On Nash-solvability of chess-like games

In 2003, the first two authors proved that a chess-likes game has a Nash equilibrium (NE) in pure stationary strategies if (A) the number n of players is at most 2, or (B) the number p of terminals is at most 2 and (C) any infinite play is worse than each terminal for every player. In this paper we strengthen the bound in (B) replacing p ≤ 2 by p ≤ 3, provided (C) still holds. On the other hand, we construct a NE-free four-person chess-like game with five terminals, which has a unique cycle, but does not satisfy (C). It remains open whether a NE-free example exists for n = 3, or for 2 ≤ p ≤ 4, or for some n ≥ 3 and p ≥ 4 provided (C) holds.