Paradox of Crosses in Association Football (Soccer) – A Game-Theoretic Explanation

Posted: 15 Sep 2017

See all articles by Sumit Sarkar

Sumit Sarkar

XLRI - Xavier School of Management

Date Written: September 12, 2017


Regression run on observations for 98 teams playing in the top tires of soccer leagues in England, Spain, Germany, France and Italy shows significant negative impact of number of crosses per game on number of non-penalty goals scored per game during the 2016-17 season. This finding is consistent with earlier studies that found negative impact of crosses on goals (Vecer, 2015; Sarkar and Chakraborty, 2017). The game theoretical model developed in this paper explains why crosses adversely affect goal-scoring. Under specific parametric conditions, there exists only a mixed strategy Nash equilibrium (MSNE). In the MSNE, Attack’s probability of playing a cross decreases with increase in their cross accuracy, heading accuracy and probability of winning aerial balls. The result is paradoxical because these very efficiency parameters increase Attack’s chance of scoring from a cross. Defence’s probability of using an offside trap increases if Attack is efficient in these parameters, which forces Attack to use crosses less frequently. In the MSNE, teams with a greater chance of scoring from crosses use crosses less frequently than teams having a smaller chance of scoring from crosses. The proposition of the game-theoretical model was subsequently verified using the data for the 2016-17 football season.

Keywords: Association Football Leagues, Regression, Game Theory, Mixed Strategy Nash Equilibrium, Probability of goal-scoring

JEL Classification: C51, C72

Suggested Citation

Sarkar, Sumit, Paradox of Crosses in Association Football (Soccer) – A Game-Theoretic Explanation (September 12, 2017). Available at SSRN: or

Sumit Sarkar (Contact Author)

XLRI - Xavier School of Management ( email )

C. H. Area (East)
Jamshedpur, Jharkhand 831001

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