Paradox of crosses in association football (soccer) – a game-theoretic explanation

Abstract

Abstract In association football, crosses from the wide areas of the pitch in the attacking third is a standard tactic for creating goal-scoring opportunities. But recent studies show that crosses adversely impact goals. Regression run in this paper on data from the premier soccer leagues of England, Spain, Germany, France and Italy for 2016–2017 season also found this inverse relation. However, there is no research that explains the reason for this inverse relation between crosses and goals. A game-theoretical model developed in this paper explains why crosses adversely affect goal-scoring. The model identifies a mixed strategy Nash equilibrium (MSNE), wherein the attacking team’s probability of playing a cross decreases with increase in their crossing accuracy, heading accuracy and probability of winning aerial balls. If the attacking team is good in terms of these parameters, the defending team’s probability of using an offside trap increases and that forces the attacking team to use crosses less frequently. In the MSNE, teams with a greater chance of scoring from crosses use the crosses less frequently than teams having a smaller chance of scoring from crosses. The theory was subsequently validated using the data of the 2016–2017 football season.