Convex vNM-Stable Sets for a Semi Orthogonal Game. Part IV: Large Economies: The Existence Theorem

Abstract

Within this paper we establish the existence of a vNM-Stable Set for (cooperative) linear games with a continuum of players. The coalitional function is generated by r+1 production (non atomic measures). r factors are given by orthogonal probabilities (cornered factors) establishing the core of the game. Factor r+1 (the centralized facotr) is represented by a nonantomic measure with carrier across the corners of the market; i.e., this factor is more abundantly available and the representing measure is not located within the core of the game. The present paper continues a series of presentations of this topic, for Part I, II, III see I1I,I2I,I3I. We focus on convex vNM-Stable Sets of the game and we present an existence theorem valid for Large Economies (the termin is not quite orthodox). There are some basic assumptions for the present model which enable us to come up with a comprehensive version of an existence theorem. However, in order to make our presentation tractable (and readable) we wisely restrict ourselves to a simplified model. As in our previous models there is a (not necessarily unique) imputation outside the core such that the vNM-Stable Set is the convex hull of this imputation and the core. Significantly, this additional imputation can be seen as a truncation of the centralized distribution, i.e., the r+1^st factor. Hence there is a remarkable similarity mutatis mutandis regarding the Characterization Theorem that holds true for the purely orthogonal case (I4I,I5I).