Essays on assignment markets: Vickrey outcome and Walrasian Equilibrium

Abstract

This thesis is devoted mainly to the study of assignment problems in two- sided markets. The thesis is divided into three parts. The underlying objective of this division is to provide different perspectives to the analysis of allocations of indivisible objects. Essentially, we explore three distinct processes to determine allocations: the Walrasian economy in which trades are regulated by a system of prices and budget constraints, the cooperative game of exchange among coalitions of the traders, and a noncooperative approach.

The second chapter of this thesis, One-seller assignment markets: core and Walrasian equilibrium is devoted to the study of markets in which there is a single seller and many buyers. The seller is the owner of multiple indivisible objects and these could be heterogeneous, e.g., houses, cars. On the other side of the market, every buyer wants to purchase a fixed amount of objects. The aim of this chapter is to study the relationship between the core of the cooperative game (see e.g., Peleg and Sudhölter, 2007) associated with the market and the set of Walrasian equilibria of the market. In general, it has been shown that in generalizations of the assignment game (Shapley and Shubik, 1972), the core and the set of payoff vectors associated with Walrasian equilibria do not coincide (see e.g., Massó and Neme, 2014). Our first result shows that the cooperative game associated with the market is buyers-submodular (Ausubel and Milgrom, 2001). As a consequence, the core is non-empty and it has a lattice structure. The most important result of this chapter offers a characterization of the coincidence between the core and the set of payoff vectors associated with the Walrasian equilibria.

In the third chapter, An implementation of the Vickrey outcome with gross- substitutes, we analyze exchanges in markets with only one seller and many buyers. The seller owns a finite set of indivisible objects on sale. In this chapter, we assume that each buyer has a valuation in terms of money for each package of objects on sale. In fact, we impose a monotonic property and the Gross-substitutes condition (see e.g., Gul and Stacchetti, 1999). The purpose of this chapter is to study the strategic behavior of agents in a non- cooperative environment under the assumption of complete information. The chapter provides a non-cooperative game or mechanism in which each

buyer makes a purchase offer to the seller. Then, the seller responds by selecting an allocation and a price for each package to be assigned. We show that every Subgame Perfect Nash equilibrium of the mechanism leads to a Vickrey outcome (see e.g., Vickrey, 1978 and Milgrom, 2004). That is, the allocation of the objects is efficient and each buyer receives a payoff equivalent to his marginal contribution to the whole market.

In the last chapter, Axioms for the minimum Walrasian equilibrium in assignment problems with unitary demands, we consider a group of buyers who wish to purchase items on sale. In particular, each buyer can acquire at most one object and has a preference over pairs formed by an indivisible object and money. Essentially, the domain of preferences considered in this chapter includes the domain of quasi-linear preferences. It is assumed that an institution will determine a rule for the assignment of objects. Of course, a fundamental question is how to allocate the objects? In previous works such as Demange and Gale (1985) and Morimoto and Serizawa (2015), it is shown that the minimum Walrasian equilibrium as an allocation rule satisfies outstanding properties such as envy-freeness and strategy- proofness. In this chapter, we provide a new characterization for the rule that selects minimum Walrasian equilibrium in both domains, general and quasi-linear preferences.