Nash oligarchies

Abstract

A bargaining solution is a Nash oligarchy if there exists a set of players $O$, the oligarchy, such that in every bargaining problem every player outside $O$ is allocated his disagreement payoff, and the oligarchs share the surplus according to the $\left|O\right|$-person Nash solution. The Nash bargaining solution corresponds to the case where $O$ contains all players, and dictatorial solutions correspond to singleton $O$s. I show that a solution satisfies three standard axioms together with a condition that balances utilitarianism and egalitarianism if and only if it is a Nash oligarchy. Strengthening the condition implies that the oligarchy contains at most two players. Strengthening it further implies a dictatorship.