This paper is the first to introduce an algorithm to compute stationary equilibria in stochastic games, and shows convergence of the algorithm for almost all such games. Moreover, since in general the number of stationary equilibria is overwhelming, we pay attention to the issue of equilibrium selection. We do this by extending the linear tracing procedure to the class of stochastic games, called the stochastic tracing procedure. From a computational point of view, the class of stochastic games possesses substantial difficulties compared to normal form games. Apart from technical difficulties, there are also conceptual difficulties,, for instance the question how to extend the linear tracing procedure to the environment of stochastic games. We prove that there is a generic subclass of the class of stochastic games for which the stochastic tracing procedure is a compact one-dimensional piecewise differentiable manifold with boundary. Furthermore, we prove that the stochastic tracing procedure generates a unique path leading from any exogenously specified prior belief, to a stationary equilibrium. A well-chosen transformation of variables is used to formulate an everywhere differentiable homotopy function, whose zeros describe the (unique) path generated by the stochastic tracing procedure. Because of differentiability we are able to follow this path using standard path-following techniques. This yields a globally convergent algorithm that is easily and robustly implemented on a computer using existing software routines. As a by-product of our results, we extend a recent result on the generic finiteness of stationary equilibria in stochastic games to oddness of equilibria.
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Paper provided by Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization in its series Research Memoranda with number
004.
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Herings,P. Jean-Jacques & Peeters,Ronald J.A.P., 2001.
"Equilibrium Selection in Stochastic Games,"
Research Memoranda
009, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization.
[Downloadable!]
Other versions:
Peeters,Ronald & Schinkel,Maarten Pieter & Herings,P. Jean-Jacques, 2001.
"Intertemporal Market Divison,"
Research Memoranda
011, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization.
[Downloadable!]
Frank H. Page, Jr., Myrna H. Wooders, 2009.
"Endogenous Network Dynamics,"
Caepr Working Papers
2009-002, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington.
[Downloadable!]