A Non-cooperative Equilibrium for Supergames: A Correction - Friedman, James W. (Review of Economic Studies, 1973)
The purpose of this note is to correct an error in my paper [1]. Under the assumptions of the paper, Proposition 3 is not, in general, true. The point at which the proof goes awry is in the use of the mapping n. n is treated in the paper as if it were a function (i.e. a point to point mapping); whereas it is in fact a correspondence (a point to set mapping). n maps points of the unit simplex into itself, using a subset of the Pareto optimal set to obtain the simplex. A given point in the payoff space may be the image of more than one point in the strategy space. Each strategy, s, which maps into a given point in the Pareto optimal set (i.e. which has a given p associated with it in the simplex), can map into a distinct (j in the simplex. Thus the Brouwer theorem cannot be used. While n is surely upper semi-continuous, it need not have convex image sets, ruling out use of the Kakutani fixed point theorem. Furthermore, it need not be lower semi-continuous, ruling out another route to the use of the Brouwer theorem. That route is to use the results of E. Michael [2], which establish that if n is lower semi-continuous, then it has a selection which is a continuous function from the unit simplex into itself. The upshot is that the axioms of [1] must be somehow strengthened in one of the