  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.
