Volume 14, Issue 1

This paper considers an extension of the Kelly criterion used in sports wagering. By recognizing that the probability p of placing a correct wager is unknown, modified Kelly criteria are obtained that take the uncertainty into account. Estimators are proposed that are developed from a decision theoretic framework. We observe that the resultant betting fractions can differ markedly based on the choice of loss function. In the cases that we study, the modified Kelly fractions are smaller than original Kelly.

The duration of matches has been a common concern in professional tennis. Governing bodies have recently begun to introduce new match formats, like Fast4, to curb match lengths yet the impact of these formats on the professional game remains poorly understood. In this paper, we develop a shot-level Monte Carlo match simulation approach for estimating the duration, points played, and upset probability given a specific match format. Our model is built on validated models of the in-play and between-play time of matches using Hawk-eye tracking data and publicly available shot-level tennis statistics. When we applied our models to a variety of match formats with serve characteristics representative of current elite players, we found that Fast4 formats had an expected duration of 60 minutes, best of 3 averaged 90 minutes, and best of 5 averaged 120 minutes. Our results also showed that longer matches favor the better player and make match outcomes more predictable. Fast4 formats had a typical upset frequency of 20% compared to 13% for best of 3 matches and 10% for best of 5 matches. The modeling approach we have developed can be a useful resource for tennis governing bodies in assessing the impact of new match formats.

In association football, crosses from the wide areas of the pitch in the attacking third is a standard tactic for creating goal-scoring opportunities. But recent studies show that crosses adversely impact goals. Regression run in this paper on data from the premier soccer leagues of England, Spain, Germany, France and Italy for 2016–2017 season also found this inverse relation. However, there is no research that explains the reason for this inverse relation between crosses and goals. A game-theoretical model developed in this paper explains why crosses adversely affect goal-scoring. The model identifies a mixed strategy Nash equilibrium (MSNE), wherein the attacking team’s probability of playing a cross decreases with increase in their crossing accuracy, heading accuracy and probability of winning aerial balls. If the attacking team is good in terms of these parameters, the defending team’s probability of using an offside trap increases and that forces the attacking team to use crosses less frequently. In the MSNE, teams with a greater chance of scoring from crosses use the crosses less frequently than teams having a smaller chance of scoring from crosses. The theory was subsequently validated using the data of the 2016–2017 football season.