$F_\sigma$ Games and Reflection in $L(\mathbb{R})$

TL;DR

This paper establishes a deep connection between the determinacy of certain $F_\sigma$ games of length $2$ and the existence of a specific transitive model of set theory with reflection properties, linking game theory and inner model theory.

Contribution

It proves that $F_\sigma$ game determinacy of length $2$ is equivalent to the existence of a transitive model of KP + AD with reflection of $1$ facts about the next admissible set.

Findings

Determinacy of $F_\sigma$ games of length $2$ is equivalent to a model existence statement.

Identifies a specific model of KP + AD with reflection properties.

Bridges the gap between game determinacy and inner model reflection principles.

Abstract

It is shown that determinacy of $F_\sigma$ games of length $\omega^2$ is equivalent to the existence of a transitive model of KP + AD which contains the reals and reflects $\Pi_1$ facts about the next admissible set.