Partizan Kayles and Misere Invertibility

TL;DR

This paper provides a complete analysis of Partizan Kayles under misere play, revealing complex invertibility properties and contributing to the understanding of misere combinatorial game theory.

Contribution

It develops a full solution including the misere monoid for Partizan Kayles and explores its implications for misere invertibility and game theory.

Findings

Complete solution for Partizan Kayles under misere play

Identification of positions with non-standard invertibility properties

Insights into the structure of misere invertible games

Abstract

The impartial combinatorial game Kayles is played on a row of pins, with players taking turns removing either a single pin or two adjacent pins. A natural partizan variation is to allow one player to remove only a single pin and the other only a pair of pins. This paper develops a complete solution for "Partizan Kayles" under misere play, including the misere monoid all possible sums of positions, and discusses its significance in the context of misere invertibility: the universe of Partizan Kayles contains a position whose additive inverse is not its negative, and, moreover, this position is an example of a right-win game whose inverse is previous-win.