Int J Game Theory
Invertibility modulo dead-ending no-
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
also have Q
as their quotient.
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 game combinatorial ensure that one of the players has
a winning strategy. The main objective of combinatorial game theory is to determine
which player has a winning strategy and what this strategy is. A basic way would be
to look at all possible moves for both players all the way until the game ends in all
By convention, Left is a female player whereas Right is a male player.
Supported by the ANR-14-CE25-0006 project of the French National Research Agency.
University of Mons-UMONS, Place du Parc 20, 7000 Mons, Belgium