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∃ℝ-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games

Authors Kristoffer Arnsfelt Hansen , Steffan Christ Sølvsten

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Kristoffer Arnsfelt Hansen
  • Aarhus University, Denmark
Steffan Christ Sølvsten
  • Aarhus University, Denmark

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Kristoffer Arnsfelt Hansen and Steffan Christ Sølvsten. ∃ℝ-Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 45:1-45:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is ∃ℝ-complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is ∃ℝ-complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Exact and approximate computation of equilibria
  • Existential Theory of the Reals
  • Stationary Nash Equilibrium
  • Perfect Information Stochastic Games


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