Extensions and failure thresholds for the affine-payoff extremality result

Determine whether the affine payoff remains extremal for the potential share on every regular conflict graph with continuous priorities, determine for which numbers of agents the affine-payoff extremality fails with two priority levels, and establish whether the finite-quantisation bound of Theorem 1 extends to orientations of conflict graphs containing directed cycles.

Background

The paper proves that, on the complete conflict graph under a total-order protocol and with at least three priority levels, every nonconstant payoff-from-rank function has potential share at most $(N-1)/N$, with equality exactly for affine payoffs. It also shows that this extremality can fail for two priority levels and that the affine payoff can be beaten at finite numbers of levels on conflict graphs other than the complete graph.

Three explicit extensions or thresholds are left unresolved: whether affine-payoff extremality holds on all regular graphs in the continuous-priority setting; the precise values of $N$ for which the two-level case $K=2$ fails; and whether the finite-level theorem extends from total-order protocols to orientations containing directed cycles.