On head-to-head results as tie-breaker and consequent opportunities for collusion

TL;DR

This paper analyzes how head-to-head tie-breaker rules in football tournaments can create collusion opportunities, increasing the risk of match-fixing and suggesting alternative tie-breaking methods to mitigate this risk.

Contribution

It provides a mathematical characterization of collusion opportunities caused by head-to-head tie-breakers and quantifies the increased risk through simulations based on a major tournament.

Findings

Head-to-head tie-breakers increase collusion risk by 11.5-14.8 percentage points.

Static scheduling cannot effectively mitigate collusion risk.

Using goal difference as the primary tie-breaker reduces collusion opportunities.

Abstract

The outcome of some football matches has benefited both teams at the expense of a third team because head-to-head results were used for breaking ties. Inspired by these examples, our mathematical analysis identifies all possible collusion opportunities caused by this particular ranking rule in a single round-robin tournament with four teams. According to a simulation model based on the 2024 UEFA European Football Championship, merely the tie-breaking rule increases the probability of reaching a situation vulnerable to collusion by between 11.5 and 14.8 percentage points. This risk can scarcely be mitigated by a static match schedule. Therefore, tournament organisers are strongly encouraged to choose goal difference as the primary tie-breaking rule, similar to the official policy of FIFA.