$\mathcal{L}\mathcal{R}$-Ending partisan rulesets
Hiroki Inazu, Shun-ichi Kimura, Koki Suetsugu
TL;DR
This work introduces $\mathcal{L}\mathcal{R}$-ending partisan rulesets, a new combinatorial game framework where two terminal positions determine the winner despite identical option sets for both players. It develops a formal algebraic apparatus—positions, disjunctive sums, outcome classes, equivalence, conjugation, and a partial order—and builds a nim-like value theory with $*L_n$, $*R_n$, $\maltese_n$, and $\bigstar_n$ that generalizes to sums via mex and nim-sum-like rules. The authors provide concrete value constructions and simplification techniques, proving that many positions reduce to canonical representatives and that sums exhibit predictable behavior akin to Sprague-Grundy theory in an LR setting. They apply this framework to Even nim and LR-ending subtraction games (notably with $S=\{2,5\}$ and general $S$ sets), uncovering periodic value patterns and LR-quotients that enable efficient outcome analysis. Overall, the paper lays foundational tools for analyzing nonstandard partisan games under two-terminal conventions and reveals a rich algebraic structure analogous to classic CGT, with potential for broad application and extension.
Abstract
In this paper, we consider $\mathcal{L}\mathcal{R}$-ending partisan rulesets as a branch of combinatorial game theory. In these rulesets, the sets of options of both players are the same. However, there are two kinds of terminal positions. If the game ends in one of the terminal positions, then a player wins and if the game ends in the other terminal position, the other player wins. We introduce notations for positions in $\mathcal{L}\mathcal{R}$-ending partisan rulesets and show their algebraic structures. We also introduce some examples of $\mathcal{L}\mathcal{R}$-partisan rulesets and show how our results can be used for analyzing the rulesets.