The n-player Hirshleifer contest

We characterize the equilibrium set of the n-player Hirshleifer contest with homogeneous valuations. A symmetric equilibrium always exists. It necessarily corresponds to multilateral peace for sufficient noise and uses finite-support randomized strategies otherwise. Asymmetric equilibria are feasible for n≥3 contestants only, and only for sufficiently small noise. In pure strategies, any asymmetric equilibrium corresponds to one-sided dominance, but there is also a variety of payoff-inequivalent mixed-strategy equilibria for small noise. For arbitrarily small noise, at least two contestants engage in cut-throat competition, while any others become ultimately inactive. Of some conceptual interest is the observation that, for n sufficiently large, the unique equilibrium is multilateral peace.