A mis\`ere play $\star$-operator

TL;DR

This paper extends the $ ext{ extsterling}$-operator concept to misère-play in impartial vector subtraction games, revealing more structure and proving convergence and other properties, thus advancing understanding of game theory under different play conventions.

Contribution

It introduces the misère-play version of the $ ext{ extsterling}$-operator and demonstrates its convergence and structural properties, expanding prior normal-play analyses.

Findings

More structure under misère-play compared to normal-play

Proved convergence of the misère $ ext{ extsterling}$-operator

Established properties of the misère version

Abstract

We study the $\star$-operator (Larsson et al. 2011) of impartial vector subtraction games (Golomb 1965). Here we extend the notion to the mis\`ere-play convention, and prove convergence and other properties; notably more structure is obtained under mis\`ere-play as compared with the normal-play convention (Larsson 2012).