Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

Incentives and Efficiency in Constrained Allocation Mechanisms

We study private-good allocation mechanisms where an arbitrary constraint delimits the set of feasible joint allocations. This generality provides a unified perspective over several prominent examples that can be parameterized as constraints in this model, including house allocation, roommate assignment, and social choice. We first characterize the set of two-agent strategy-proof and Pareto efficient mechanisms, showing that every mechanism is a "local dictatorship." For more than two agents, we leverage this result to provide a new characterization of group strategy-proofness. In particular, an N-agent mechanism is group strategy-proof if and only if all its two-agent marginal mechanisms (defined by holding fixed all but two agents' preferences) are individually strategy-proof and Pareto efficient. To illustrate their usefulness, we apply these results to the roommates problem to discover the novel finding that all group strategy-proof and Pareto efficient mechanisms are generalized serial dictatorships, a new class of mechanisms. Our results also yield a simple new proof of the Gibbard-Satterthwaite Theorem

Essays in Market Design

abstract: I study the design of two different institutions to evaluate the welfare implications of counterfactual policies. In particular, I analyze (i) the problem of assigning students to colleges (majors) in a centralized admission system; and (ii) an auction where the seller can use securities to determine winner’s payment, and bidders suffer negative externalities. In the former, I provide a novel methodology to evaluate counterfactual policies when the admission mechanism is manipulable. In the latter, I determine which instrument yields the highest expected revenue from the class of instruments that combines cash and equity payments.Dissertation/ThesisDoctoral Dissertation Economics 201

A Field Experiment on Course Bidding at Business Schools

Allocation of course seats to students is a challenging task for registrars' offices in universities. Since demand exceeds supply for many courses, course allocation needs to be done equitably and efficiently. Many schools use bidding systems where student bids are used both to infer preferences over courses and to determine student priorities for courses. However, this dual role of bids can result in course allocations not being market outcomes and unnecessary efficiency loss, which can potentially be avoided with the use of an appropriate market mechanism. We report a field experiment done at the University of Michigan Business School in Spring 2004 comparing its typical course bidding mechanism with the alternate Gale-Shapley Pareto-dominant market mechanism. Our results suggest that using the latter could vastly improve efficiency of course allocation systems while facilitating market outcomes.

Pick-an-object Mechanisms

We introduce a new family of mechanisms for one-sided matching markets, denoted pick-an-object (PAO) mechanisms. When implementing an allocation rule via PAO, agents are asked to pick an object from individualized menus. These choices may be rejected later on, and these agents are presented with new menus. When the procedure ends, agents are assigned the last object they picked. We characterize the allocation rules that can be sequentialized by PAO mechanisms, as well as the ones that can be implemented in a robust truthful equilibrium. We justify the use of PAO as opposed to direct mechanisms by showing that its equilibrium behavior is closely related to the one in obviously strategy-proof (OSP) mechanisms, but includes commonly used rules, such as Gale-Shapley DA and Top Trading Cycles, which are not OSP-implementable. We run laboratory experiments comparing truthful behavior when using PAO, OSP, and direct mechanisms to implement different rules. These indicate that individuals are more likely to behave in line with the theoretical prediction under PAO and OSP implementations than their direct counterparts

Ties matter: improving efficiency in course allocation by introducing ties

We study the course allocation system at Nanyang Technological University, where students submit strict preferences for courses and courses have implicit preferences for students. This formulates a many-to-many matching problem. We show the inefficiencies of the current mechanism and propose new competing mechanisms called Pareto-improving draft and dictatorship mechanisms, which introduce ties into students' preferences. Our mechanisms generate (group) stable and Pareto-efficient allocations, and the dictatorship mechanism can be implemented truthfully. Simulations on real data show that introducing ties into students' preferences can significantly improve efficiency, and the draft mechanism outperforms the dictatorship mechanism despite that the former is non-strategyproof

Dominant and Nash Strategy Mechanisms for the Assignment Problem

In this paper, I examine the problem of matching or assigning a fixed set of goods or services to a fixed set of agents. I characterize the social choice correspondences that can be implemented in dominant and Nash strategies when transfers are not allowed. This is an extension of the literature that was begun by Gibbard (1973) and Satterthwaite (1975), who independently proved that if a mechanism is nonmanipulable it is dictatorial. For the classes of mechanisms that are described in the paper, the results imply that the only mechanisms that are implementable in dominant and Nash strategies are choice mechanisms that rely only on ordinal rankings. I also describe a subclass of mechanisms that are Pareto optimal. In addition, the results explain the modeling conventions found in the literature - that when nontransfer mechanisms are studied individuals are endowed with ordinal preferences, and when transfer mechanisms are studied individuals are endowed with cardinal preferences