A comparison of penalty shootout designs in soccer

TL;DR

This paper compares different penalty shootout rules in soccer, introduces an improved Adjusted Catch-Up Rule, and proposes a complexity measure to evaluate penalty mechanisms.

Contribution

It introduces the Adjusted Catch-Up Rule, which outperforms existing rules, and provides a complexity measure for penalty shootout mechanisms.

Findings

Adjusted Catch-Up Rule outperforms other rules

Catch-Up Rule is not fairer than Alternating Rule

A complexity measure for penalty mechanisms is proposed

Abstract

Penalty shootout in soccer is recognized to be unfair because the team kicking first in all rounds enjoys a significant advantage. The so-called Catch-Up Rule has been suggested recently to solve this problem but is shown here not to be fairer than the simpler deterministic Alternating (ABBA) Rule that has already been tried. We introduce the Adjusted Catch-Up Rule by guaranteeing the first penalty of the possible sudden death stage to the team disadvantaged in the first round. It outperforms the Catch-Up and Alternating Rules, while remains straightforward to implement. A general measure of complexity for penalty shootout mechanisms is also provided as the minimal number of binary questions required to decide the first-mover in a given round without knowing the history of the penalty shootout. This quantification permits a two-dimensional evaluation of any mechanism proposed in the future.