Journal of Quantitative Analysis in Sports A Potential Function Approach to the Flow of Play in Soccer David R. Brillinger Statistics Department University of California, Berkeley Abstract There is a growing literature on the statistical analysis of data from association-football/soccer games, seasons or groups of seasons. In contrast this paper is concerned with a single play, that is a sequence of successful passes. The play studied contained 25 passes and ended in a goal for Argentina in World Cup 2006. One question addressed is how to describe analytically the spatial-temporal movement of such a particular sequence of passes. The basic data are points in the plane, successively joined by straight lines. The resulting figure represents the trajectory of the moving soccer ball. The approach of this study is to develop a useful potential function, a concept arising from physics and engineering. In particular the potential function leads to a regression model that may be fit directly by linear least squares. The resulting potential function may be used for simple description, summary, comparison, simulation, prediction, model appraisal, bootstrapping, and employed for estimating quantities of interest. The purpose illustrated here is to simulate play in a game where the ball goes back and forth between two teams each having their own potential function.1. Introduction The 2006 World Cup included some grand moments. One of the most spectacular was the 25-pass scoring play of Argentina in the Serbia-Montenegro (S-M) game on 16 June. The shot that ended the play was a goal scored by Cambiasso, but some 8 players worked hard to get the ball into position for his shot. The play has been described as: “one of the all time great World Cup goals”, “the play of the tournament”, “a joy forever”, “a glorious goal”, “mesmerizing”, and “a string of pearls”. Not long after the game, sketches of the path of the ball started to appear in newspapers (e.g. Expressen, Sweden) and magazines (e.g. Cambio, Columbia). Purposes of this paper are to develop an analytic description and model for such a play and to explore its uses, e.g. for simulation of plays where the ball changes sides. The play began in the Argentine half of the field with Maxi passing back to Heinze. The sequence of players involved then was: Mascherano, Riquelme, Maxi, Sorin, Maxi, Sorin, Mascherano, Riquelme, Ayala, Maxi, Mascherano, Maxi, Sorin, Maxi, Cambiasso, Riquelme, Mascherano, Sorin, Saviola, Riquelme, Saviola, Cambiasso, Crespo, Cambiasso with Cambiasso scoring. Maxi was involved 6 times, while Requilme, Masherano and Sorin contributed 4 passes each. Videos of the goal may be found at YouTube, see YouTube (2006a, 2006b). Figure 1 below provides the estimated locations of where the passes initiated during the play. (How the estimation was carried out will be described below.) The locations are denoted by small circles. Straight lines join them in order of time. The track is meant to represent the path of the ball being played about the field as seen from above. One notes that the ball generally moved towards the S-M goal with passes going off in many directions and some back passes being made. There are no very short passes, the shortest being 5.6m . There is a growing literature on the statistical modeling of aspects of soccer matches. One highly quoted study was performed by Reep and Benjamin (1968). They investigated data on goal scoring and lengths of passing sequences from 3213 games. They summarize the counts of successful passes by the negative binomial distribution and, for example, conclude that “it takes 10 shots to score 1 goal.” Hughes, M. and Franks, I. (2005) describe the Reep and Benjamin paper as a “landmark”, but in a discussion of the ‘long-ball game’ versus ‘direct play’ complain about various of Reep and Benjamin’s conclusions and their impact.Figure 1. The trajectory of the play. The ball ends in the S-M goal represented by a box on the right hand side of the figure. The circles represent the positions of players initiating passes. Lee (1997) fits Poisson models for the number of goals scored in games for the 1995-96 season to assess the strengths of various teams. Karlis and Ntzoufras (2003) fit bivariate Poissons to the pairs, (X,Y), of goals scored where X is the number of goals scored by one team and Y by the other. Hirotsu and Wright have a succession of papers. One paper, Hirotsu and Wright (2002), modeled the progress of play in a game as a continuous time Markov process with four states. The four states were: each team is in possession of the ball, and when each team scores a goal. Another paper, Hirotsu and Wright (2006) applies game theory to develop effective strategies. Brillinger (2006a) viewed the results of games as ordinal (win, tie, loss) and fit a model for such data to the Norwegian League results for the 2003 season. A setup of quite a different type altogether is that of Kozlov et al. (1993). Theyconsider an abstract version of a soccer match. The field is infinite with the usual two goals. The path of the ball is planar Brownian motion. They consider the variance of the number of goals and discuss its dependence on the width of the goal for example. It appears that most of the existing published papers study whole games, tournaments, seasons, or groups of seasons. The initial purpose of this paper was to study that one play in that one game, but the purpose went on to include using the results to develop a flexible model including changes of possession, variable play lengths There is concern about focusing on a highly unusual play, on an outlier. It was unusual, it was highly exciting, it contained an unusual number of passes, and it lead to an important goal. Now in statistical data analyses an outlier is to be noted and studied. The analysis should split with a part dropping or weighting down the outlier, and a part looking into it. In DeVeaux et al (2006) one can read, page 534, “An analysis of the nonoutlying points, along with a separate discussion of the outliers, is often more informative, and can reveal important aspects of the data.” Briefly, there are things to be learned by analyzing outliers. The path in Figure 1 may be viewed as a realization of a stochastic process described by the time ti at which the i-th pass was initiated and (x(ti),y(ti)) the location where