Modelling tactical changes in association football using a Markov game.

This article focuses on modeling tactical changes in association football (soccer) matches using a Markov game framework. The pitch is discretized into nine zones, and states are defined by ball location, team possession, and score difference. The study develops both a Markov decision process (MDP) model, where one team optimizes its winning probability, and a two-person zero-sum Markov game model, where both teams simultaneously optimize their tactics. Using play-by-play data from 306 matches in the Japan Professional Football League (J.League), transition rates between states are estimated via log-linear models, enabling calculation of expected staying times, goal distributions, and winning probabilities. The models identify optimal timing and types of tactical changes (e.g., substitutions or formation shifts) that can maximize a team’s probability of winning, accounting for the opposing team’s potential tactical responses. Results demonstrate the applicability of the approach to real matches and suggest managerial use in in-play decision-making.