Modeling Tactical Changes of Formation in Association Football as a Zero-Sum Game

Although tactical decisions made by managers during a match of team sports are very important, there have been few quantitative analyses which include the effect of interaction between both teams' decisions, because of the complexity of the problem where one team's decision will affect the other team's. A game theoretic approach can be useful for tackling this type of problem.This paper proposes a game theoretic approach to modeling tactical changes of formation in an association football match. We assume probabilities of scoring and conceding a goal follow Poisson distributions and use a regression model to evaluate the means of the distributions. These means represent the offensive strength for scoring a goal and defensive propensity to concede a goal in terms of a team's formation, i.e. a combination of the number of each type of outfield player on the pitch, and are estimated by means of the maximum likelihood method. We then develop a mathematical formulation with which we can calculate the probability of the home team winning the match, and use it to analyse tactical changes of the teams' formations, modeling the football match as a zero-sum game, in which the gain in probability of one team winning is equal to the loss in probability of the other team winning. We demonstrate how the managers' decisions affect the probability of winning the match using real data of the Japan professional football league, by showing four cases of the quality of both managers' decisions, depending on whether they each use their best or worst strategies.There still remains some uncertainty and longer observational studies will be required for a complete analysis, but this method can help to evaluate quantitatively the quality of tactical decisions made by managers.