Game matching number of graphs

We study a competitive optimization version of ${\alpha }^{\mathrm{\prime }}(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted $\text{Max}(G)$ and $\text{Min}(G)$, respectively. We show that always $\mathrm{\mid }\text{Max}(G)-\text{Min}(G)\mathrm{\mid }\le 1$. We obtain a sufficient condition for $\text{Max}(G)={\alpha }^{\mathrm{\prime }}(G)$ that is preserved under cartesian product. In general, $\text{Max}(G)\ge \frac{2}{3}{\alpha }^{\mathrm{\prime }}(G)$, with equality for many split graphs, while $\text{Max}(G)\ge \frac{3}{4}{\alpha }^{\mathrm{\prime }}(G)$ when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, $\text{Max}(G)\ge n\mathrm{/}3$, and there are such examples with $\text{Max}(G)\le 7n\mathrm{/}18$. For an $n$-vertex path or cycle, the answer is roughly $n\mathrm{/}7$.