Volume 12, Issue 3

This paper considers the problem of determining optimal substitution times in soccer. We review the substitution rule proposed by Myers (Myers, B. R. 2012. “A Proposed Decision Rule for the Timing of Soccer Substitutions.” Journal of Quantitative Analysis in Sports 8: Article 9.) and provide a discussion of the results. An alternative analysis is then presented that is based on Bayesian logistic regression. We find that with evenly matched teams, there is a goal scoring advantage to the trailing team during the second half of a match. In addition, we provide a different perspective with respect to the substitution guidelines advocated by Myers (Myers, B. R. 2012. “A Proposed Decision Rule for the Timing of Soccer Substitutions.” Journal of Quantitative Analysis in Sports 8: Article 9.). Specifically, we observe that there is no discernible time during the second half when there is a benefit due to substitution.

In this rejoinder we summarize the differences between Myers, B. R. 2012. “A Proposed Decision Rule for the Timing of Soccer Substitutions.” Journal of Quantitative Analysis in Sports 8: Article 9 and Silva, R. M. and T. B. Swartz. 2016. “Analysis of Substitution Times in Soccer.” Journal of Quantitative Analysis in Sports 12(3):113–122. We also extend the model of Silva and Swartz (2016).

Sports forecasting models – beyond their interest to bettors – are important resources for sports analysts and coaches. Like the best athletes, the best forecasting models should be rigorously tested and judged by how well their performance holds up against top competitors. Although a number of models have been proposed for predicting match outcomes in professional tennis, their comparative performance is largely unknown. The present paper tests the predictive performance of 11 published forecasting models for predicting the outcomes of 2395 singles matches during the 2014 season of the Association of Tennis Professionals Tour. The evaluated models fall into three categories: regression-based, point-based, and paired comparison models. Bookmaker predictions were used as a performance benchmark. Using only 1 year of prior performance data, regression models based on player ranking and an Elo approach developed by FiveThirtyEight were the most accurate approaches. The FiveThirtyEight model predictions had an accuracy of 75% for matches of the most highly-ranked players, which was competitive with the bookmakers. The inclusion of career-to-date improved the FiveThirtyEight model predictions for lower-ranked players (from 59% to 64%) but did not change the performance for higher-ranked players. All models were 10–20 percentage points less accurate at predicting match outcomes among lower-ranked players than matches with the top players in the sport. The gap in performance according to player ranking and the simplicity of the information used in Elo ratings highlight directions for further model development that could improve the practical utility and generalizability of forecasting in tennis.

We consider the traveling tournament problem (TTP) and the traveling umpire problem (TUP). In TTP, the task is to design a double round-robin schedule, where no two teams play against each other in two consecutive rounds, and the total travel distance is minimized. In TUP, the task is to find an assignment of umpires for a given tournament such that every umpire handles at least one game at every team’s home venue and an umpire neither visits a venue nor sees a team (home or away) too often. The task is to minimize the total distance traveled by the umpires. We present a combined approximation for this problem, when the number of umpires is odd. We therefore first design an approximation algorithm for TTP and then show how to define an umpire assignment for this tournament such that a constant-factor approximation for TUP is guaranteed.