When the M-optimal match is being chosen, it s a dominant strategy for the men to report their true preferences

Transcription

1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 19 November Last Thursday, we proved that the set of stable matchings in a given marriage market form a lattice, and then we went very quickly through a bunch of results on truthful revelation. I want to go back and give an example of two of the key results. Recall two of our claims: When the M-optimal match is being chosen, it s a dominant strategy for the men to report their true preferences When the M-optimal match is being chosen, if there are more than one stable matching, some woman can gain by misrepresenting her preferences There are more complicated examples of this in both the book and the Roth (1982) paper, but all it takes is a very simple one. Go back to our example of two men and two women, where w 1 m1 w 2 ; w 2 m2 w 1 ; m 2 w1 m 1 ; and m 1 w2 m 2. So there are two stable matchings: one where the men get their top choices, one where the women do. If we run the men-proposing deferred acceptance algorithm (with the true preferences), there is just one step. Both men propose to their favorite woman; both women tentatively accept; there are no rejections, so the algorithm ends at the men-optimal stable matching. But now suppose woman 1 misrepresents her preferences by saying that only man 2 is acceptable. That is, rather than reporting her true preferences m 2 m 1 w 1, she reports m 2 w 1 m 1. What happens if we run the algorithm? In the first round, man 1 proposes to woman 1, but she rejects him; and man 2 proposes to woman 2, which she tentatively accepts. Man 1 was rejected, so in round two, he proposes to his second choice, woman 2; since she prefers him to man 2, she says yes to man 1, and rejects man 2. So in the third round, man 2, newly single, proposes to woman 1; and the algorithm ends, with 1 matched to 2 and 2 matched to 1. So woman 1, by shortening her preference list, ends up with her favorite (man 2) instead of man 1. Obviously, when the women report their true preferences, the men have no reason to lie, since they each get their first choice. But now suppose that woman 1 reports m 2 w 1 m 1 (and woman 2 reports the truth, m 1 m 2 w 2 ). We want to show that truthful reporting is still optimal for both men. Which means, neither one can get his first choice of woman by lying about his preferences. To see this, first note that man 1 can t get woman 1 regardless of what he says, since she reported him as unacceptable. So the only question is, can man 2 get woman 2? And the answer is no. Whatever he says, man 1 is going to get rejected by woman 1 in the first round, and propose to woman 2 in the second round; and woman 2 prefers man 1. So whatever preferences man 2 reports, the best he can do is to end up with woman 1, which is what he gets if he tells the truth. 1

2 So that reinforces our claim that the proposing side has a dominant strategy of truthful revelation, while the other side can typically gain by misrepresenting preferences. There were a bunch of other results elaborating a bit on this, as well as another negative result: there is mapping from preferences to stable matchings that makes it a dominant strategy for everyone to report truthfully. From there, we move on to... many-to-one matching. The classic application considered here is the matching of graduating medical students to residency programs at hospitals. (I should give you some institutional details here, but I m not going to; Al Roth s website has nice overviews if you re interested.) Another classic application is that matching of applicants to schools college applications, or public schools (as in school choice programs in Boston and New York in recent years). In public school examples, it might be reasonable to assume that schools don t care that much about which students they get, that is, don t really have meaningful preferences. In the case of applicants to colleges, it s probably reasonable to suppose that colleges have preferences over individual students, but don t really care that much about which combinations they get. On the other hand, when hospitals are taking in a small group of residents, they may care very much about combinations they may want diversity of specializations, or demographic diversity, or whatever. So an important question becomes, how do we formulate hospital preferences over combinations of applicants in a way that makes the problem tractable? It turns out, there are a few different ways. Roth uses the concept of responsive preferences, which we ll define in a bit. For matching with money, Kelso and Crawford use a gross substitutes condition, very similar to the one we put on bidder preferences over objects, and show that things work out very well that way. The Hatfield and Milgrom paper gives a unifying framework for modeling matching both with and without money, and give a substitutes condition that works well in both cases. As it happens, I like the later formulations more, but for today, we ll use Roth s setup. Also, for today, I ll be mostly following the treatment in the Roth and Sotomayor book, so I ll refer to the two sides as Colleges and Students, as in the college application model. The model: A set of colleges C = {C 1, C 2,..., C n }, with space for q 1, q 2,..., q n students A set of students S = {s 1,..., s m } Students preferences are like in the one-to-one model: each student has strict preferences over C {s}, with those colleges which are better than remaining alone referred to as acceptable College c starts off with preferences P (c) over students S {c}, so some students are acceptable (better than unfilled quota) and some are not College c is then allowed to have any preferences over groups of students such that their preference between two particular students (or between a student and nobody) does not depend on who else they have 2

3 That is, if P (c) is college c s preference list over individual students, and P # (c) its preferences over groups of students (of size q c or less), then for any X S and s, s / X, X {s} P # (c) X {s } if and only if s P (c) s. This is what is meant by responsive preferences the preference over outcomes responds to the preference over individual students Now, a matching is what you think it is, but the definition is a little trickier. Formally, a matching µ is a function from C S to the set of unordered, possibly redundant sets of elements of C S such that µ(s) = 1 for every student s, and µ(s) C {s} µ(c) = q C for every college C, and if r denotes the number of students in µ(c), then µ(c) contains q C r copies of C µ(s) = C if and only if s µ(c) Now, we give two definitions of stability, then show they re the same A matching is pairwise stable if it is not blocked by any individual (student or college) or any student-college pair (s, C) A matching µ is blocked by a coalition A (where A can contain multiple colleges and multiple students) if there is another matching µ such that for every s A, µ (s) A µ (s) s µ(s) s µ (C) implies s A µ(c), so every college in A only takes new students from A, although it can keep some of its old students who are not in A µ (C) C µ(c) so A can be thought of a deviating coalition of students and colleges, but those colleges can still hang onto a subset of the students they were already matched to under µ A matching is group stable if it is not blocked by any coalition And then, what gives us tractability: Lemma. A matching is group stable if and only if it is pairwise stable. So we ll refer to pairwise stability as just stability. 3

4 Proof: suppose a matching µ was pairwise stable but not group stable. (The other direction is trivial.) Pick a college C A. Since C strictly prefers its new match µ to its old match µ, there must be some student s in its new match that is strictly preferred to some student s in its old match. (If this was not the case, we could generate a contradiction using the fact that preferences are responsive and transitive.) But this means that college C and student s block the matching µ. (Note that group and pairwise stability are only equivalent under responsive preferences. If we allowed for more general college preferences, we would need to consider blocking coalitions consisting of one college and a group of students, but those would still be the only coalitions we had to worry about. That s part of why many-to-one is still so much easier than manyto-many under many-to-many, since there would be overlaps, we d have to worry about coalitions of multiple agents from each side of the market.) The definition of group stability suggests a relationship between stable matchings and the core of the game Not sure whether you guys covered the core in the firstyear sequence The core is a solution concept from cooperative game theory, which is the analysis of games where payoffs are a function of how players group themselves into coalitions In this context, for any two feasible outcomes x and y, we say x dominates y if there exists a coalition of players S such that every member of S strictly prefers x to y the members of the coalition can achieve x by trading among themselves And we define the core as the set of undominated outcomes, that is, the set of outcomes where no coalition can all do strictly better by leaving the game together In a marriage market, individual rationality of an outcome just means there are no coalitions of size 1 that can improve on the outcome by staying alone And stability just means there are no coalitions of size 1 or 2 that can In a marriage market, these are the only coalitions that matter, so the core of a marriage market is exactly the set of stable matchings In the college admissions model, things are a little more complicated In order to improve on a given outcome, some college may want to keep some of the students it s already matched to, and add some others it s not The problem: under the definition above, the students it s keeping can t be a part of the deviating coalition, because they don t get strictly higher payoffs 4

5 But the college can t achieve its higher payoff without those students, so we have a problem (If we had money, there would be no problem the college could just pay those students $1 more each and include them in the coalition) But without money, we need to define a new type of core We will say a matching µ weakly dominates µ if there is a coalition A C S such that for all s A, µ (s) A, and for C A, if s µ (C), s A µ (s) s µ(s) for all s A, µ (C) C µ(c) for all C A There is some s or C in A that does strictly better So weak dominance just allows coalitional deviations which are just weak Pareto-improvements to the coalition members And the core defined by weak domination C W (P ) is the set of matchings which are not weakly dominated by any other matching Since weak dominance is easier, more matchings are ruled out, so C W (P ) C(P ) the usual core (With strict preferences, the two are equivalent for marriage markets, but not for college admissions markets) And for college admissions markets, with strict preferences, the weak core is equal to the set of stable matchings 5

6 Roth and Sotomayor go on to consider the algorithm used since 1951 to match medical interns to hospitals. It feels a little bit like the deferred acceptance algorithm with hospitals proposing, but it s not the same It does similarly leads to a stable matching Students and hospitals submit strict rank-ordered lists over individuals (in particular, hospitals do not report preferences over groups of students, just individual students and how many spots they have) The lists are edited to remove hospitals from a student s list which don t find that student acceptable, and vice versa The first step: look for student-hospital pairs where the student ranked the hospital first, and the hospital ranked the student among their top q choices, where q is their capacity (so that functionally, the student was one of their top choices) If there are any such pairs, this round ends; if not, look for pairs where the student ranked the hospital second, and the hospital ranked the student among their top choices. If there are none, look for pairs where the student ranked the hospital third. Keep going till you find some. At that point, those students are tentatively matched to those hospitals, and are guaranteed of doing at least that well Now, shorten those students preference list to exclude every hospital they like less than their tentative match; and similarly, cross those students off of hospitals preference lists who the student ranks lower This means that some hospitals lose some of their top q choices, so there are new students within the top tier of their list; return to the top, looking for new tentative matches When there are no new tentative matches, the algorithm ends Consider the following example, from Roth and Sotomayor (p 139). Hospital 1 has a quota of 2, the other two hospitals have quotas of 1: s 1 : H 3 H 1 H 2 s 2 : H 2 H 1 H 3 s 3 : H 1 H 3 H 2 s 4 : H 1 H 2 H 3 H 1 : s 1 s 2 s 3 s 4 H 2 : s 1 s 2 s 3 s 4 H 3 : s 3 s 1 s 2 s 4 6

7 In stage 1 of the algorithm, we note none of the hospitals top choices (s 1 and s 2 for H 1, s 1 for H 2, s 3 for H 3 ) rank that hospital first. s 1 and s 2 both rank H 1 second, and s 3 ranks H 3 second, so the first tentative match is H 1 to {s 1, s 2 }, and H 3 to s 3. Given that, we shorten the preference lists: s 1 : H 3 H 1 s 2 : H 2 H 1 s 3 : H 1 H 3 s 4 : H 1 H 2 H 3 H 1 : s 1 s 2 s 3 s 4 H 2 : s 2 s 4 H 3 : s 3 s 1 s 4 In the next round, s 2 and H 2 are now mutual first choices, so they tentatively match. We adjust preferences to s 1 : H 3 H 1 s 2 : H 2 s 3 : H 1 H 3 s 4 : H 1 H 2 H 3 H 1 : s 1 s 3 s 4 H 2 : s 2 s 4 H 3 : s 3 s 1 s 4 In the next round, s 3 and H 1 tentatively match, so H 3 loses s 3, and we adjust preferences to s 1 : H 3 H 1 s 2 : H 2 s 3 : H 1 s 4 : H 1 H 2 H 3 H 1 : s 1 s 3 s 4 H 2 : s 2 s 4 H 3 : s 1 s 4 In the next round, s 1 and H 3 match, so H 1 loses s 1, and we adjust preferences to s 1 : H 3 s 2 : H 2 s 3 : H 1 s 4 : H 1 H 2 H 3 H 1 : s 3 s 4 H 2 : s 2 s 4 H 3 : s 1 s 4 7

8 In the final round, H 1 grabs s 4, leaving H 1 matched to s 3 and s 4, H 2 with s 2, and H 3 with s 1, and leaving every student at their top choice of hospital A couple positive results about this algorithm: Theorem. This algorithm always leads to a stable matching (with respect to the reported preferences) Theorem. This algorithm gives every hospital its k highest-ranked achievable students (an even stronger condition than it being the hospital-optimal stable matching) This means a hospital-optimal stable matching exists. matching exists. Similarly, a student-optimal stable It turns out, many of the nice results from marriage markets extend to the students in a college admissions model, but not to the colleges For example, in a marriage market, we found that the men-optimal stable matching was weakly Pareto optimal for the men: there was no other individually rational matching, stable or not, that the men all strictly preferred Here, the student-optimal stable matching is weakly Pareto optimal for the students; but the same need not hold for the hospitals Take the example we just did, where the hospital-optimal stable matching turned out to be H 1 matched to s 3 and s 4, H 2 with s 2, and H 3 with s 1 Compare this to the alternative matching of H 1 to s 2 and s 4, H 2 to s 1, and H 3 to s 3, which is individually rational Under the new matching, hospitals 2 and 3 now get their favorite student, so they re strictly better off; H 1 prefers s 2 to s 3, so with any responsive preferences, it prefers {s 2, s 4 } to {s 3, s 4 }, so all three hospitals do strictly better 8

9 On to strategic concerns. Recall in the marriage model, when the deferred acceptance algorithm was being run with the men proposing, the men had a dominant strategy to report their true preferences, while the women did not Here, again, this extends for the students but not for the hospitals Theorem. When the student-optimal stable matching is being implemented, it is a dominant strategy for students to report their true preferences Theorem. No stable matching mechanism exists which makes it a dominant strategy for all hospitals to report their true preferences (Roth and Sotomayor use the example we used earlier, and showed that Hospital 1 could gain by misreporting.) They also negate the coalitional result from the marriage model: In a college application model, there can be a coalition of agents who all do strictly better by jointly misreporting their preferences. 9

10 Roth and Sotomayor mention one complication to the intern matching program: married couples If a husband and wife are finishing medical school at the same time and want jobs in the same city, this ends up being extremely difficult to accomodate The old system (up to the 1980s) was: the couple would submit two preference lists, one for each of them, and specify a leading member. The leading member would be matched in the usual way. Then the other member would have their list shortened by removing all jobs that were too far away, and then matched in the same large area as the spouse. Not that great a system, since they were not guaranteed jobs in the same city, just the same part of the country, even if two jobs in the same city might have been achievable As a result, many couples just interviewed for jobs outside the centralized match Roth and Sotomayor give a result that basically, this problem is hard. If you think of a couple as two students with joint preferences over joint outcomes, then in the hospital-intern problem with couples, a stable matching may not exist. One objection to the centralized matching procedure was from some hospitals, particularly in rural locations, that had trouble filling positions, and felt they might be getting screwed by the algorithm Recall that in a marriage model, we saw that the same set of individuals ended up married under all stable matchings; all that changed was who they were married to Here, we get even stronger results for rural hospitals: Theorem. When preferences are strict, the set of students hired, and the set of positions filled, is the same in any stable matching. Theorem. When preferences are strict, any hospital that does not fill its quota in some stable matching, gets the same set of students in any stable matching. By the way, I keep insisting preferences are strict. In fact, we can weaken this a little. What is required is that preferences over individuals are strict. That is, students have strict preferences over hospitals, and hospitals have strict preferences over individual students. However, hospitals are allowed to have indifferences between sets of students. (For example, a hospital could be indifferent between getting its first and fourth choice students, or its second and third.) In the intern matching algorithm, recall, hospitals only rank individual students, not combinations, so this wouldn t even be detected! 10

11 Turning back to the set of stable matchings, it turns out the lattice results from before do extend to this setting We said that colleges must have strict preferences among individual students, but are allowed to have indifferences among collections of students This turns out not to matter: Roth and Sotomayor give a lemma that colleges will always have strict preferences among the collections of students they might get in a stable matching (That is, in the example I gave before: there cannot be one stable matching that gives a particular college its first and fourth choices, and another that gives it its second and third.) In fact, if there are two stable matchings µ and µ and college C prefers µ to µ, then it prefers every student in µ(c) to every student in µ (C) µ(c) Thus, the college s preferences over individuals are all that really matter they specify the college s preferences over relevant outcomes Theorem. If all colleges prefer µ to µ, then all students prefer µ to µ Corollary. So the college-optimal match is the worst for the students, and vice versa Similarly, given two stable matchings µ and µ, we can define their supremum and infimum as before, using one side s preferences The sup and inf, then, turn out to also be matchings, and also stable So the set of stable matchings is closed under sup and inf, and therefore a lattice I ll stop here for today. Thursday, we introduce money. 11