Discrete Public Goods with Incomplete Information

We investigate a simultaneous discrete public good provision game with incomplete information. To use the terminology of Admati and Perry (1991), we consider both contribution and subscription games. In the former, contributions are not refunded if the project is not completed, while in the latter they are. In the presence of complete information about individuals' valuations for the public good, the difference between the equilibrium outcomes of a subscription game and a contribution game is not significant. However, there is both casual evidence from the fund-raising literature and experimental evidence that subscription games are ``superior '', i.e., a refund increases the chance of providing the good given that it is efficient to do so. Our analysis shows that this is indeed the case in the presence of incomplete information. We compute a symmetric equilibrium for the subscription game and show that it is not necessarily efficient. This inefficiency stems from the difficulties arising in coordinating to overcome the free-rider problem in the presence of incomplete information. Although it is well known that informational disparities impose limits on the efficiency of outcomes, the novel feature of our analysis is to explicitly model the resulting trade-off --- when deciding how much to contribute towards the public good --- between increasing the likelihood of provision and creating incentives for free-riding by the other player. Moreover, we show that for the contribution game, ``contributing zero'' is the only equilibrium for a given range of the fixed cost of provision and for a family of distributions.