Stable matching. Carlos Hurtado. July 5th, Department of Economics University of Illinois at Urbana-Champaign

Transcription

1 Stable matching Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign July 5th, 2017 C. Hurtado (UIUC - Economics) Game Theory

2 On the Agenda 1 Introduction 2 The Model 3 Existence of stable matching 4 Optimal Matching 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

3 Introduction On the Agenda 1 Introduction 2 The Model 3 Existence of stable matching 4 Optimal Matching 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

4 Introduction Introduction Today we will learn the subject of stable matching introduced in 1962 by David Gale and Lloyd Shapley. Stable matching became the starting point of a rich literature on matching problems in two-sided markets - workers and employers - interns and hospitals - students and universities Today we will learn the Gale and Shapley s basic model of matching men to women, the concept of stable matching, and an algorithm for finding it. We will NOT learn several variations of the model: - the case in which there are more men than women. - the case in which bachelorhood is not the worst outcome. - the case of many-to-one matchings (e.g., many students to one college). - matchings in a single-gender population. C. Hurtado (UIUC - Economics) Game Theory 1 / 12

5 Introduction Introduction The study of matching began at the end of the nineteenth century, with the introduction of residency requirements for recent medical school graduates. Over the years, the residents played an increasingly important role in the staffs of hospitals, and hospitals began competing with each other for the best medical school graduates. By 1944, medical students beginning their third year of medical school were interviewed for residency positions. medical schools and hospitals agreed in 1951 to adopt a formal system for matching graduating medical students with hospitals, beginning the following year. The system works as follows: 1 Medical students rank the hospitals at which they were interviewed. 2 At the same time, hospitals rank the students that they interviewed. 3 Each hospital also announces the maximum number of residents that it can hire. 4 The data are collected by a special national resident matching program. 5 NRMP inputs the data into an algorithm. The algorithm takes into account the preferences of both the hospitals and the medical students with an attempt to arrive at a "best fit" C. Hurtado (UIUC - Economics) Game Theory 2 / 12

6 Introduction Introduction In 1962, David Gale and Lloyd Shapley published a paper defining the matching problem and the concept of "stable matching" In that paper, they also proved that stable matchings always exist, and spelled out an algorithm for computing stable matchings. Several questions: 1 What is the best definition of a stable placement of candidates for residency with open positions? 2 Do such stable placements always exist? 3 If so, how can they be found? 4 Can a hospital (or a candidate for residency) obtain a more satisfactory placement by submitting a preference ordering over candidates that differs from the honest preference ordering? C. Hurtado (UIUC - Economics) Game Theory 3 / 12

7 The Model On the Agenda 1 Introduction 2 The Model 3 Existence of stable matching 4 Optimal Matching 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

8 The Model The Model We will only consider the simple case in which the number of residency candidates equals the number of hospitals, and each hospital is seeking only one resident. Such a situation better describes the matching of men and women in married couples, and we will therefore use the language of that metaphor in analyzing this problem. Definition Let X be a set. A preference relation over X is a binary relation satisfying the following properties: For every x y, either x y or y x (complete) x x (irreflexive) If x y, and y z, then x z (transitive) You can show that every complete, irreflexive, and transitive relation is asymmetric. That is, if x y, then x y if and only if y x. Note: A preference relation, as we have defined here, represents strict preferences (not possibility of indifference) C. Hurtado (UIUC - Economics) Game Theory 4 / 12

9 The Model The Model Definition A matching problem is given by: A natural number n representing the number of men and the number of women in a population (thus, we assume that the number of women equals the number of men). Every woman has a preference relation over the set of men. Every man has a preference relation over the set of women. The set of women will be denoted by W. An element in that set is denoted by w. The set of men will be denoted by M. An element in that set is denoted by m. Definition A matching µ is a one-to-one function from the set M W to itself such that: µ(µ(x)) = x for all x M W µ(m, ) {m} W and µ(, w) M {w} for all m M and w W C. Hurtado (UIUC - Economics) Game Theory 5 / 12

10 The Model The Model A matching is a collection of n pairs {(w 1, m 1), (w 2, m 2),, (w n, m n)} such that {m 1, m 2,, m n} = M and {w 1, w 2,, w n} = W. If a pair (w, m) is included in a matching, then we say that the man m is matched to the woman w (and vice versa). We will sometimes denote matchings using the letters A, B, C, etc. An individual blocks a matching µ if it matches him to someone he/she seems unacceptable. Definition A pair (m, w) blocks µ if they are not matched in µ but each strictly prefers the other to their matches under µ. A matching µ is stable if it is not blocked by any individual or couple (m, w) C. Hurtado (UIUC - Economics) Game Theory 6 / 12

11 The Model The Model Example: - W = {w 1, w 1, w 3, w 4} - M = {m 1, m 1, m 3, m 4} - p(m 1) = w 4, w 3, w 2, w 1 p(m 2) = w 2, w 1, w 3, w 4 p(m 3) = w 2, w 4, w 3, w 1 p(m 4) = w 1, w 4, w 3, w 2 - p(w 1) = m 2, m 1, m 3, m 4 p(w 2) = m 3, m 1, m 4, m 2 p(w 3) = m 4, m 1, m 3, m 2 p(w 4) = m 4, m 3, m 2, m 1 Consider the matching µ - {(m 1, w 1), (m 2, w 2), (m 3, w 3), (m 4, w 4)} Notice that w 3 would prefer m 1 over m 3 AND m 1 would prefer w 3 over w 1. Definition matching A is Stable if in every case that a man (woman) prefers another woman (men) to the partner under A, that woman (man) prefers the partner to whom she (he) is matched under A. The matching µ {(m 1, w 3), (m 2, w 1), (m 3, w 2), (m 4, w 4)} Is stable C. Hurtado (UIUC - Economics) Game Theory 7 / 12

12 Existence of stable matching On the Agenda 1 Introduction 2 The Model 3 Existence of stable matching 4 Optimal Matching 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

13 Existence of stable matching Existence of stable matching Theorem To every matching problem there exists a stable matching Proof: Consider the following algorithm: 1 Stage 1(a): Every man goes to stand in front of the house of the woman he most prefers. 2 Stage 1(b): Every woman asks the man whom she most prefers from among the men standing in front of her house, if there are any, to wait, and dismisses all the other men. 3 Stage 2(a): Every man who was dismissed by a woman in the first stage goes to stand in front of the house of the woman he most prefers from among the women who have not previously dismissed him (i.e., the woman who is second on his list). 4 Stage 2(b): Every woman asks the man whom she most prefers from among the men standing in front of her house, if there are any (including the man whom she told to wait in the previous stage), to wait, and dismisses all the other men. 5 Repeat 3 and 4 until there is one man standing in front of every woman s house. C. Hurtado (UIUC - Economics) Game Theory 8 / 12

14 Existence of stable matching Existence of stable matching Example: - W = {w 1, w 1, w 3, w 4} - M = {m 1, m 1, m 3, m 4} The matching algorithm reaches the stable matching µ {(m 1, w 3), (m 2, w 1), (m 3, w 2), (m 4, w 4)} - p(m 1) = w 4, w 3, w 2, w 1 p(m 2) = w 2, w 1, w 3, w 4 p(m 3) = w 2, w 4, w 3, w 1 p(m 4) = w 1, w 4, w 3, w 2 - p(w 1) = m 2, m 1, m 3, m 4 p(w 2) = m 3, m 1, m 4, m 2 p(w 3) = m 4, m 1, m 3, m 2 p(w 4) = m 4, m 3, m 2, m 1 C. Hurtado (UIUC - Economics) Game Theory 9 / 12

15 Optimal Matching On the Agenda 1 Introduction 2 The Model 3 Existence of stable matching 4 Optimal Matching 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

16 Optimal Matching Optimal Matching Example: - W = {w 1, w 1, w 3, w 4} - M = {m 1, m 1, m 3, m 4} - p(m 1) = w 1, w 2, w 3, w 4 p(m 2) = w 4, w 2, w 3, w 1 p(m 3) = w 4, w 3, w 1, w 2 Man propose first: {(m 1, w 1), (m 2, w 2), (m 3, w 3), (m 4, w 4)} Women propose first: {(m 1, w 4), (m 2, w 1), (m 3, w 2), (m 4, w 3)} Very different! In this example there are (at least) two stable matches. p(m 4) = w 1, w 4, w 3, w 2 - p(w 1) = m 2, m 3, m 1, m 4 p(w 2) = m 3, m 1, m 2, m 4 p(w 3) = m 4, m 1, m 2, m 3 p(w 4) = m 1, m 4, m 2, m 3 C. Hurtado (UIUC - Economics) Game Theory 10 / 12

17 Optimal Matching Optimal Matching Definition A stable matching µ is M-optimal if every man weakly prefers his match in µ to his match in any other stable matching. A stable matching µ is W-optimal if every woman prefers her match in µ to her match in any other stable matching. Theorem If the preferences of each man and each woman are strict, then the matching produced by the matching algorithm in which men make the proposals is M-optimal and the matching that is produced in this algorithm when women make the proposals is W -optimal. C. Hurtado (UIUC - Economics) Game Theory 11 / 12

18 Exercises On the Agenda 1 Introduction 2 The Model 3 Existence of stable matching 4 Optimal Matching 5 Exercises C. Hurtado (UIUC - Economics) Game Theory

19 Exercises Exercises Consider the following marriage market: - W = {w 1, w 1, w 3, w 4} - M = {m 1, m 1, m 3, m 4} Preferences Determine the M-optimal stable matching. Determine the W-optimal stable matching - p(m 1) = w 1, w 2, w 3, w 4 p(m 2) = w 2, w 1, w 4, w 3 p(m 3) = w 1, w 4, w 3, w 2 p(m 4) = w 4, w 1, w 2, w 3 - p(w 1) = m 2, m 3, m 1, m 4 p(w 2) = m 3, m 4, m 1, m 2 p(w 3) = m 4, m 1, m 3, m 2 p(w 4) = m 3, m 2, m 4, m 1 C. Hurtado (UIUC - Economics) Game Theory 12 / 12