  The 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.
