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,确定型N人的最短路径游戏在纯度和固定式策略中具有纳什电子里程,如果它是边缘对称(即(u,v)是每次(v,u)的动作,除了进入终点顶点的移动之外,每次移动的时间长度对每个玩家来说都是积极的。这两个条件都很重要,尽管这仍然是一个开放的问题,是否存在着无NE的2人非尖端对称游戏,其长度是积极的。我们为无NE的2人边缘对称游戏提供了非肯定的范例。我们还考虑了终点游戏的特殊案例(短程游戏中,只有终点动作为非零长度,可能为负长度),并证明边缘对称 n人终端游戏在纯度和静止策略中总是有纳什的平衡。此外,我们证明,边缘对称2人终端游戏有一个统一的(次称完美)纳什平衡,只要任何无限的游戏都比两个玩家的终点更差。