Simple Stochastic Games with Almost-Sure Energy-Parity Objectives are in NP and coNP

We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative $$\omega $$ ω -regular condition: The maximizer aims to avoid running out of energy while simultaneously satisfying a parity condition. We show that the corresponding almost-sure problem, i.e., checking whether there exists a maximizer strategy that achieves the energy-parity objective with probability 1 when starting at a given energy level k, is decidable and in $$\mathsf {NP}\cap \mathsf {coNP}$$ NP ∩ coNP . The same holds for checking if such a k exists and if a given k is minimal.