Sigma-algebra structure of unravelable payoff sets

Determine whether, for a fixed game tree T on a set A, the collection of payoff sets P ⊆ [T] for which the Gale–Stewart game (T, P) is unravelable (meaning that for every k ∈ ℕ there exists a k-covering (π, ·): T′ → T with π−1(P) clopen in [T′]) forms a σ-algebra on [T], i.e., is closed under complementation and countable unions.

Background

The paper introduces Martin’s notion of unravelability for Gale–Stewart games (T, P): for each k, there must exist a k-covering (π, ·): T′ → T such that π−1(P) is clopen. It also defines universal unravelability, which requires unravelability to persist under all coverings T′ → T.

A key result proved in the paper is that, for a fixed tree T, the universally unravelable payoff sets form a σ-algebra. In contrast, Martin’s original argument proceeds via unravelability alone and a transfinite induction over the Borel hierarchy.

The authors note that it is unclear whether the class of unravelable payoff sets (as opposed to universally unravelable ones) also forms a σ-algebra, which would parallel their σ-algebra result for universal unravelability.