Treasure game

Abstract

A prize is located at an unknown point on an island. In each period, each of n players searches a subset of the as yet unsearched portion of the island. If one player alone finds the prize he wins it and the game ends. Players have a per-period discount factor and a search cost proportional to area searched. Efficient symmetric Markov perfect equilibria are characterized when search is observable. Equilibria for n ≥ 2 exhibit two types of inefficiency: a tragedy of the commons (for small islands) and free riding (for large islands). For n ≥ 3, equilibrium properties are non-monotonic: players may be better off searching larger islands, and larger islands may take less time to search. When search is unobservable and players are sufficiently impatient, multi-player search can be efficient. The model is very general: applications include R&D races, team production, and extraction of exhaustible resources.