Determining the best strategy for changing the configuration of a football team
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Determining the best strategy for changing the configuration of a football team N Hirotsu and M Wright* Lancaster University, Lancaster, UK This paper proposes a dynamic programming (DP) approach to find the optimal substitution strategy for a football match, which maximises the probability of winning or the expected number of league points, supported by real data of the English Premier League. We use a Markov process model to evaluate the offensive and defensive strengths of teams by means of maximum likelihood estimators. We develop a DP formulation to derive quantitatively the optimal substitution strategy of a team, in relation to the number required of each type of outfield player. We demonstrate how this approach may help to determine how many of each type of player should start a match and be substituted during a match. We also show how the expected league points would increase if the optimal strategy were followed. Journal of the Operational Research Society (2003) 54, 878–887. doi:10.1057/palgrave.jors.2601591 Keywords: decision; dynamic programming; Markov process; soccer; sports; substitution
Introduction Modelling an association football match is a topic of interest to operational research (OR) workers,1 and the task of finding optimal tactics can be assisted by the application of OR techniques. In a previous paper, we proposed a Markov process model for a football match, considering the propensity not only to score or concede goals but also to gain or lose possession.2 Further, we demonstrated how this approach may help to determine when to make a tactical change using a simple example. In this paper, this analysis is developed considerably, with the help of real English Premier League data of the 1999– 2000 season obtained by Opta Index Ltd, the official statisticians of the Premier League. Here, we focus on a typical team in the Premier League and analyse its offensive and defensive strengths based on the combination of the number of each type of outfield player on the pitch. Using these strengths, we quantitatively derive the optimal substitution strategy, in relation to the number required of each type of player, by means of dynamic programming (DP). We can apply the DP formulation for maximising the probability of winning, the probability of drawing, or the expected number of league points. In general, DP is a useful method for analysing decisionmaking in sports.3 For example, Washburn4 addresses the optimal time to ‘pull’ the goalie in ice hockey by DP. Clarke and Norman5 propose a method to determine when to rush a ‘behind’ in Australian Rules football. They also propose *Correspondence: M Wright, Department of Management Science, Lancaster University, Lancaster LA1 4YX, UK. E-mail: [email protected]
some DP models for cricket.6 We apply it to determine the timing of substitution and tactical change during a football match.2
The Markov process model In the previous paper,2 we define
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