Nash equilibrium problems with congestion costs and shared constraints

Generalized Nash equilibria (GNE) represent extensions of the Nash solution concept when agents have shared strategy sets. This generalization is particularly relevant when agents compete in a networked setting. In this paper, we consider such a setting and focus on a congestion game in which agents contend with shared network constraints. We make two sets of contributions: (1) Under two types of congestion cost functions, we prove the existence of the primal generalized Nash equilibrium. The results are provided without a compactness assumption on the constraint set and are shown to hold when the mappings associated with the resulting variational inequality are non-monotone. Under further assumptions, the local and global uniqueness of the primal and primal-dual generalized Nash equilibrium is also provided. (2) We provide two distributed schemes for obtaining such equilibria: a dual and a primal-dual algorithm. Convergence of both algorithms is analyzed and preliminary numerical evidence is presented with the aid of an example.