The tracing procedure: A Bayesian approach to defining a solution forn-person noncooperative games

Abstract

The paper proposes a Bayesian approach to selecting a particular equilibrium points ^* of any given finiten-person noncooperative game Γ as solution for Γ. It is assumed that each playeri starts his analysis of the game situation by assigning a subjective prior probability distributionp _j to the set of all pure strategies available to each other playerj. (The prior distributionsp _j used by all other playersi in assessing the likely strategy choice of any given playerj will be identical, because all these playersi will compute this prior distributionp _j from the basic parameters of game Γ in the same way.) Then, the players are assumed to modify their subjective probability distributionsp _j over each other's pure strategies systematically in a continuous manner until all of these probability distributions will converge, in an appropriate sense, to a specific equilibrium points ^* of Γ, which, then, will be accepted as solution.

A mathematical procedure, to be called thetracing procedure, is proposed to provide a mathematical representation for this intellectual process of convergent expectations. Two variants of this procedure are described. One, to be called thelinear tracing procedure, is shown to define a unique solution in “almost all” cases but not quite in all cases. The other variant, to be called thelogarithmic tracing procedure, always defines a unique solution in all possible cases. Moreover, in all cases where the linear procedure yields a unique solution at all, both procedures always yield the same solution. For any given game Γ, the solution obtained in this way heavily depends on the prior probability distributionsp ₁,...,p _n used as a starting point for the tracing procedure. In the last section, the results of the tracing procedure are given for a simple class of two-person variable-sum games, in numerical detail.