Parametrized Inexact-ADMM based coordination games: A normalized Nash equilibrium approach

Generalized Nash equilibrium problems are single-shot Nash equilibrium problems, whereby the decisions of all agents are coupled through a shared constraint. Such games are generally challenging to solve as they might give rise to a very large number of solutions. In this context, spanning many equilibria can be interesting to provide meaningful interpretations. In the literature, to compute equilibria, equilibrium problems are classically reformulated as optimization problems, potential games, relaxed and extended games. Applications of these reformulations to an economic dispatch problem under perfect and imperfect competition are provided. Unfortunately, these approaches only enable to describe a very limited part of the equilibrium set. To fill that gap, relying on normalized Nash equilibrium as solution concept, we provide a parametrized decomposition algorithm inspired by the Inexact-ADMM to span many more equilibrium points. Complexifying the setting, we consider an information structure in which the agents can withhold some local information from sensitive data, resulting in private coupling constraints. The convergence of the algorithm and deviations in the players’ strategies at equilibrium are formally analyzed. In addition, the algorithm can be used to coordinate the agents on one specific equilibrium with desirable properties at the system level. The coordination game is formulated as a principal-agent problem, and a procedure is detailed to compute the equilibrium that minimizes a secondary cost function capturing system-level properties. Finally, the Inexact-ADMM is applied to a cellular resource allocation problem, exhibiting better convergence rate than vanilla ADMM, and to compute equilibria that achieve both system-level efficiency and maximum fairness.