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

42 Pages Posted: 12 Feb 2021 Last revised: 10 Jul 2022

See all articles by Steffen Eibelshäuser

Steffen Eibelshäuser

Goethe University Frankfurt, Faculty of Economics and Business Administration, Department of Management and Applied Microeconomics

Victor Klockmann

University of Würzburg - Business Administration & Economics; Max Planck Institute for Human Development - Center for Humans and Machines

David Poensgen

Goethe University Frankfurt, Department of Management and Applied Microeconomics

Alicia von Schenk

University of Würzburg - Business Administration & Economics; Max Planck Institute for Human Development - Center for Humans and Machines

Date Written: July 9, 2022

Abstract

This paper develops the logarithmic stochastic tracing procedure for finite discounted stochastic games. 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 that gradually transform priors into equilibrium beliefs. Harsanyi and Selten interpret this as a process of strategic Bayesian reasoning, making it a well-founded tool for equilibrium selection. We prove two main results: First, there exists a unique solution path which is smooth, interior, isolated, and of finite length. Second, the limiting solution path is consistent with the linear procedure whenever the latter is well-defined. Our method allows the computation of stationary equilibria of any stochastic game via homotopy continuation; a ready-to-use implementation is publicly available.

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 (July 9, 2022). Available at SSRN: https://ssrn.com/abstract=3748830 or http://dx.doi.org/10.2139/ssrn.3748830

Steffen Eibelshäuser (Contact Author)

Goethe University Frankfurt, Faculty of Economics and Business Administration, Department of Management and Applied Microeconomics ( email )

Germany

Victor Klockmann

University of Würzburg - Business Administration & Economics ( email )

Sanderring 2
Wuerzburg, 97070
Germany

Max Planck Institute for Human Development - 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

University of Würzburg - Business Administration & Economics ( email )

Sanderring 2
Wuerzburg, D-97070
Germany

Max Planck Institute for Human Development - Center for Humans and Machines ( email )

Berlin
Germany

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