Game Theory 101: Soccer Penalty Kicks

William Spaniel explores the strategic dynamics of penalty kicks in soccer through the lens of game theory. He simplifies the scenario to two primary strategies for both the kicker and the goalie: kicking left or right. With the assumption of superhuman players, the kicker must aim in the opposite direction of the goalie’s dive to score. Spaniel illustrates this using a payoff matrix that highlights the outcomes of different strategies, concluding that there are no pure strategy Nash Equilibria, prompting the need for a mixed strategy approach.

By applying mixed strategy algorithms, Spaniel determines that both the kicker and the goalie should kick or dive left and right with equal probability when both are equally skilled. However, when the kicker is less proficient, the analysis shifts. The kicker, who misses half the time when aiming left, must adjust his strategy accordingly. The results show that the goalie will dive left one-third of the time and right two-thirds, while the kicker will kick left two-thirds of the time and right one-third. This unexpected outcome arises because the goalie anticipates the kicker's stronger right side, leading the kicker to favor his weaker side to maintain equilibrium.

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