AI 中文总结
研究组合博弈论中玩法规则，提出变换使错序玩法和可弃权玩法能视为正常玩法，揭示了错序玩法推广与可弃权玩法推广之间存在对偶性。
AI 中文摘要
在组合博弈论中，玩法规则的选择是该理论的一个基本方面。研究最广泛的两种规则是正常玩法（最后一步行动的玩家获胜）和错序玩法（最后一步行动的玩家失败）。另一种著名的变体是可弃权玩法。在这类博弈中，游戏期间总共最多可使用一次弃权：一旦使用了弃权，此后双方玩家都不得再弃权。此外，一旦到达终局位置，就不再允许弃权。对于错序玩法，有时会以与正常玩法相同的递归方式定义SG值，但将终局位置的值设为1。正常玩法的博弈也可以考虑SG值。在这项工作中，我们提出了一些变换，使错序玩法和可弃权玩法都能被视为正常玩法。因此，我们表明错序玩法的一种推广和可弃权玩法的一种推广之间存在某种对偶性。
英文摘要
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.
Comments7 pages, 6 figures
