A duality of mis\`{e}re 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 mis\`{e}re 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 mis\`{e}re 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 mis\`{e}re 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 mis\`{e}re play and a generalization of play with a pass.