LP Games with Committee Control:behavior of the core under direct market replication

Abstract

Some optimization situations, in which several independent decision makers are involved, can be modeled as cooperative games with side payments. This is the case of those economic production situations in which economic agents pool resources to produce finished goods which can be sold at a given market price. Since Owen (1975) defined linear production games arising from situations in which the production process is linear, several generalizations of the original model have been proposed. Generalizations of Owen�s model are given in Dubey and Shapley (1984), Granot (1986) and Curiel, Derks and Tijs (1989). Owen (1975) makes use of the duality theory of linear programming to obtain equilibrium price vectors and to prove the non-emptiness of the core. He shows that in linear production games, a part of the core elements can be found by using shadow prices for the resources. When each player receives a payoff that equals the value of his/her resource bundle under the shadow price, this forms a core element in the production game. Moreover he shows that, under replication, in the non-degenerated case, the core converges finitely to the set of competitive equilibria. Afterwards, Samet and Zemel (1984) obtain a necessary and sufficient condition for finite convergence, under the equivalent approach of refinement. Here, we consider the generalization proposed by Curiel et al. (1989): linear production games with committee control. In this model, it is assumed that resources are controlled by committees of players. Now, each coalition has access to certain parts of the total resource bundle. The access to each of these parts is governed by a simple game. When each of these simple games have a non-empty core, there is a possibility to assign a part of the resource bundle to each of the players by using core allocations of each of the control games and to assign to each player the shadow value of the part of the resource bundle he/she obtained. This payoff vector is once again a core allocation of the associated production game. Following Owen�s previous work, our goal is to study the limiting behaviour of the economy when the number of players of each type increases. To be specific, the central problem we tackle in this paper is the convergence of the set sequence composed by the core of the LP-games obtained when the player set is successively refined. We concentrate on the relation between the limit of the cores of the r-refinements and the set of corecompetitive- equilibria or shadow-price-based core allocations, which is defined as the set of all payoff vectors obtained by the process described before. First, we face the problem of extending the Curiel et al. model, i.e., we describe how to define the LP-game with committee control which arises when the player set is refined (or equivalently replicated). The refinement we have considered is based on the definition of a �direct market generated by a game� introduced by Shapley and Shubik (1969) and it is suitable when players are activity owners. Then, we proceed on the analysis of the limiting behaviour of the core. We give necessary conditions for convergence, as well as for finite convergence, to the set of core-competitive-equilibria.