Repeated Contests with Toughness

Abstract

A set of players fights in a repeated standard all-pay auction for a fixed prize in each period. Each player is rational or tough; a tough player always exerts an exogenous, high "tough effort". We fully characterize the unique type-symmetric equilibrium. Rational players may exert tough efforts to appear tough. A rational player has strictly positive payoff only if she is monopolizing reputation for toughness. In a reputational oligopoly, a fierce war of attrition to become the monopolist may yield overdissipation. In a reputational monopoly instead, overdissipation never happens. Thus, reputational concerns may increase or decrease expected aggregate effort. The monopolist mixes between a non-tough effort to cash in on her reputation today and the tough effort to boost her reputation. While a monopoly may last arbitrarily long, an oligopoly has an endogenous maximum duration. Applications include turf wars, conflicts, and litigation.

JEL Classification: C72, D82, D83