  In this short note we give an example of a four-person finite positional game
with perfect information that has no positions of chance and no Nash equilibria
in pure stationary strategies. The corresponding directed graph has only one
directed cycle and only five terminal positions.
  It remains open: (i) if the number $n$ of the players can be reduced from $4$
to $3$, (ii) if the number $p$ of the terminals can be reduced from $5$ to $4$,
and most important, (iii) whether it is possible to get a similar example in
which the outcome $c$ corresponding to all (possibly, more than one) directed
cycles is worse than every terminal for each player.
  Yet, it is known that (j) $n$ cannot be reduced to $2$, (jj) $p$ cannot be
reduced to $3$, and (jjj) there can be no similar example in which each player
makes a decision in a unique position.
  Keywords: stochastic, positional, chess-like, transition-free games with
perfect information and without moves of chance; Nash equilibrium, directed
cycles (dicycles), terminal position.
