Computing Perfect Stationary Equilibria in Stochastic Games

Peixuan Li, Chuangyin Dang, P.J.J. Herings

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Abstract

The notion of stationary equilibrium is one of the most crucial solution concepts in stochastic games. However, a stochastic game can have multiple stationary equilibria, some of which may be unstable or counterintuitive. As a refinement of stationary equilibrium, we extend the concept of perfect equilibrium in strategic games to stochastic games and formulate the notion of perfect stationary equilibrium (PeSE). To further promote its applications, we develop a differentiable homotopy method to compute such an equilibrium. We incorporate vanishing logarithmic barrier terms into the payoff functions, thereby constituting a logarithmic-barrier stochastic game. As a result of this barrier game, we attain a continuously differentiable homotopy system. To reduce the number of variables in the homotopy system, we eliminate the Bellman equations through a replacement of variables and derive an equivalent system. We use the equivalent system to establish the existence of a smooth path, which starts from an arbitrary total mixed strategy profile and ends at a PeSE. Extensive numerical experiments further affirm the effectiveness and efficiency of the method.
Original languageEnglish
Place of PublicationTilburg
PublisherCentER, Center for Economic Research
Number of pages41
Volume2023-006
Publication statusPublished - 20 Feb 2023

Publication series

NameCentER Discussion Paper
Volume2023-006

Keywords

  • stochastic games
  • stationary equilibria
  • perfectness
  • logarithmic barrier
  • DIFFERENTIABLE HOMOTOPY method

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