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: (i) there 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. © 2011 Elsevier B.V. All rights reserved.
Boros, E., Elbassioni, K., Gurvich, V., & Makino, K. (2012). On Nash equilibria and improvement cycles in pure positional strategies for Chess-like and Backgammon-like n-person games. Discrete Mathematics, 312(4), 772–788. https://doi.org/10.1016/j.disc.2011.11.011