Why Chess and Backgammon can be solved in pure positional uniformly optimal strategies

We study existence of (subgame perfect) Nash equilibria (NE) in pure positional strategies in finite positional n-person games with perfect information and terminal payoffs. However, these games may have moves of chance and cycles. Yet, we assume that All Infinite Plays Form One Outcome a∞, in addition to the set of Terminal outcomes VT . This assumption will be called the AIPFOOT condition. For example, Chess or Backgammon are AIPFOOT games, since each infinite play is a draw, by definition. All terminals and a∞ are ranked arbitrarily by the n players. It is well-known that in each finite acyclic positional game, a subgame perfect NE exists and it can be easily found by backward induction, suggested by Kuhn and Gale in early 50s. In contrast, there is a two-person game with only one cycle, one random position, and without NE in pure positional strategies. In 1912, Zermelo proved that each two-person zero-sum AIPFOOT game without random moves (for example, Chess) has a saddle point in pure strategies. Yet, there are cycles in Chess. Zermelo’s result can be extended in two directions: (i) Each two-person (not necessarily zero-sum) AIPFOOT game without random moves has a (not necessarily subgame perfect) NE in pure positional strategies; although, the similar statement remains a conjecture for the n-person case. (ii) Each two-person zero-sum AIPFOOT game (which might have random moves) has saddle point in pure positional uniformly optimal strategies. Surprisingly, to prove (ii), it appears convenient to treat such a game (for example, Backgammon or even Chess) as a stochastic game with perfect information.