Using a Markov process model of an association football match to determine the optimal timing of substitution and tactical decisions

Abstract A football match is modelled as a four-state Markov process. A log-linear model, fed by real data, is used to estimate transition probabilities by means of the maximum likelihood method. This makes it possible to estimate the probability distributions of goals scored and the expected number of league points gained, from any position in a match, for any given set of transition probabilities and hence in principle for any match. This approach is developed in order to estimate the optimal time to change tactics using dynamic programming, either by making a substitution or by some other conscious change of plan. A simple example of this approach is included as an illustration.