Nonnegative voluntary implementation beyond the sufficient condition

Characterize nonnegative voluntary implementations of payment flows beyond the sufficient condition that coalition worths are nonnegative and average coalition worth does not decrease when a player joins.

Background

The paper proves that every efficient allocation can be implemented by a local, budget-balanced, voluntary flow when signed payments are permitted. For equal-residual sharing, it gives a sufficient condition for the existence of a voluntary flow with nonnegative entrant and incumbent payments: nonnegative coalition values together with nondecreasing average coalition worth under coalition expansion.

The authors explicitly leave unresolved the broader characterization of when nonnegative voluntary implementations exist, including cases in which the sufficient average-worth condition fails but a nonnegative voluntary flow may nevertheless be feasible.