1. Market Design and Analysis Lecture 1 Lecturer: Ning Chen ( 陈宁 ) Email: ningc@ntu.edu.sg

2. 2 Class Information  Focuses  economic models and solution concepts  computational aspects  incentive analysis

3. 3 References  Roth, Sotomayor, Two-Sided Matching: A Study in Game-Theoretical Modeling and Analysis, Cambridge Press, 1990.  Nisan, Roughgarden, Tardos, Vazirani, Algorithmic Game Theory, Cambridge Press, 2007.  Internet (Google, Wikipedia, etc.)  Fuhito Kojima (http://sites.google.com/site/fuhitokojimaeconomics/)http://sites.google.com/site/fuhitokojimaeconomics/

4. 4 Let’s start !

5. 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. 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. 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. 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. 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. 10 Internet Advertising keywords sponsored links

11. 11 Matching Markets Design  Mathematical models  Economic solution concepts  computational issues  mathematical properties  economic properties

12. 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

13. 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

14. 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’.

15. Matching  Example. V1V1 V2V2

16. 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. 17 Let’s start (again) !

18. 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. 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. 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. 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. 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. 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. 24 Stable Matching  Theorem 1.2. [Gale & Shapley’1962] For any stable marriage problem, there always is a stable matching.

25. 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. 26 Gale-Shapley (Deferred-Acceptance) Algorithm  Initially all men and women are free  While there is a man m who is free and hasn’t proposed to every woman  choose such a man m arbitrarily  let w be the highest ranked woman in m’s preference list to whom m hasn’t 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. 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. 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. 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 hasn’t proposed yet.  (O4) the sequence of women to whom m proposes gets worse and worse (in terms of his preference list).

30. 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. 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 m’w w’

32. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 45 Conway’s 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. 46 Conway’s 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. 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. 48 Conway’s 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. 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. 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. 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. 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, it’s unique.  Lower bound and greatest lower bound (denoted by inf X) are defined similarly.

53. 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. 54 Lattice  Examples.

55. 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. 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. 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. 58 Linear Structure of Stable Matchings  Theorem 2.10. (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.