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Invertibility modulo dead-ending no-$$\mathcal {P}$$P-universes

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In combinatorial game theory, under normal play convention, all games are invertible, whereas only the empty game is invertible in misère play. For this reason, several restricted universes of games… Click to show full abstract

In combinatorial game theory, under normal play convention, all games are invertible, whereas only the empty game is invertible in misère play. For this reason, several restricted universes of games were studied, in which more games are invertible. Here, we study combinatorial games under misère play, in particular universes where no player would like to pass their turn. In these universes, we prove that having one extra condition makes all games become invertible. We then focus our attention on a specific quotient, called $${\mathcal {Q}_{\mathbb {Z}}}$$QZ, and show that all sums of universes whose quotient is $${\mathcal {Q}_{\mathbb {Z}}}$$QZ also have $${\mathcal {Q}_{\mathbb {Z}}}$$QZ as their quotient.

Keywords: invertibility modulo; modulo dead; dead ending; ending mathcal; mathcal mathbb; mathcal universes

Journal Title: International Journal of Game Theory
Year Published: 2018

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