Optimal Provision of a Discrete Public Good: Linear Equilibria in the Private-Information Subscription Game

We analyze a symmetric Bayesian game in which two players individually contribute to fund a discrete public good; contributions are refunded if they do not meet a threshold set by the seller of the good. We provide a general characterization of symmetric equilibrium strategies that are continuous and nonconstant over the set of values for which the good has a positive chance of provision. Piecewise-linear strategies are our special focus. We characterize the distributions of players' private values that can support a continuous piecewise-linear symmetric equilibrium, and we calculate such equilibria for these distributions. Allowing the seller to charge a nonrefundable entry fee before players make their private contributions, we show these piecewise-linear equilibria can maximize the seller's expected utility, which may include an altruistic component, over all incentive compatible selling mechanisms.