Refinement of the equilibrium of public goods games over networks: Efficiency and effort of specialized equilibria

Recently Bramoulléand Kranton presented a model for the provision of public goods over a network and showed the existence of a class of Nash equilibria called specialized equilibria wherein some agents exert maximum effort while all other agents free ride. We examine the welfare, effort and cost of specialized equilibria in comparison to other equilibria. Our main results show that the welfare of a particular specialized equilibrium, which we call influential equilibrium, approaches the maximum welfare among all equilibria as the concavity of the benefit function tends to unity. For forest networks a similar result holds wherein the welfare of a particular specialized equilibrium, called frugal equilibrium, approaches maximum welfare among all equilibria as the concavity approaches zero. Moreover, without any such concavity conditions, for any network there always exists a specialized equilibrium that requires the maximum total weighted effort amongst all equilibria, for any set of positive weights. If the network is a forest, the frugal equilibrium also incurs the minimum total cost amongst all equilibria. For a class of networks called well-covered forests we show that all welfare maximizing equilibria are specialized and all equilibria incur the same total cost. Thanks to these results, we argue that specialized equilibria may be considered a refinement of the notion of the equilibrium of the public goods game. The above results would be of relevance to a principal or a policy maker interested in ascertaining equilibria that result in the optimum of the above criteria, since they identify the specific equilibrium that attains the optimum and show that there is no loss of optimality in restricting attention to specialized equilibria. Additionally we show several results about the structure and efficiency of equilibria that would also be of relevance to a policy maker.