Convex vNM–Stable Sets for a Semi Orthogonal Game. Part V: All Games have vNM–Stable Sets

Within this paper we conclude the treatise of vNM-Stable Sets for (cooperative) linear production games with a continuum of players. The paper resumes a series of presentations of this topic, for Part I, II, III, IV, see IMW 483, IMW 498, IMW 500, IMW 534. The framework has been outlined previously. The coalitional function is generated by r+1 "production factors'' (non atomic measures). r factors are given by orthogonal probabilities ("cornered'' production factors) establishing the core of the game. Factor r+1 (the "central'' production factor) is represented by a nonantomic measure with carrier "across the corners'' of the market. I.e., this factor is available in excess and the representing measure is no element of the core of the game. Generalizing our set-up, we assume now that the "central'' production factor is represented by an arbitrary measure not necessarily of step function character. Then the existence theorem is achieved by an approximation procedure. Again it turns out that there is a (not necessarily unique) imputation outside of the core which, together with the core generates the vNM-Stable Set as the convex hull. Significantly, this additional imputation can be seen as a truncation of the "central'' distribution, i.e., the r+1st production factor. Hence, there is a remarkable similarity mutatis mutandis regarding the Characterization Theorem that holds true for the "purely orthogonal case'' Rosenmüller and Shitoviz (2000). This justifies to use the term "Standard vNM-Stable Set'' in the presence of a central production factor.