We consider finite $n$-person deterministic graphical (DG) games. These games are modelled by finite directed graphs (digraphs) $G$ which may have directed cycles and, hence, infinite plays. Yet, it is assumed that all these plays are equivalent and form a single outcome $c$, while the terminal vertices $V_T = \{a_1, \ldots, a_p\}$ form $p$ remaining outcomes. We study the existence of Nash equilibria (NE) in pure stationary strategies. It is known that NE exist when $n=2$ and may fail to exist when $n > 2$. Yet, the question becomes open for $n > 2$ under the following extra condition: (C) For each of $n$ players, $c$ is worse than each of $p$ terminal outcomes. In other words, all players are interested in terminating the play, which is a natural assumption. Moreover, Nash-solvability remains open even if we replace (C) by a weaker condition: (C22) There exist no two players for whom $c$ is better than (at least) two terminal outcomes. We conjecture that such two players exist in each NE-free DG game, or in other words, that (C22) implies Nash-solvability, for all $n$. Recently, the DG games were extended to a wider class of the DG multi-stage (DGMS) games, whose outcomes are the strongly connected components (SCC) of digraph $G$. Merging all outcomes of a DGMS game that correspond to its non-terminal SCCs we obtain a DG game. Clearly, this operation respects Nash-solvability (NS). Basic conditions and conjectures related to NS can be extended from the DG to DGMS games: in both cases NE exist if $n=2$ and may fail to exist when $n > 2$; furthermore, we modify conditions (C) and (C22) to adapt them for the DGMS games. Keywords: $n$-person deterministic graphical (multi-stage) games, Nash equilibrium, Nash-solvability, pure stationary strategy, digraph, directed cycle, strongly connected component.

翻译：我们考虑的是有限的美元确定性图形游戏(DG) 。 我们研究的是 Nash equilibria (NE) 在纯粹的固定战略中的存在。 众所周知, 当美元=2美元时NE就存在, 当2美元游戏时可能不存在。 然而, 在以下额外条件下,所有游戏都相当于美元,形成一个单一结果 $c$, 而终端的顶点为$V_T = $a_1,\ldots, a_pv 美元代表剩余结果 。 换句话说, 所有玩家都有兴趣结束游戏, 这是一种自然假设。 此外, 纳什- solviriga (NNE) 在美元=2美元时就存在 NE, 而当2美元游戏为2美元时可能不存在。 然而,这个问题在以下额外条件下, 美元为2美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 货币, 货币, 货币, 美元, 货币, 货币, 货币, 两种, 货币, 货币, 货币, 两种, 货币, 货币, 货币, 货币, 货币, 货币, 两种, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 两种, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 货币, 等, 等, 等, 等, 。, 等, 等, 等, 等, 等, 等, 。, 。, 。,,,,