Volume 11, Issue 3

Many games and sports, including races, involve outcomes in which competitors are rank ordered. In some sports, competitors may play in multiple events over long periods of time, and it is natural to assume that their abilities change over time. We propose a Bayesian state-space framework for rank ordered logit models to rate competitor abilities over time from the results of multi-competitor games. Our approach assumes competitors’ performances follow independent extreme value distributions, with each competitor’s ability evolving over time as a Gaussian random walk. The model accounts for the possibility of ties, an occurrence that is not atypical in races in which some of the competitors may not finish and therefore tie for last place. Inference can be performed through Markov chain Monte Carlo (MCMC) simulation from the posterior distribution. We also develop a filtering algorithm that is an approximation to the full Bayesian computations. The approximate Bayesian filter can be used for updating competitor abilities on an ongoing basis. We demonstrate our approach to measuring abilities of 268 women from the results of women’s Alpine downhill skiing competitions recorded over the period 2002–2013.

In this paper we provide a method for calculating the implied volatility of the outcome of a sports game. We base our analysis on Stern’s stochastic model for the evolution of sports scores (Stern, H. S. 1994. “A Brownian Motion Model for the Progress of Sports Scores.” Journal of the American Statistical Association 89:1128–1134.). Using bettors’ point spread and moneyline odds, we extend the model to calculate the market-implied volatility of the game’s score. The model can also be used to calculate the time-varying implied volatility during the game using inputs from real-time, online betting and to identify betting opportunities. We illustrate our methodology on data from Super Bowl XLVII between the Baltimore Ravens and the San Francisco 49ers and show how the market-implied volatility of the outcome varied as the game progressed.

In baseball, heat maps, which visualize a batter’s ability across regions in and around the strike zone, play an important role in baseball commentary and scouting reports. We represent the stochastic process underlying these heat maps as a spatial Gaussian field with isotropic covariance. Spatial interpolation (kriging) relies on the assumption of a known covariance function, but in reality the parameters of the covariance are typically estimated from the data. Our simulation study, based on a known covariance function, helps to understand and explain the spatial dependence of the process and allows us to produce improved heat maps.

We critically examine a number of flaws in the current procedure for the final draw of the FIFA World Cup™: imbalance (the eight groups are not of the same competitive level), unfairness (some teams have a greater chance than others of ending up in a tough group), and uneven distribution (all the possible outcomes of the draw are not equally likely). These flaws result from the way FIFA has decided to enforce the geographic constraints that they put on the draw. We explain how, by building eight pots by level organized in an S-curve, and drawing first a continental distribution of the groups and then the teams, we can enforce the geographic constraints without sacrificing balance, fairness, and even distribution. As a result, we describe a new tractable draw procedure that produces eight balanced and geographically diverse groups, is fair to all teams, and gives equally likely outcomes.

Soccer is as popular as ever, and the sport attracts significant attention from spectators, sponsors, media, and academics. One aspect of the sport that has received relatively little attention, is the effect of the playing surface on the sporting performance of a team. In particular, this paper is concerned with measuring the performance of teams that switch from playing their games on natural grass to playing their games on artificial turf. It is shown that teams, on average, achieve improved results after switching, and that this, at least in part, can be explained by an increased home field advantage.