Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games

CentER Discussion Paper Series No. 1998-4

30 Pages Posted: 2 Dec 1998

Date Written: 1998

Abstract

Harsanyi and Selten (1988) have proposed a theory of equilibrium selection that selects a unique Nash equilibrium for any non-cooperative N-person game. The heart of their theory is given by the tracing procedure, a mathematical construction that adjusts arbitrary prior beliefs into equilibrium beliefs. The tracing procedure plays an important role in the definition of risk-dominance for Nash equilibria. Although the term "procedure" suggests a numerical approach, the tracing procedure itself is a non-constructive method. In this paper we propose a homotopy algorithm that generates a path of strategies. By employing lexicographic pivoting techniques it can be shown that for the entire class of non-cooperative N-person games the path converges to an approximate Nash equilibrium, even when the starting point or the game is degenerate. The outcome of the algorithm is shown to be arbitrarily close to the beliefs proposed by the tracing procedure. Therefore, the algorithm does not compute just any Nash equilibrium, but one with a sound gametheoretic underpinning. Like other homotopy algorithms, it is easily implemented on a computer. To show our results we apply methods from the theory of simplicial algorithms and algebraic geometry.

JEL Classification: C63, C72

Suggested Citation

Herings, P. Jean-Jacques and van den Elzen, Antoon H., Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games (1998). CentER Discussion Paper Series No. 1998-4, Available at SSRN: https://ssrn.com/abstract=138083 or http://dx.doi.org/10.2139/ssrn.138083

P. Jean-Jacques Herings (Contact Author)

Tilburg University ( email )

Department of Econometrics and Operations Research
P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 4668797 (Phone)
5000 LE (Fax)

HOME PAGE: http://https://sites.google.com/view/jean-jacques-herings/home

Antoon H. Van den Elzen

Tilburg University, CentER ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 466 3178 (Phone)
+31 13 466 3019 (Fax)