Intersection Games and Bernstein Sets

TL;DR

This paper explores the determinacy of classical intersection games on the real line when the target set is a Bernstein set, highlighting the impact of non-measurable sets on game outcomes.

Contribution

It analyzes the determinacy of key intersection games specifically for Bernstein sets, connecting set theory and game theory in a novel way.

Findings

Determinacy results depend on the set's construction

Bernstein sets lead to non-determined games

Links between axiom of choice and game outcomes

Abstract

The Banach-Mazur game, Schmidt's game and McMullen's absolute winning game are three quintessential intersection games. We investigate their determinacy on the real line when the target set for either player is a Bernstein set, a non-Lebesgue measurable set whose construction depends on the axiom of choice.