Finite Dimensional Variational Inequalities and Nash Equilibria

Abstract

In this chapter, we lay the foundation for turning our focus from dynamic optimization, which has been the subject of preceding chapters, to the notion of a dynamic game. To fully appreciate the material presented in subsequent chapters, we must in the present chapter review some of the essential features of the theory of finite-dimensional variational inequalities and static noncooperative mathematical games. Today many economists and engineers are exposed to the notion of a game-theoretic equilibrium that we study in this chapter, namely Nash equilibrium. Yet, the relationship of such equilibria to certain nonextremal problems known as fixed-point problems, variational inequalities and nonlinear complementarity problems is not widely understood. It is the fact that, as we shall see, Nash and Nash-like equilibria are related to and frequently equivalent to nonextremal problems that makes the computation and qualitative investigation of such equilibria so tractable. Although the static games discussed in this chapter are really steady states of dynamic games, we are, for the most part, indifferent in this chapter to any underlying dynamics. We also comment that readers familiar with finite-dimensional variational inequalities and static Nash games may wish to skip this chapter.