Mathematical Economics: Some simple, useful examples

1. Mathematical Economics:Some simple, useful examples Herb Scarf Whitney Humanities Center February 25, 2009

7. Preferences

8. A Stable Marriage(with the same Preferences)

9. The Roommate Problem If the assignments are {Alan, Bob}, {Chuck, Dan} then Bob and Chuck will object. If the assignments are {Alan, Chuck}, {Bob, Dan} then Bob and Alan will object. If the assignments are {Alan, Dan}, {Bob, Chuck} then Alan and Chuck will object. There is no stable assignment.

10. Polygamous Matching An important generalization of the Matching Problem arises when the two sets to be matched don't have the same size. For example, the two sets can consist of a large number of Residents (R) who are to be assigned to a small number of Hospitals (H). Each Hospital has a limited number of places. For simplicity let us assume that the number of places at the various Hospitals is equal to the total number of Residents. We assume that each Resident ranks all of the Hospitals, and each Hospital ranks all of the Residents. The problem is to assign each Resident to a Hospital in a stable fashion.

11. Stability The assignment is stable if there is no Resident R and Hospital H such that Resident R prefers Hospital H to the hospital to which she is assigned, and Hospital H prefers Resident R to one of the residents who has been assigned to H. Is there a stable assignment? Theorem: Yes! (Gale, Shapley, 1962) This includes the Marriage Problem as a special case.

12. Observations

13. Applications These techniques have been used for many years in the National Residents Matching Program (MRMP). They have been used in the assignments of students to schools in New York, Boston, Budapest and Singapore. And in college admissions in Turkey. In the allocation of graduating students from Hebrew Union College to congregations. The assignment of students to houses at MIT. Matching advertisers to advertising slots on Google. I know of applications to the assignment of packets on the Internet.

14. Now I would like to chat with you about a variation in the basic model that has been used to discuss, among other things, kidney transplants. The issue is to assign the kidneys offered by willing donors to recipients in need of a functioning kidney. The problem is that the friendly donor’s kidney may be not be suitable for the particular recipient and may have to be exchanged for the kidney of another donor. “A Kidney Exchange Clearinghouse in New England,” Alvin E. Roth, Tayfun Sönmez, M. Utku Ünver (2004) The model to be discussed is called the House Problem. It is very simple, but it is a serious introduction to the economics of exchange.

15. The House Problem O-1 H-1 O-2 H-2 O-3 H-3 O-4 H-4 O-5 H-5

16. How to Continue?

18. The Solution O-1 H-1 O-2 H-2 O-3 H-3 O-4 H-4 O-5 H-5

21. Some Reflections