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

Martin Castillo-Quintana, Gianfranco Miranda-Romero

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.

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

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.
Paper Structure (31 sections, 12 theorems, 37 equations, 3 figures, 4 tables)

This paper contains 31 sections, 12 theorems, 37 equations, 3 figures, 4 tables.

Table of Contents

  1. The Strategic Problem
  2. The Model
  3. Setup
  4. Preliminary Results
  5. The Type-to-Type Mapping
  6. Examples of Equilibrium Multiplicity
  7. Example 1: Exponential Distribution
  8. Example 2: Uniform Distribution
  9. Example 3: Pareto Distribution
  10. Two Flavors of Multiplicity
  11. Characterization of Multiplicity
  12. The Integral Identity
  13. The Hazard Potential
  14. Boundary Conditions
  15. Classification
  16. ...and 16 more sections

Key Result

Lemma 1

Let $(\sigma_1, \sigma_2)$ be an equilibrium. Wherever $\sigma_1$ and $\sigma_2$ are finite, they are (i) non-decreasing and (ii) continuous.

Figures (3)

  • Figure 1: Equilibrium strategies for $F\sim \mathrm{Exp}(\lambda)$ with $\gamma=1/3$.
  • Figure 2: Equilibrium strategies for $F\sim U(0,1)$ with $\gamma=2$.
  • Figure 3: Equilibrium strategies for $F\sim\mathrm{Pareto}(\underline{\theta},\alpha)$ with $\underline{\theta}=1$, $\alpha=1$, $\underline{\theta}_1=2$.

Theorems & Definitions (27)

  • Remark 1: Pure Strategies
  • Lemma 1: Monotonicity and Continuity; fudenberg1986theory
  • Lemma 2: Strict Monotonicity in the Interior
  • Lemma 3: At Most One Side Concedes at Zero; hendricks1988war
  • Lemma 4: At Most One Side Fights Forever
  • Lemma 5: Differentiability
  • Lemma 6: ODE Characterization; amann1996asymmetric
  • Definition 1: Admissible Type-to-Type Function
  • Proposition 1: Sufficiency
  • Lemma 7: Integral Identity
  • ...and 17 more