A General Public Goods Provision Problem With Many Agents

In Mailath & Postlewaite(1990), there is a famous asymptotic inefficiency theorem about the provision of the public goods. That theorem asserts that the probability that the public goods will be provided (i.e. the probability that the public goods being provided under the second-best mechanism) approaches to zero as the number of the agents in the economy tends to infinity although the probability that the public goods should be provided (i.e. the probability that the public goods being provided under the first-best mechanism) approaches to 1 as the number of the agents in the economy tends to infinity. In this paper, we will relax the constraint "individual rationality" in Mailath & Postlewaite(1990) to a weaker constraint named "consensus rate constraint" (i.e. a mechanism can be implemented if and only if some proportional of (not all of) the agents agree upon that mechanism, we name this proportion as "consensus rate1") and find that the asymptotical inefficiency theorem will not hold for sure. Indeed, in this paper, it has been proved that if the required consensus rate is lower than some positive number (which is determined by the parameter of the model and is between zero and 1), then the probability that the public goods will be provided in the second best mechanism will tend to 1 as the size of the economy goes to infinity, and if the consensus rate is larger than that number, then the probability that the public goods will be provided in the second best mechanism will go to 0 as the size of the economy goes to infinity.(In Mailath&Postlewaite(1990), the consensus rate is 1, and hence (as a special case here), the probability that the public goods will be provided goes to 0 as the size of the economy goes to infinity.)...