Game-theoretic variants of cardinal invariants

TL;DR

This paper explores game-theoretic versions of key cardinal invariants of the continuum, analyzing strategies and characterizations for invariants like the reaping, bounding, dominating, and null ideal additivity numbers.

Contribution

It introduces and studies game-theoretic variants of several classical cardinal invariants, providing new characterizations and insights into their strategic properties.

Findings

Characterization of winning strategies for Player I and II in tallness games.

New relationships between game-theoretic variants and classical invariants.

Insights into the structure of ideals on alculus.

Abstract

We investigate game-theoretic variants of cardinal invariants of the continuum. The invariants we treat are the reaping number $\mathfrak{r}$, the bounding number $\mathfrak{b}$, the dominating number $\mathfrak{d}$, and the additivity number of the null ideal $\operatorname{add}(\mathsf{null})$. We also consider games, called tallness games, defined according to ideals on $\omega$ and characterize that each of Player I and Player II has a winning strategy.