Multiplicity of Equilibria in the War of Attrition with Two-Sided Asymmetric Information

TL;DR

This paper analyzes the causes of multiple equilibria in the war of attrition with asymmetric information, showing how type distribution properties influence equilibrium uniqueness and the effectiveness of refinements.

Contribution

It characterizes the sources of equilibrium multiplicity and demonstrates the conditions under which refinements select unique equilibria, especially highlighting the role of type support bounds.

Findings

Unbounded type support leads to persistent equilibrium multiplicity.

Refinements like payoff perturbations and behavioral types select unique equilibria only with bounded support.

Type distribution properties determine the form of equilibrium multiplicity.

Abstract

The war of attrition with two-sided asymmetric information is a foundational model in political economy, yet it generically admits a continuum of perfect Bayesian equilibria. This paper characterizes the sources of equilibrium multiplicity. We identify conditions on the type distribution that determine which form of multiplicity arises: when the lower limit of the hazard potential -- the integral of the hazard rate normalized by type -- diverges, the free parameter is the relative aggressiveness of strategies; when that limit is finite, the free parameter is the mass of types conceding immediately. We prove that the Amann-Leininger payoff perturbation and the introduction of behavioral types -- two seemingly distinct refinements -- are mathematically equivalent and succeed in selecting a unique equilibrium if and only if the type support is bounded. For unbounded supports, multiplicity persists. These results provide guidance for applied theorists: choosing distributions with bounded support ensures existing refinements deliver unique predictions.