Group Efforts when Performance Is Determined by the “Best Shot”

The achievements of a group frequently depend on the efforts of just a few members but represent a public good benefit to all members. This paper investigates such a situation: the private provision of a public good whose level is determined as the maximum effort made by a group member. Members’ costs of effort are envisioned as either commonly known or privately known. With perfect information, symmetric equilibria are in mixed strategies. For symmetric games, any number of players may be active and we characterize the unique equilibrium in which active contributors use the same strategy. An increase in the number of active players causes each active player to stochastically reduce his contributions and the distribution of the realized level of the public good shifts leftward, reducing equilibrium payoffs to inactive players. When information is private, we focus on the symmetric equilibrium, which is in pure strategies. Now an increase in the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Stochastically increasing the distribution of players’ costs increases players’ contribution strategy, with a resulting decrease in interim payoffs; and increasing ex ante heterogeneity of players’ costs can increase payoffs. Whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency. JEL Codes: H41, D61, D82