2015 Theses Doctoral

# Design and Analysis of Matching and Auction Markets

Auctions and matching mechanisms have become an increasingly important tool to allocate scarce resources among competing individuals or firms. Every day, millions of auctions are run for a variety of purposes, ranging from selling valuable art or advertisement space in websites to acquiring goods for government use. Every year matching mechanisms are used to decide the public school assignments of thousands of incoming high school students, who are competing to obtain a seat in their most preferred school. This thesis addresses several questions that arise when designing and analyzing matching and auction markets.

The first part of the dissertation is devoted to matching markets. In Chapter 2, we study markets with indivisible goods where monetary compensations are not possible. Each individual is endowed with an object and has ordinal preferences over all objects. When preferences are strict, the Top-Trading Cycles (TTC) mechanism invented by Gale is Pareto efficient, strategy-proof, and finds a core allocation, and is the only mechanism satisfying these properties. In the extensive literature on this problem since then, the TTC mechanism has been characterized in multiple ways, establishing its central role within the class of all allocation mechanisms. In many real applications, however, the individual preferences have subjective indifferences; in this case, no simple adaptation of the TTC mechanism is Pareto efficient and strategy-proof. We provide a foundation for extending the TTC mechanism to the preference domain with indifferences while guaranteeing Pareto efficiency and strategy-proofness. As a by-product, we establish sufficient conditions for a mechanism (within a broad class of mechanisms) to be strategy-proof and use these conditions to design computationally efficient mechanisms.

In Chapter 3, we study several questions associated to the Random Priority (RP) mechanism from a computational perspective. The RP mechanism is a popular way to allocate objects to agents with strict ordinal preferences over the objects. In this mechanism, an ordering over the agents is selected uniformly at random; the first agent is then allocated his most-preferred object, the second agent is allocated his most-preferred object among the remaining ones, and so on. The outcome of the mechanism is a bi-stochastic matrix in which entry (i,a) represents the probability that agent i is given object a. It is shown that the problem of computing the RP allocation matrix is #P-complete. Furthermore, it is NP-complete to decide if a given agent i receives a given object a with positive probability under the RP mechanism, whereas it is possible to decide in polynomial time whether or not agent i receives object a with probability 1. The implications of these results for approximating the RP allocation matrix as well as on finding constrained Pareto optimal matchings are discussed.

Chapter 4 focuses on assignment markets (matching markets with transferable utilities), such as labor and housing markets. We consider a two-sided assignment market with agent types and stochastic structure similar to models used in empirical studies, and characterize the size of the core in such markets. The value generated from a match between a pair of agents is the sum of two random productivity terms, each of which depends only on the type but not the identity of one of the agents, and a third deterministic term driven by the pair of types. We allow the number of agents to grow, keeping the number of agent types fixed. Let n be the number of agents and K be the number of types on the side of the market with more types. We find, under reasonable assumptions, that the relative variation in utility per agent over core outcomes is bounded as O^*(1/n^{1/K}), where polylogarithmic factors have been suppressed. Further, we show that this bound is tight in worst case, and provide a tighter bound under more restrictive assumptions.

In the second part of the dissertation, we study auction markets. Chapter 5 considers the problem faced by a procurement agency that runs an auction-type mechanism to construct an assortment of products with posted prices, from a set of differentiated products offered by strategic suppliers. Heterogeneous consumers then buy their most preferred alternative from the assortment as needed. Framework agreements (FAs), widely used in the public sector, take this form; this type of mechanism is also relevant in other contexts, such as the design of medical formularies and group buying. When evaluating the bids, the procurement agency must consider the trade-off between offering a richer menu of products for consumers, versus offering less variety, hoping to engage the suppliers in a more aggressive price competition. We develop a mechanism design approach to study this problem, and provide a characterization of the optimal mechanisms. This characterization allows us to quantify the optimal trade-off between product variety and price competition, in terms of suppliers' costs, products' characteristics, and consumers' characteristics. We then use the optimal mechanism as a benchmark to evaluate the performance of the Chilean government procurement agency's current implementation of FAs, used to acquire US $2 billion worth of goods per year. We show how simple modifications to the current mechanism, which increase price competition among close substitutes, can considerably improve performance.

## Subjects

## Files

- Saban_columbia_0054D_12821.pdf application/pdf 1.14 MB Download File

## More About This Work

- Academic Units
- Business
- Thesis Advisors
- Sethuraman, Jay
- Weintraub, Gabriel
- Degree
- Ph.D., Columbia University
- Published Here
- July 10, 2015