12 - Nash equilibria: the variational approach

Summary

Non-cooperative game theory is a branch of game theory for the resolution of conflicts among players (or economic agents), each behaving selfishly to optimize their own well-being subject to resource limitations and other constraints that may depend on the rivals' actions. While many telecommunication problems have traditionally been approached by using optimization, game models are being increasingly used; they seem to provide meaningful models for many applications where the interaction among several agents is by no means negligible, for example, the choice of power allocations, routing strategies, and prices. Furthermore, the deregulation of telecommunication markets and the explosive growth of the Internet pose many new problems that can be effectively tackled with game-theoretic tools. In this chapter, we present a comprehensive treatment of Nash equilibria based on the variational inequality and complementarity approach, covering the topics of existence of equilibria using degree theory, global uniqueness of an equilibrium using the P-property, local-sensitivity analysis using degree theory, iterative algorithms using fixed-point iterations, and a descent approach for computing variational equilibria based on the regularized Nikaido–Isoda function. We illustrate the existence theory using a communication game with QoS constraints. The results can be used for the further study of conflict resolution of selfish agents in telecommunication.

Introduction

The literature on non-cooperative games is vast. Rather than reviewing this extensive literature, we refer the readers to the recent survey [20], which we will use as the starting point of this chapter.