Using a Markov process model of an association football match to determine the optimal timing of substitution and tactic
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Using a Markov process model of an association football match to determine the optimal timing of substitution and tactical decisions N Hirotsu* and M Wright Lancaster University, Lancaster, UK 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. Journal of the Operational Research Society (2002) 53, 88–96. DOI: 10.1057=palgrave=jors=2601254 Keywords: football; Markov process; soccer; sports; decision; tactics
Introduction Association football (soccer) has been quantitatively analysed by a number of researchers.1 Moroney2 and Pollard et al3 claim to show that the number of goals scored by a team is best described by a negative binomial distribution. However, Maher4 suggests that a Poisson distribution is more appropriate. He shows that a relatively simple model gives a reasonably good fit to data obtained from real matches. Home-team advantage is another well-analysed topic for football. Courneya and Carron5 summarise research which statistically demonstrates the existence of home-team advantage. Clarke and Norman6 suggest that the geographical distance between two teams significantly affects the home-team advantage. The statistical evaluation of the strength of football teams has been analysed by Lee,7 based on the results of the English Premier League in the 1995–96 season. He estimates the offensive and defensive capabilities for each team using the maximum likelihood method, assuming a Poisson distribution for scoring goals. Dixon and Coles8 also evaluate the strength of teams for making profit in the football betting market. They introduce a time-dependent effect to a Poisson regression model based on Maher’s model. They estimate their maximum likelihood parameters using English league and cup football data from 1992 to 1995 and they claim that, using their betting strategy, it is possible to have a positive return. Dixon and Robinson9 analyse the *Correspondence: N Hirotsu, Department of Management Science, Lancaster University, Lancaster LA1 4YX, UK. E-mail: [email protected]
change in the rate of scoring goals for the football spread betting market, where prices are updated during a match. They incorporate the rate of scoring goals by considering the process as a time inhomogeneous birth process and develop a complicated statistical model to predict the outcome of the matc
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