The likelihood of inefficiency, a prisoner's dilemma and suboptimality in games of binary voluntary provision of public goods

This paper focuses on one possible measure for the severity of the problems of inefficiency, suboptimality and the Prisoner's Dilemma in voluntary public-good provision. The proposed measure is the probability of the emergence of the respective problem in an impartial environment where all feasible combinations of the parameters of the voluntary public good provision game are equally likely. These probabilities are derived using the game of voluntary binary contributions to the provision of public goods recently analyzed by Gradstein and Nitzan (1990). The severity of the three problems (Prisoner's Dilemma, suboptimality and inefficiency) is computed, respectively for games with up to N=12, N=6 and N=4 players. The decreasing order of N reflects the increasing complexity of the problems and, in turn, computation of their likelihood. It turns out that the likelihoods of all three problems are increasing with the number of players, N. More importantly, social optimality is more likely than suboptimality when N≤2, efficiency is more likely than inefficiency when N≤3 and no Prisoner's Dilemma is more likely than a Prisoner's Dilemma when N≤5.