Consider an n-person stochastic game with Borel state space S, compact metric action sets A <Subscript>1</Subscript>,A <Subscript>2</Subscript>,…,A <Subscript> n </Subscript>, and law of motion q such that the integral under q of every bounded Borel measurable function depends measurably on the initial state x and continuously on the actions (a <Subscript>1</Subscript>,a <Subscript>2</Subscript>,…,a <Subscript> n </Subscript>)...</subscript></subscript></subscript></subscript></subscript></subscript>

We prove that a two-person, zero-sum stochastic game with arbitrary state and action spaces, a finitely additive law of motion and a bounded Borel measurable payoff has a value.

Every n-person stochastic game with a countable state space, finite action sets for the players and bounded, upper semi-continuous payoffs has an -equilibrium for every >0.

There exists a Nash equilibrium (-Nash equilibrium) for every n-person stochastic game with a finite (countable) state space and finite action sets for the players if the payoff to each player i is one when the process of states remains in a given set of states Gi and is zero otherwise.

We show that if y is an odd integer between 1 and $2^{n}-1$, there is an $n\times n$ bimatrix game with exactly y Nash equilibria (NE). We conjecture that this $2^{n}-1$ is a tight upper bound on the number of NEs in a "nondegenerate" $n\times n$ game.