On an N-person noncooperative Markov game with a metric state space

We consider a noncooperative N-person discounted Markov game with a metric state space, and define the total expected discounted gain. Under some conditions imposed on the objects in the game system, we prove that our game system has an equilibrium point and each player has his equilibrium strategy. Moreover in the case of a nondiscounted game, the total expected gain up to a finite time can be obtained, and we define the long-run expected average gain. Thus if we impose a further assumption for the objects besides the conditions in the case of the discounted game, then it is proved that the equilibrium point exists in the nondiscounted Markov game. The technique for proving the nondiscounted case is essentially to modify the objects of the game so that they become objects of a modified Markov game with a discounted factor which has an equilibrium point in addition to the equilibrium point of the discounted game.