Pairwise-stability and Nash equilibria in network formation

Suppose that individual payoffs depend on the network connecting them. Consider the following simultaneous move game of network formation: players announce independently the links they wish to form, and links are formed only under mutual consent. We provide necessary and sufficient conditions on the network link marginal payoffs such that the set of pairwise stable, pairwise-Nash and proper equilibrium networks coincide, where pairwise stable networks are robust to one-link deviations, while pairwise-Nash networks are robust to one-link creation but multi-link severance. Under these conditions, proper equilibria in pure strategies are fully characterized by one-link deviation checks. We thank William Thomson, an associate editor and two anonymous referees for their suggestions that led to substantial improvements. We also thank Sjaak Hurkens, Bettina Klaus, Jordi Massó and Giovanni Neglia for helpful conversations. The first author gratefully acknowledges the financial support from the Spanish Ministry of Education and FEDER through grant SEJ2005-01481/ECON, the Fundación BBVA and the Barcelona Economics Program of XREA. The second author is grateful to the Netherlands Organization for Scientific Research (NWO) for its support under grant VIDI-452-06-013.