Instant Exit from the Asymmetric War of Attrition

Two players engage in a costly fight. A conceding player yields a prize of privately-known value to her opponent. As is well known, this classic game exhibits multiple equilibria. Perturbing the war of attrition—for instance, by imposing a time limit or allowing for the small probability of players who are restricted to fight forever—yields a unique equilibrium. In this paper, I examine the properties of this unique equilibrium in the limit, as such perturbations are allowed to vanish, hence selecting an equilibrium from the classic war of attrition. I employ a measure of stochastic strength to order the distributions from which players’ prize valuations are drawn. This measure is based on the relative-hazard rates of the distributions in their upper tails. In the selected equilibrium, the stochastically weaker player always exits at the beginning of the game, even though her realized prize valuation may well be greater. Thus a war of attrition is resolved immediately, in favor of a player who is merely perceived to be stronger ex ante, rather than actually stronger ex post. 1. Wars of Attrition In a classic two-player war of attrition, the combatants compete to win an indivisible prize. At a player’s disposal is a single weapon: Her time of exit. The first player to quit concedes the prize to her opponent. Fighting is costly, and hence is worthwhile only if a player expects her opponent to quit in the near future. These features ensure that the war of attrition serves as a stylized representation of many important economic phenomena, including labor-market negotiations, the voluntary provision of public goods, macroeconomic stabilization, the adoption of technological standards, and political lobbying. An analysis of these scenarios must ask two simple questions: Who will win? When will the war of attrition end? Early answers to these questions suffered from the presence of multiple equilibria. A number of authors, most notably Fudenberg and Tirole (1986), Kornhauser, Rubinstein, and Wilson (Acknowledgements are omitted from this review copy. This paper is based on Myatt (1999).) I give a fuller account of such applications, as well as the related literature, in Sections 2 and 7. Consider the simplest complete-information war of attrition: The costs incurred by the players are directly proportional to the length of the war, and their valuations for the prize are commonly known. In his study of this game, Maynard Smith (1974) examined a symmetric mixed-strategy equilibrium, in which each player exits with a constant hazard rate. There are, however, many other equilibria. For instance, it is an equilibrium for one player to fight forever, while her opponent quits at the beginning.