Undoubtedly, a fundamental part of a soccer coach’s job is to make strategic decisions – sometimes driven by rationale, other times by intuition. Now, an interdisciplinary group of researchers from Italy and Belgium have developed a game-theoretical approach that could help to decide whether a team should play defensively or offensively. But does it work?
By Daniele Gambarelli, Gianfranco Gambarelli and Dries Goossens
Arena Pernambuco, Recife, Brazil, June 20th 2014, 11am. In two hours, the Chilean referee, Enrique Osses, will blow his whistle for the opening game in Group D at the World Cup, between Italy and Costa Rica. Coaches Cesare Prandelli (Italy) and Jorge Luis Pinto (Costa Rica) must reveal their formations and initial tactics to their players. The time has come to choose whether to play offensively in an attempt to qualify immediately for the next round, or to adopt a more defensive formation, thus reducing the risk of a defeat.
Could game theory help coaches make the right decision?
Game theory is the science of competitive tactics. It is defined by mathematical models describing strategies designed to maximize wins and minimize losses. In the soccer world, game-theoretical approaches have been used to measure a player’s contribution to the success of their team, or to model the competitive dynamics between kickers and goalkeepers during a penalty kick. Researchers from Ghent University and the University of Bergamo have now developed a general game-theoretical model that provides insights into the choice of tactics that can be adopted at the start of a match. They applied the model to the Italy–Costa Rica match as a case study, and their results were recently published in the Journal of Quantitative Analysis in Sports.
According to the leading bookmakers prior to the start of the match, the odds were 61% in favor of a win for Italy versus 15% for a win for Costa Rica, and 24% for a draw. However, these probabilities are dependent on the strategies chosen by both coaches. Based on both teams’ track records over the previous year and on relevant scientific literature, the researchers calculated the strategy-dependent outcome probabilities of a win, draw or loss for each team (as shown in the table below).
Based on Table 1, the expected number of points each team would collect can be calculated for each possible combination of strategies. For instance, in the case of both teams playing a defensive formation (and taking into account that a win results in 3 points, with 1 point being awarded for a draw), the expected pay-off for Italy is: 3×0.56 + 1×0.34 = 2.02. The expected pay-off for Costa Rica is: 1×0.34 + 3×0.10 = 0.64. Table 2 shows all expected outcomes; with the first number in each pair being the expected pay-off for Italy, and the second for Costa Rica.
Let us now consider the problem from the point of view of Italy. If Italy chooses a defensive approach, they would obtain 2.02 points if Costa Rica also opts for defense, or 2.47 points if Costa Rica chooses an offensive approach. If, on the other hand, Italy chooses an offensive approach, they would obtain 2.07 or 2.12 points, in each respective case. Therefore, the Italian team does not have a clearly preferable option.
Considering things from the point of view of Costa Rica, a defensive approach is certainly preferable because it leads to a better expected pay-off than an offensive approach, no matter how Italy chooses to play. Returning to Italy and bearing in mind the above, the best strategy under these conditions would be offensive, leading to an expected pay-off of 2.07 points, instead of 2.02.
Therefore, the researchers’ game theoretical approach advises a defensive strategy for Costa Rica and an offensive one for Italy.
Which strategies did the coaches actually choose?
Contrary to this advice, the coaches of both teams opted for defensive initial formations and tactics. The teams took to the field and played very cautiously, to the extent that in the first 30 minutes the only noteworthy action was a header following a corner kick by Costa Rica, which landed two meters outside the goalmouth. The situation remained more or less unchanged until the 44th minute, when Bryan Ruiz Gonzàlez scored for Costa Rica.
This goal, as well as the fact that Italy now only had one half left to turn the tide, changed the probabilities to 27% for a win for Italy, 33% for a draw and 40% for a win for Costa Rica. Apart from changes to the expected pay-offs, however, the Costa Rican goal did not alter the initial strategic advice. Defending remains a dominant strategy for Costa Rica because, no matter what the Italians do, the expected pay-offs are still higher if Costa Rica assumes a defensive formation. Italy does not have a dominant strategy, but given that Costa Rica has a clear interest to defend, they should opt to attack.
In the second half, the Italian coach modified the formation of the Italian team, making it more offensive. These changes, however, turned out to be too little too late and did not change the final result: the match ended 1-0 to Costa Rica. Eventually, Costa Rica qualified for the next round (and even reached the quarter-finals), whilst Italy did not survive the group stage.
Despite the questionable decision made by Cesare Prandelli, it seems likely that game theoretical models alone would have difficulty guiding a team through the World Cup too. Nevertheless, the application of game theoretical approaches could provide coaches with strategic advice, and perhaps persuade them to reconsider their strategy if this advice contradicts their intuition.
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