Essays on multi-sided assignment markets

Abstract

This dissertation covers the study of assignment problems in a game theoretical framework, focusing on multi-sided assignment games and stability notions.

In Chapter 2, we provide some preliminaries on assignment markets and assignment games. We give some needed definitions and crucial results with their proof.

In Chapter 3, a generalization of the classical three-sided assignment market is considered, where value is generated by pairs or triplets of agents belonging to different sectors, as well as by individuals. For these markets we represent the situation that arises when some agents leave the market with some payoff by means of a generalization of Owen (1992) derived market. Consistency with respect to the derived market, together with singleness best and individual anti-monotonicity, axiomatically characterize the core for these generalized three-sided assignment markets. When one sector is formed by buyers and the other by two different type of sellers, we show that the core coincides with the set of competitive equilibrium payoff vectors.

In Chapter 4, we consider a multi-sided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite graph on the set of agents, (b) a basic coalition is formed by agents from different connected sectors, and (c) the worth of a basic coalition is the addition of the weights of all its pairs that belong to connected sectors. We provide a sufficient condition on the weights to guarantee balancedness of the related multi-sided assignment game. Moreover, when the graph on the sectors is cycle-free, we prove the game is strongly balanced and the core is described by means of the cores of the underlying two-sided assignment games associated with the edges of this graph. Moreover, once selected a spanning tree of the cycle-free graph on the sectors, the equivalence between core and competitive equilibria is established.

In Chapter 5, we focus on two-sided assignment games. Solymosi and Raghavan (2001) characterizes the stability of the core of the assignment game by means of a property of the valuation matrix. They show that the core of an assignment game is a von Neumann-Morgenstern stable set if and only if its valuation matrix has a dominant diagonal. Their proof makes use of some graph-theoretical tools, while the present proof relies on the notion of buyer-seller exact representative in Núñez and Rafels (2002).

In Chapter 6, we study von Neumann-Morgenstern stability for three-sided assignment games. Since the core may be empty in this case, we focus on other notions of stability such as the notions of subsolution and von Neumann-Morgenstern stable sets. The dominant diagonal property is necessary for the core to be a stable set, and also sufficient in case each sector of the market has only two agents. Furthermore, for any three-sided assignment market, we prove that the union of the extended cores of all mu- compatible subgames, for a given optimal matching mu, is the core with respect to those allocations that are compatible with that matching, and it is always non-empty.