Polynomial-time algorithms for classic game-theoretic problems (parity, mean-payoff, simple stochastic, Shapley)

Determine whether parity games, mean-payoff games, Condon’s simple stochastic games, and Shapley’s stochastic games admit polynomial-time algorithms, given that these problems are known to lie in NP ∩ coNP.

Background

The paper highlights that the Tarski fixed point problem subsumes several longstanding problems in game theory and verification, notably parity games, mean-payoff games, Condon’s simple stochastic games, and Shapley’s stochastic games. These problems are central due to their applications and their unusual complexity status: they lie in NP ∩ coNP but lack known polynomial-time algorithms.

This longstanding question motivates studying the query complexity of Tarski fixed points, since advances there can inform or impact algorithms for these game-theoretic problems.

References

These problems have important applications in verification and semantics—for example, parity games are linear-time equivalent to μ-calculus model checking—and have also captivated complexity theorists due to their distinctive complexity status: they are among the few natural problems known to lie in NP ∩ coNP, yet whether {they} admit polynomial-time algorithms remains a notorious open problem.

— The Mystery Deepens: On the Query Complexity of Tarski Fixed Points  (2604.00268 - Chen et al., 31 Mar 2026) in Section 1 (Introduction)

In contrast, for a general stochastic game the number of pure stationary strategies is exponential in the number of states, and whether the value is computable in polynomial time is a long-standing open problem.

— Values of Absorbing Recursive Games Express All Real Algebraic Numbers  (2609.11583 - Asadi et al., 10 Sep 2026) in Section 7, Discussion, paragraph “Expressive power versus computational complexity”

Parity games admit quasipolynomial-time algorithms \citep{CaludeEtAl2017}, but whether they can be solved in polynomial time remains a longstanding open problem.

— Linear Programming Representations and Strongly Polynomial Algorithms for Robust Markov Decision Processes  (2610.02131 - Zhong et al., 1 Oct 2026) in Section 7.1, “Dependence on the discount factor,” following Corollary 7.1