A Social Coalitional Equilibrium Existence Lemma

Abstract

The purpose of this paper is to construct an abstract model of a society in which each member can cooperate with others by forming a coalition, but at the same time can be influenced by the members outside the coalition. A new concept of equilibrium, called here a social coalitional equilibrium, is proposed, and a sufficient condition for its existence provided. The social coalitional equilibrium may be considered a synthesis of the Nash equilibrium (a noncooperative solution concept) and the core (a cooperative solution concept). The paper provides a broadly applicable mathematical tool for proving existence of equilibrium for models of labor-managed market economies. THE PURPOSE OF THE PRESENT PAPER is to construct an abstract model of a society in which each member can cooperate with others by forming a coalition, but at the same time can be influenced by the members outside the coalition. A new concept of equilibrium, called here a social coalitional equilibrium, is proposed, and a sufficient condition for its existence provided. A coalition structure is endogenously determined in equilibrium. When the model is suitably specified, it describes noncooperative behavior of the members, and the social coalitional equilibrium is reduced to the Nash equilibrium. Given another extreme specification, the model is reduced to a cooperative game without sidepayments in characteristic function form, and the set of social coalitional equilibria corresponds precisely to the core. The need for study of the present model arose originally out of the author's recent investigation of labor-managed market economies, both the socialistic version and the capitalistic version, in [10, 11, 12]. Indeed, the model and the concept of social coalitional equilibrium were obtained by abstracting and generalizing the formal aspect of those studies. The socialistic labor-managed market economy has long been the object of study of various authors; see, e.g., Vanek [19]. The capitalistic labor-managed market economy seems to reflect the institutional aspect of the modern capitalistic society better than the economies formulated by other available general equilibrium models. It differs, for example, from the standard private ownership economy of Arrow and Debreu [1] (see also Debreu [6, Sec. 5.5]) in that (i) the ownership and control of a firm are separated; (ii) no particular firm is given a priori, but instead, the list of potential firms is given as a primitive datum; and (iii) the behavioral principle of the economic agents involves both noncooperative behavior and cooperative behavior. Because of the above feature (ii), any general equilibrium model of this economy has to formulate the economic mechanism that underlies the formation of firms in equilibrium. As