Convex vNM–Stable Sets for a Semi Orthogonal Game. Part I: epsilon–Relevant Coalitions

We consider (cooperative) linear production games with a continuum of players. The coalitional function is generated by r + 1 "production factors" that is, non atomic measures defined on an interval. r of these are orthogonal probabilities which, economically, can be considered as "cornered" production factors. The r+1th measure involved has positive mass "across the carriers" of the orthogonal probabilities. That is, there is a "non–cornered" (or "central") production factor available throughout the market. We consider convex vNM–Stable Sets of this game. Depending on the size of the central measure, we observe cases in which a vNM–Stable Set is uniquely defined to be either the core or the convex hull of the core plus a unique additional imputation. We observe other situations in which a variety of vNM–Stable Sets exists. Within this first part we will present the coalitions that are necessary and sufficient for dominance relations between imputations. In the context of the "purely orthogonal" production game this question is answered in a rather straightforward way by the "Inheritance Theorem" established in [3]. However, once orthogonality is abandoned one has to establish prerequisites about epsilon–relevant coalitions. Thus, this first part centers around the formulation of a generalized "Inheritance Theorem". As a consequence, based on the Inheritance Theorem, we provide conditions for the core to be a vNM–Stable Set whenever the central commodity is available in abundance.