Proportional individual rationality and the provision of a public good in a large economy

This paper studies the public good provision problem in which a non-excludable public good can be provided and payments can be collected from agents only if the proportion of agents who obtain nonnegative interim expected utilities from the public good provision mechanism weakly exceeds a prespecified ratio α. We call this requirement “α proportional individual rationality”. We identify a key threshold such that if α is less than this threshold, then efficiency obtains asymptotically. If α is greater than the threshold, then inefficiency obtains asymptotically. In addition, we obtain the convergence rate of the probability of provision to its efficient/inefficient level. Moreover, as a methodological contribution of this paper, we propose the standard deviation of an agent's interim expected provision as a measure of the agent's influence in a mechanism. We find that as the economy becomes large, an agent's influence in any sequence of anonymous mechanisms converges to zero, and thus any sequence of anonymous feasible mechanisms must converge to a constant mechanism. We obtain uniform bounds for those rates of convergence.