Modelling tactical changes in association football using a Markov game

Abstract

We model tactical changes in association football as a Markov game. The pitch is discretised into nine zones and the states of the Markov game are defined according to the zone in which the ball is located in play, the team in possession, and the score. We first model tactical changes in a Markov decision process framework, wherein one team maximises their probability of winning. Then, we model tactical changes in a two-person zero-sum Markov game framework, wherein both teams maximise their probability of winning. Fundamental to our modelling is the notion that tactical changes impact upon transition rates. We verify the models using data from matches in a season of the Japan Professional Football League. We define a change in transition rates that can be realised by changes in tactics, and illustrate an example of optimal tactical changes when both teams can vary their tactics. The models we develop in the paper can support managers who are considering important decisions about substitutions and changes to formation, for example, when a match is in-play.