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We consider both discrete and continuous time finite state-action stochasticgames. In discrete time stochastic games, it is known that a stationaryBlackwell-Nash equilibrium (BNE) exists for a single controller additive reward(SC-AR) stochastic game which is a special case of a general stochastic game.We show that, in general, the additive reward condition is needed for theexistence of a BNE. We give an example of a single controller stochastic gamewhich does not satisfy additive reward condition. We show that this exampledoes not have a stationary BNE. For a general discrete time discountedstochastic game we give two different sets of conditions and show that astationary Nash equilibrium that satisfies any set of conditions is a BNE. Oneof these sets of conditions weakens a set of conditions available in theliterature. For continuous time stochastic games, we give an example that doesnot have a stationary BNE. In fact, this example is a single controllercontinuous time stochastic game. Then, we introduce a continuous time SC-ARstochastic game. We show that there always exists a stationary deterministicBNE for continuous time SC-AR stochastic game. For a general continuous timediscounted stochastic game we give two different sets of conditions and showthat a Nash equilibrium that satisfies any set of conditions is a BNE.

Blackwell-Nash Equilibrium for Discrete and Continuous Time Stochastic Games