Exact worst-case normalized inefficiency for four and general player counts

Determine whether $4/7$ is the exact supremum of the normalized inefficiency ratio $(V^\ast-\bar V)/(V^\ast-V_{\min})$ for four players, and determine the corresponding supremum for general numbers of players.

Background

The paper proves that the price of no agreement is unbounded relative to the minimum efficiency gap . Numerical optimization for four-player cycling equilibria repeatedly produces a normalized inefficiency ratio of 4/7, but the computations do not establish that this value is optimal.

The unresolved problem is to characterize the exact supremum, first for four players and then for arbitrary player counts, rather than relying on numerical evidence from a restricted family of candidate cycles and payoff boxes.

References

Whether $4/7$ is the exact supremum of $(V\ast-\bar V)/(V\ast-V_{\min})$ for $n=4$, and what it is for general $n$, we have not determined.

— Does the grand coalition form? Persistence, arrival, and the role of the sharing rule in a dynamic process of nested binding agreements  (2608.17766 - Heitzig, 18 Aug 2026) in Section 'The price of no-agreement', immediately after Proposition