Power, Status and the Stability of Hierarchies

We present a game-theoretic model of social hierarchy formation and provide a notion of stability and weak stability for hierarchies. We characterize both notions and establish the existence of weakly stable hierarchies. We then analyze three applications. In the first application, we consider a society divided into different classes of individuals in which nobility and wealth are the main drivers of status, and players have a preference for similarity and nobility. We study how the stable hierarchy changes as one class of individuals grows richer. In our second application we study how factions with different status form within a political party when members primarily care about the policy preferences of the faction they belong to. We demonstrate that stability favors factions with extreme policies at the expense of factions with moderate policies. The third application studies an environment in which players are de ned by a set of observable attributes, and the common preferences solely depend on these attributes. We show that although status is always increasing in attribute values, the complicated knock-on effects imply that status might not increase as the weight of an attribute increases even for individuals that rank highest in the said attribute.