Fairness–stability trade-offs in knockout pairings for the 48-team FIFA World Cup

Question

The 2026 World Cup has 12 groups and advances the 8 best third-placed teams. Which 8 groups contribute a third-placed team is only known after the group stage, and Annex C of the regulations fixes, for each of the 495 possible sets, which group-winner slot each third-placed team goes to. The paper does not forecast matches. It asks what this table does as a rule: does it expose third-placed teams from different groups to the eight slots evenly, and when one qualifier changes, how many of the other seven are reshuffled?

Results

On the full rematch-free space of 88 admissible group–slot edges, the minimum possible exposure deviation is exactly 1/308, by an integer-counting argument, and a 495-row mechanism attains it.
Fixing one group, the scenarios containing it form the Johnson graph J(11,4). Delsarte's inequalities for that scheme give a closed-form bound on the edge boundary of any label class, which sums to: every exposure-optimal mechanism changes at least 26,248 adjacent-scenario comparisons. The best feasible construction so far has 35,194; the gap is open.
Under equal scenario weights, two deterministic alternatives improve on the official table (one keeps exposure fixed and cuts instability by 4.46%). Under 380 non-uniform weightings this dominance does not always hold, and the paper says so.

What I did myself

I completed the formulation, proofs, computation and verification myself. The numerical parts (integer bounds, Johnson-scheme parameters, the Delsarte linear programme) were recomputed with an independent exact-arithmetic script, which also showed that the Delsarte bound used is already the LP optimum for the relevant class sizes.

Status and limits

English manuscript prepared for a sports-analytics journal; not submitted. Before it goes anywhere I am doing two things: having the two core theorems read by people with no stake in the paper, and re-deriving the 1/308 bound on a whiteboard without the draft in front of me. If either step fails, the paper waits.