On guarantees, vetoes, and random dictators

A mechanism guarantees a certain welfare level to its agents, if each of them can secure that level against unanimously adversarial others. How high can such a guarantee be, and what type of mechanism achieves it? In the n-person probabilistic voting/bargaining model with p deterministic outcomes a guarantee takes the form of a probability distribution over the ranks from 1 to p. If n ≥ p, the uniform lottery is shown to be the only maximal (unimprovable) guarantee. If n < p, combining (variants of) the familiar random dictator and voting by veto mechanisms yields a large family of maximal guarantees: it is exhaustive if n = 2 and almost so if p ≤ 2n. Voting rules à la Condorcet or Borda, even in probabilistic form, are ruled out by our worst case viewpoint.