On the non-uniformity of the 2026 FIFA World Cup draw

TL;DR

This paper analyzes the non-uniformity in the 2026 FIFA World Cup draw process, quantifies its extent, and proposes an integer programming method to evaluate and improve fairness in team allocations.

Contribution

It provides a detailed quantification of non-uniformity in the draw and introduces an integer programming approach to assess and enhance fairness.

Findings

Official draw is optimal under four non-uniformity measures.

Non-uniformity can be reduced by treating top teams more equally.

Recursive backtracking is intractable for large-scale analysis.

Abstract

The group stage of a sports tournament is often made more appealing by introducing additional constraints in the group draw that promote an attractive and balanced group composition. For example, the number of intra-regional group matches is minimised in several World Cups. However, under such constraints, the traditional draw procedure may become non-uniform, meaning that the feasible allocations of the teams into groups are not equally likely to occur. Our paper quantifies this non-uniformity of the 2026 FIFA World Cup draw for the official draw procedure, as well as for 47 reasonable alternatives implied by all permutations of the four pots and two group labelling policies. We show why simulating with a recursive backtracking algorithm is intractable, and propose a workable implementation using integer programming. The official draw mechanism is found to be optimal based on four measures of non-uniformity. Nonetheless, non-uniformity can be more than halved if the organiser aims to treat the best teams drawn from the first pot equally.