Comment: 33 pagesThe guarantee of an anonymous mechanism is the worst case welfare an agent can secure against unanimously adversarial others. How high can such a guarantee be, and what type of mechanism achieves it? We address the worst case design question in the n-person probabilistic voting/bargaining model with p deterministic outcomes. If n is no less than p the uniform lottery is the only maximal (unimprovable) guarantee
there are many more if p>
n, in particular the ones inspired by the random dictator mechanism and by voting by veto. If n=2 the maximal set M(n,p) is a simple polytope where each vertex combines a round of vetoes with one of random dictatorship. For p>
n>
2, we show that the dual veto and random dictator guarantees, together with the uniform one, are the building blocks of 2 to the power d simplices of dimension d in M(n,p), where d is the quotient of p-1 by n. Their vertices are guarantees easy to interpret and implement. The set M(n,p) may contain other guarantees as well
what we can say in full generality is that it is a finite union of polytopes, all sharing the uniform guarantee.