We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative 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 NP ∩ coNP . The same holds for checking if such a k exists and if a given k is minimal.

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

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