Designing Fair Tiebreak Mechanisms: The Case of FIFA Penalty Shootouts

Abstract

In the current FIFA penalty shootout mechanism, a coin toss decides which team will kick first. Empirical evidence suggests that the team taking the first kick has a higher probability to win a shootout. We design sequentially fair shootout mechanisms such that in all symmetric Markov-perfect equilibria each of the skill-balanced teams has exactly 50% chance to win whenever the score is tied at any round. Consistent with empirical evidence, we show that the current mechanism is not sequentially fair and characterize all sequentially fair mechanisms. Taking additional desirable properties into consideration, we propose and uniquely characterize a practical mechanism.

Making the Rules of Sports Fairer

“…Before reaching either of these two shaded states, the order of kicking switches to YX -at YX (1,1) in the first case, and YX (0,0) in the second case. Only when X fails to score and Y does score at XY (0-1) in the first round does the order of kicking not switch (it stays XY ), because by rule 2, X is still advantaged since it did not score when Y did.…”

“…The Catch-Up Rule differs from the "behind first, alternating order" mechanism of Anbarci, Sun, andÜnver [1], which depends on the current score: If one team is behind, it kicks first on the next round; if the score is tied, the order of kicking alternates (i.e., switches from the previous round). By comparison, the Catch-Up Rule depends only on the performances of the teams on the previous round.…”

“…Anbarci, Sun, and Unver [1] model the optimal spot for the kicker to aim at to balance the probability of the goalie stopping a goal (if the ball is aimed close to the center of the net) and the probability of the ball going outside the net (if the ball is aimed close to a post on the side). By contrast, we offer no such model but assume instead that the probability of a successful kick depends only on whether the kicker kicks first or second on a round, implicitly assuming that where to aim has been optimally chosen and so not modeling this aspect of the game.…”

Making the Rules of Sports Fairer

“…Observe that if both teams score on both rounds, play ends at XY (2-2) in the "behind first, alternating order" mechanism of Anbarci et al [1], which depends 198 on the current score: If one team is behind, it kicks first on the next round; if the 199 score is tied, the order of kicking alternates (i.e., switches from the previous round).…”

“…Anbarci et al [1] 211 model the optimal spot for the kicker to aim at to balance the probability of the 212 goalie stopping a goal (if the ball is aimed close to the center of the net) and the 213 probability of the ball going outside the net (if the ball is aimed close to a post on the 214 side). By contrast, we offer no such model but assume instead that the probability 215 of a successful kick depends only on whether the kicker kicks first or second on a 216 round, implicitly assuming that where to aim has been optimally chosen and not 217 modeling this aspect of the game.…”