Equilibrium Selection in Stochastic Games

Game theory, stochastic games, Equilibrium selection, Linear tracing procedure, Correlated beliefs

Equilibrium Selection in Stochastic Games.

In this paper a selection theory for stochastic games is developed. The theory itself is based on the ideas of Harsanyi and Selton to select equilibria for games in standard form. We introduce several possible definitions for the stochastic tracing procedure, an extension of the linear tracing procedure to the class of stochastic games. We analyze the properties of these alternative definitions. We show that exactly one of the proposed extensions ois consistent with the formulation of Harsanyi-Selten for games in standard form and captures stationarity.microeconomics ;

Stationary Equilibria in Stochastic Games: Structure, Selection, and Computation

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.mathematical economics and econometrics ;

Heterogeneous Interacting Economic Agents and Stochastic Games

Stochastic games offer a rich mathematical structure that makes it possible to analyze situations with heterogeneous and interacting economic agents. Depending on the actions of the economic agents, the economic environment changes from one period to another. We focus on stationary equilibrium, the simplest form of behavior that is consistent with rationality. Since the number of stationary equilibria abound, we present the stochastic tracing procedure, a method to select equilibria. Since stationary equilibria are difficult to characterize analytically, we also present a numerical algorithm by which they can be computed. The algorithm is constructed in such a way that the equilibrium selected by the stochastic tracing procedure is computed. We illustrate the usefulness of this approach by showing how it leads to new insights in the theory of dynamic oligopoly.microeconomics ;

symposium articles: A differentiable homotopy to compute Nash equilibria of n -person games

The literature on the computation of Nash equilibria in n-person games is dominated by simplicial methods. This paper is the first to introduce a globally convergent algorithm that fully exploits the differentiability present in the problem. It presents an everywhere differentiable homotopy to do the computations. The homotopy path can therefore be followed by several numerical techniques. Moreover, instead of computing some Nash equilibrium, the algorithm is constructed in such a way that it computes the Nash equilibrium selected by the tracing procedure of Harsanyi and Selten. As a by-product of our proofs it follows that for a generic game the tracing procedure defines a unique feasible path. The numerical performance of the algorithm is illustrated by means of several examples.Computation of equilibria - Noncooperative game theory - Tracing procedure.