Abstract
Our purpose is to obtain necessary conditions that the optimal pure strategies for a minimizer u and a maximizer υ must satisfy. By a static game is meant a so-called continuous game, and by a dynamic game an ordinary differential game.
We obtained previously the necessary conditions for the continuous game such that the pay-off function F(u, υ) and the inequaiity constraints g(u, υ)≤0 are functions of only u and υ. The obtained conditions were Kuhn-Tucker conditions for the game. They were derived by considering a pair of Lagrangian functions. In this paper we extend these conditions to the case when there exist equality constraints f(x, u, υ)=0 where x is a state vector. We prove also a duality theorem for the game.
Next, we consider a differential game with inequality constraints g(u, υ)≤0, and use a technique with a pair of Lagrangian functions as we did for the static game. Following Berkowitz, we base our analysis on the calculus of variation in obtaining the necessary conditions for the optimal minimizer u0(t) and maximizer υ0(t) in correspondence to Euler equations, Clebsch condition etc. for simple minimization.
In addition, the maximum principle for the game is derived in the case when no inequality constrains exist.
Finally we study relations between a static problem and a dynamic problem, namely, between Kuhn-Tucker conditions and Euler, Clebsch conditions. We conclude that the former conditions are indeed derived from the latter conditions.
