6. Nash and Saddle-Point Equilibria of Infinite Dynamic Games

6.1 IntroductionThis chapter discusses properties and derivation of Nash and saddle-point equilibria in infinite dynamic games of prescribed fixed duration. The analysis is first confined to dynamic games defined in discrete time, and with a finite number of stages, and then extended to differential games. Some results for infinite-horizon formulations are also presented, primarily for affine-quadratic structures.Utilization of the two standard techniques of optimal control theory, viz. the minimum principle and dynamic programming, leads to the so-called open-loop and feedback Nash equilibrium solutions, respectively. These two different Nash equilibria and their derivation and properties (such as existence and uniqueness) are discussed in Section 6.2, and the results are also specialized to affine-quadratic games, as well as to two-person zero-sum games.When the underlying information structure for at least one player is dynamic and involves memory, a plethora of (so-called informationally nonunique) Nash equilibria with different sets of cost values exists, whose derivation entails some rather intricate analysis—not totally based on standard techniques of optimal control theory. This derivation, as well as several important features of Nash equilibria under closed-loop perfect state information pattern, are discussed in Section 6.3, first within the context of a scalar three-person two-stage game (cf. subsection 6.3.1) and then for general dynamic games in discrete time (cf. subsection 6.3.2).