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

We consider n -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 c , in addition to the standard outcomes a 1 , … , a m formed by the terminal positions. (For example, in the case of Chess or Backgammon n = 2 and c is a draw.) These m + 1 outcomes are ranked arbitrarily by n 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: ere are no random positions; (ii) there are no directed cycles; (iii) the ïnfinite outcome” c is ranked as the worst one by all n players; (iv) n = 2 ; (v) n = 2 and the payoff is zero-sum.