We set up the basic model of two-sided, one-to-one matching

Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 18 To recap Tuesday: We set up the basic model of two-sided, one-to-one matching Two finite populations, call them Men and Women, who want to match to mates from the other group (or stay alone) Strict preferences over the other group or being alone We defined a matching, as a pairing up of (some) men and women We said a matching was stable if nobody is paired up who would rather be alone no pair (m, w) would both rather be with each other than with their assigned mate (note how helpful strict preferences are in simplifying these conditions...) We proved the result from Gale and Shapley, who proved the existence of a stable matching for any set of preferences, by providing an algorithm that finds one (step through the deferred acceptance algorithm) We showed that not only does the deferred acceptance algorithm end on a stable matching, but when the men are proposing, it ends on the stable matching that every man in the population weakly prefers to every other stable matching (the men-optimal stable matching) And we showed that if all the men unanimously (weakly) prefer a stable matching µ to another stable matching µ, then all the women weakly prefer µ to µ So all the players on one side of the market have aligned preferences (over stable matchings, and roughly they all agree on which matching is best); but their preferences are opposed to the preferences of the players on the other side of the market (if there is more than one stable matching) We wrapped up by introducing the notion of a lattice, and claiming that the set of stable matchings formed a lattice, but we didn t have time to prove it, so that s where we ll pick up today 1

Lattice Theorem Tuesday, we got nearly all the way through proving the Lattice Theorem. For a general set with a partial order, we can define the meet of two points as their least upper bound that is, the point (if it exists) which is lower than every point which is higher than both x and y And we can define the join as the greatest lower bound the point (if it exists) which is higher than every point which is lower than both x and y A lattice is any set, with a partial order, which is closed under meet and join that is, a partially-ordered set is a lattice if for any two points in their set, the subset of points that are greater than both has a minimal point, and the subset of points which are less than both has a maximal point What we re trying to show is that with the partial order provided by the preferences of the men that is, one matching is weakly greater than another if every man weakly prefers it the set of stable matchings of a given marriage market turns out to be a lattice Formally, take some marriage market, and let µ and µ be two stable matchings. Define a new mapping λ : M W M W, such that λ(m) is whichever of µ(m) and µ (m) is preferred by m λ(w) is whichever of µ(w) and µ (w) is least-preferred by w So we can think of λ as the pairwise max of µ and µ, from the mens perspective (and the pairwise min from the womens perspective). Then it turns out that λ is not just an arbitrary map, but it s a matching; and it turns out to be a stable matching. We could do the same thing with a new mapping ν which matches men to the worse of µ(m) and µ (m), and women to the better of µ(w) and µ (w), and this would also be a stable matching This means that for any two stable matchings µ and µ, sup(µ, µ ) and inf(µ, µ ) are also stable matchings so the set of stable matchings is closed under sup and inf, and is therefore a lattice which gives us a bunch of nice mathematical structure 2

Last class, we managed to slog through the proof that λ is a matching that is, that w = λ(m) if and only if m = λ(w). We still need to show it s a stable matching. That is, we need to show that if µ and µ are stable matchings, then λ is individually rational and unblocked. Individual rationality follows from individual rationality of µ and µ. Whoever you match to under λ, you must have matched to under either µ or µ ; so if those are IR, so is λ. So now, suppose there was a pair (m, w) that blocked λ. If m prefers w to λ(m), then by definition, he prefers w to both µ(m) and µ (m). And if w prefers to m to λ(w), then she prefers to m to at least one of µ(w) and µ (w) If she prefers m to µ(w), and he prefers her to µ(m), then they would have blocked µ; similarly with µ in the other case So λ is a stable matching It s not hard to show that it s the minimal matching that is men-preferred to both µ and µ. Every man just does exactly as well under λ as under his favorite of µ and µ ; so be any other matching (stable or not) which is preferred by all men to µ and µ, has to be bigger than λ. So λ is the least upper bound of µ and µ ; since it s a stable matching, the set of stable matchings is closed under the supremum operator And we can do the same with ν, the infimum And so we learn that the set of stable matchings is a lattice (It actually has some additional properties as a lattice technically, it s a complete, distributive lattice.) Roth and Sotomayor give an example, taken from Knuth (where the lattice theorem was proved first, I think), of a marriage market (four men, four women, and a set of preferences) which has 10 stable matchings, and shows how they are arranged according to the partial order: µ 1 {µ 2, µ 3 } µ 4 {µ 5, µ 6 } µ 7 {µ 8, µ 9 } µ 10 While you could come up with an example where there are more than two matchings in an indifference class, the rest of this structure has to hold in a lattice for any indifference class with more than one point, there must be a single point in the next group up, and in the next group down otherwise, there would not be a well-defined sup or inf 3

Another cool result. Roth and Sotomayor give this as a corollary of a different result, but it follows pretty directly from the lattice result we just proved. Take two matching µ and µ, and let M and M be the set of men who match to some woman (don t end up alone) under each, and W and W likewise Let λ be the meet of µ and µ, as defined above; let M λ and W λ be the men, and women, who end up matched up λ Now, if m matches under µ, it s because he prefers his mate to being alone; so if he matches under µ, he matches under λ. Similarly if he matches under µ. So M λ = M M But by the same logic, if a woman w ends up alone at either µ or µ, that s her worse result, so she ends up alone at λ, so W λ = W W This means that M λ is at least as big as the bigger of M and M, and W λ is at least as small as the smaller of W and W But the men who match under µ, match to the women who match under µ, so M and W are the same size; same with M and W So M λ max{ M, M } min{ M, M } = min{ W, W } W λ But λ is a matching, so the same number of men and women match under λ; so M λ = W λ. Which means that max{ M, M } = min{ M, M }, which means M = M And so the same number of men match to women under µ and µ or any two stable matchings have the same number of couples Which is surprising, but we re not done yet Since M λ = M M, M λ M ; and since W λ = W W, W λ W = M ; so M λ M = W W λ = M λ so M λ = M = W λ But M λ = M M it s the union of M and another set the same size as M, but it s the same size as M which means that M and M must have complete overlap that is, M = M And similarly, W λ is the same size as W, but it s the intersection of W and another set the same size, so W = W 4

So the set of men who end up married is the same under any two stable matchings; and the set of women who end up married is the same for any two stable matchings So if we look at any two stable matchings, the same men end up matched, and the same women end up matched; all that s different is who matches to whom. Roth and Sotomayor give a cool result called the decomposition lemma. Take any two stable matchings µ and µ Let M µ be the set of men who strictly prefer µ, and M µ the set who strictly prefer µ Similarly, let W µ and W µ be the set of women who prefer each matching Then the men in M µ match with the women in W µ, under both µ and µ (Since preferences are strict, the men who are indifferent between them must get the same mate in either; same with the women; so the men who are indifferent match with the women who are indifferent. We already knew that the men who are alone under µ are also alone under µ, same with the women we proved that earlier today.) Roth and Sotomayor (ch 2.4-2.5) give show some additional comparative statics basically, comparing outcomes in one marriage market to those in another marriage market with a very similar structure (change one side s preferences, or add one woman to the market, things like that); they also spend some time on what breaks down and what still works when preferences are not strict 5

They also point out that we can write any matching (stable or not) as an M W matrix of 1 s and 0 s where a 1 in place (m, w) indicates that that man and that woman match to each other For simplicity, they assume that M = W and everyone is acceptable to everyone, so we don t have to worry about people staying single Feasibility is simply the requirement that j x mj = 1 and i x iw = 1 each man matches to only one woman, each woman matches to only one man Stability is the constraint that for any (m, w), x mj + x iw + x mw 1 j mw i wm since this says that either m and w match to each other; or the man matches to someone he likes more than w; or the woman matches to someone she likes more than m So stable matchings are exactly the set of all integer matrices satisfying the constraints j x mj = 1 for every m i x iw = 1 for every w j m w x mj + i w m x iw + x mw 1 for every (m, w) x mw 0 for every (m, w) We can take away the constraint of integer values, and consider C, the convex polyhedron consisting of all points in R M W that satisfy these four constraints. It turns out that the integer points are exactly the extreme points of this polyhedron; and so the set of stable matchings are characterized as the set of extreme points corners of this polyhedron Chapter 3 of the textbook also offers algorithms for calculating every stable matching, every achievable pair, and so on But I want to move on to strategic questions 6

So far, we ve focused on mapping preferences to outcomes, without worrying about how we know everyone s preferences That is, everything we ve done so far has been for the full information case where everyone s preferences are common knowledge, and we just need to get from preferences to a matching But now suppose that peoples preferences are private information is it an equilibrium for them to truthfully reveal them? (We haven t really taken a stand yet on how we re implementing a stable matching. That is, if we talk about running the Gale-Shapley algorithm, we could actually get all the men and women in a big room and have the men start proposing; or, we could have everyone write down their preference list, hand it to someone (say, Al Roth), and let him run the Gale-Shapley algorithm himself and just tell everyone who they ended up with. In the latter case, the strategic question is, can we expect truthful revelation? Is reporting your true preferences to the matchmaker an equilibrium, or even a dominant strategy? In the former case, the strategic question is, will players play the way we would expect them to, that is, according to their true preferences? Or will they imitate a player with different preferences? But because they re outcome-equivalent, the questions are the same. So we answer both of them at once.) 7

The headline results on strategic concerns: Suppose we are using the men-proposing deferred acceptance algorithm, run using the players reported preferences Then it is a dominant strategy for the men to report their true preferences. But it is not for women. In fact, as long as there is more than one stable matching, there will always be at least one woman who can gain by misrepresenting her preferences (assuming she knows everyone else s preferences and everyone else is telling the truth) And this problem is not particular to the rule of using the Gale-Shapley algorithm to choose a stable matching: There is no rule for mapping preferences to stable matchings under which it is always a best-response for every player to report truthfully A lot of this comes from the paper by Al Roth, The Economics of Matching: Stability and Incentives, and is covered as well in the book They do point out that there are ways of mapping preferences to Pareto-efficient outcomes which make it an equilibrium for everyone to report their true types, but that these rules do not always lead to a stable matching An example of such a rule: rank the men in some order (say, alphabetically). Match the first guy to his first choice. Match the second guy to his first choice of the remaining women. Match the third guy to his first choice of the remaining women. And so on. It s clear it s a dominant strategy for the men to report truthfully And womens reports don t matter, so they can t gain by misreporting And it s clear that this outcome is Pareto-efficient any change would leave some man strictly worse off But this outcome is not necessarily a stable matching in fact, it doesn t even necessarily lead to an individually rational matching (some women might rather be alone than with the man who chose them) But the more interesting question choosing a stable matching for each set of preferences cannot be done without creating an incentive for someone to misreport They give a theorem: if, under the true preferences, there is more than one stable matching; then under any mechanism that selects a stable matching, someone can gain by misrepresenting their preferences (if they know everyone s preferences and everyone else reports truthfully). 8

The proof is simple. Suppose that at some true set of preferences P, the mechanism selects a matching µ which is different from µ W. (If there are more than one stable matching, µ M µ W ; if the mechanism selects µ W, make the same argument for the men.) Let w be some woman who strictly prefers µ W to µ. Now suppose she misrepresents her preferences by truncating the list of acceptable mates at µ W (w); that is, she truthfully reports the top of her preference list, but claims that anyone worse than her best achievable mate is unacceptable Let P be the original preferences, and P the manipulated preferences If µ W was stable under P, it must also be stable under P there are fewer pairs that could potentially block it So now there s a stable matching under P in which w matches to µ W (w) But we know that if w matches under any stable matching, she matches under all stable matchings So in any stable matching under P, she matches to someone But the only men she claimed are acceptable are the ones she likes at least as much as µ W (w) So by misreporting her preferences, she matches to someone she likes at least as much as her best achievable mate under P (Having done the proof, we can skip the example where a woman can gain by misreporting her type when the men-optimal stable match is chosen: three men, three women, all mates acceptable: P (m 1 ) : 2 1 3 P (m 2 ) : 1 2 3 P (m 3 ) : 1 2 3 P (w 1 ) : 1 3 2 P (w 2 ) : 3 1 2 P (w 3 ) : 1 2 3 If we run Gale-Shapley with the men proposing, we find the men-optimal stable match x : m 1 w 2, m 2 w 3, m 3 w 1. If we run it with the women proposing, we find the women-optimal stable match y : m 1 w 1, m 2 w 3, m 3 w 2. But now suppose woman 1 lies about her preferences, and claims P (w 1 ) : 1 2 3. Under this new set of preferences (everyone else reports P, but w 1 reports P ), there is a single stable matching, y So by misreporting her preferences, woman 1 can force the outcome y instead of x) 9

Back to the men we want to show it s a dominant strategy to report truthfully. We need a couple of lemmas to get there. Lemma 1. Let µ be any individually rational matching w.r.t. strict preferences P, and let M be the set of men who prefer µ to µ M. If M is nonempty, there is a pair (m, w) that blocks µ such that m M M and w µ(m ). First case: µ(m ) µ M (M ). Pick w µ(m ) µ M (M ), that is, w is some woman who matches to a man m M under µ but to a man m / M under µ M. m prefers w to µ M (m ), so for µ M to be stable, w must prefer m to m. But since m / M, m does not prefer µ to µ M, and since he matches to different women and preferences are strict, he must prefer w to whoever he matches to under µ. So (m, w) block µ. Second case: µ(m ) = µ M (M ) = W Consider the men-proposing deferred acceptance algorithm, and let w be the last woman in W to receive a proposal from an acceptable member of M. Since she s in W, her match under µ is some man in M, call him m We claim that w must have already rejected m before she got her last proposal from an acceptable member of M How do we know this? We know m prefers w to µ M (m ), so he would propose to her first under the deferred acceptance algorithm. Since he eventually proposes to (and ends up with) µ M (m ), he must have proposed to w already, and been rejected But if w only rejects m when she gets her last acceptable proposal from a man in M, then m s proposal to µ M (m ) would have come later, which contradicts how we chose w So w rejected m before she got her last proposal from an acceptable member of M ; this means that by the time that last proposal happened, she must have been holding onto a proposal from someone else, let s call him m. (So what we know: at some point, m proposed to w, and was (at some point) rejected; at some point, m proposed to w, and was still tentatively accepted when µ M (w) proposed to her, and that s who she ended up with.) Now, m prefers w to µ M (m), since he only proposed to µ M (m) after w rejected him. And we know m / M, since he later proposed to µ M (m) W, and we assumed w was the last woman in W to receive an acceptable proposal from a man in M. So since m / M, he doesn t prefer µ to µ M, so he prefers w to his match under µ. But we saw that w rejected m and was later engaged to m, so she must prefer m to m, which is her match under µ So m and w prefer each other to their matches under µ, so they block µ. 10

Lemma 2. (Limits on Successful Manipulation) Let P be true preferences, and let P differ from P in that some coalition C of men and women misstate their preferences. Then there is no matching µ which is stable under P, which is strictly preferred to every stable matching under P by every member of C. Suppose some subset C = M W misstate their preferences, and all are strictly better off under µ than under any stable matching under P If µ is not individually rational under the true preferences, someone is matched to an unacceptable mate; that person must be in C, since everyone else is reporting their true preferences; but that person does worse than under any stable matching under P ; so µ is IR under P. By definition, µ(m) m µ M (m) for every m M If M is nonempty, apply the blocking lemma to (M, W, P ). µ is an individually rational matching and some men prefer µ to µ M, so there is a pair (m, w) that block µ, with m M M and w µ(m ), where M is the set of men who prefer µ to µ M under P Since m does not prefer µ to µ M under the true preferences, he can t be in M, so he s reporting his true preferences under P Let m = µ(w). Since m prefers w to µ M (m ), for µ M to be stable, w must prefer µ M (w) to m. So w prefers any stable match to µ, so she s not in W, so she s also stating her true preferences under P So m and w block µ under the true preferences, but report their preferences truthfully, so they block µ under P. We assumed M was nonempty; if it s empty, W must be nonempty, and the symmetric argument works. 11

This lemma s pretty powerful. Among its corollaries: If we re selecting the men-optimal stable matching, truthful reporting is a dominant strategy for men. Why? Pick a man m. Let P denote m s true preferences, along with whatever preferences everyone else is reporting. Let C = {m}, and P denote m s misreported preferences, along with everyone else s reported preferences. Limits on Successful Manipulation says that if everyone in C prefers µ to their best stable match under P, µ can t be stable under P So if µ gives m a better match than µ M, it s not stable under P Similarly: There is no individually rational matching that all men strictly prefer to µ M. To see this, let C = M, and let each man report that only one woman is acceptable, so the only stable matching is for each to get his choice The last lemma says that if this is individually rational and every man does strictly better than µ M, it s not stable under the reported preferences 12

So that s the men. On the other hand, when the men-optimal stable match is being chosen, the women can potentially gain a lot by misreporting For example: let µ W be the woman-optimal stable match under the true preferences Suppose the men play their dominant strategy (report their true preferences), and the women truncate their lists below µ W (w); that is, every woman w who matches to a man µ W (w) under the woman-optimal stable match, reports that that man is the lowest she d be willing to go, that is, anyone below him on her preference list is unacceptable. This is an equilibrium, and it leads to µ W. (In fact, it s a strong equilibrium no coalition of women can all do strictly better by jointly misreporting.) (There are also equilibria which lead to any stable matching µ under the true preferences.) It turns out: when the M-optimal stable matching is being chosen, lying about your first choice is a dominated strategy for the women; and putting men on your list who aren t actually acceptable is dominated as well. But that s all we really know. (No other strategy is dominated; so any other strategy is a best-response to something.) The book then goes on to talk a bit about strategies when the players are really playing a Bayesian game, that is, other players preferences are private information; but they re not able to say very much Basically, dominant and dominated strategies extend so given incomplete information, the men can still report truthfully, and the women still shouldn t lie about their top choice but they can t really say much else To me, this is the important direction to extend things: looking at how much information is required to successfully manipulate the outcome. If a mechanism is only manipulable by someone who knows everyone else s true preferences and the market is pretty big, that (seems to me) makes it pretty robust. On the other hand, if you need very little information to successfully gain from misreporting, that makes it a lot less robust. I m being vague here I think some work has already been done in this direction, but I think it s an interesting question. (When we do many-to-one matching next week, we ll see a result on big markets that as the number of players on both sides gets big, the fraction of players who could gain from manipulation gets small.) 13