Market Design and Analysis Lecture 1 Lecturer: Ning Chen ( ) 2 Class Information Focuses economic models and solution concepts computational aspects incentive analysis 3 References Roth, Sotomayor, Two-Sided Matching: A Study in Game-Theoretical Modeling and Analysis, Cambridge Press, Nisan, Roughgarden, Tardos, Vazirani, Algorithmic Game Theory, Cambridge Press, Internet (Google, Wikipedia, etc.) Fuhito Kojima (http://sites.google.com/site/fuhitokojimaeconomics/)http://sites.google.com/site/fuhitokojimaeconomics/ 4 Lets start ! 5 Motivating Example School Admission Students have preferences over different schools and departments every student goes to one school/department Schools and departments also have preferences over students school/department seats are limited How to decide the admission process globally? 6 Motivating Example American Hospital- Intern Market Medical students work as interns at hospitals. In the US more than 20,000 medical students and 4,000 hospitals are matched through a clearinghouse, called NRMP (National Resident Matching Program). Doctors and hospitals submit preference rankings to the clearinghouse, who uses a specified rule to decide who works where. What is a good way to match students and hospitals? 7 Motivating Example Kidney Exchange Medical transplant matches kidney donors and patients A successful transplant must have compatible blood test there are four blood types: O, A, B, AB O patients can receive kidneys from O donors A patients can receive kidneys from O or A donors B patients can receive kidneys from O or B donors AB patients can receive kidneys from all donors Kidney exchange: match two (or more) incompatible donor-patient pairs and swap donors. How to find efficient exchanges? 8 NTU Class Registration System Students submit a preference for UE and PE courses (up to five). Courses have implicit priorities over students according to, e.g., year of study. How to assign students to courses so that most students are happy? 9 Matching Markets Input: Two heterogeneous sets of agents form a two- sided market Output: Set up matches between agents of different sides Other examples dormitory allocation marriage job market advertising market (TV, newspaper, Internet) 10 Internet Advertising keywords sponsored links 11 Matching Markets Design Mathematical models Economic solution concepts computational issues mathematical properties economic properties Graph Definition 1.1. A graph G = (V, E) consists of V: a non-empty set of vertices (or nodes) E: a set of pairs of distinct elements of V called edges. Two vertices u and v are called adjacent (or neighbors) if (u,v) E. Example V = {v 1, v 2, v 3, v 4, v 5 } E = {(v 1,v 2 ), (v 1,v 3 ), (v 1,v 5 ), (v 2,v 4 ), (v 3,v 5 )} v5v5 v4v4 v1v1 v3v3 v2v2 Bipartite Graph A graph G = (V, E) is called bipartite if its vertex set V can be partitioned into two disjoint sets V 1 and V 2 such that every edge in the graph connects a vertex in V 1 and a vertex in V 2. That is, there are no edges in V 1 and V 2. V1V1 V2V2 Matching Given a bipartite graph G = (V 1,V 2 ; E), a matching of G is a subset of edges E such that for any e, e E, they do not have the same endpoints. The number of edges in E, i.e. |E|, is called the size of the matching E. Matching Example. V1V1 V2V2 Maximum and Perfect Matching A matching E of a bipartite graph G is called maximum if it has the largest size of all matchings of G. In a given a bipartite graph G = (V 1,V 2 ; E), if |V 1 |=|V 2 |=n and the maximum matching E of G has size n, then E is called a perfect matching. V1V1 V2V2 17 Lets start (again) ! 18 Gale-Shapley Marriage Model There are a set of men M and set of women W, where |M| = |W| = n. Each man m has a strict preference over women in W (denoted by > m ). Each woman w has a strict preference over men in M (denoted by > w ). m1m1 w2w2 w1w1 m2m2 w 1 > w 2 m 1 > m 2 19 Gale-Shapley Marriage Model Preferences are required to be complete: any two alternatives can be compared strict: strict preference over any two alternatives transitive: if w 1 > m w 2 and w 2 > m w 3, then w 1 > m w 3 20 A Marriage Problem Question: How to match men and women in M and W such that everyone is happy with the solution? w 1 > w 2 m 1 > m 2 m1m1 m2m2 w2w2 w1w1 21 Blocking Pair A matching of an instance (M,W) is a set of disjoint edges, denoted by f: M W, i.e., f(m) is the woman matched to m M. Given a matching f, we say a man-women pair (m,w) is a blocking pair if w > m f(m) and m > w f(w). w f(m)m f(w) 22 Blocking Pair A matching of an instance (M,W) is a set of disjoint edges, denoted by f: M W, i.e., f(m) is the vertex matched to m M. Given a matching f, we say a man-women pair (m,w) is a blocking pair if w > m f(m) and m > w f(w). w 1 > w 2 m 1 > m 2 w2w2 w1w1 m1m1 m2m2 (m 1,w 1 ) is a blocking pair 23 Stable Matching A matching f is called stable if there it has no blocking pair. Questions: Does a stable matching always exist? If yes, how to find one? What mathematical / economic properties it has? 24 Stable Matching Theorem 1.2. [Gale & Shapley1962] For any stable marriage problem, there always is a stable matching. 25 Example m 3 > m 1 > m 2 > m 4 m 3 > m 4 > m 1 > m 2 m 1 > m 4 > m 2 > m 3 m 4 > m 1 > m 3 > m 2 m1m1 m2m2 m3m3 m4m4 w2w2 w1w1 w4w4 w3w3 w 1 > w 2 > w 3 > w 4 w 2 > w 1 > w 3 > w 4 w 3 > w 2 > w 4 > w 1 26 Gale-Shapley (Deferred-Acceptance) Algorithm Initially all men and women are free While there is a man m who is free and hasnt proposed to every woman choose such a man m arbitrarily let w be the highest ranked woman in ms preference list to whom m hasnt proposed yet m proposes to w if w is free, then (m,w) become engaged else, w is currently engaged to m if w prefers m to m, then m remains free if w prefers m to m, then (m,w) become engaged and m becomes free 27 Example m 1 > m 2 > m 3 m 1 > m 3 > m 2 m 1 > m 2 > m 3 \\ \ w 1 > w 2 > w 3 \ m1m1 m2m2 m3m3 w2w2 w1w1 w3w3 \ w 1 > w 3 > w 2 28 Analysis of the Algorithm To prove correctness of an algorithm analyze convergence of the algorithm (i.e., show that the algorithm will always terminate) analyze correctness of the algorithm (i.e., show that the algorithm always generates the desired outcome) 29 Analysis of the Algorithm Observations For any woman w, (O1) w remains engaged from the point at which she receives her first proposal. (O2) the sequence of partners to which w is engaged gets better and better (in terms of her preference list) For any man m, (O3) if m is free at some point in the execution of the algorithm, then there is a woman to whom m hasnt proposed yet. (O4) the sequence of women to whom m proposes gets worse and worse (in terms of his preference list). 30 Analysis of the Algorithm Lemma 1.3. G-S algorithm returns a perfect matching in finite steps. Proof. By observations O1 and O3. Theorem 1.4. G-S algorithm returns a stable matching. 31 Analysis of the Algorithm Proof. Let f be a matching returned by the algorithm. Assume that (m,w) is a blocking pair, where (m,w),(m,w) f. That is, m prefers w to w and w prefers m to m. In the algorithm, m last proposal was to w (by definition). Then if m has proposed to w or not? if yes, since the sequence of partners of w only increases (O2), w will be matched to a man better than m if not, by the algorithm, m should propose to w before w (O4) A contradiction. m mw w 32 G-S Algorithm Women Propose \ \ \ w 1 > w 2 > w 3 w 1 > w 3 > w 2 \ w 1 > w 2 > w 3 m1m1 m2m2 m3m3 w2w2 w1w1 w3w3 \ m 1 > m 2 > m 3 m 1 > m 3 > m 2 m 1 > m 2 > m 3 33 Which Stable Matching is Better? w 1 > w 2 w 2 > w 1 m 2 > m 1 m 1 > m 2 m1m1 m2m2 w2w2 w1w1 m1m1 m2m2 w2w2 w1w1 m1m1 m2m2 w2w2 w1w1 GS algorithm: men propose GS algorithm: women propose 34 Stable Matching by G-S For any man m, let best(m) be the best woman matched to m in all possible stable matchings. Theorem 1.5. Gale-Shapley algorithm, when men propose, returns a stable matching, where for any man m, m is matched to best(m). Implications: different orders of free men picked do not matter for any men m 1 m 2, best(m 1 ) best(m 2 ) 35 Stable Matching by G-S Proof. Assume otherwise that some men m are matched worse than their best(m). Then m must be rejected by best(m) in the course of the algorithm. Consider the first moment in the algorithm in which some man, say m, is rejected by w = best(m). The rejection of m by w because either m proposed but was turned down (w prefers her current partner) or w broke her engagement to m in favor of a better proposal. In either cases, at this moment w is engaged to a man m whom she prefers to m, i.e., m > w m. m m w =best(m) 36 Stable Matching by G-S In GS algorithm, because m is the first man who is rejected by best(m), at that moment m hasnot been rejected by best(m) when he is engaged to w. By O4, this implies that w m best(m) m m w =best(m) 37 Stable Matching by G-S In GS algorithm, because m is the first man who is rejected by best(m), at that moment m hasnot been rejected by best(m) when he is engaged to w. By O4, this implies that w m best(m) By definition of best(m), consider the stable matching f where m is matched to w=best(m). Assume that m is matched to w in f. Hence, best(m) m w Hence, w > m w (note that w w) Contradiction to the fact that f is stable. m m w w f f =best(m) 38 Men/Women Optimal Stable Matching For any two stable matchings f and f, denote f(m) > m f(m) if m prefers his partner in f than f f(m) m f(m) if either f(m) > m f(m) or f(m) = f(m) f > M f if f(m) m f(m) for all m M and f(m) > m f(m) for at least one man m. f M f if f(m) m f(m) for all m M f(w) > w f(w), f(w) w f(w), f > W f and f W f are defined similarly. 39 Men/Women Optimal Stable Matching Definition. A stable matching f is called men-optimal if for any other stable matching f, we have f M f. A stable matching f is called women-optimal if for any other stable matching f, we have f W f. For any stable matching f and any man m M, we have best(m) m f(m). Theorem 2.1. Gale-Shapley algorithm, when men propose, it returns a men-optimal stable matching; when women propose, it returns a women-optimal stable matching. 40 Men/Women Optimal Stable Matching Theorem 2.2. Men (women)-optimal stable matching is unique. Proof. Assume f and f are two men-optimal stable matchings, then f M f and f M f. Hence, for any man m M, we have f(m) m f(m) and f(m) m f(m), i.e., f(m) = f(m); this implies f = f. 41 Women-Pessimal Stable Matching Theorem 2.3. For any two stable matchings f and f, f > M f if and only if f > W f. Proof. Assume that f > M f, we will show f > W f. Assume otherwise that there is a woman w W such that f(w) > w f(w); let m = f(w). By the assumption, w = f(m) > m f(m). Hence, (m,w) is a blocking pair for f, a contradiction. f(w) f(w)m f(m) w = f f 42 Women-Pessimal Stable Matching Theorem 2.3. For any two stable matchings f and f, f > M f if and only if f > W f. Corollary 2.4. Men-optimal stable matching is women-pessimal; women-optimal stable matching is men-pessimal. 43 Pointing Function (sup) Given two stable matchings f and f, define a mapping g (denoted by f v f) as follows: for each man m M, assign him more preferred partner g(m) = f(m) if f(m) m f(m) g(m) = f(m) if f(m) > m f(m) for each woman w W, assign her less preferred partner g(w) = f(w) if f(w) w f(w) g(w) = f(w) if f(w) < w f(w) 44 Pointing Function (sup) Is it possible that g(m) = g(m) for two different men? Is it possible that g(m) = w, but g(w) m? If g is a matching (i.e., the answers to the above two questions are NO), can it be unstable? 45 Conways Lattice Theorem Theorem 2.5. If f and f are two stable matchings, then g = f v f is a stable matching as well. Proof. We first show that g is a matching. Assume g(m) = w, and wlog f(m) = g(m). Hence, w > m f(m). If g(w) m, i.e., g(w) = f(w), then m > w f(w). Thus, (m,w) is a blocking pair for f, a contradiction. That is, g(m) = w g(w) = m. Because |M| = |W|, this implies that g(w) = m g(m) = w. f(w) m f(m) w f f 46 Conways Lattice Theorem Theorem 2.5. If f and f are two stable matchings, then g = f v f is a stable matching as well. Proof. We next show that g is a stable matching. Assume (m,w) is a blocking pair for g. Then w > m g(m) and m > w g(w) The former implies w > m f(m) and w > m f(m) Therefore, (m,w) blocks f if g(w) = f(w), or f if g(w) = f(w), a contradiction. 47 Pointing Function (inf) Given two stable matchings f and f, define a mapping h (denoted by f f) as follows: for each man m M, assign him less preferred partner h(m) = f(m) if f(m) m f(m) h(m) = f(m) if f(m) < m f(m) for each woman w W, assign her more preferred partner h(w) = f(w) if f(w) w f(w) h(w) = f(w) if f(w) > w f(w) 48 Conways Lattice Theorem Theorem 2.5. If f and f are two stable matchings, then g = f v f and h = f f both are stable matchings. By the definition of g and h g > M f > M h, g > M f > M h h > W f > W g, h > W f > W g 49 Example m 4 > m 3 > m 2 > m 1 m 3 > m 4 > m 1 > m 2 m 2 > m 1 > m 4 > m 3 m 1 > m 2 > m 3 > m 4 m1m1 m2m2 m3m3 m4m4 w2w2 w1w1 w4w4 w3w3 w 1 > w 2 > w 3 > w 4 w 2 > w 1 > w 4 > w 3 w 3 > w 4 > w 1 > w 2 w 4 > w 3 > w 2 > w 1 50 w2w2 w1w1 w4w4 w3w3 f 1 m 1 m 2 m 3 m 4 f 2 m 2 m 1 m 3 m 4 f 3 m 1 m 2 m 4 m 3 f 4 m 2 m 1 m 4 m 3 f 5 m 3 m 1 m 4 m 2 f 6 m 2 m 4 m 1 m 3 f 7 m 3 m 4 m 1 m 2 f 8 m 4 m 3 m 1 m 2 f 9 m 3 m 4 m 2 m 1 f 10 m 4 m 3 m 2 m 1 f 2 v f 3 = f 1 f 2 f 3 = f 4 f 5 v f 6 = f 4 f 5 f 6 = f 7 f 8 v f 9 = f 7 f 8 f 9 = f 10 f2f2 f5f5 f8f8 f3f3 f6f6 f9f9 f1f1 f4f4 f7f7 f 10 51 Lattice Consider a set S which contains n elements with a partial order , which satisfies antisymmetry property: if a b and a b, then a = b. For any a,b S, if a b, we say a is greater than or equal to b if a b, we say a is smaller than or equal to b 52 Lattice Upper bound: An upper bound of any subset X of S is an element a S such that for all b X, we have a b. Denote by sup X the least upper bound of X if an upper bound exists. That is a = sup X if a is an upper bound of X and there is no other upper bound a of X such that a a. By the antisymmetry property, if sup X exists, its unique. Lower bound and greatest lower bound (denoted by inf X) are defined similarly. 53 Lattice Definition. A lattice is a partially ordered set S, where any two elements a,b S, have a sup, denoted by a v b, and have an inf, denoted by a b. A lattice S is complete if each of its subset has a sup and an inf in S. In particular, if lattice S if complete, then there is a sup S and inf S. 54 Lattice Examples. 55 Distributive Lattice Definition. A lattice S is distributive if for any a,b,c S, the following two facts hold a (b v c) = (a b) v (a c) a v (b c) = (a v b) (a v c) Theorem 2.6. (Conway) The set of all stable matchings forms a distributive lattice. Theorem 2.7. (Blair) Every finite distributive lattice equals the set of stable matchings of some marriage market. 56 Linear Structure of Stable Matchings For any given market, a matching f can be written by a matrix A = (a mw ) |M|x|W| (called configuration matrix),where a mw = 1 if f(m) = w a mw = 0 otherwise Let j >m w a mj denote the sum over all women j W that man m prefers to woman w i >w m a iw denote the sum over all men i M that woman w prefers to man m 57 Linear Structure of Stable Matchings Theorem 2.9. (Vande Vate) A matching is stable if and only if its configuration matrix is an integer matrix satisfying the following set of constraints: 1) j a mj = 1 for all m M 2) i a iw = 1 for all w W 3) j >m w a mj + i >w m a iw + a mw 1, for all m M and w W 4) a mw 0, for all m M and w W 58 Linear Structure of Stable Matchings Theorem (Vande Vate) Let C be the convex polyhedron of the solutions to the linear constraints (1)-(4). Then the extreme points of the linear constraints (1)-(4) corresponds precisely to the stable matchings. Implication: the stable matching that maximizes a linear objective function can be computed by linear programming.