A connections model with decreasing returns link-formation technology

We study a connections model where the strength of a link depends on the amount invested in it and is determined by an increasing strictly concave function. The revenue from investments in links is the value (information, contacts, friendship) that the nodes receive through the network. First, assuming that links are the result of investments by the node-players involved, there is the question of stability. We introduce and characterize a notion of marginal equilibrium, where all nodes play locally best responses, and identify different marginally stable structures. This notion is based on weak assumptions about node-players' information and is necessary for Nash equilibrium and for pairwise stability. Second, efficient networks in absolute terms are characterized. Efficiency and stability are shown to be incompatible, but partial subsidizing is shown to be able to bridge the gap.