Various Diamond Properties in Combinatorial Game Theory

TL;DR

This work addresses when combinatorial game positions admit simple algebraic values by introducing a unified framework, the $\Diamond_\mathcal{A}$-property with $\mathcal{A}\in\{\mathbb{Z},\mathbb{D}\}$, for sets closed under options. It unifies and extends existing diamond-type properties, showing under mild conditions that values are restricted to $\mathbb{Z}$, $\mathbb{D}$, or integer/number pairs $\{x|y\}$, via Translation and Simplicity principles applied in the Conway algebra. The authors connect these results to several diamond variants and demonstrate that Yashima game on bipartite graphs yields only integer-pair values, highlighting a case where complex game trees collapse to simple expressions. This advances the algebraic toolkit for analyzing impartial/partizan games and offers practical simplifications for evaluating game positions, with potential algorithmic benefits. $

Abstract

We investigate conditions under which positions in combinatorial games admit simple values. We introduce a unified diamond framework, the $\Diamond_A$-property ($A\in\{\mathbb{Z},\mathbb{D}$), for sets of positions closed under options. Under certain conditions, this framework guarantees that all values are integers, dyadic rationals, or pairs $\{m|n\}$ (on $\mathbb{Z}$ or $\mathbb{D}$). As an application, we establish that every position in \textsc{Yashima} game on bipartite graphs has an integer pair value.

Figures (6)

• Figure 1: Yashima game
• Figure 2: Diamond property
• Figure 3: $\Diamond_\mathbb{Z}$-property and $\Diamond_\mathbb{D}$-property
• Figure 4: Diamond properties in Definition \ref{['def:dia-Zs']}
• Figure 5: Diamond properties in Definition \ref{['def:dia-Ds']}

Theorems & Definitions (24)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3: Translation Theorem
• Proposition 2.4: Simplicity Theorem
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9
• Definition 2.10