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

Invertibility modulo dead-ending no- $$\mathcal {P}$$ P -universes 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 $${\mathcal {Q}_{\mathbb {Z}}}$$ Q Z , and show that all sums of universes whose quotient is $${\mathcal {Q}_{\mathbb {Z}}}$$ Q Z also have $${\mathcal {Q}_{\mathbb {Z}}}$$ Q Z as their quotient. 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

International Journal of Game Theory, Volume 47 (3) – Jun 5, 2018
13 pages

Publisher
Springer Journals
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

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 $${\mathcal {Q}_{\mathbb {Z}}}$$ Q Z , and show that all sums of universes whose quotient is $${\mathcal {Q}_{\mathbb {Z}}}$$ Q Z also have $${\mathcal {Q}_{\mathbb {Z}}}$$ Q Z as their quotient.

### Journal

International Journal of Game TheorySpringer Journals

Published: Jun 5, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create folders to

Export folders, citations