On Nash-solvability of n-person graphical games under Markov and a-priori realizations

We consider finite graphical n-person games with perfect information that have no Nash equilibria in pure stationary strategies. Solving these games in stationary mixed strategies, we introduce probability distributions in all non-terminal positions. The corresponding probability distributions on the set of plays can be defined in two different ways called the Markov and a-priori realizations. The former one guarantees existence of a uniform best response for each player in every situation. Nevertheless, Nash equilibrium may fail to exist even in stationary mixed strategies. The classical Nash’s theorem is not applicable, because in this case limit distributions and expected payoffs may be discontinuous. Although a-priori realizations do not share many nice properties of the Markov ones (for example, the existence of uniform best responses) but in return, Nash’s theorem is applicable. We illustrate both realizations in details by two examples with 2 and 3 players. We also survey some general results related to Nash-solvability, in pure and mixed stationary strategies, of stochastic n-person games with perfect information and n-person graphical games among them.