Strategy-Proofness and Singleton Cores in Generalized Matching Problems

We introduce and study the class of generalized matching problems. Two subclasses of this class are marriage problems (Gale and Shapley 1962) and the housing market (Shapley and Scarf 1974). We search for strategy-proof solutions to generalized matching problems. We show that if the core is a singleton and is stable for all problems then it is sstrategy-proof as a solution. We also show that on the class of problems with a non-empty core therre exists a Pareto efficient, individually rational, and sstrategy-proof solution only if the core is a singleton for all problems. Furthermore if such a solution exists, it is the core.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100985/1/ECON427.pd

Implementation in Generalized Matching Problems

We search for (Nash) implementable solutions on a class of one-to-one matching problems which includes both the housing market (Shapley and Scarf 1974) and marriage problems (Gale and shapley 1962). We show that the core correspondence is implementable. We furthermore show that any solution that is Pareto-efficient, individually rational, and implementable is a supersolution of the core correspondence. That is, the core correspondence is the minimal solution that is Pareto-efficient, individually rational, and implementable. A corollary of the independent interest in the context of the housing market is that the core correspondence is the only single-valued solution that is Pareto-efficient, individually rational, and implementable.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100984/1/ECON426.pd

Strategy-Proofness in Many-to-One Matching Problems

We search for strategy-proof solutions in the context of (many-to-one) matching problems (Gale and Shapley 1962). In this model, whenever the firms can hire as many workers as they want (the capacities are unlimited) the stable set is a singleton. There exists a Pareto efficient, individually rational, and strategy-proof mathicng rule if and only if the capacities are unlimited. Furthermore, whenever the capacities unlimited, the matching rule which selects the unique stable matching is the only matching rule that is Pareto efficient, individually rational, and strategy-proof.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100986/1/ECON428.pd

House Allocation with Existing Tenants: An Equivalence

In this paper we analyze two house allocation mechanisms each of which is designed to eliminate inefficiencies in real-life house allocation problems where there are both existing tenants and newcomers. The first mechanism chooses the unique core allocation of a "sister" exchange economy which is constructed by assigning each existing tenant her current house and randomly assigning each newcomer a vacant house. The second mechanism -top trading cycles mechanism- first chooses an ordering from a given distribution and next determines the final outcome as follows: Assign first agent her top choice, next agent her top choice among remaining houses and so on, until someone demands house of an existing tenant who is still in the line. At that point modify the queue by inserting her at the top and proceed. Similarly, insert any existing tenant who is not already served at the top of the queue once her house is demanded. Whenever a loop of existing tenants forms, assign each of them the house she demands and proceed. Our main result is that the core based mechanism is equivalent to an extreme case of the top trading cycles mechanism which orders newcomers before the existing tenants.

Pairwise Kidney Exchange

The theoretical literature on exchange of indivisible goods finds natural application in organizing the exchange of live donor kidneys for transplant. However, in kidney exchange, there are constraints on the size of feasible exchanges. Initially, kidney exchanges are likely to be pairwise exchanges, between just two patient-donor pairs, as these are logistically simpler than larger exchanges. Furthermore, the experience of many American surgeons suggests to them that preferences over kidneys are approximately 0-1, i.e. that patients and surgeons should be largely indifferent among healthy donors whose kidneys are compatible with the patient. This is because, in the United States, transplants of compatible live kidneys have about equal graft survival probabilities, regardless of the closeness of tissue types between patient and donor. We show that, although the pairwise constraint eliminates some potential exchanges, there is a wide class of constrained-efficient mechanisms that are strategy-proof when patient-donor pairs and surgeons have 0-1 preferences. This class of mechanisms includes deterministic mechanisms that would accomodate the kinds of priority setting that organ banks currently use to allocate cadaver organs, as well as stochastic mechanisms that allow distributive justice issues to be

School Admissions Reform in Chicago and England: Comparing Mechanisms by Their Vulnerability to Manipulation

In Fall 2009, officials from Chicago Public Schools changed their assignment mechanism for coveted spots at selective college preparatory high schools midstream. After asking about 14,000 applicants to submit their preferences for schools under one mechanism, the district asked them re-submit their preferences under a new mechanism. Officials were concerned that "high-scoring kids were being rejected simply because of the order in which they listed their college prep preferences" under the abandoned mechanism. What is somewhat puzzling is that the new mechanism is also manipulable. This paper introduces a method to compare mechanisms based on their vulnerability to manipulation. Under our notion, the old mechanism is more manipulable than the new Chicago mechanism. Indeed, the old Chicago mechanism is at least as manipulable as any other plausible mechanism. A number of similar transitions between mechanisms took place in England after the widely popular Boston mechanism was ruled illegal in 2007. Our approach provides support for these and other recent policy changes involving matching mechanisms.

Efficient Kidney Exchange: Coincidence of Wants in a Markets with Compatibility-Based Preferences

Patients needing kidney transplants may have donors who cannot donate to them because of blood or tissue incompatibility. Incompatible patient-donor pairs can exchange donor kidneys with other pairs only when there is a “double coincidence of wants.” Developing infrastructure to perform three-way as well as two-way exchanges will have a substantial effect on the number of transplants that can be arranged. Larger than three-way exchanges have less impact on efficiency. In a general model of type-compatible exchanges, the size of the largest exchanges required to achieve efficiency equals the number of types.Economic

A Kidney Exchange Clearinghouse in New England

Economic