Memoryless sufficiency for other objectives

Determine whether memoryless team strategies suffice for limit-sure winning under objectives other than reachability, including Büchi, co-Büchi, and parity conditions, for teams without shared randomness.

Background

The paper establishes the complexity of limit-sure reachability for teams whose members use independent randomisation and relies on the fact that memoryless team strategies are sufficient for reachability. The authors note that limit-sure winning is already well understood for Büchi, co-Büchi, and parity conditions in two-player games, but their reachability proof does not directly extend to those objectives.

The unresolved issue is whether the memoryless-strategy sufficiency property continues to hold for other objectives in the team setting without shared randomness. Establishing or refuting this property would clarify whether analogous finite-dimensional and complexity-theoretic techniques can be applied beyond reachability.