  We consider graphical $n$-person games with perfect information that have no
Nash equilibria in pure stationary strategies. Solving these games in mixed
strategies, we introduce probabilistic distributions in all non-terminal
positions. The corresponding plays can be analyzed under two different basic
assumptions: Markov's and a priori realizations. The former one guarantees
existence of a uniformly best response of each player in every situation.
Nevertheless, Nash equilibrium may fail to exist even in mixed strategies. The
classical Nash theorem is not applicable, since Markov's realizations may
result in the limit distributions and effective payoff functions that are not
continuous. The a priori realization does not share many nice properties of the
Markov one (for example, existence of the uniformly best response) but in
return, Nash's theorem is applicable. We illustrate both realizations in
details by two examples with $2$ and $3$ players and also provide some general
results.
