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On the Complexity of Computing Markov Perfect Equilibrium in General-Sum Stochastic Games

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Xiaotie Deng, Ningyuan Li, David Mguni, Jun Wang, Yaodong Yang

Computer Science and Game Theory · Submitted 4 Sep 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we derive that computing an approximate Markov Perfect Equilibrium (MPE) in a finite-state discounted Stochastic Game within the exponential precision is \textbf{PPAD}-complete.

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arXiv ID2109.01795 (version 2)
Pith IDLZY4IKEC, Pith's permanent record of this paper and its review
CitationDavid Mguni, Jun Wang, Ningyuan Li, Xiaotie Deng, Yaodong Yang. "On the Complexity of Computing Markov Perfect Equilibrium in General-Sum Stochastic Games." arXiv:2109.01795 (2021). https://pith.science/paper/LZY4IKEC
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Similar to the role of Markov decision processes in reinforcement learning, Stochastic Games (SGs) lay the foundation for the study of multi-agent reinforcement learning (MARL) and sequential agent interactions. In this paper, we derive that computing an approximate Markov Perfect Equilibrium (MPE) in a finite-state discounted Stochastic Game within the exponential precision is \textbf{PPAD}-complete. We adopt a function with a polynomially bounded description in the strategy space to convert the MPE computation to a fixed-point problem, even though the stochastic game may demand an exponential number of pure strategies, in the number of states, for each agent. The completeness result follows the reduction of the fixed-point problem to {\sc End of the Line}. Our results indicate that finding an MPE in SGs is highly unlikely to be \textbf{NP}-hard unless \textbf{NP}=\textbf{co-NP}. Our work offers confidence for MARL research to study MPE computation on general-sum SGs and to develop fruitful algorithms as currently on zero-sum SGs.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Similar to the role of Markov decision processes in reinforcement learning, Stochastic Games (SGs) lay the foundation for the study of multi-agent reinforcement learning (MARL) and sequential agent interactions. background 1.00 0.04 0.02 asserted 1.00
s1 In this paper, we derive that computing an approximate Markov Perfect Equilibrium (MPE) in a finite-state discounted Stochastic Game within the exponential precision is \textbf{PPAD}-complete. result 0.96 0.96 0.92 asserted 1.00
s2 We adopt a function with a polynomially bounded description in the strategy space to convert the MPE computation to a fixed-point problem, even though the stochastic game may demand an exponential number of pure strategies, in the number of states, for each agent. method 0.99 0.92 0.05 asserted 0.96
s3 The completeness result follows the reduction of the fixed-point problem to {\sc End of the Line}. method 0.90 0.82 0.07 asserted 1.00
s4 Our results indicate that finding an MPE in SGs is highly unlikely to be \textbf{NP}-hard unless \textbf{NP}=\textbf{co-NP}. interpretation 0.94 0.81 0.12 hedged possibility 0.98
s5 Our work offers confidence for MARL research to study MPE computation on general-sum SGs and to develop fruitful algorithms as currently on zero-sum SGs. interpretation 0.76 0.25 0.07 hedged possibility 0.72

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Original abstract

Similar to the role of Markov decision processes in reinforcement learning, Stochastic Games (SGs) lay the foundation for the study of multi-agent reinforcement learning (MARL) and sequential agent interactions. In this paper, we derive that computing an approximate Markov Perfect Equilibrium (MPE) in a finite-state discounted Stochastic Game within the exponential precision is \textbf{PPAD}-complete. We adopt a function with a polynomially bounded description in the strategy space to convert the MPE computation to a fixed-point problem, even though the stochastic game may demand an exponential number of pure strategies, in the number of states, for each agent. The completeness result follows the reduction of the fixed-point problem to {\sc End of the Line}. Our results indicate that finding an MPE in SGs is highly unlikely to be \textbf{NP}-hard unless \textbf{NP}=\textbf{co-NP}. Our work offers confidence for MARL research to study MPE computation on general-sum SGs and to develop fruitful algorithms as currently on zero-sum SGs.

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  • Like us, the authors of [ref] also use the Brouwer function $f_$ (more commonly known as Nash's Brouwer function and originally designed for linear utilities); however, unlike us, they do not establish or use any special pseudo-linear structure on the value functions.

    The Complexity of Infinite-Horizon General-Sum Stochastic Games · Complexity of NE in $$s · 0.97

Cites it as background 6 papers
  • A well-known solution concept that describes the equilibrium of NF games is Nash Equilibrium (NE) [ref] .

    Measuring the Non-Transitivity in Chess · Preliminaries and Notations · 0.99

  • Solving Nash equilibrium (NE) [ref] in multi-player games has become a central interest in a variety of fields including but not limited to economics, computer science and artificial intelligence.

    On the Convergence of Fictitious Play: A Decomposition Approach · Introduction · 0.98

  • When both agents achieve zero exploitability, they reach a Nash equilibrium (NE) [ref] , a classical solution concept from Game Theory [ref] .

    Neural Auto-Curricula · Introduction · 0.99

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