On Nash-solvability of finite $n$-person deterministic graphical games; Catch 22

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.