Various Diamond Properties in Combinatorial Game Theory

We investigate conditions under which positions in combinatorial games admit simple values. We introduce a unified diamond framework, the ${\mathrm{◊}}_A$-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 $\{m\mathrm{\mid }n\}$ (on $\mathbb{Z}$ or $\mathbb{D}$). As an application, we establish that every position in $\text{textsc}Yashima$ game on bipartite graphs has an integer pair value.