Efficiency of football betting markets: the economic significance of trading strategies

Transcription

1 Accounting and Finance 45 (25) Efficiency of football betting markets: the economic significance of trading strategies Philip Gray, Stephen F. Gray, Timothy Roche UQ Business School, University of Queensland, St Lucia, 472, Australia Abstract Prior studies of the Australian Rugby League betting market report a degree of predictability well in excess of that attributable to chance. However, two important recent changes in the structure of the market facilitate an unambiguous assessment of the statistical significance of predictability and the economic significance of returns to betting strategies. The present paper reexamines the efficiency of the Australian Rugby League betting market under the revised market structure. In addition, a set of measures of the economic significance of trading strategies are developed and implemented. Relative to prior studies, the out-of-sample success of the predictive model has diminished notably under the revised market structure; although a naïve strategy betting on home underdogs still performs significantly better than can be attributed to chance. Simulation experiments suggest that the documented level of predictability from several strategies generates economically significant returns. Key words: economic significance; football betting; market efficiency; probit; sports betting JEL classification: G1, G14 doi: /j x x 1. Introduction The study of market efficiency is concerned with the ability of individuals to quickly and accurately incorporate relevant information into the market prices of assets. In an efficient market, prices are an unbiased estimate of the true asset value. That is not to say that assets are never incorrectly valued, but rather that there are no systematic biases that can be exploited to yield consistent abnormal (risk-adjusted) returns. The difficulty in testing efficiency in financial markets is that the true asset The authors gratefully acknowledge the suggestions of an anonymous referee and financial support from a UQ Business School Summer Research grant. Received 27 February 24; accepted 21 May 24 by Robert Faff (Editor). C AFAANZ, 25. Published by Blackwell Publishing.

2 27 P. Gray et al. / Accounting and Finance 45 (25) value is never revealed. Share value, for example, is a complex function of firmspecific and macroeconomic factors that are difficult (if not impossible) to measure and forecast. In contrast, sports betting markets are often idyllic for testing efficiency. Prior to the event (game, horse race etc.), all possible outcomes and their pay-offs are known. Expectations are readily observable (by way of spreads, odds, or relative support rates). Most importantly, the true value is revealed, without ambiguity, at the conclusion of the event. In such ideal conditions, efficiency can be easily examined from both statistical and economic perspectives. In football betting, for example, the pointspread should be an unbiased estimate of the actual outcome of the game. If systematic biases exist in the market s ability to incorporate information into spreads, betting strategies can be formulated to exploit these biases and earn economically significant profits, therefore providing evidence of the inefficiency of the market. In a prior study of the Australian Rugby League (ARL) betting market, Brailsford et al. (1995) document success rates to naïve trading strategies using publicly available information well in excess of those that could be attributed to chance. For example, over the period, betting on home underdogs in the ARL has a per cent success rate. A more sophisticated trading strategy based on a probit model has a success rate over 75 per cent. While these findings strongly suggest market inefficiencies, there are two attributes of the market structure over the Brailsford et al. (1995) sample period that are important when interpreting their results. First, during their sample period, it was not possible to place spread bets on individual games. Spreads were quoted for each game, but bettors had to correctly predict the outcome in all eight games in the round (a system known as Pick the Winners). While Brailsford et al. (1995) argue that there is no reason to believe spreads set for Pick the Winners are in any way biased, the fact remains that their simulated trading strategies could not have been implemented. Second, the Brailsford et al. (1995) dataset did not permit an examination of the economic significance of betting on ARL games. This is because dividends were not known at the time of placing the bet, but rather were calculated after the game (as explained in Section 2). Subsequent to Brailsford et al. (1995), two important changes have occurred in the (legal) ARL betting market that: (i) are likely to increase the efficiency of spreads on individual games; and (ii) appreciably enhance our ability to examine the economic significance of trading strategies. SportsTAB, a government-backed betting house, commenced pointspread betting on individual games in As a consequence, the importance of setting unbiased spreads for each game is now paramount and the strong biases documented by Brailsford et al. (1995) are unlikely to persist. In addition, betting strategies such as those originally examined by Brailsford et al. (1995) are now genuinely implementable. The second important change is that dividends are no longer calculated in a parimutuel manner after the completion of games. Instead, a fixed dividend is established at the time the bet is placed, permitting an unambiguous examination of the economic significance of betting strategies. C AFAANZ, 25

3 P. Gray et al. / Accounting and Finance 45 (25) Together, these changes present a unique opportunity to conduct a robust investigation of the efficiency of the ARL football betting market. The initial objective of the present paper is to reexamine the efficiency of the ARL betting market under the revised market structure. Furthermore, the present paper develops and implements a set of measures of the economic significance of various betting strategies. The more recent dataset ( ) following the introduction of SportsTAB betting on individual games guarantees both that betting strategies are readily implementable and that the returns to such strategies can be reliably measured. The remainder of the present paper is structured as follows. Section 2 details the important recent changes in the structure of the ARL betting market that motivate this paper. Section 3 describes the data and predictive model used to develop betting strategies. Section 4 reports the findings in terms of both statistical and economic significance of betting strategies, and Section 5 concludes the paper. 2. Recent changes in betting market structure Prior to the introduction of SportsTAB in 1998 pointspreads for all ARL games were set by the government-operated FootyTAB. However, spread betting on individual ARL games was unavailable. 1 Rather, bettors played a system known as Pick the Winner, under which they attempted to correctly select the winners (after adjusting for the pointspread) in all eight games in a particular round. While FootyTAB had incentives to set unbiased spreads, the consequences of an occasional error were minimal because correctly selecting the winners in the remaining games was a difficult task (see Brailsfordet al., 1995). In contrast, with betting now allowed on individual games, Sportstab will ideally set an unbiased spread for each game to achieve equal support on either team, therefore maximising their commission. As a result, the bias in recent spreads is likely to be lower than documented by Brailsford et al. (1995). The mechanism by which dividends are calculated has also changed. Under FootyTAB, spreads were set on the Tuesday before the weekend games and remained fixed. Calculated in parimutuel fashion, dividends were a function of the total prize pool (less FootyTAB s 25 per cent commission) and the support rate for the winning team. Bettors did not know their dividend until after the completion of all games in the week. In contrast, SportsTAB establishes the dividend at the time the bet is placed. Spreads are set mid-week with an opening margin of $A1.85 for each team (i.e. an 85 per cent return on a $A1 bet). Dividends on new bets change in the lead up to kick-off as new information becomes available. It is important to understand the institutional structure of Australian football betting in the context of international markets. In US football markets, the pay-off to a winning bet is fixed and the spread fluctuates to reflect changing information. In such markets, 1 Other betting houses, both legal and illegal, took bets on individual games, but no data from these sources are available. C AFAANZ, 25

4 272 P. Gray et al. / Accounting and Finance 45 (25) the spread at any point in time is expected to be an unbiased estimate of the game outcome. Under SportsTAB, while the opening spread is expected to be unbiased at the time it is set, it obviously becomes stale as time to kick-off draws near. Instead, changing expectations are captured by revisions in quoted dividends. The empirical analysis in the present paper utilizes opening spreads and dividends set by SportsTAB. Although, to some extent, opening spreads become stale as conditions change in the lead up to kick-off (e.g. match-day weather, player injuries etc.), any bias introduced by the market s use of fixed pointspreads is unlikely to be systematic across the sample. In any event, the use of opening dividends is a valid means of determining the economic significance of betting strategies because the forecasting model developed in Section 3 incorporates conditioning information that is available at the time opening dividends are established. Betting strategies can be formulated at the time of the opening dividend and, unlike the Brailsford et al. (1995) study, returns to trading can be objectively calculated. 3. Data and methodology Data on all ARL games from 1998 to 22 were obtained from a variety of sources. All 22 data were downloaded directly from the official SportsTAB website ( while 2 1 data were obtained from the renowned fan website World of Rugby League ( Data for were hand collected from the Sydney Morning Herald archives. Because our predictive model conditions on home team status, final-series (playoff) games are excluded because most are played at neutral venues. In total, there are 988 regular-season games during Following the innovation of Brailsford et al. (1995) and Gray and Gray (1997), a probit model is employed to forecast game outcomes. Let Y i = 1 if the home team beats the spread and Y i = otherwise. In the probit model, Y i = 1ifY i > and Y i = otherwise. Maximum-likelihood estimation is used to calibrate the following model (Y i ) to the observed data (Y i): Y i = α + β 1 SPREAD i + β 2 DOG i + β 3 HL4 i + β 4 AL4 i + β 5 HFA i + β 6 AFA i + ε i. (1) DOG is a dummy variable that takes the value of unity if the home team is an underdog. The pointspread (SPREAD), which is positive (negative) if the home team is an underdog (favourite), is included to capture any bias associated with the degree of favouritism. HL4 and AL4 are variables reflecting recent performance, defined as the number of wins (relative to the spread) in the home and away teams last four games. The season for and against record of the home (HFA) and away (AFA) teams is included as a longer-term performance measure. C AFAANZ, 25

5 P. Gray et al. / Accounting and Finance 45 (25) Results 4.1. Statistical significance of predictability Summary statistics on pointspreads and actual game margins show a clear underestimation of the home-ground advantage. On average, home teams receive 3.6 points start, but win by 4.91 points (the difference in means is statistically significant at better than 1 per cent). Surprisingly, this bias is more than double that reported by Brailsford et al. (1995) during the period, suggesting that the home-ground factor continues to be underpriced by the market. In contrast, favourites appear correctly priced: on average, they give up 7.96 points start and win by 7.75 points (a statistically insignificant difference). Table 1 reports maximum likelihood estimates of probit model (1). Because Y i is defined relative to the home team, model (1) can be used to predict the probability that the home team will beat the spread. The most prominent finding is the positive marginal impact of underdog status (DOG): home underdogs are significantly more likely to beat the spread. Other conditioning information, such as the degree of favouritism (SPREAD), recent performance (HL4, AL4), and season-to-date Table 1 Estimates of probit model Variable Coefficient p-value Intercept (α) SPREAD (β 1 ).44.7 DOG (β 2 ) HL4(β 3 ) AL4(β 4 ) HFA (β 5 ) AFA (β 6 ).2.75 Correct predictions 56.9% <.1 The table shows estimates for probit model (1): Y i = α + β 1 SPREAD i + β 2 DOG i + β 3 HL4 i + β 4 AL4 i + β 5 HFA i + β 6 AFA i + ε i. DOG is a dummy variable that takes the value of unity if the home team is an underdog. The pointspread (SPREAD), which is positive (negative) if the home team is an underdog (favourite), is included to capture any bias associated with the degree of favouritism. HL4 and AL4 are variables reflecting recent performance, defined as the number of wins (relative to the spread) in the home and away teams last four games. The season for and against record of the home (HFA) andaway(afa) teams is included as a longer-term performance measure. In all games, the pointspread and underdog status are defined from the home team s perspective. The sample consists of 826 regular-season games during The p-value on correct predictions adopts a Binomial test on the null hypothesis that the predictability is 5 per cent. C AFAANZ, 25

6 274 P. Gray et al. / Accounting and Finance 45 (25) performance (HFA, AFA), appears to be accurately incorporated into the pointspread. 2 In terms of predictive ability, model (1) correctly forecasts the outcome in 56.9 per cent of all games. This level of predictability falls short of the 6 per cent success rate reported by Brailsford et al. (1995), consistent with the expected impact of the significant change in market structure. 3 To summarize, there is some evidence of statistical inefficiency in the ARL betting market. Simple conditioning information such as knowledge of the home team and underdog status is significantly related to game outcomes. Relative to prior findings, however, the predictive ability of the probit model appears to have diminished. This is likely to be a result of the revised market structure that now permits betting on individual games at fixed odds. Naturally, the findings suggest several betting strategies: for example, betting on home teams and/or underdogs. The next section documents the economic significance of returns to such strategies Economic significance of betting strategies Analysis of the economic significance of inefficiency is based on the assumption that bets are placed according to the opening spreads set by SportsTAB at the opening dividend. With default opening margins of $A1.85, the breakeven success rate is 54.1 per cent. 4 Table 2 reports the success rates and returns to a number of betting strategies. Three naïve strategies are tested on the basis of biases documented above. Simple strategies such as betting strictly on home teams or underdogs are possible in all games, yet neither trading rule is profitable. The statistical predictability documented in Section 4.1 does not translate into economic profits. While success rates are above 5 per cent, they do not reach the breakeven rate. In contrast, betting whenever the home team is the underdog has a success rate of per cent and generates significant profits (7.54 per cent). This opportunity arises in nearly 37 per cent of all ARL games during , further supporting the economic significance of returns. The probit model allows the implementation of more sophisticated trading rules. For example, in each game, a bet can be placed on the team forecasted to have the highest probability of winning. Alternatively, filter rules can be implemented under which bets are placed only when the probit probability of winning exceeds 2 The home-team for-and-against record (HFA) is statistically significant at the 5 per cent level, but the magnitude of β 5 is economically negligible. That is, a one standard deviation increase in HFA has a negligible impact on the model s prediction of the game outcome. 3 In other results (not reported), the precise model of Brailsford et al. (1995) was estimated for data after the change in market structure. The predictive accuracy of that model was around 53 per cent. 4 The breakeven success rate follows from solving for p in (p.85) + (1 p) 1 =, where p is the probability of success. C AFAANZ, 25

7 P. Gray et al. / Accounting and Finance 45 (25) Table 2 Economic significance: returns to betting strategies Strategy Number of bets Success p-value Average p-value (% of sample) rate (%) return (%) Naïve Strategies Bet on underdogs 988 (1.%) Bet on home teams 988 (1.%) Bet on home underdogs 363 (36.74%) < Probit-based strategies Bet on probit prediction In-sample 826 (1.%) 56.9 < Out-of-sample 34 (1.%) Only bet when probit probability exceeds 54.1% In-sample 471 (57.2%) < Out-of-sample 185 (6.86%) Only bet when probit probability exceeds 57.5% In-sample 266 (32.2%) 6.9 < Out-of-sample 12 (39.47%) Only bet when probit probability exceeds 6.% In-sample 157 (19.1%) < Out-of-sample 79 (25.99%) Only bet when probit probability exceeds 65. In-sample 35 (4.24%) Out-of-sample 31 (1.2%) Only bet when probit probability exceeds 7.% In-sample 4 (.48%) 1. < <.1 Out-of-sample 5 (1.64%) This table shows the success rates of and returns to both naïve and probit-based betting strategies. For naïve (probit-based) betting strategies, the in-sample data consists of 988 (826) regular-season games during Out-of-sample results are based on probit estimates using 522 games during , applied to 34 games during The significance of the success rate is the probability of obtaining the observed success rate if the probability of success is 5 per cent. The significance of the trading return compares the observed success rate to the breakeven rate of 54.1 per cent. some arbitrary cut-off level. The distribution of forecasted probabilities of beating the spread under probit model (1) shows that, while games with extremely high probabilities of beating the spread are rare, there are numerous games with forecasted probabilities in the 55 7 per cent range. 5 We examine a number of filter rules involving these games. Two sets of results are reported in Table 2: (i) in-sample results estimate the probit model and evaluate trading profits over the entire sample (826 games); 5 The distribution of probit probabilities (not shown to preserve space) is available on request. C AFAANZ, 25

8 276 P. Gray et al. / Accounting and Finance 45 (25) and (ii) out of-sample results estimate the probit model over (522 games) then analyse trading rules during the 21 2 seasons (34 games). 6 Table 2 reports that several probit-based betting strategies generate out-of-sample success rates significantly higher than can be attributed to chance. Betting on the team predicted by the probit model, which is possible in all games, has an out-of-sample success rate of per cent. Using the model to implement filter rules generates even higher success rates. For example, betting on teams with a better than 6 per cent chance of beating the spread (an opportunity that arises in nearly 26 per cent of all games) produces a success rate of per cent. Because all probit-based strategies succeed more often than the breakeven rate, they result in positive returns. A betting strategy using the 6 per cent probit filter returns 7.72 per cent over the out-of-sample period. The economic significance of these returns is judged using a binomial test where the actual success rate is compared to the breakeven rate. In stark contrast to Brailsford et al. (1995), out-of-sample returns are insignificant for all probit-based strategies. This might reflect the heightened importance of setting unbiased spreads after the introduction of betting on individual games with fixed odds in Our final analysis aims to illustrate the practical implications of success rates documented in Table 2 to bettors adopting such strategies. Using a simulation experiment, a set of measures are developed to assess the economic significance of various betting strategies. We assume an initial bankroll of $1 and simulate the performance of a hypothetical bettor following various betting strategies. In the simulation, the success rate of each strategy is assumed to be the out-of-sample performance from Table 2. Similarly, the frequency with which bets are placed is calibrated to Table 2. For example, a home-underdog betting strategy has a per cent chance of success and opportunities arise in per cent of games. We assume a 15-game season; therefore, 55 home-underdog bets are available in a single season. A bet is placed on each available game at the opening dividend of $1.85. The performance over the season is simulated and the procedure is repeated 1 times. Table 3 reports several probabilities of interest to bettors; namely, the probability the bettor will be financially ruined during a single season, the probability of making a loss over a single season, and the probability that the initial bankroll will be doubled in a single season. The mean and median season-end closing bankroll over 1 simulations are also calculated, as well as 95 per cent intervals for the closing balance. In separate simulations, we calculate the expected time (expressed in number of seasons) to double the initial bankroll. 7 Two forms of bets are considered: (i) the bettor places fixed wagers of $5; and (ii) the wager is a fixed proportion of the accumulated bankroll at the time each bet is placed. For the latter, the optimal 6 Bets are not placed on the first four games each season because the probit model conditions on the most recent four games. The reduced sample for probit-based betting strategies is 826 games over This calculation again incorporates the number of available betting opportunities under a particular strategy. Betting on the home team or the team predicted by the probit model can be undertaken in all games. Probit filter-based bets arise less frequently. C AFAANZ, 25

9 P. Gray et al. / Accounting and Finance 45 (25) Table 3 Implications of success rates Betting strategy Home Home Bet on Bet Bet Bet team underdog Probit when when when prediction p > 54.1% p > 6% p > 65% Fixed amount wagers Mean ($) ,11 Median ($) , % intervals [,1918] [563,1858] [,2288] [428,2185] [58,169] [62,136] Pr(ruin) (%) Pr(loss) (%) Pr(double bankroll) (%) E (time to double infinite bankroll) Fixed proportion wagers Mean ($) Median ($) % intervals [56,27] [379,3853] [586,1884] [296,5492] [44,398] [876,1129] Pr(ruin) (%) Pr(loss) (%) Pr(double bankroll) (%) E(time to double bankroll) The table shows statistics of interest to bettors relating to various betting strategies. The initial bankroll is $1. Fixed amount wagers are $5 per bet, while fixed proportion wagers are according to the Kelly ratio. Pr(ruin), Pr(loss), and Pr(double bankroll) are the probabilities of bankruptcy, a loss, and doubling the initial bankroll over the course of a 15 game season. The E(time to double bankroll) is how many 15-game seasons are required to double the initial bankroll. The calculations take into account the frequency with which bets can be placed under the various strategies. All results follow from 1 simulations. proportion is determined using the Kelly ratio (Kelly, 1956). 8 The Kelly ratio is optimal in that it maximizes the exponential rate of growth and achieves a particular target bankroll in the minimum time (see Beaudoin et al., 22). For fixed $5 bets, Table 3 favours a probit-based strategy betting whenever the probability of success exceeds the breakeven 54.1 per cent. It results in the highest mean closing bankroll ($1319), has the shortest time to doubling the initial bankroll (2.49 seasons), and exposes the bettor to a negligible probability of ruin (.29 per cent). Betting strictly on home underdogs also performs well. Interestingly, 8 In this context, the Kelly ratio f = ((p win $1.85) 1)/.85, where p win is the probability of winning under a particular strategy. C AFAANZ, 25

10 278 P. Gray et al. / Accounting and Finance 45 (25) Home team.2 Home underdog.12 Proportion of simulations Probit prediction.1 p>.541 Proportion of simulations p>.6.3 p> Proportion of simulations Figure 1 Distribution of closing bankroll (fixed wagers). The figure shows the distribution of the closing bankroll to six betting strategies over 1 simulated seasons. Under each strategy, the probability of success and frequency with which bets are placed is calibrated to the out-of-sample results in Table 2. The initial bankroll is assumed to be $1 and wagers are fixed at $5. C AFAANZ, 25

11 P. Gray et al. / Accounting and Finance 45 (25) the breakeven filter strategy outperforms the home-underdog strategy despite having a lower out-of-sample success rate. This is attributable to the fact that the former strategy allows betting in more games each season (6.86 per cent vs per cent). To further illustrate the risk of each betting strategy, Figure 1 graphs the distribution of season-end closing bankroll (under fixed $5 wagers) over the 1 simulations. The attractiveness of the breakeven filter and home-underdog strategies is apparent. In contrast, home team and probit prediction strategies show a high dispersion of final bankroll, with considerable mass of the distribution below the initial bankroll (65 per cent and 41 per cent, respectively). Probit-based filter strategies implemented only when there is a high probability of success have notably less variation, which is attributable to the fact that they allow bets in relatively few games each season. Table 3 also suggests that proportional wagers according to the Kelly ratio outperform fixed dollar wagers. In most cases, the median season-end closing bankroll exceeds that under fixed wagers, and the probability of doubling the initial bankroll in a single season increases dramatically. For example, given a $1 initial bankroll, the breakeven filter strategy has a median closing bankroll of $1377, representing a 38 per cent return over a season. An added advantage of a wager proportional to the bankroll is that the bettor is never ruined. The expected time to double the initial bankroll is notably faster under proportional wagers. On the more successful strategies, the dollar amount bet increases rapidly as the bankroll rises. For several strategies (e.g. probit prediction and p > 65 per cent), the expected time to double bankroll is slower under proportional betting. These strategies have modest success rates (in the vicinity of 55 per cent: see Table 2), so if luck has it that the initial bankroll is eroded, the dollar amount wagered also decreases, meaning it will take longer to recover losses. Figure 2 shows the distribution of season-end closing bankroll under proportional wagers. To enhance the clarity in these graphs and comparability across betting strategies, extreme outliers from the right tail of the distribution were deleted. 9 Unlike Figure 1, the graphs under proportional wagers do not exhibit the spikes at zero closing bankroll, because it is impossible to be ruined using the Kelly ratio. Proportional wagers, therefore, appear beneficial across the board. 5. Conclusions Prior research on the efficiency of Australian football betting markets suggests a degree of predictability using publicly available conditioning information well in excess of what can reasonably be attributed to chance (Brailsford et al., 1995). However, the economic significance of this predictability is unclear because the 9 Specifically, and of the 1 simulations with closing bankrolls exceeding $4 were excluded from the graphs of the home-underdog and breakeven filter strategies, respectively. Therefore, the performance of home-underdog and breakeven filter strategies is even more impressive than these graphs suggest. C AFAANZ, 25

12 28 P. Gray et al. / Accounting and Finance 45 (25) Home team.12 Home underdog.12.1 Proportion of simulations Probit prediction.1 p>.541 Proportion of simulations p>.6.6 p> Proportion of simulations Figure 2 Distribution of closing bankroll (proportional wagers). The figure shows the distribution of the closing bankroll to six betting strategies over 1 simulated seasons. Under each strategy, the probability of success and frequency with which bets are placed is calibrated to the out-of-sample results in Table 2. The initial bankroll is assumed to be $1 and wagers are proportional to the bankroll as determined by the Kelly ratio. C AFAANZ, 25

13 P. Gray et al. / Accounting and Finance 45 (25) structure of the betting market at the time did not allow the implementation of the required betting strategies. The present paper reexamines the efficiency of Australian football betting markets after two recent changes in the market structure. Bettors can now place bets on individual games and dividends are fixed at the time of the bet. Therefore, it is possible to make an unambiguous assessment of the statistical significance of predictability and the economic significance of returns to betting strategies. The results suggest that, while statistical biases remain in relation to home ground advantage and underdog status, the predictive ability of the probit model has diminished notably since the revision in the structure of the betting market. Similarly, out-of-sample success rates to betting strategies are lower than previously reported. However, the success rates still appear sufficient to generate profits that bettors would regard as economically significant. With proportional wagers determined according to the Kelly ratio, a naïve strategy betting on home underdogs generates a (median) return of 21 per cent over a typical season. A sophisticated strategy that bets only when the probit model suggests a likelihood of success exceeding the 54.1 per cent breakeven rate returns 37.7 per cent over a typical season. Under this strategy, there is no possibility of ruin and the bettor expects to double the initial bankroll every 1.2 seasons. References Beaudoin, D., R. Insley, and T. B. Swartz, 22, Studying the bankroll in sports gambling, proceedings of the Sixth Australian Conference on Mathematics and Computers in Sport, Bond University, Australia, 1 3 July. Brailsford, T. J., S. A., Easton, P. Gray, and S. F. Gray, 1995, The efficiency of Australian football betting markets, Australian Journal of Management 2, Gray, P., and S. F. Gray, 1997, Testing market efficiency: evidence from the NFL sports betting market, Journal of Finance 52, Kelly, J. L., 1956, A new interpretation of information rate, Bell Systems Technical Journal 35, C AFAANZ, 25