Elongation of the Core in an Assignment Game

Abstract : A two sided matching model is a game in which there are two types of agents, and the essential coalitions are singletons and doubletons containing one agent of each type. Over the years, they have become an important part of economic theory. One reason for this is that they have been deemed worthy models of economic markets with indivisible goods. Gale and Shapley modeled college admissions as a matching market. Shapley and Shubik adopted a similar model for their housing market. Crawford and Knoer defined labor markets in these terms. In all of these instances, the relevant solution concept is that of the core which is the set of economic allocations where no coalition of agents can improve their lot on their own. Herein lies another reason for the study of these games; the fact that their cores have many nice properties. For instance, their cores are always nonempty. Much of the literature relate relatively simple algorithms which calculate core points. Another important idea is that in some of these games, the core and the set of economic equilibria are equivalent and both are generally sets of positive measure. This contrasts other economic models (with completely divisible goods) in which the set of equilibria is a proper subset of the core and is of measure zero. Thus, given a sets of agents' preferences, there is usually some latitude in choosing equilibrium prices. Intuitively, the amount of flexibility would be measured by the core's volume. This paper quantifies this last notion in the setting of Shapley and Shubik's housing market.