On Nash-solvability in pure stationary strategies of the deterministic nn-person games with perfect information and mean or total effective cost

We study existence of Nash equilibria (NE) in pure stationary strategies in nn-person positional games with no moves of chance, with perfect information, and with the mean or total effective cost function.We construct a NE-free three-person game with positive local costs, thus disproving the conjecture suggested in Boros and Gurvich (2003). Still, the following four problems remain open: Whether NE exist in all two-person games with total effective costs such that(I) all local costs are strictly positive or (II) there are no dicycles of the total cost zero?Whether NE exist in all nn-person games with the terminal (transition-free) cost functions, provided all dicycles form a unique outcome cc and(III) assuming that cc is worse than any terminal outcome or (IV) without this assumption?For n=3n=3 the case (I) (and hence (II)) is answered in the negative. This is the main result of the present paper. For n=2n=2 the case (IV) (and hence (III)) was answered in the positive earlier.We briefly survey the above and some other negative and positive results on Nash-solvability in pure stationary strategies for the games under consideration.