On Nash equilibria and improvement cycles in pure positional strategies for Chess-like and Backgammon-like nn-person games

We consider nn-person positional games with perfect information modeled by finite directed graphs that may have directed cycles, assuming that all infinite plays form a single outcome cc, in addition to the standard outcomes a1,…,ama1,…,am formed by the terminal positions. (For example, in the case of Chess or Backgammon n=2n=2 and cc is a draw.) These m+1m+1 outcomes are ranked arbitrarily by nn players. We study existence of (subgame perfect) Nash equilibria and improvement cycles in pure positional strategies and provide a systematic case analysis assuming one of the following conditions:(i) there are no random positions; (ii) there are no directed cycles; (iii) the ïnfinite outcome” cc is ranked as the worst one by all nn players; (iv) n=2; (v) n=2 and the payoff is zero-sum.