Pivotal Players and the Characterization of Influence

Abstract

A player inuences a collective outcome if his actions can change the probability of that outcome. He is �-pivotal if this change exceeds some threshold �. We study inuence in general environments with N players and arbitrary sets of signals. It is shown that inuence is maximized when players' signals are identically distributed and the outcome is determined according to simple majority rule.This leads to the surprising conclusion that majority rules already contain the maximal number of pivotal players. From this we derive a tight bound on average inuence, as well as a tight bound on the number of �-pivotal players, which is independent of N. This analysis is relevant to problems where players' inuence is a key consideration in determining their strategic behavior. The applications we consider include the problem of designing a mechanism for the provision of public goods in the spirit of Mailath and Postlewaite (1990), partnership games, games with production complementarities, and cooperation in a noisy prisoner's dilemma.