A statistical theory of optimal decision-making in sports betting

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Optimality in “moneyline” wagering

For example, Theorem 2 is highly related to the area of no profitable bet presented in the aforementioned paper. In general, I see a lot of overlap between the two papers, both analysing betting decisions both from a theoretical point and based on real-world data. The author could also describe how the current manuscript is different from the aforementioned paper, e.g. by using point spreads and by analysing quantiles.

I am sorry that we cannot be more positive on this occasion, but hope that you appreciate the reasons for this decision. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

If one bets on the Pacers, the Pacers would have to win outright or lose by no more than three points for the bettor to win. A four point victory by the Heat (four point loss by the Pacers) would equal a tie and the money bet by the NBA gambler is returned to him. The general patterns in Figure 1, that home underdogs perform better relative to the point spread than home favorites and that outcomes tend to be less extreme than predicted by extreme over/unders, are well-documented in the literature. The literature in this area is quite vast (search “NFL betting” in Google Scholar to see what I mean). I do not find the incremental contribution of this paper to be compelling enough for PLOS-ONE.

The significance of this may be exacerbated by the high noise level on the target random variable, and the low ceiling on model accuracy that this imposes. The theoretical results presented here, despite seemingly straightforward, have eluded explication in the literature. The central message is that optimal wagering on sports requires accurate estimation of the outcome variable’s quantiles. For the two most common types of bets—point spread and point total—estimation of the 0.476, 0.5 (median), and 0.524 quantiles constitutes the primary task of the bettor (assuming a standard commission of 4.5%). For a given match, the bettor must compare the estimated quantiles to the sportsbook’s proposed value, and first decide whether or not to wager (Theorem 2), and if so, on which side (Theorem 1). To that end, the goal of this paper is to provide a statistical framework by which the astute sports bettor may guide their decisions.

Optimal estimation of the margin of victory

Wagering is cast in probabilistic terms by modeling the relevant outcome (e.g. margin of victory) as a random variable. Together with the proposed sportsbook odds, the distribution of this random variable is employed to derive a set of propositions that convey the answers to the key questions posed above. This permitted the estimation of the 0.476, 0.5, and 0.524 quantiles over subsets of congruent matches.

The bookmaker establishes the lines at which betting commences and then moves the line as bets are wagered on both sides of the line. Bettors typically pay the bookmaker $11 to win $10, providing the bookmaker a commission profit if money on both sides of the bet are balanced. ballybet usa Because of this commission, commonly called the “vig” or “juice”, bettors must win 52.38% of their bets to break even. A winning percentage greater than 52.38% insures a profit for the bettor. In order to overcome the discrete nature of the margins of victory and point totals, kernel density estimation was employed to produce continuous quantile estimates. The KernelDensity function from the scikit-learn software library was employed with a Gaussian kernel and a bandwidth parameter of 2.

• Define an “excess error” as a wager that is placed on the side that does not maximize the expected profit of a moneyline wager.
• Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.
• In over-under betting, a positive expected profit is only possible if the sportsbook total τ is less than the(ϕo1+ϕo)-quantile, or greater than the(11+ϕu)-quantile, of t.
• Matches were stratified into 24 subsamples defined by the value of the sportsbook total.
• I have now heard back from an expert reviewer on your paper titled “Statistical Principles of Optimal Decision-Making in Sports Wagering.” The reviewer expressed that the work does not adequately add to the existing literature on this topic.
• The author would also like to acknowledge the effort of the reviewers, in particular Fabian Wunderlich, for providing many helpful comments and critiques throughout peer review.

The findings presented here suggest that conventional regression may be a sub-optimal approach to guiding wagering decisions, whose optimality relies on knowledge of the median and other quantiles. The presence of outliers and multi-modal distributions, as may be expected in sports outcomes, increases the deviation between the mean and median of a random variable. In this case, the dependent variable of conventional regression is distinct from the median and thus less relevant to the decision-making of the sports bettor.

The sportsbook’s proposed spread (or point total) effectively delineates the potential outcomes for the bettor (Theorem 3). This finding underscores the importance of not wagering on matches in which the sportsbook has accurately captured the median outcome with their proposition. In such matches, the minimum error rate is lower bounded by 47.6%, the maximum error rate is upper bounded by 52.4%, and the excess error rate (Theorem 4) is upper bounded by 4.8%. One may intuit that the goal of the sports bettor is to produce a closer estimate of the median outcome than the sportsbook. However, an important consequence of Theorem 4 is that estimators of the median outcome in sports betting need not be more precise than the sportsbook’s proposition in order to achieve a positive expected profit.

The classic paper by Kelly 25 provides the theory for optimizing betsize (as a function of the likelihood of winning the bet) and can readily be applied to sports wagering. The Kelly bet sizing procedure and two heuristic bet sizing strategies are evaluated in the work of Hvattum and Arntzen 26. The work of Snowberg and Wolfers 27 provides evidence that the public’s exaggerated betting on improbable events may be explained by a model of misperceived probabilities. Wunderlich and Memmert 28 analyze the counterintuitive relationship between the accuracy of a forecasting model and its subsequent profitability, showing that the two are not generally monotonic. Despite these prior works, idealized statistical answers to the critical questions facing the bettor, namely what games to wager on, and on what side to bet, have not been proposed.

Moreover, it is interesting to see a manuscript in the domain of forecasting/sports betting that is able to answer relevant questions without the use of a concrete forecasting model. The theory is explained intuitively and is easy to follow, the results are well-explained and graphically represented. I’d like to express my respect for the effort done by the author with regard to this manuscript. Please see my several critical comments as an effort to further improve the manuscript. Theorem 1 To maximize the expected profit of a wager, one should bet on the home team if and only if the spread is less than the -quantile of m. A positive expected profit is only possible if the spread is less than the(ϕh1+ϕh)-quantile, or greater than the(11+ϕv)-quantile of m.

The analysis of sportsbook point spreads performed here indicates that only a single point deviation from the true median is sufficient to allow one of the betting options to yield a positive expectation. On the other hand, realization of this potential profit requires that the bettor correctly, and systematically, identify the side with the higher probability of winning against the spread. Forecasting the outcomes of sports matches against the spread has been elusive for both experts and models 6, 36. Due to the abundance of historical data and user-friendly statistical software packages, the employment of quantitative modeling to aid decision-making in sports wagering 37 is strongly encouraged. The following suggestions are aimed at guiding model-driven efforts to forecast sports outcomes.

Indeed, the stratifications showing this trend were home favorites, agreeing with the idea that the sportsbooks are exploiting the public’s bias for wagering on the favorite 23. Indeed, the decisions made by sportsbooks to set the offered odds and payouts have been previously analyzed 13, 23, 24. On the other hand, arguably less is known about optimality on the side of the bettor.

The log likelihood test statistics have a chi-square distribution with one degree of freedom. This section collects any data citations, data availability statements, or supplementary materials included in this article. I would like to underline that – in my opinion – the manuscript has merit and I am very confident that it will be worth publishing if several revisions are made to the manuscript.

Similarly, the theoretical limits on wagering accuracy, and under what statistical conditions they may be attained in practice, are unclear. In general, financial markets of limited attention or visibility are expected to exhibit more inefficient asset prices. Such markets thus require eventual, increased trading to resolve these inefficiencies. In this study, we consider this financial framework within the sports gambling market for National Football League (NFL) games. We consider whether NFL games garnering less public attention are more likely to see larger movements in their betting lines. We consider games with smaller television audiences, with kickoff times shared with other contests, and between teams with smaller fanbases, to be indicative of less visibility.

Importantly, it is not an objective of this paper to propose or analyze the utility of any specific predictors (“features”) or models. Nevertheless, the paper concludes with an attempt to distill the presented theorems into a set of general guidelines to aid the decision making of the bettor. Define as the bettor’s estimate of t, and as the CDF of the sampling distribution of . The following result may be proven by replacing with in the Proof of Theorem 4.

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. Participant privacy or use of data from a third party—those must be specified. The author would like to thank Ed Miller and Mark Broadie for fruitful discussions during the preparation of the manuscript.

The average closing line based on the favorite score minus the underdog score was 5.89 and actual difference in score in the NBA games in the sample was 5.38. For the entire sample of games the underdog won 49.86% of the games, indicating that a strategy of betting the underdog was a fair bet, based on the log likelihood ratio test. In the sports gambling world, an over/under or totals wager is a bet that is won or lost depending upon the combined score of both teams in a game. A bookmaker will predict the combined score of the two teams and bettors will bet that the actual number of points scored in the game will be higher or lower than that combined score. For example, in an NBA game of the Miami Heat versus the San Antonio Spurs the over/under for the score of the game was set at 195.

In over-under betting, a positive expected profit is only possible if the sportsbook total τ is less than the -quantile, or greater than the -quantile, of t. Define an “excess error” as a wager that is placed on the side that does not maximize the expected profit of a moneyline wager. Any estimator that satisfies minimizes the probability of an excess error.