A duality of misère games and play with a pass

Abstract

In combinatorial game theory, the choice of play convention is a fundamental aspect of the theory. The two most widely studied conventions are normal play, in which the player who makes the last move wins, and misère play, in which the player who makes the last move loses. Another well-known variant is play with a single shared pass. In such games, at most one pass may be used in total during the game: once the pass has been used, neither player may pass thereafter. Moreover, once a terminal position has been reached, passing is no longer allowed. Recall that, for misère play, one sometimes defines SG values (or Sprague-Grundy values) in the same recursive manner as in normal play but assigns the terminal position the value 1. Also, SG values can be considered for normal-play games with a pass. In this work, we propose transformations that allow both misère play and play with a pass to be treated as normal-play games. As a consequence, we show that there is a certain duality between a generalization of misère play and a generalization of play with a pass.