The Complexity of Infinite-Horizon General-Sum Stochastic Games

Authors Yujia Jin, Vidya Muthukumar, Aaron Sidford

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Yujia Jin
  • Stanford University, CA, USA
Vidya Muthukumar
  • Georgia Institute of Technology, Atlanta, GA, USA
Aaron Sidford
  • Stanford University, CA, USA


The authors thank Constantinos Daskalakis, Noah Golowich, and Kaiqing Zhang for kindly coordinating on uploads to arXiv. The authors also thank Aviad Rubinstein and anonymous reviewers for helpful feedback. Part of this work was conducted while the authors were visiting the Simons Institute for the Theory of Computing.

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Yujia Jin, Vidya Muthukumar, and Aaron Sidford. The Complexity of Infinite-Horizon General-Sum Stochastic Games. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 76:1-76:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the complexity of computing stationary Nash equilibrium (NE) in n-player infinite-horizon general-sum stochastic games. We focus on the problem of computing NE in such stochastic games when each player is restricted to choosing a stationary policy and rewards are discounted. First, we prove that computing such NE is in PPAD (in addition to clearly being PPAD-hard). Second, we consider turn-based specializations of such games where at each state there is at most a single player that can take actions and show that these (seemingly-simpler) games remain PPAD-hard. Third, we show that under further structural assumptions on the rewards computing NE in such turn-based games is possible in polynomial time. Towards achieving these results we establish structural facts about stochastic games of broader utility, including monotonicity of utilities under single-state single-action changes and reductions to settings where each player controls a single state.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • complexity
  • stochastic games
  • general-sum games
  • Nash equilibrium


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