An incompleteness problem in Harsanyi's general theory of games and certain related theories of non-cooperative games

Abstract

The theory of games recently proposed by John C. Harsanyi in ‘A General Theory of Rational Behavior in Game Situations’, (Econometrica, Vol. 34, No. 3) has one anomalous feature, viz., that it generates for a special class of non-cooperative games solutions which are not equilibrium points. It is argued that this feature of the theory turns on an argument concerning the instability of weak equilibrium points, and that this argument, in turn, involves appeal to an unrestricted version of a postulate subsequently included in the theory in restricted form. It is then shown that if this line of reasoning is permitted, then one must, by parity of reasoning, permit another instability argument. But, if both of these instability arguments are permitted in the construction of the theory, the resultant theory must be incomplete, in the sense that there will be simple non-cooperative games for which such a theory cannot yield solutions. This result is then generalized and shown to be endemic to all theories which have made the equilibrium condition central to the treatment of non-cooperative games. Some suggestions are then offered concerning how this incompleteness problem can be resolved, and what one might expect concerning the postulate structure and implications of a theory of games which embodies the revisions necessitated by a resolution of this problem.