Wost Case in Voting and Bargaining

TL;DR

This paper investigates the worst-case welfare guarantees in probabilistic voting and bargaining mechanisms, identifying optimal guarantees and their geometric structure depending on the number of agents and outcomes.

Contribution

It characterizes the structure of maximal worst-case guarantees in voting/bargaining models, revealing geometric properties and specific mechanisms that achieve these guarantees.

Findings

Uniform lottery is the only maximal guarantee when n ≥ p.

For p > n, guarantees include those inspired by random dictator and veto mechanisms.

The set of guarantees forms a union of polytopes with shared uniform guarantee.

Abstract

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.