Journal of Systems Science and Complexity, in press



Science of winning soccer: Emergent pattern-forming dynamics in Association Football

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Luís Vilar, Duarte Araújo, Keith Davids & Yaneer Bar-Yam

Introduction

An important aim of sports science is to improve understanding of strategic performance and success in team competition [1]. Quantitative analyses can provide feedback to players and coaches, allowing them to enhance their performance and interpretation of the activity beyond what can be achieved by personal observation [2-4]. Traditional analysis of performance in team sports has examined behavior through reporting cumulative data on discrete actions in a Who [did]-What-Where-When fashion [5]. However, in team sports, each player's behavior is dependent on the locations and interactions of other players (both teammates and opponents), and the locations of the ball and the goal. Therefore, research should consider the behavior of multiple players and the emergent nature of performance. The performance depends on pattern-forming dynamics, i.e., on the dynamic physical relationships each player establishes with his/her teammates and opponents [6-9].



Quantitative analysis of interpersonal coordination has largely been limited to the spatio-temporal patterns of coordination between attacker and defender in one versus one (henceforth 1v1) dyadic system sub-phases of team sports [9-13]. These studies have considered how one attacker carrying the ball breaks local symmetry with the immediate defender to perform a successful pass, dribble or shot. In team sports it is reasonable to expect that an analysis of such 1v1 dynamics is not sufficient because multiplayer interactions are important in determining success and failure. Nevertheless, analyses that go beyond considering the players' 1v1 interactions in many multiplayer team competitions, including basketball, rugby-union and association football (commonly known as soccer), are limited in number.



Here we consider team defense and offense within a general framework that characterizes the dynamic stability and instability of team interactions. When sequences of offensive and defensive actions and reactions maintain stability, no advantage results. Opportunities for scoring arise when offensive players engage in actions that destabilize the defensive response. In order to destabilize defensive systems, offensive teams displace their players and the ball irregularly to promote a cascade of local instabilities in their opponents' defense. Rather than considering how this might be done by individual player actions and responses, we consider measures of collective action. We hypothesize that team advantage in defense and offense can be quantified, at a first approximation, by the relative advantage in the number of players in a local area. In particular, we expect that instability is likely when the offensive team establishes local offensive numerical superiority (e.g., 2v1 or 3v2, so that there is one more attacker than defender) near the ball's location. Conversely, the defending team could attempt to maintain stability in local sub-systems by, for example, increasing the presence of defenders adjacent to the ball. This framework suggests that we can simply analyze dynamic stability or instability of the offensive and defensive sub-phases of the game to identify effective or ineffective performance. An analysis of the pattern-forming dynamics in different sub-areas of play in team sports may explain how the players coordinate their actions to maintain or disrupt system stability.



Association football is a game played between 11-player teams on an approximately 105m by 65m field, with teams attacking in opposite directions. Each team attempts to kick the ball into the opposing team's goal, while preventing the ball from entering its own goal. Only one player on each team, the goalkeeper, is allowed to use hands to intercept the ball and then only inside the (defended) goal-area. Compared with team sports in which all players use their hands to control the ball and the field is significantly smaller (40m by 20m), such as basketball or handball, association football is a defensive game characterized by a low number of goals scored. The low scoring constrains the strategic adoption of different systems of play, and arises from the chosen systems self-consistently. The systems of play are generally defined by the starting formations of play: the number of players in the team's back, midfield and front lines (e.g., 4:4:2, 4:3:3, etc.) [14]. However, systems of play only describe the global organization of each team. During the game, players move and interact with one another, constantly changing the team's spatial structure. Coordination between players to achieve performance objectives (i.e., scoring or preventing goals) through dribbling, passing and tackling arise under spatial organization constraints that affect those actions [9].



In this paper we investigate collective pattern-forming dynamics in association football by examining how team coordination emerges in one actual competitive match. The study of only a single match suggests our analysis has limited claim to generality. However, a single match is considered adequate as a measure of the relative strength of two teams, as evidenced by single match playoffs. Thus, the measures we observe that consistently distinguish the winning from losing team may indeed reflect team capability.  We will show that it is possible to perform an analysis of stability and instability consistent with our intuitive hypothesis and game outcome.



In order to focus on the offensive and defensive actions we consider a new definition of the area of play. Rather than considering the entire field, we define the area of play as that area circumscribed by the location of the 20 outfield (non-goalkeeper) players. We then identify the offensive and defensive sub-areas within this area of play to characterize how the teams organize themselves dynamically during the game. In order to do this, we dynamically track a spatial frame that moves with the players. We use this novel method to follow the game dynamics as a first step toward understanding collective offensive and defensive risk and security in team game performance. We show how a strategy emerges from teams' interactions and results in arrangements of players across the field which create local stabilities and instabilities in specific sub-areas. We find that teams allocate more players than their opponents in sub-areas closer to their own goal to ensure higher security. The focus on security is consistent with the defensive nature of low scoring games where conceding even a single goal could easily result in losing the game. By analyzing the unpredictability of teams' numerical relationships in each sub-area of play, we identify the regions where more transitions between stable and unstable modes of coordination occur, enabling us to describe each team's competitive performance profiles. We also observe differences between team strategies: for example, while both adopted a 4:3:3 distribution of backfield, midfield and front field players, significant differences are observable.



The analysis shows greater scoring opportunities for team A due to more frequent dominance in the key offensive areas. Equivalently, team A was more successful in maintaining defensive stability near its goal areas. Consistent with our analysis as well as expectations that a single goal advantage could well determine the winner, team A won the match examined here by a score of 2 to 1.



Methods

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Data collection

Twenty-eight male professional players participated in an association football match in the English Premier League in October 2010. The total duration of the match was 95 minutes and 29 seconds. The performance of all participants was monitored using a multiple-camera match analysis system [ProZone3, ProZone Holdings Ltd, Leeds, UK]. Movements of twenty outfield players (goalkeepers were excluded) from the two competing teams were recorded during the entire game, using eight cameras positioned at the top of the stadium. Video files were synchronized and 10Hz frames were obtained by automated processing [15, 16]. This procedure yielded two-dimensional player displacement coordinates. Excluding the out-of-bound locations to which players went during play, the effective playing area was 68m wide (from x = -34m to x = 34m) and 105m long (from y = -52.5m to y= 52.5m). Teams switch sides of the field halfway through the game. To facilitate visualization, we inverted player displacements for the second half of the game, so team A would always attack toward positive coordinate values and team B would always attack toward negative coordinate values (see Figure 1).

























































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Data analysis

We excluded the time the game was stopped and the players abandoned their standard positions, such as for injuries (8 minutes 52 seconds), goal celebrations (2 minutes 41 seconds) or substitutions (1 minutes 35 seconds). From the location of the 20 outfield players in each increment of time (recorded frame), we calculated the area of play using a convex hull computation (i.e., the minimal convex area containing all outfield players). The analysis considered the distribution of players in this dynamically adaptive area of play, changing from frame to frame during the game time.



We calculated two longitudinal (goal to goal) vectors that divided the area of play into three channels: right (0% to 25%), center (26% to 74%) and left channels (75% to 100%). We also calculated three lateral (side line to side line) vectors that divided the area of play into two segments for the right and left channels: back (0% to 50%) and front segments (51% to 100%); and three segments for the center channel: back (0% to 25%), midfield (26% to 74%) and front segments (75% to 100%). The interaction of channels and segments led to the construction of seven sub-areas of play [17]: right-back (RB), center-back (CB) and left-back (LB); center-middle (CM); right-front (RF), center-front (CF) and left-front (LF)  (see Figure 1). Only the center-middle sub-area was the same for team A and team B. The remaining performance sub-areas of each team had an opposing relationship: the center-back sub-area of team A was the center-front sub-area of team B (and vice-versa); the right-back and left-back sub-areas of play of team A were left-front and right-front sub-areas of play of team B, respectively (and vice-versa).



In each frame we calculated the number of players from each team inside the different sub-areas of play and the difference between the players of team A and team B, or the net team numerical advantage (a disadvantage for negative values). Frequency histograms were plotted for both of these variables. We also computed the uncertainty of the team numerical advantage across sub-areas using Shannon's entropy. This measure characterizes the variability of number of players in each region.

Results

We begin by considering the coordination between players of opposing teams based on the numerical relationships established in opposing sub-areas of play. Second, we explore the uncertainty in numbers across sub-areas of play. Finally, we compare directly the spatial patterns of the two teams. The results reveal aspects of team behavior that may give rise to distinct performance profiles.

 

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​Inter-team coordination tendencies
   
The numerical advantage in every sub-area is shown in Figure 2. There is a notational symmetry between opposite sub-areas of play for the two teams. For example, a +1-player advantage in the left-back sub-area for team A means that team A had one player more in its left-back sub-area of play than team B had in its right-front.





































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Results show a pattern of focus on defensive stability. For example, the most likely team numerical advantage in center defensive areas is +1. This defensively stable pattern of one more defender than attacker in the center-back sub-areas is present for nearly half of the playing time (47% of match time in the CB of team A, and 44% of match time in CB of team B). Each team tries to secure those regions against dominance by the opposition. Considering the adjacent possibilities of 0 or +2, we see that team A has a higher likelihood of +2 than 0 defenders, while team B has a nearly an equal number. This suggests that team B is either not as defensively oriented, or not as successful in achieving defensive stability, consistent with the victory of team A in the match. We can also frame this advantage of numbers as an offensive superiority. For team A an equal 0-player and +1-player pattern occurred in its center-front region (CF vs. CB) with a frequency of 21% and 6%, respectively, while for team B a 0-player and +1-player patterns occurred in its own CF region (CB vs. CF) with a frequency of only 13% and 2%, respectively. The longer time during which team A had numerical equality or superiority in its center-front areas suggests the occurrence of instabilities in the center-back sub-area of play of team B, and a winning offensive advantage for team A.



While team A had greater defensive dominance than B in the center, when we consider the two wings we see that team A had only a slightly greater numerical advantage in its RB area than team B did in its own RB area, if any, and team A had less of an advantage in its LB area than team B had in its own LB area.  Numerically, the +1 defender pattern was held by team A in its RB area  (RB vs. LF) 43% of the time, while it was held by team B in its RB area (LF vs. RB) for almost the same percentage of the time, 42%. In the left channel, the advantage is to team B; the +1 defender pattern was held by team A in its LB area (LB vs. RF) for 35% of match time, while it was held by team B in its LB area (RF vs. LB) for 42% of match time. Team A's prevalence of defensive security in the center areas rather than the wings suggests a slightly lower importance assigned to wing defense than center defense. This small difference in prioritization may have contributed to the successful strategy of team A.



Unpredictability of inter-team coordination



Figure 3 shows the uncertainty/variability of team numerical relationships in each sub-area of play during the match. The highest uncertainty is in the center-middle sub-areas of play with a value of 2.51. The uncertainty reflects a flatter distribution in the team numerical advantage. Such a distribution reflects dynamic shifts of players into and out of the sub-area from adjacent sub-areas that change the relative number of players of the teams over time. The center-middle area has boundaries to all other sub-areas, and can be both the origin and target of shifting player movements. Changes in this sub-area have less significance for either offense or defense than other sub-areas, which are more likely to be the source of attacks on the goal. Therefore, teams may choose not to maintain as close control over the balance of players in this area. Instead, they can use the center-middle sub-area as a reservoir of players to move to the locations of greatest need, allowing the variability in that area to enable the higher priority stabilization of the other areas.



































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The next highest uncertainty areas are the two symmetrically defined center-front and center-back areas. These areas are the primary attack and defense areas. From Figure 2 we can see that this uncertainty is primarily a variation of the number of excess defenders. Rarely does a team dominate its CF area, though according to our hypothesis about stability and instability such an advantage is likely to be a major tactical objective.



We can compare the uncertainty between the two teams in their corresponding sub-areas. The largest difference is the lower uncertainty in team A's CB (team B's CF), 1.98, in comparison to team B's CB (team A's CF), 2.13. This shows team A in the center-back sub-area of play was more predictable in its defensive dominance than team B was in its defensive area. From Figure 2, we see that indeed, team A was better at limiting times in which team B had an offensive advantage in this area or even an equal number of players. Correspondingly, team A had more occasions with a greater than or equal number of players in its primary offensive area. This would predict that team A would be more successful due to a greater reliability of its defensive pattern than team B, and indeed, team A is the victor in this match.



Team A's right-back sub-area uncertainty is 1.78, which is more predictable than team B's at 1.89. The only pair of corresponding defensive (back) zones for which uncertainty predicts an advantage for team B is each team's LB zone. Uncertainty in A's LB (B's RF) was 1.84, while uncertainty in B's LB (A's RF) was 1.78.



 

Internal team coordination



Figures 4 and 5 show the frequency distribution of the number of players of team A and B respectively in each area of play.  Both Team A and Team B prioritized the defensive rather than offensive sub-areas of play. The mode number of players present for each team is 2 in each team's center-back area of play (for 44% of match time for Team A and 41% for Team B), 2 in the center-middle, (39% of the time for Team A and 38% for Team B), and 1 in the center-forward (52% of the time for Team A and 50% for Team B).

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Results for both also show a higher importance accorded to center channels relative to the wings. Team A allocated 2 players to the center-back for 44% of match time, but allocated 2 players to the left-back and right-back only 26% of the time for each. Likewise, team B allocated 2 players to the center-back for 41% of match time, but allocated 2 players to the left-back and right-back only 29% and 20% of match time, respectively. However, the teams differed in their left-right symmetry; team A seems to have placed greater importance on the right-back than the left-back (with 1 player there 43% of the time versus 37%), while team B had an approximately equal amount of time with no player in either wing. Team B did have a higher percentage of two players in its right-back sub-area, which may be a response to the larger number of players in the right-front area of team A.