A Generalized Assignment Game

Resumen

We study the interaction of a finite number of sellers and buyers in a market. The fact that agents in these markets belong, from the outset, to one of two disjoint sets, sellers and buyers, and the bilateral nature of exchange that exists, allows us to treat them as �two-sided matching markets�. The proposed game is a natural extension of the Assignmnet Game (Shapley and Shubik, 1972) to the case where each seller owns a set of possibly different objects instead of only one indivisible object, and each buyer wants to buy at most one object. Sotomayor (1999) studies a special case of our model where all the objects a seller owns are equal. The game is a many-to-one matching model with money. An outcome of the game specifies a matching between buyers and sellers and a vector of prices. The worth of a potential transaction is given by a nonnegative real number associated with each possible pair of a buyer and an object. The set of sellers in our model can also be seen as multiproduct firms with one vacancy per division and the set of buyers as workers, where salaries are determined as part of the outcome of the game. Two natural questions are what kinds of partnerships we can expect to observe and how agents will divide their gain. The answers to these questions involve the choice of an appropriate concept of equilibrium that in these kinds of games is called stability. There are two different ways of defining stability in a many-to-one matching model: considering deviations only of pairs of agents (pairwise stability) or deviations of groups of agents (group stability). We define a new concept of pairwise stability that takes into account the fact that the objects that one seller owns are possibly different, which implies that the gain that a buyer and a seller can share is no longer the same, but depends on the object bought. We also propose a definiton of group stability adapted to our framework. This is the concept of stability we will concentrate our results on, since it is the more adequate concept for a many-to-one matching model. In contrast with the previous models, we prove that group stability is sufficient but not necessary for pairwise stability. We prove existence of both pairwise and group stable outcomes and study the structure of the group stable set. We show that the group stable set is the Cartesian Product of the set of group stable payoffs and the set of optimal matchings, and that it forms a complete lattice with an optimal group stable payoff for each side of the market. We also show that the core of a given market equals the set of group stable outcomes. We relate the group stable set with the set of competitive equilibria and compare our results with the Assignment Game.