The Logarithmic Stochastic Tracing Procedure: A Homotopy Method for Computing and Selecting Stationary Equilibria of Stochastic Games

32 Pages Posted: 12 Feb 2021 Last revised: 26 Apr 2021

See all articles by Steffen Eibelshäuser

Steffen Eibelshäuser

Goethe University Frankfurt, Department of Management and Applied Microeconomics

Victor Klockmann

Goethe University Frankfurt; Max Planck Society for the Advancement of the Sciences - Center for Humans and Machines

David Poensgen

Goethe University Frankfurt, Department of Management and Applied Microeconomics

Alicia von Schenk

Goethe University Frankfurt; Max Planck Society for the Advancement of the Sciences - Center for Humans and Machines

Date Written: December 14, 2020

Abstract

This paper presents the logarithmic stochastic tracing procedure, a homotopy method for the computation and selection of stationary equilibria of any finite discounted stochastic game. It generalizes both the logarithmic tracing procedure (Harsanyi and Selten, 1988), which is defined only for normal form games, and the linear stochastic tracing procedure (Herings and Peeters, 2004), which is guaranteed to be well-defined only for generic games. Similar in spirit, our method defines a family of auxiliary games from prior beliefs; a path of equilibria of these is traversed until an equilibrium of the original game is reached. Harsanyi and Selten interpret this process as one of strategic Bayesian reasoning, in which priors are gradually transformed into equilibrium beliefs. This interpretation also applies to our procedure, making it a suitable tool for equilibrium selection.

Because existence of a smooth, interior, and isolated solution path is guaranteed, the present algorithm is well-suited for the computation of stationary equilibria via numerical continuation methods. A ready-to-use implementation is publicly available; we report computational performance in this paper.

Keywords: Stochastic game, Tracing, Homotopy method, Equilibrium computation, Equilibrium selection

JEL Classification: C73, C63

Suggested Citation

Eibelshäuser, Steffen and Klockmann, Victor and Poensgen, David and von Schenk, Alicia, The Logarithmic Stochastic Tracing Procedure: A Homotopy Method for Computing and Selecting Stationary Equilibria of Stochastic Games (December 14, 2020). Available at SSRN: https://ssrn.com/abstract=3748830 or http://dx.doi.org/10.2139/ssrn.3748830

Steffen Eibelshäuser (Contact Author)

Goethe University Frankfurt, Department of Management and Applied Microeconomics ( email )

Germany

Victor Klockmann

Goethe University Frankfurt ( email )

Theodor-W.-Adorno-Platz 4
Frankfurt am Main, 60323
Germany

Max Planck Society for the Advancement of the Sciences - Center for Humans and Machines ( email )

Lentzeallee 94
Berlin, 14195
Germany

David Poensgen

Goethe University Frankfurt, Department of Management and Applied Microeconomics ( email )

Germany

Alicia Von Schenk

Goethe University Frankfurt ( email )

Theodor-W.-Adorno-Platz 4
Frankfurt am Main, 60323
Germany

Max Planck Society for the Advancement of the Sciences - Center for Humans and Machines ( email )

Lentzeallee 94
Berlin, 14195
Germany

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