On the Existence of Pairwise Stable Weighted Networks

In network theory, Jackson and Wolinsky introduced a now widely used notion of stability for unweighted network formation called pairwise stability. We prove the existence of pairwise stable weighted networks under assumptions on payoffs that are similar to those in Nash's and Glicksberg’s existence theorem (continuity and quasi concavity). Then, we extend our result, allowing payoffs to depend not only on the network, but also on some game-theoretic strategies. The proof is not a standard application of tools from game theory, the difficulty coming from the fact that the pairwise stability notion has both cooperative and noncooperative features. Last, some examples are given and illustrate how our results may open new paths in the literature on network formation.