Volume 12, Issue 1

We present a methodology for finding globally optimal knockout tournament designs when partial information is known about the strengths of the players. Our approach involves maximizing an expected utility through a Bayesian optimal design framework. Given the prohibitive computational barriers connected with direct computation, we compute a Monte Carlo estimate of the expected utility for a fixed tournament bracket, and optimize the expected utility through simulated annealing. We demonstrate our method by optimizing the probability that the best player wins the tournament. We compare our approach to other knockout tournament designs, including brackets following the standard seeding. We also demonstrate how our approach can be applied to a variety of other utility functions, including whether the best two players meet in the final, the consistency between the number of wins and the player strengths, and whether the players are matched up according to the standard seeding.

In this paper, we consider fantasy football, an increasingly-popular online game based on the actual, on-the-field performances of players in the National Football League. It is estimated by the Fantasy Sports Trade Association that in 2011 there were 35 million people in the US and Canada playing fantasy sports online. About 85 percent of all fantasy sports participants play fantasy football, most of whom have their games set up in major media websites such as Yahoo!, ESPN, MSN, and NFL. Numerous websites specialize in reporting NFL games, providing preseason rankings, fantasy points projections, team and player statistics, and expert draft opinions. However, despite the vast popularity of the game, the intensive analysis by experts, and various online tools that offer prediction for the values of players, to the best of our knowledge, there is no method that provides a comprehensive optimization strategy for the entire Fantasy Football season. We set out to develop such a methodology that predicts team and player performance based on the rich historical data, and builds a mixed-integer optimization model using such predictions for the draft selection as well as weekly line-up management that incorporates the entire objective of winning a fantasy football season. Numerical tests of our model show promising performance.

This paper treats the problem of modeling and forecasting the outcomes of NBA basketball games. First, it is shown how the benchmark model in the literature can be extended to allow for heteroscedasticity and estimation and testing in this framework is treated. Second, time-variation is introduced into the model by introducing a dynamic state space model for team strengths. The in-sample results based on eight seasons of NBA data provide weak evidence for heteroscedasticity, which can lead to notable differences in estimated win probabilities. However, persistent time variation is only found when combining the data of several seasons, but not when looking at individual seasons. The models are used for forecasting a large number of regular season and playoff games and the common finding in the literature that it is difficult to outperform the betting market is confirmed. Nevertheless, a forecast combination of model based forecasts with betting odds can lead to some slight improvements.

Bill James’ discovery of a Pythagorean formula for win expectation in baseball has been a useful resource to analysts and coaches for over 30 years. Extensions of the Pythagorean model have been developed for all of the major professional team sports but none of the individual sports. The present paper attempts to address this gap by deriving a Pythagorean model for win production in tennis. Using performance data for the top 100 male singles players between 2004 and 2014, this study shows that, among the most commonly reported performance statistics, a model of break points won provides the closest approximation to the Pythagorean formula, explaining 85% of variation in season wins and having the lowest cross-validation prediction error among the models considered. The mid-season projections of the break point model had performance that was comparable to an expanded model that included eight other serve and return statistics as well as player ranking. A simple match prediction algorithm based on a break point model with the previous 9 months of match history had a prediction accuracy of 67% when applied to 2015 match outcomes, whether using the least-squares or Pythagorean power coefficient. By demonstrating the striking similarity between the Pythagorean formula for baseball wins and the break point model for match wins in tennis, this paper has identified a potentially simple yet powerful analytic tool with a wide range of potential uses for player performance evaluation and match forecasting.