We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is (u,v) is a move whenever (v,u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in which only terminal moves have nonzero length, possibly negative) and prove that edge-symmetric n-person terminal games always have Nash equilibria in pure and stationary strategies. Furthermore, we prove that an edge-symmetric 2-person terminal game has a uniform (subgame perfect) Nash equilibrium, provided any infinite play is worse than any of the terminals for both players.

翻译：我们证明了如果确定性n人最短路径博弈是边对称的（即当(v,u)是移动时，(u,v)也是移动，除非移动进入终点顶点），且每个玩家的每一步移动长度均为正，则该博弈存在纯平稳策略纳什均衡。这两个条件都是必要的，尽管具有正步长的非边对称2人游戏是否存在无纳什均衡的情况仍是一个未解问题。我们提供了具有非正步长的边对称2人博弈中无纳什均衡的实例。我们还考虑了终点博弈的特殊情况（只有终点移动具有非零长度（可能为负）的最短路径博弈），并证明了边对称n人终点博弈总是存在纯平稳策略纳什均衡。此外，我们证明了如果对两个玩家而言任意无限游戏都劣于任意终点，则边对称2人终点博弈具有一致（子博弈完美）纳什均衡。