Dual Selection Games

TL;DR

This paper explores the concept of duality in selection games, establishing a general theorem that relates winning strategies in dual pairs of such games through coinitiality conditions.

Contribution

It introduces a broad theorem linking dual selection games via coinitiality, extending known dualities like the Rothberger and point-open games.

Findings

Established a general duality theorem for selection games.

Connected specific examples like Rothberger and point-open games.

Provided conditions under which strategies in dual games correspond.

Abstract

Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if $\{\operatorname{ran}{f}:f\in\mathbf C(\mathcal R)\}$ is coinitial in $\mathcal A$ with respect to $\subseteq$, where $\mathbf C(\mathcal R)=\{f\in(\bigcup\mathcal R)^{\mathcal R}:R\in\mathcal R\Rightarrow f(R)\in R\}$ collects the choice functions on the set $\mathcal R$, then $G_1(\mathcal A,\mathcal B)$ and $G_1(\mathcal R,\neg\mathcal B)$ are dual selection games.