Football (soccer) is a strategic and dynamic game. In every match, the two teams are pitched into direct opposition for a period of play of 90 minutes’ notional duration. Each team starts the match with one goalkeeper and ten outfield players. Team managers or coaches are at liberty to deploy their outfield players in any formation of their choosing, and to adjust their team’s formation and style of play at any stage of the match. The immediate objectives of the two teams throughout the match are symmetric: each team attempts to score goals and prevent its opponent from scoring. Accordingly there is a high level of interdependence: both teams’ strategies have implications for both teams’ probabilities of scoring and conceding goals.
Several researchers have noted the usefulness or potential of professional sports for testing economic hypotheses concerning strategic behaviour and risk taking based on non-experimental data (Walker and Wooders, 2001; Chiappori, Levitt and Groseclose, 2002; Palacios-Huerta, 2003). In many sports, only two teams or players are involved in each contest. The contest takes place within a specific and clearly defined time-frame. Strategies and payoffs are relatively simple to identify. Outcomes matter, because large sums of money are at stake for the winners and losers. Large data sets are readily available, and new data are being continually generated.
In this paper, we examine the extent to which the strategic behaviour of football teams throughout the course of matches can be rationalized in accordance with game-theoretic principles of optimizing strategic behaviour by independent agents when payoffs are interdependent. The model that is developed in this paper has two important antecedents in the economics literature. First, in an unpublished working paper, Palomino, Rigotti and Rustichini (2000) (henceforth, PRR) develop a dynamic game-theoretic model, in which the two teams choose continuously between a defensive and an attacking formation. PRR show that teams tend to play more defensively when they are leading than they play when they are either level or trailing, especially during the later stages of matches. Second, a similar model has been used by Banerjee, Swinnen and Weersink (2007) (henceforth, BSW) to examine the impact on team strategies in the National Hockey League of a recent change in the league points scoring system affecting matches that were tied at the end of regular time.
The PRR and BSW models contain several simplifying assumptions that are either controversial or counterfactual, or which limit the generality of their analysis. PRR rely upon an unproven assumption that if one team adopts an attacking strategy and the other adopts a defensive strategy, the goal scoring rate of the attacking team increases by more than that of the defending team. BSW relax this assumption, by introducing a parameter allowing for comparative advantage in either attack or defense. Both PRR and BSW consider the special case in which the attacking and defensive strengths of the two teams are the same; and most of PRR’s results are based on a counterfactual assumption that the payoff structure at the end of the match is zero-sum. PRR are unable to account for what appears to be a strong empirical regularity, that after controlling for team quality and duration effects, the scoring rates of both teams when the scores are level tend to be lower than those of teams that are trailing, and not significantly different from those of teams that are leading.
Both PRR and BSW restrict the teams’ strategic choices to two options: attack and defense. Neither considers a second strategic dimension: the choice between a violent and non-violent style of play. Teams that play violently commit foul play and other acts of violence or aggression in an attempt to disrupt or sabotage the opposing team. By doing so, however, they incur an increased risk of being subject to disciplinary sanction, which may involve the dismissal of a player from the field of play for the remainder of the match. By allowing the teams to exercise choices concerning the level of violence or foul play, the model that is developed in this paper incorporates the link between strategic behaviour and disciplinary sanction.
In the model that is developed below, we assume that the available strategic choices are discrete: teams choose between ‘defensive’ and ‘attacking’ formations, and between ‘non-violent’ and ‘violent’ styles of play. These strategic choices influence the probabilities of scoring and conceding goals at the current stage of the match, and the probabilities that players are dismissed. Using numerical simulations based on a dynamic game-theoretic model incorporating a broad spectrum of structural assumptions, we show that the optimal strategic choices at each stage of the match are dependent on the current difference in scores and on the amount of time that has elapsed. The numerical simulations provide a simple and flexible apparatus for exploring the effects on strategic behaviour and outcomes of changes in any of the structural assumptions.
We subject the theoretical model to empirical scrutiny, using data on the timings of goals and player dismissals from more than 12,000 English professional league matches played between 2001 and 2007. We find a high level of consistency between the theoretical model, and observed behaviour and outcomes reflected in the coefficients of a set of empirical hazard functions for the conditional arrival rates of player dismissals and goals.