Mechanism Design without Money. Lecture 7. M-optimal and W-optimal matchings.
A stable matching is M-optimal if every man likes it at least as well as any other stable matching; that is, a stable matching is M-optimal if for every other stable matching we have for all (from now on we will write ).
Similarly, is W-optimal if for every other stable matching we have .
Theorem (GS 1962): When all men and women have strict preferences, the matching produced by the DA algorithm with men-proposing is the M-optimal stable matching. Symmetrically, the matching produced by the DA algorithm with women-proposing is the W-optimal stable matching.
Proof: Look at the first man being rejected by an “achievable” woman . If is unacceptable to , we reach contradiction. If not, and the man that chose over in that step block any stable matching in which and are matched.
Theorem (Knuth, 1976): When all agents have strict preferences, the common preferences of the two sides of the markets are opposed on the set of stable matchings. That is, If and are both stable matchings then:
if and only if .
Proof: Let and be two stable matchings such that . Suppose for some woman . Then . But for man we also have . So the pair block , which was supposedly stable.
When preferences are strict, for any two matchings and let be the matching such that if and otherwise, and if and otherwise. In an opposite way define the matching .
The Lattice theorem (Conway): When and are stable matchings, then and are both stable matchings.
The stable matchings are:
A more complicated example with 10 matches in a lattice in Roth and Sotomayor (1990), pp. 37-38.
Pareto – no one gets hurt, and someone improves their situation
Basic optimality criterion
Obviously, both and are optimal with respect to the entire population (otherwise, there would be a blocking pair).
Thm: is Pareto efficient with respect to the men That is, no other matching (even non stable) is weakly preferred by all men.
Also known as the “college admissions” model.
A market is a finite set of hospitals , a finite set of doctors , and preferences .
For every doctor , is a strict order over .
For every hospital , is a strict order over .
A hospital has responsive preferences with quota if its preference between any two sets of doctors that differ by only one doctor is determined by some constant order over , and it prefers any set of size or less to any set of size or more.
Lemma: When hospitals have responsive preferences, a matching is stable if and only if it is group-stable.
When hospitals have responsive preferences we can consider the related marriage marketin which each hospital is split into mini-hospitals with demand of one doctor (quota of 1).
Lemma: A matching is stable if and only if the corresponding matchings of the related marriage market are stable.
Some, but not all, of the results still hold.
For example, a stable matching still exists (modification of DA algorithm), and it takes some work, but the lattice structure is also there.
However, weak Pareto optimality is lost. For example:
Here the only stable matching is given by:
All hospitals prefer the (non-stable) matching :
~1900: Internships introduced
1944: Contracts signed two years before internship begins.
1946: Transcripts released only at the end of junior year.
1945-1951: Students holding offers. Offers with 12 hours deadline.
1951: Trial run of a centralized matching algorithm.
1952: First use of the NIMP algorithm to preform a match.
1962: Gale and Shapley (first?) prove existence of stable matching.
Late 1980’s and early1990’s: The market gradually becomes more complex. Lack of confidence in the algorithm.
1995: NRMP board decides to design a new algorithm.
1998: First run of the Roth-Peranson algorithm.
We mentioned that the trial-run algorithm was rejected because it was prone to strategic behavior.
What can we say about strategic behavior when using stable mechanisms?
Direct revelation mechanisms: doctors and hospitals reveal their private information (their preferences over matchings), and the mechanism selects a matching accordingly.
We will be interested in direct revelation mechanisms that output stable matchings.
Reminder 1: a matching is called stable if it is individually rational and pairwise stable.
Reminder 2: for every set of preferences, a stable matching exists. Furthermore, we can point to two special matchings: the doctor-optimal stable matching and the hospital-optimal stable matching.
Reminder 3: One way for constructing those two matchings is by running (e.g. on a computer) a deferred-acceptance algorithm.
When using the doctor-optimal stable mechanism, a hospital may be able to manipulate the outcome by submitting different preferences.
The doctor-optimal stable matching is .
If submits instead the preference , then the only stable matching is .
Theorem: There exists no stable mechanism for which truthful revelation of the preference is dominant strategy.
Proof: Use the last example. If the stable matching that the mechanism selects for truthful revelation is and not , then one of the doctors can manipulate.
Theorem: When using the doctor-optimal stable mechanism, it is a (weakly) dominant strategy for the doctors to state their true preferences.
Proof: Follow the doctor that can cheat. Use independence of order.
Theorem: If hospitals all have quota of one (that is, in one-to-one matching) and the doctor-optimal stable mechanism is being used, all outcomes of Nash equilibria in weakly undominated strategies are stable with respect to the true preferences. Furthermore, every stable matching is the outcome of some Nash equilibrium.
Proof: The second part of the theorem is easy (doctors submit preference truthfully), and hospitals according to the stable matching. The proof of the first part is omitted.
Theorem: When using the hospital-optimal stable mechanism, truthful revelation is not a dominant strategy for the hospitals.
Intuition: Hospital with more than one position is like a group of men and they can deviate together in order to induce a weakly better allocation to all of them.
Theorem: If some hospitals have quota above one and the mechanism is either the doctor-optimal stable mechanism or the hospital optimal stable mechanism, not every outcome that corresponds to a Nash equilibrium in weakly undominated strategies is necessarily stable with respect to the true preferences.
It seems like many of the results are negative. Why is it then that the NIMP algorithm “survived” and the Roth-Peranson algorithm (to be discussed later) is also considered a success?
Answer 1: Imperfect information makes truthful revelation more appealing (Roth and Rothblum, 1999).
Answer 2: NRMP data and computational experiments show that in fact the set of stable matchings is small, and only very few participants can manipulate the results.
Answer 1: Correlation. Note that if one side has perfectly correlated preferences, then there exists only one stable matching, and there is no incentive to misrepresent the true preference.
Answer 2: Short ROLs (rank order lists) do not allow long rejection chains (in DA algorithm).
It turns out the second answer is quite sufficient to explain the results.
Intuition: Rejecting a man will cause a rejection cycle that might or might not return to the original woman. When no woman is popular enough there is an increasingly big chance that the rejection cycle will end with a man who stays single, or with a woman who was single before.
Corollary 1: The game induced by the men-proposing stable mechanism has an equilibrium in which, in expectation, fraction of the strategies are truthful.
Corollary 2: For every , if is large enough the game has an -approximate Nash equilibrium in which everybody is truthful.
Kojima and Pathak (2009) – Extension to many-to-one markets.
Azevedo and Leshno (working paper, 2011) – A model with continuum of students.
Lee (working paper, 2013) – With bounded underlying cardinal utility, utility gains from preference misrepresentation are less than for fraction of the population.
Ashlagi, Kanoria, and Leshno (working paper, 2013) – Small number of stable matches in large uniform and uneven markets.
The problem with allocation of doctors to rural hospitals.
Theorem: When preferences are strict, the set of doctors employed and positions filled is the same at every stable matching.
Theorem: When preferences are strict, any hospital that does not fill its quota at some stable matching is assigned precisely the same set of students at every stable matching.
With time and progress, couples became a problem for the NRMP.
The “leading member” adjustment didn’t work very well…
This was one of the main reasons the original NRMP algorithm had to be replaced.
It turns out that when couples are present, the set of stable matchings may be empty.
Even when stable matchings do exist, there does not have to be a side-optimal stable matching.
Clearinghouses currently using the algorithm (Taken from Roth’s slides):
Why does the Roth-Peranson algorithm works?
When is the set of stable matchings (with couples) non-empty?
Kojima, Pathakand Roth (working paper) – Using a model similar to Immorlica and Mahdian (2003) and Kojima and Pathak (2009), if the number of couples is then a stable matching exists (and can be reached by the Roth and Peranson algorithm).
Intuition: Instead of bounding using the probability that there is no rejection cycle (and all rejection chains end with a hospital that had a vacant position). For couples bound using also that there is no rejection path from one member of the couple to the other.
Ashlagi, Bravermanand Hassidim (working paper) – Better bounds and slightly more sophisticated algorithm insure existence of stable matching even if the number of couples is for any .
They also provide a counterexample for the case in which the number of couples is .
Roommate problems, multi-sided matching
Many-to-one with discrete money and substitutable preferences (Crawford and Knoer, 1981; Kelso and Crawford, 1982)
Many-to-many with responsive preferences (Roth, 1984)
Matching with contracts (Hatfield and Milgrom, 2005)
Many-to-many matching with contracts (Echenique and Oveido, 2006)
Matching in supply chains (Ostrovsky, 2008)
Matching in networks with bilateral contracts (Hatfield, Kominers, Nichifor, Ostrovsky and Westkamp, working paper)
Matching with minimum quotas, regional caps, etc. (Biro, Fleiner, Irving and Manlove, 2010, Kamada and Kojima, 2013)
Roth and VandeVate (1990) – Random paths to stability
Jackson and Watts (2002)
Ausubel and Milgrom (2000) on package bidding