuracy the behaviour manifested in the FLB. It was previously stated that explaining the FLB is a very complicated task with numerous different influences likely to play a role. Figure 6 shows that although the model predicts the correct direction of behaviour, the equilibrium subjective win probabilities deviate from the objective probabilities in a more extreme manner than the empirical data suggests. However, this was also to be expected. Through calculating equilibrium results, the model has assumed that all bettors 41

43 develop the same expected emotions regarding their bets, yet it is not controversial to say that this cannot be the case. There will inevitably exist a proportion of bettors who do not follow the predictions of this model for a variety of reasons. For example, some bettors with a high level of initial wealth, or pure leisure bettors, may not experience the same extent of emotions such as disappointment from losing a bet (even on a big favourite) that this model predicts. It is also possible that slight changes to the functional form and allowing for more flexibility in the risk attitude of bettors could provide even more successful results. Having said this, the evidence suggests an approach which incorporates expected emotions and as a result assumes individuals attempt to maximise both monetary reward and emotional satisfaction could be an important influence in explaining the decision making process of a significant number of bettors. With such evidence provided, the assumption that bettors experience varying emotions on different types of bets transforms from an intuitive proposal to an empirical finding which should no longer be disregarded when discussing the FLB. As a result, a serious alternative potential explanation for the existence of the FLB, both in the betting market for men s professional tennis and in other markets, has been provided in this paper. VIII. CONCLUDING REMARKS There has existed for a number of decades a keen interest in the anomaly found in sports markets known as the favourite-longshot bias. Despite the longitude of this interest and the high volume of theories and models it has produced there is still no widespread acceptance as to which provides the best explanation for this empirical regularity. It would seem there are three main reasons for this. Firstly, despite a number of successful results, existing theories are insufficient to provide a satisfactory explanation of the FLB either by themselves or in combination with other theories. Secondly, achieving a set of sufficient theories is a very complicated task made more challenging through the fact that it is extremely difficult to obtain data on individual betting patterns, rather than the aggregate data used in most papers (including this one) which is often readily available. Thirdly, the majority of papers have focused on the racetrack betting market. In order for a sufficient collection of theories to be widely accepted, it is a necessity that they can be successfully applied to more than one sports market. Different findings in a variety of sports markets should provide clues to human (and bookmaker) behaviour within these markets, and should therefore be considered when developing a theory to explain the FLB. 42

44 This paper happens to contribute to the pursuit of these goals. Through studying the largely unexplored market of men s professional market, comprehensive evidence has been found that the FLB exists in this market. The key result here showed that the odds provided by the bookmaker appear to provide better returns for bets on favourites than for bets on longshots. The relationship between odds and return appear to be non-linear, with returns to heavy-longshots providing significantly worse returns. In attempting to explain the FLB, a supply-side model in which bookmakers effectively trim the odds of longshots when they expect there to be insider trading (which is more likely in low-profile events) was tested. Contrary to a previous paper by Forrest and McHale (1997), evidence was found to suggest that such a theory may play a role in explaining the FLB in this market. The result of a series of econometrics tests appeared to show that in the latter rounds of tournaments (which are relatively high profile) the FLB was mitigated to a certain degree. This matched the discussed theory that suggested high-profile events are less likely to be susceptible to insider trading and hence the FLB should be less prominent. In response to the belief that existing theories on the FLB are insufficient, the final part of this paper explored an alternative explanation involving the application of a model incorporating expected emotions. As the concluding remarks of Section VII suggested, such a model presented encouraging results to suggest that emotions play an important role in the decisions of bettors. The conclusion that can be drawn from the results is that a positive step towards a fuller understanding of the stated research question has been achieved. Once the presence of the FLB was necessarily established, it was shown that its underlying causes may not be limited to studying the preferences of bettors. This means future studies in this market and others should adopt a modified research question that is inclusive of the impact market makers (bookmakers) appear to have in distorting market prices. Additionally the work conducted in Section VII provided a new potential explanation as to why bettors consistently bet on longshots even though the expected return is relatively lower than betting on favourites. There appears no reason why this last finding may not be applied to other sports markets in which a positive FLB has also been found. The model could also potentially partially explain the lack of a FLB in some markets. For example, as Table 1 in Section II showed, there are no big longshots or favourites when betting in the baseball market, with bookmaker odds for two teams relatively similar compared to other sports. 43

45 Therefore, the expected emotions arising from betting on the two different teams is likely to also be relatively similar and therefore play less of a role in betting on these games. This could therefore explain why there was the absence of a positive FLB in the study by Woodland and Woodland (2004) (although a negative FLB was found, this may be the result of a different influence). Further work could also be conducted to test formally if the model in this paper supports findings in other sports markets, as well as in financial markets, given the widespread role emotions appear to play in decision making in this context. Several features of this paper as well as others in the field give light to the suggestion that future research on this topic should look to shift the focus to examining data on individual bettors rather than purely studying aggregate data. This could be done through experimental tests, including studies in the field of neuro-economics, and would be particularly useful in improving the understanding of the emotional processes experienced by bettors. Additionally, empirical studies conducted with more inclusive data should also be undertaken. For example, the paper by Feess, Mueller and Schumacher (2011) used a unique data set which not only contained bookmaker odds and results, but also the individual demographics of bettors such as age and gender, which are assigned to each bet. The use of this data allowed the authors to achieve convincing results on betting patterns amongst men and women. Further studies of this nature which assign betting data to individual bettors would allow some of the strict assumptions present in most studies to be relaxed and achieve more realistic and therefore convincing results on the FLB. 44

46 APPENDICES Appendix 1. Retirement and Walkover Matches In order to determine whether matches which had the result Retired or Walkover should be excluded from the analysis, a comparison of results between the relevant subsamples was conducted. Table 1.1 represents this 18. Regressor(s) (1) (2) (3) (4) Probability odds (data including Retirement/Walkovers) (0.0356) Probability Odds (without Retirements/Walkovers) (0.0339) Probability Odds (Retirements/Walkovers Only) (0.3789) Probability Odds (With Retirements/Walkovers, Dummy & IT) (0.0317) Dummy variable (= 1 for Retirements/Walkovers, = 0 for all other bets) (0.0997) Interaction Term (Probability Odds * Dummy) (0.3800) Table 1.1: Table showing the regression coefficients that show that there is a significantly different relationship between return and odds for completed matches, compared to those which had the result Retired or Walkover. Regression (1) is analogous to the regression in Equation (1.1) (see below), except with the addition of Retirement/Walkover matches. (1.1) Return = β 0 + β 1 ProbabilityOdds + u i Regression (2) is identical to the regression that is presented in Equation (2) of this paper. Regression (3) involved once again the same regression equation, but only including Retirement/Walkover matches. The result displayed for this regression is not significant 19. Regression (4) then tests directly for any significant difference in the 18 Numbers in brackets represent standard errors 19 t-statistic =

47 relationship between return and odds for the different sub-samples of matches. The Interaction Term is only significant at an 11% level 20. This finding was already suggested by Regression (3), which displays high variability in its results in comparison to Regressions (1) and (2), and makes a conclusion difficult to make. Table 1.2 presents the confidence intervals for the coefficients on Probability Odds in Regression Equations (1) (3) from Table 1.1: Regression Equation Slope Coefficient 95% Confidence Interval (1) (2) (3) Table 1.2: Table showing the 95% confidence intervals of the Probability Odds coefficients in the Regression Equations in Table 1.1. The evidence suggests that the presence of Retirement/Walkover matches in the analysis may bias the coefficient on Probability Odds downwards, given the possible negative relationship that exists within these matches. However, this cannot be said to be a significant relationship, given the high standard errors associated with Regression (3). Any potential for betting on longshots being better in Retirement/Walkover matches could be explained by the evidence in Table 1.3 below: Type of match # of matches % of Matches Won by Longshots Completed 22, Retirements/Walkovers Table 1.3: Table showing the percentage of matches won by a longshot for each type of match. Table 1.3 shows that the proportion of matches won by outsiders is significantly higher in Retirement/Walkover matches, compared to the success rate of outsiders in completed matches. This could explain why betting on outsiders could be more favourable in these matches, as the frequency of success is higher. An important aspect of incomplete matches which is important in deciding on the inclusion or exclusion of these matches is the possibility of insider information amongst betters about potential retirements/walkovers. It is possible that bookmakers may believe in some cases that insider information amongst bettors exists about a match that will be a 20 t-statistic =

48 walkover/retirement in favour of the longshot (or favourite). Under such beliefs, the rational decision of bookmakers would be to trim the relative odds of the longshot (favourite) if the belief is that the favourite (longshot) will retire. One way to test for this is to compare the relationship between rankings and odds in complete and incomplete matches. It is proposed here that on average the larger the difference in ranking between players, the larger the difference in the odds between the two players. In other words, this means the price of the favourite is lower and the price of the longshot is higher. By conducting a series of separate regressions (see below) of the difference in odds between two players (difodds), the odds of the favourite (oddsfav) and the odds of the longshot (oddslong) against the difference in ranking between players (difrank), the proposed idea can be further explored. (1.2) difodds = B 0 + B 1 difrank (1.3) oddsfav = B 0 + B 1 difrank (1.4) oddslong = B 0 + B 1 difrank If the proposed idea is correct, the slope coefficient should be significantly positive in Equation (1.2), significantly negative in Equation (1.3), and significantly positive in Equation (1.4). To test whether bookmakers believe insider information exists about a future incomplete match (or even on average predict these results themselves), one can compare the regression results of complete matches, versus the two types of incomplete matches (the favourite retires and the longshot retires). The regression results are shown in Table 1.4 below 21 : Type of Match oddsfav oddslong difodds Complete Fav Retires Long Retires Table 1.4: Table showing regression slope coefficients for Equations (1.2) - (1.4) for each type of match. Looking first to the results for complete matches, the sign on each of the slope coefficients is as predicted. This suggests that is true that larger differences in rankings significantly predict larger differences in odds between two players in a match. It is now interesting to compare these results with those for incomplete matches. 21 All coefficients are significant at the 5% level 47

49 For matches where the favourite retired, the slope coefficient from the oddsfav regression is significantly more negative than in the complete matches regression. This would suggest that in matches where the favourite retired, the price of the favourite is comparatively lower relative to the difference in ranking. This is a confusing result, as one would expect that in these matches if the bookmaker on average predicts the retirement/walkover (or believes insider information exists), it should trim the relative odds of the longshot. A similarly confusing result is presented in the oddslong regression for the same type of match, where the relative odds of the longshot are higher than in completed matches. One of the only feasible explanations for this result may be that favourites are better at hiding injuries, making it more difficult for both bookmakers and bettors to predict when a retirement will occur. In comparison, the results for the matches where the longshot retired follow a more intuitive path. The comparative price of the favourite relative to difference in ranking is lower in these matches, and the comparative price of the longshot is higher. The evidence presented suggests that bookmakers are not able to accurately predict retirements in a high proportion of matches. As a result, they could instead adjust their average prices in all matches to reflect the possibility that a retirement may occur. Given that the percentage of longshot wins is higher in incomplete matches compared to complete matches, average prices of the bookmaker should increase the price of longshots and lower the price of favourites. If this proposal is significant, one would expect that there should be no longshot bias when incomplete matches are included in the statistical analysis, yet the results of Regression (1) in Table 1.1 show this is not the case. It can therefore be seen that data originating from Retirement/Walkover matches may not be representative of the data set as a whole and conflicting evidence has been presented to how the market predicts the results of these particular matches, which introduces further uncertainty as to the reliability of this subsample. Its inclusion in the data analysis could potentially lead to biased conclusions being drawn and introduce uncertainty that would cloud the reliability of any results, which provides the explanation for why it was excluded from the analysis in Section V. 48

50 Appendix 2. Dynamics of the proposed model The proposed model in this paper takes the following form: (2.1) U = q W U W + q L U L (2.2) U = q W (U c,w + U g,w ) + q L (U c,l + U g,l ) (2.3) U = q w [r + {r (rq w )} 2 ] + q L [ 1 { 1 ( q L )} 2 ] This function can be divided into two parts: the Win part (q W U W ), which is associated with the expected utility gain from winning a bet; and the Lose part (q L U L ), which is associated with the expected utility loss from losing a bet. Within each part exists a division between consumption utility (U c,w and U c,l ) and gain-loss utility (U g,w and U g,l ), an idea that is discussed further in Section VII. In order to check that the utility function follows the intuition for this model, the firstorder derivatives for each part should be derived. It is postulated that q W U W takes the following form: (2.4) q W U W = q w [U c,w + U g,w ] (2.5) q W U W = q w [r + {r (rq w )} 2 ] This can be written: (2.6) q W U W = q w r + q W {r (rq W )} 2 U c,w is synonymous with classical EU theory, therefore the derivative of utility with respect to the belief of winning (q w ) should be greater than zero (increasing), and this indeed is the case for all values of q w : U c,w q W = r > 0 Equally, it is simple to show that the derivative of q L is decreasing in this part of the utility function. On the other hand, the intuition behind this model suggests that as the belief of winning increases, the extent of an expected positive emotional response (such as elation) to winning is decreasing. Therefore, the first derivative of U g,w should be less than zero. In the majority of cases, this is what is found. However, when the belief of winning is below one-third the function is increasing. This is discussed further at the end of this appendix. 49

51 The proof of this derivative is outlined below. The gain-loss section of the Win part can be simplified as follows: U g,w = q W (r rq W ) 2 = q W [r 2 2(r 2 )q W + (rq W ) 2 ] = q W [1 2q W + q 2 W ]r 2 = (q W 2q 2 W + q 3 W )r 2 The resulting derivative is: U g,w = 1 4q q W + 3q 2 W < 0 W = (3q W 1)(q W 1) < 0 Since (q W 1) < 0, this can be written as: = 3q W 1 > 0 = q W > 1 3 Therefore U g,w is decreasing in q W for all values above one-third, which supports the intuition of the model to a large extent. It is also simple to show that the derivative of q L is decreasing for all values below two-thirds. Having matched the intuition behind the model with the Win part of the utility function, the Lose part becomes the focus of attention: (2.7) q L U L = q L (U c,l + U g,l ) (2.8) q L U L = q L [ 1 { 1 ( q L )} 2 ] This can be written: (2.9) q L U L = q L q L { 1 ( q L )} 2 With U c,l being analogous to classical EU theory, the derivative of utility with respect to the belief of losing (q L ) should be less than zero, and this indeed is the case for all values of q L : U c,l q L = 1 < 0 50

52 Equally, it is easy to observe the derivative of q W is be increasing in this part of the utility function. The intuition of the model suggests that as the belief of losing increases, the extent of the expected negative emotional response (such as disappointment) to losing is decreasing. Therefore, the first derivative of U g,l should be greater than zero (as utility losses become less negative). The expected result is largely found in this model, except for when the belief of losing falls below one-third. A discussion of this is undertaken at the end of this Appendix, and the algebraic proof is shown below. The gain-loss section of the Lose part can be simplified as follows: U g,l = q L (q L 1) 2 = q L (q 2 L 2q L + 1) = q 3 L + 2q 2 L q L The resulting derivative is: U g,l = 3q 2 q L + 4q L 1 > 0 L = ( 3q L + 1)(q L 1) > 0 = 3q L 1 > 0 = q L > 1 3 Therefore U g,l is increasing in q L for all values above one third, and decreasing for values of q L that are less than one-third. It is also simple to show that the derivative of q W is decreasing for all values below two-thirds. Explanation for why the dynamics of the model do not completely follow the initial expected intuition First it is important to state that when the gain-loss utility section of the model is taken in isolation i.e. it is not weighted by the beliefs of the bettor, then the original intuition is perfectly represented in the model. This means looking at {r (rq w )} 2, which is clearly always decreasing in q w, and {q L 1} 2 which is clearly always increasing (becoming less negative) in q L. However, the reason for the first-order derivatives taking on different properties than was initially expected is that this gain-loss utility is weighted by the beliefs the bettor 51

53 holds. In the extremities of when the belief of winning is above two-thirds (the belief of losing is below one-third) the sign of the first-order derivative changes. Looking first to the Win part of the utility function, the results showed that gainloss utility is decreasing in q W for all values above one-third, and increasing in values below one-third. It may be the case that as a bettor s belief of winning becomes so small, any potential expected emotions (as large as they may be) become less significant as the bettor s subjective probability of that particular outcome occurring is so small. Therefore the change in the first-order derivative may be due to this effect. Turning attention to the Lose part of the utility function, a similar argument can be made. The results found that gain-loss utility is increasing in q L for all values above one third, and decreasing for values of q L that are less than one-third. Similar to the argument above, this may be reconciled through the fact that although the degree of the emotional response may be very large, the chance of it occurring is so small that the bettor may attached small significance to it in his decision making process. Appendix 3. Algebraic proof of the solution to the proposed model The algebraic derivations displayed below outline the process of solving the model for the variable r, which represents the return to a bet. U = q w [r + {r rq W } 2 ] + q L [ 1 {q L 1} 2 ] = q w [r + {r r(1 q L )} 2 ] + q L [ 1 {q L 1}{q L 1}] = q w [r + (rq L ) 2 ] + q L [ 1 {q L 1}{q L 1}] = q w [r + (rq L ) 2 ] + q L [ 1 {q 2 L 2q L + 1}] = q w [r + (rq L ) 2 ] + q L [2q L q 2 L 2}] = [r + (rq L ) 2 ] q L [r + (rq L ) 2 ] + q L [2q L q 2 L 2] = [r + (rq L ) 2 ] + q L [2q L q 2 L 2 r (rq L ) 2 }] = r + (rq L ) 2 + 2q L 2 q L 3 2q L rq L (rq L ) 2 q L = r + (rq L ) 2 + 2q L 2 q L 3 2q L rq L (rq L ) 2 q L = r 2 (q L 2 q L 3 ) + r(1 q L ) + (2q L 2 q L 3 2q L ) For an equilibrium state to exist, bettors should be indifferent between betting on the two options and not betting at all. Therefore, the expected modified utility from betting given 52

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