New math ensures fair daily assignments for up to 12 players

In a new paper accepted to SAGT 2026, mathematicians Terrence Adams and Erel Segal-Halevi tackle the problem of assigning n indivisible items to n players repeatedly, each day, with fairness guarantees after every single day rather than every n days. They propose two balance conditions on latin squares—structures where each label appears exactly once per row and column. The first condition is always satisfiable but weak; the second, stronger condition ensures that the cumulative assignment is always 'proportional up to one item' (PROP1) in a strong ordinal sense, meaning each player receives at least one of their top items early on.

However, the authors prove that this strong balance condition can only be satisfied for n ≤ 12, and cannot be satisfied for many larger values, including all n > 108. For the weaker balance condition, which guarantees ordinal proportionality up to two items (PROP2), it remains an open question whether it can be satisfied for all n. The work bridges combinatorics and game theory, with implications for scheduling, resource allocation, and fair division algorithms that require continuous fairness over time.