These lecture notes are based on a short course on misère quotients offered at the Weizmann Institute of Science in Rehovot, Israel, in November 2006. They include an introduction to impartial games, starting from the beginning; the basic misère quotient construction; a proof of the Guy--Smith--Plambeck Periodicity Theorem; and statements of some recent results and open problems in the subject.
We consider the abstract structure of the monoid M of misère impartial game values. Several new results are presented, including a proof that the group of fractions of M is almost torsion-free; a method of calculating the number of distinct games born by day 7; and some new results on the structure of prime games. Also included are proofs of a few older results due to Conway, such as the Cancellation Theorem, that are essential to the analysis but whose proofs are not readily available in the literature. Much of the work presented here was done jointly with John Conway and Dan Hoey, and I dedi…
We consider misère Nim as a normal-play game obtained from Nim by removing the terminal position. While explicit formulas are known for the Sprague-Grundy functions of Nim and Welter's game, no explicit formula is known for that of misère Nim. All three of these games can be considered as position restrictions of Nim. What are the differences between them? We point out that Nim and Welter's game are saturated, but misère Nim is not. Moreover, we present explicit formulas for the Sprague-Grundy functions of saturations of misère Nim, which are obtained from misère Nim by adjoining some moves.
This paper addresses several significant gaps in the theory of restricted misère play (Plambeck, Siegel 2008), primarily in the well-studied universe of dead-ending games, $\mathcal{E}$ (Milley, Renault 2013); if a player run out of moves in $X\in \mathcal E$, then they can never move again in any follower of $X$. A universe of games is a class of games which is closed under disjunctive sum, taking options and conjugates. We use novel results from absolute combinatorial game theory (Larsson, Nowakowski, Santos 2017) to show that $\mathcal{E}$ and the universe $\mathcal{D}\subset \mathcal{E}$ o…
We introduce and motivate the study of hypergraphical clustering games of mis-coordination. For two specific variants we prove the existence of a pure Nash equilibrium and provide bounds on the price of anarchy as a function of the cardinality of the action set and the size of the hyperedges.
We determine the misère equivalence classes of Nim positions under two equivalence relations: one based on playing disjunctive sums with other impartial games, and one allowing sums with partizan games. In the impartial context, the only identifications we can make are those stemming from the known fact about adding a heap of size 1. In the partizan context, distinct Nim positions are inequivalent.
Absolute combinatorial game theory was recently developed as a unifying tool for constructive/local game comparison (Larsson et al. 2018). The theory concerns {\em parental universes} of combinatorial games; standard closure properties are satisfied and each pair of non-empty sets of forms of the universe makes a form of the universe. Here we prove that there is an infinite number of absolute misère universes, by recursively expanding the dicot misère universe and the dead-ending universe. On the other hand, we prove that normal-play has exactly two absolute universes, namely the full space, a…
When normal and misère games are played on bi-type binary Galton-Watson trees (with vertices coloured blue or red and each having either no child or precisely $2$ children), with one player allowed to move along monochromatic edges and the other along non-monochromatic edges, the draw probabilities equal $0$ unless every vertex gives birth to one blue and one red child. On bi-type Poisson trees where each vertex gives birth to Poisson$(λ)$ offspring in total, the draw probabilities approach $1$ as $λ\rightarrow \infty$. We study such \emph{novel} versions of normal, misère and escape games on …
