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Introduction to Market Design, Econ 2056 and Business 2150:

Professors Al Roth and Peter Coles, Spring 2006

Some useful websites to find the syllabus, handouts (like these slides), and links to related papers: http://my.harvard.edu/icb/icb.do?course=fas-ec2056&pageid=tk.page.ec2056.generalinfo http://kuznets.fas.harvard.edu/~aroth/alroth.html Assignment: One final paper. The objective of the final paper is to study an existing market or an environment with a potential role for a market, describe the relevant market design questions, and evaluate how the current market design works and/or propose improvements on the current design. Handouts: Some handouts will be put in the Handout section of the course webpage, and some will have links from the syllabus.

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Recommended Texts: Roth, Alvin E. and Marilda Sotomayor Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, Econometric Society Monograph Series, Cambridge University Press, 1990. (get the paperback edition☺ Milgrom, Paul "Putting Auction Theory to Work" by Paul Milgrom (Churchill Lectures), Cambridge University Press, 2004 Klemperer, Paul "Auctions: Theory and Practice" (Toulouse Lectures), Princeton University Press, 2004.

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As introduction to some of the main themes of the course this morning, Peter and I are going to flash before your eyes many things that we’ll go over in more detail as the course unfolds.

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Methodological theme: The economic environment evolves, but it is also designed. Entrepreneurs and managers, legislators and regulators, lawyers and judges, all get involved in the design of economic institutions. Recently, economists in general and game theorists in particular, have started to take a substantial role in economic design.

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From: “Jonathan Strange & Mr Norrell: A Novel” by Susanna Clarke (The novel describes a world in which magic used to exist, but no longer does…) “Mr Segundus wished to know, he said, why modern magicians were unable to work the magic they wrote about…. “The President of the…society…explained that the question was a wrong one. ‘It presupposes that magicians have some sort of duty to do magic—which is clearly nonsense. You would not, I imagine, suggest that it is the task of botanists to devise more flowers? Or that astronomers should labour to rearrange the stars? Magicians…study magic which was done long ago. Why should anyone expect more?’ ”

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Designing markets, instead of just studying existing markets, will also require us to study markets in different ways.

We need to understand how markets work well enough to fix them when they’re broken. Design also comes with a responsibility for detail. Designers can’t be satisfied with simple models that explain the general principles underlying a market; they have to be able to make sure that all the detailed parts function together. Responsibility for detail requires the ability to deal with complex institutional features that may be omitted from simple models.

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Game theory, the part of economics that studies the “rules of the game,” provides a framework with which design issues can be addressed. But dealing with complexity will require new tools, to supplement the analytical toolbox of the traditional theorist.

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Game Theory, experimentation, and computation, together with careful observation of historical and contemporary markets (with particular attention to the market rules), are complementary tools of Design Economics Computation helps us find answers that are beyond our current theoretical knowledge. Experiments play a role

• In diagnosing and understanding market failures, and successes

• In designing new markets • In communicating results to policy

makers

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A rough analogy may help indicate how the parts of this course hang together. Consider the design of suspension bridges. Their simple physics, in which the only force is gravity, and all beams are perfectly rigid, is beautiful and indispensable. But bridge design also concerns metal fatigue, soil mechanics, and the sideways forces of waves and wind. Many questions concerning these complications can’t be answered analytically, but must be explored using physical or computational models. These complications, and how they interact with that part of the physics captured by the simple model, are the concern of the engineering literature. Some of this is less elegant than the simple model, but it allows bridges designed on the same basic model to be built longer and stronger over time, as the complexities and how to deal with them become better understood.

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In this class, the simple models will be models of matching, and of auctions. In recent years there have been some great advances in the theory of each of these, that brings them much closer together. A lot of these theoretical insights have come from the difficulties faced in designing complex labor markets and auctions (e.g. labor markets in which there may be two-career households, and auctions in which bidders may wish to purchase packages of goods). (Paul Milgrom and his colleagues have led the way on much of the recent theoretical work.)

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Some markets I’ve had a hand in designing or redesigning (with many coauthors, see the syllabus):

Labor markets: • The market for new American and Canadian doctors

(The National Resident Matching Program) • The market for Gastroenterologists

o This is a medical market for more senior docs that is trying to get reorganized as we speak.

• The market for new economists o I’m presently the chair of an AEA committee to

suggest changes in the market—suggestions welcome!

School Choice systems: • The New York City high school choice system

o Our system has been in operation since 2003. • The Boston Public Schools choice system

o going into operation in 2006 • Harvard FAS freshman seminar assignments

o new algorithm since Fall 2005 Kidney Exchange • The New England Program for Kidney Exchange

opened in 2005.

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A number of those markets experienced market failures that were cured through the design of centralized clearinghouses.

We’ll study both the clearinghouses, and the

market failures (including recent market failures such as in the markets for law clerks, gastroenterologists, and college football bowls, that will suggest new questions about how both decentralized and centralized markets work).

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Loosely speaking, we’ll observe three kinds of market failures that can sometimes be cured by the introduction of an appropriate clearinghouse (and can sometimes be avoided by the adoption of appropriate rules for decentralized transactions).

These are:

1. failure to provide sufficient thickness, so that agents can consider many possible transactions

2.congestion: failure to provide enough

time for agents to consider sufficiently many transactions

3.failure to provide appropriate incentives

so that it is safe for agents to reveal the information needed for efficient matching.

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We can see each of these problems in the history

of the American market for new doctors. Since 1998, the vast majority of jobs for new

physicians in the U.S. (about 20,000 per year) are filled by a clearinghouse whose design I directed.

It updated and replaced an older clearinghouse, which in turn replaced a chaotic, decentralized market, that had suffered from severe coordination failures.

Coordination failures turn out to be quite common in professional, entry-level labor markets. (And the newly designed clearinghouse has in fact been adopted by entry level labor markets in a number of other professions, since its adoption by American physicians.)

To understand the design problem, we have to

first understand the history of these various markets, and the failures they experienced.

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HISTORY OF THE AMERICAN MARKET FOR NEW PHYSICIANS

1900-1945 UNRAVELLING OF APPOINTMENT

DATES 1945-1950 CHAOTIC RECONTRACTING 1950-197x HIGH RATES OF ORDERLY

PARTICIPATION ( 95%) in centralized clearinghouse 197x-198x DECLINING RATES OF PARTICIPATION (85%) particularly among the growing

number of MARRIED COUPLES 198x-present Married couples return following changes in

algorithm to accomodate couples and other kinds of match variations

1995-98 Market experienced a crisis of confidence with

fears of substantial decline in orderly participation; Design effort commissioned—to design and compare alternative matching algorithms—Roth-Peranson clearinghouse algorithm adopted, and employed

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Figure 1: Stages and Transitions Observed in the Markets of Table 1

Stage 1: UNRAVELING

Stage 2: UNIFORM DATES ENFORCED

Stage 3: CENTRALIZED MARKET CLEARING PROCEDURES

Stage 4: UNRAVELING BEFORE THE

CENTRALIZED PROCEDURES ARE EMPLOYED

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A Selection of Markets with Timing Problems Market Stage Postseason College Football Bowls 3 Entry Level Legal Labor Markets: Federal court clerkships 2,1,2 American law firms 1 Canadian articling positions Toronto 4 Vancouver 4,1 Alberta (Calgary and Edmonton) 3 Entry Level Business School Markets: New MBA's occasional stage 1 New Marketing Professors 1 Other Entry Level Labor Markets: Japanese University graduates 2 Other Two Sided Matching: Fraternity Rush 1 Sorority Rush 3

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Entry Level Medical and Health Care Labor Markets: American first year medical 3 Postgraduate positions Canadian first year positions 3 U.K. regional markets for pre-registration positions: Edinburgh 3 Cardiff 3 Birmingham 4,1 Newcastle 4,1 Sheffield 3 or 4,1 Cambridge 3 London Hospital 3 American specialty residencies and fellowships (about 30 markets) 3,4 (some 1) Clinical Psychology Internships 2,3 Dental Residencies (5 specialties + 2 general programs) 3 Optometry Residencies 1 and 3 Osteopathic Residencies 3 Pharmacy Residencies 3

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So, when the American medical market began to experience a crisis of confidence in the mid 1990’s, it deserved, and received serious concern. (The various national medical student organizations issued resolutions, and eventually Ralph Nader’s organization got involved.) There was a great deal of talk about whether the market served students’ interests well, as well as to whether they could “game the system” or find ways to do “end runs” around the market. In 1995, the NRMP board of directors commissioned the design of a new algorithm, and a study comparing different ways of organizing the match.

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A SIMPLE MODEL OF TWO-SIDED MATCHING

PLAYERS: Firms = {f1,..., fn} Workers = {w1,..., wp} # positions q1,...,qn PREFERENCES (complete, transitive, wages in job description): P(fi) = w3, w2, ... fi ... [w3 P(fi) w2] P(wj) = f2, f4, ... wj ... An OUTCOME of the game is a MATCHING: x: F∪W F∪W such that x(f) = w iff x(w) = f, and for all f and w |x(f)| is less than or equal to qf either x(w) is in F or x(w) = w. A matching x is BLOCKED BY AN INDIVIDUAL k if k prefers being single to being matched with x(k), i.e. [kP(k) x(k)] A matching x is BLOCKED BY A PAIR OF AGENTS (f,w) if they each prefer each other to x, i.e. [w P(f) w' for some w' in x(f) or w P(f) f if |x(f)| < qf ] and f P(w) x(w)] A matching x is STABLE if it isn't blocked by any individual or pair of agents.

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Stable and Unstable (Centralized) Mechanisms Market Stable Still in use (halted unraveling) NRMP yes yes (new design in ’98)

Edinburgh ('69) yes yes

Cardiff yes yes

Sororities yes yes

Cambridge no yes

London Hospital no yes

Birmingham no no

Edinburgh ('67) no no

Newcastle no no

Sheffield no no

Medical Specialties yes yes (~30 markets, 1 recent failure)

Canadian Lawyers yes yes (except B.C. since ‘96)

Dental Residencies yes yes (5 ) (no 2)

Osteopaths (< '94) no no

Osteopaths (> '94) yes yes

Pharmacists yes yes

Reform rabbis yes (first used in ‘97-98)

Clinical psych yes (first used in ‘99)

So stability looks like an important feature of a centralized labor market

clearinghouse.

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Priority Matching (an unstable system)

Edinburgh, 1967 No longer in use Birmingham 1966, 1971, 1978 " " " " Newcastle 1970's " " " " Sheffield 196x " " " " In a priority matching algorithm, a 'priority' is defined for each firm-worker pair as a function of their mutual rankings. The algorithm matches all priority 1 couples and removes them from the market, then repeats for priority 2 matches, priority 3 matches, etc. E.g. in Newcastle, the priorities for firm-worker rankings were organized by the product of the rankings, (initially) as follows: 1-1, 2-1, 1-2, 1-3, 3-1, 4-1, 2-2, 1-4, 5-1... This can produce unstable matchings -- e.g. if a desirable firm and worker rank each other 4th, they will have such a low priority (4x4=16) that if they fail to match to one of their first three choices, it is unlikely that they will match to each other. (e.g. the firm might match to its 15th choice worker, if that worker has ranked it first...) After 3 years, 80% of the submitted rankings were pre-arranged 1-1 rankings without any other choices ranked. This worked to the great disadvantage of those who didn't pre-arrange their matches.

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Deferred Acceptance Algorithm, with Firms making offers

(roughly the Gale-Shapley 1962 version) 1 a. Each firm f (with qf positions) offers them to its qf 1st choice applicants. b. Each applicant rejects any unacceptable offers and, if more than one acceptable offer is received, "holds" the most preferred. . . . k a. Any firm that had one or more offers rejected at step k-1 makes new offers to its most preferred acceptable applicants (up to the number of rejections received). b. Each applicant holds her most preferred acceptable offer to date, and rejects the rest. STOP: when no further offers are made, and match each applicant to the firm (if any) whose offer she is holding. Theorem: 1. the resulting matching is stable.

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Quick Homework: (Not to hand in, just to check your understanding …) Prove the following result (you only need to apply the definition of stability…) Theorem 2.13(Knuth) When all agents have strict preferences (i.e. no one is indifferent between any two possible mates), the common preferences of the two sides of the market are opposed on the set of stable matchings: if µ and µ' are stable matchings, then all men like µ at least as well as µ' if and only if all women like µ' at least as well as µ. That is, µ >M µ' if and only if µ' >W µ.