Vladimir A. Gurvich - Professor of Applied Mathematics and Computer Science, Rutgers Center for Operations Research, Business School, Rutgers, the State University of New Jersey, USA.
Address: 100, Rockafeller Road, Piscataway, NJ, 08854, USA.  
E-mail: gurvich@rutcor.rutgers.edu, vladimir.gurvich@gmail.com

      In this paper we give an example of a finite positional game with perfect information and without moves of chance (a chess-like game) that has no Nash equilibria in pure stationary strategies. In this example the number n of players is 4, the number p of terminals is 5; furthermore, there is only one directed cycle. On the other hand, it is known that a chess-like game has a Nash equilibrium (NE) in pure stationary strategies if (A) n £ 2, or (B) p £ 3 and (C) any infinite play is worse than each terminal for every player. It remains open whether a NE-free chess-like game exists for n = 3, or when 2 £ p £ 4, or can such a game satisfy (C) for some n and p.