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

Invertibility modulo dead-ending no- $$\mathcal {P}$$ P -universes Int J Game Theory https://doi.org/10.1007/s00182-018-0629-7 ORIGINAL PAPER Invertibility modulo dead-ending no-P-universes Gabriel Renault Accepted: 30 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 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 Q , and show that all sums of universes whose quotient is Q also have Q as their quotient. Z Z 1 Introduction A combinatorial game is a two-player game with no chance and perfect information. The players, called Left and Right, alternate moves until one player is unable to move. The last player to move loses the game under the misère play convention, while that same player would win under normal play convention. In this paper, we are only studying ﬁnite combinatorial games. The conditions that make a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Game Theory Springer Journals

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

, Volume OnlineFirst – Jun 5, 2018
13 pages

Publisher
Springer Berlin Heidelberg
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Economics; Economic Theory/Quantitative Economics/Mathematical Methods; Game Theory, Economics, Social and Behav. Sciences; Behavioral/Experimental Economics; Operations Research/Decision Theory
ISSN
0020-7276
eISSN
1432-1270
D.O.I.
10.1007/s00182-018-0629-7
Publisher site
See Article on Publisher Site

### Abstract

Int J Game Theory https://doi.org/10.1007/s00182-018-0629-7 ORIGINAL PAPER Invertibility modulo dead-ending no-P-universes Gabriel Renault Accepted: 30 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 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 Q , and show that all sums of universes whose quotient is Q also have Q as their quotient. Z Z 1 Introduction A combinatorial game is a two-player game with no chance and perfect information. The players, called Left and Right, alternate moves until one player is unable to move. The last player to move loses the game under the misère play convention, while that same player would win under normal play convention. In this paper, we are only studying ﬁnite combinatorial games. The conditions that make a

### Journal

International Journal of Game TheorySpringer Journals

Published: Jun 5, 2018

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