Two-Sided Matching. Terence Johnson. December 1, University of Notre Dame. Terence Johnson (ND) Two-Sided Matching December 1, / 47

Two-Sided Matching Terence Johnson University of Notre Dame December 1, 2017 Terence Johnson (ND) Two-Sided Matching December 1, 2017 1 / 47

Markets without money What do you do when you can t use money to solve your problems? Terence Johnson (ND) Two-Sided Matching December 1, 2017 2 / 47

Markets without money What do you do when you can t use money to solve your problems? Pure matching: heterosexual men and women marrying in a small town, students matching to universities; the NRMP Allocating publicly owned goods: seats in public schools, publicly owned housing; the NYC/Boston matches Morally repugnant transactions: kidneys, livers, hearts; the New England Kidney Exchange Terence Johnson (ND) Two-Sided Matching December 1, 2017 2 / 47

Markets without money What do you do when you can t use money to solve your problems? Pure matching: heterosexual men and women marrying in a small town, students matching to universities; the NRMP Allocating publicly owned goods: seats in public schools, publicly owned housing; the NYC/Boston matches Morally repugnant transactions: kidneys, livers, hearts; the New England Kidney Exchange We can t auction off seats to schools, or open markets for body parts: the incentives that would generate would both be unacceptable to society Terence Johnson (ND) Two-Sided Matching December 1, 2017 2 / 47

Markets without money Disclaimer: I m going to talk mostly about the heterosexual marriage market in exposition and examples, because it highlights the incentives for cheating and stability, making it easier to appreciate what is going on. I don t really think this is a prescriptive way for matching men to women, or that heterosexual marriage markets are particularly interesting, or anything like that. Terence Johnson (ND) Two-Sided Matching December 1, 2017 3 / 47

Markets without money When money can t be exchanged, what is the generalization of the second price auction or Vickrey auction? Terence Johnson (ND) Two-Sided Matching December 1, 2017 4 / 47

Markets without money When money can t be exchanged, what is the generalization of the second price auction or Vickrey auction? i.e.: Does there exist a way of inducing agents to reveal their preferences that gives them a dominant strategy to be honest, and implements the best or efficient outcome in some sense? Terence Johnson (ND) Two-Sided Matching December 1, 2017 4 / 47

Markets without money Our previous solution concepts have been competitive equilibrium, Nash equilibrium, or dominant strategy equilibrium. Our goal was to implement the efficient outcome. This doesn t seem appropriate for matching. Think about the men and women in the small town. Their partners alone aren t really the issue, it s the overall suitableness of a match for everyone in the town. We want a group equilibrium notion, not equilibrium agent-by-agent. Terence Johnson (ND) Two-Sided Matching December 1, 2017 5 / 47

Stability Consider an economy composed of agents i = 1, 2,..., N who have preferences u i (x), where x is some outcome like who gets a good at what price or who is matched with whom. Terence Johnson (ND) Two-Sided Matching December 1, 2017 6 / 47

Stability Consider an economy composed of agents i = 1, 2,..., N who have preferences u i (x), where x is some outcome like who gets a good at what price or who is matched with whom. An outcome x is stable if there is no subset of agents who prefer another outcome x, who could achieve that outcome amongst themselves. The set of stable outcomes of an economy is called the core. Terence Johnson (ND) Two-Sided Matching December 1, 2017 6 / 47

Stability Consider an economy composed of agents i = 1, 2,..., N who have preferences u i (x), where x is some outcome like who gets a good at what price or who is matched with whom. An outcome x is stable if there is no subset of agents who prefer another outcome x, who could achieve that outcome amongst themselves. The set of stable outcomes of an economy is called the core. Is the outcome of a second price auction stable? Is the Vickrey outcome always stable? Is the outcome of a perfectly competitive economy in the core? Terence Johnson (ND) Two-Sided Matching December 1, 2017 6 / 47

Stability In a matching market, an outcome is stable is no two people want to cheat and have an affair. Terence Johnson (ND) Two-Sided Matching December 1, 2017 7 / 47

Stability In a matching market, an outcome is stable is no two people want to cheat and have an affair. In the marriage market: for each man m married to woman w and each man m married to w, (i) neither m and w prefer each other to m and w, and (ii) neither w and m prefer each other to m and w. Terence Johnson (ND) Two-Sided Matching December 1, 2017 7 / 47

Stability In a matching market, an outcome is stable is no two people want to cheat and have an affair. In the marriage market: for each man m married to woman w and each man m married to w, (i) neither m and w prefer each other to m and w, and (ii) neither w and m prefer each other to m and w. But does such a matching always exist? How do we find it? What are the incentives that such a system would provide to participants? Terence Johnson (ND) Two-Sided Matching December 1, 2017 7 / 47

Marriage Markets: model There are two finite and disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w m } of women. Terence Johnson (ND) Two-Sided Matching December 1, 2017 8 / 47

Marriage Markets: model There are two finite and disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w m } of women. Each man has ordered, transitive, and complete preferences over the women, and each woman has ordered, transitive, and complete preferences over the men. The ordered list for a man is represented as P(m i ) = (w j1, w j2,..., m i, w jl,...w jm ), where m i appears at the point in the list where the man would rather be matched to himself than be matched to any of the partners w jl,..., w jm. Terence Johnson (ND) Two-Sided Matching December 1, 2017 8 / 47

Marriage Markets: model There are two finite and disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w m } of women. Each man has ordered, transitive, and complete preferences over the women, and each woman has ordered, transitive, and complete preferences over the men. The ordered list for a man is represented as P(m i ) = (w j1, w j2,..., m i, w jl,...w jm ), where m i appears at the point in the list where the man would rather be matched to himself than be matched to any of the partners w jl,..., w jm. If w appears earlier in P(m i ) than w, we say w mi w. Terence Johnson (ND) Two-Sided Matching December 1, 2017 8 / 47

Marriage Markets: model There are two finite and disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w m } of women. Each man has ordered, transitive, and complete preferences over the women, and each woman has ordered, transitive, and complete preferences over the men. The ordered list for a man is represented as P(m i ) = (w j1, w j2,..., m i, w jl,...w jm ), where m i appears at the point in the list where the man would rather be matched to himself than be matched to any of the partners w jl,..., w jm. If w appears earlier in P(m i ) than w, we say w mi w. A marriage market is a triple, (M, W, P), where P is the set of all the preferences of all the agents Terence Johnson (ND) Two-Sided Matching December 1, 2017 8 / 47

Marriage Markets Note that there are no utility functions: this is an ordinal theory Terence Johnson (ND) Two-Sided Matching December 1, 2017 9 / 47

Marriage Markets Note that there are no utility functions: this is an ordinal theory Note also that there s no calculus here: it s not even clear what we would want to maximize Terence Johnson (ND) Two-Sided Matching December 1, 2017 9 / 47

Marriage Markets Note that there are no utility functions: this is an ordinal theory Note also that there s no calculus here: it s not even clear what we would want to maximize Even if we knew what to maximize, that outcome might not be stable Terence Johnson (ND) Two-Sided Matching December 1, 2017 9 / 47

Matchings Definition A matching µ is a one-to-one mapping from the set of men to the set of women so that each agent is matched at most once to someone else, and that other person is matched to the agent (µ(µ(x)) = x). The mate of x is denoted µ(x). Definition A matching µ can be blocked or improved upon if there exists some man m and some woman w such that µ(m) w, but m prefers w to his current partner and w prefers m to her current partner. Terence Johnson (ND) Two-Sided Matching December 1, 2017 10 / 47

Matchings Definition A matching µ is a one-to-one mapping from the set of men to the set of women so that each agent is matched at most once to someone else, and that other person is matched to the agent (µ(µ(x)) = x). The mate of x is denoted µ(x). Definition A matching µ can be blocked or improved upon if there exists some man m and some woman w such that µ(m) w, but m prefers w to his current partner and w prefers m to her current partner. If a match can be blocked or improved upon, it is unstable; otherwise, it is stable. Terence Johnson (ND) Two-Sided Matching December 1, 2017 10 / 47

Examples Let P(m 1 ) = (w 1, w 2 ), P(m 2 ) = (w 2, w 1 ), P(w 1 ) = (m 1, m 2 ), and P(w 2 ) = (m 2, m 1 ). Then each agent has a mutually preferred partner, so there is a unique stable matching: µ = {(m 1, w 1 ), (m 2, w 2 )}, or ( ) m1 m µ = 2 w 1 w 2 Let P(m 1 ) = (w 1, w 2 ), P(m 2 ) = (w 2, w 1 ), P(w 1 ) = (m 2, m 1 ), and P(w 2 ) = (m 1, m 2 ). Then the men and women have exactly opposed preferences, so there are two stable matchings: µ 1 = {(m 1, w 1 ), (m 2, w 2 )} and µ 2 = {(m 1, w 2 ), (m 2, w 1 )}, or ( ) m1 m µ 1 = 2 w 1 w 2 and ( ) m1 m µ 2 = 2 w 2 w 1 So there might be many stable matches. Terence Johnson (ND) Two-Sided Matching December 1, 2017 11 / 47

Examples Let M = W = 3, and P(m 1 ) = (w 2, w 1, w 3 ) P(w 1 ) = (m 1, m 3, m 2 ) P(m 2 ) = (w 1, w 3, w 2 ) P(w 2 ) = (m 3, m 1, m 2 ) P(m 3 ) = (w 1, w 2, w 3 ) P(w 3 ) = (m 1, m 3, m 2 ) The following matching is unstable: ( w1 w µ = 2 w 3 m 1 m 2 m 3 ) because (m 1, w 2 ) can block this allocation: w 2 m1 µ(m 1 ) = w 1, and m 1 w2 µ(w 2 ) = m 2. The match however, is stable. ( ) µ w1 w = 2 w 3 m 1 m 3 m 2 Terence Johnson (ND) Two-Sided Matching December 1, 2017 12 / 47

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine Terence Johnson (ND) Two-Sided Matching December 1, 2017 13 / 47

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine There are four people trying to match into rooms, a, b, c, and d. Terence Johnson (ND) Two-Sided Matching December 1, 2017 13 / 47

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine There are four people trying to match into rooms, a, b, c, and d. P(a) = b, c, d P(b) = c, a, d P(c) = a, b, d P(d) = anything Terence Johnson (ND) Two-Sided Matching December 1, 2017 13 / 47

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine There are four people trying to match into rooms, a, b, c, and d. P(a) = b, c, d P(b) = c, a, d P(c) = a, b, d P(d) = anything Person d is everyone s last choice, and each of the other people is someone s first choice. Terence Johnson (ND) Two-Sided Matching December 1, 2017 13 / 47

Instability in one-sided markets Then the match ( c a b d ) is blocked by (c, a). The match ( a d b c is blocked by (b, c). The match ( b a d c is blocked by (a, b). That s all the matches: this market is fundamentally unstable. ) ) Terence Johnson (ND) Two-Sided Matching December 1, 2017 14 / 47

The Gale-Shapley Algorithm Definition The Gale-Shapley algorithm is the procedure where Terence Johnson (ND) Two-Sided Matching December 1, 2017 15 / 47

The Gale-Shapley Algorithm Definition The Gale-Shapley algorithm is the procedure where In the first round, (i) each man proposes to his most preferred woman, (ii) each woman then conditionally accepts her most preferred man among those who proposed to her, (iii) all the men whose proposal was not accepted remain single. In each subsequent round, (i) each single man proposes to his most preferred woman whom he has not yet proposed to, (ii) each woman retains her most preferred man among those who proposed to her this period and her current mate, (iii) all the men whose proposal was not conditionally accepted or who were dumped are single. Note that men never revisit women who have previously rejected them. Terence Johnson (ND) Two-Sided Matching December 1, 2017 15 / 47

Marriage Markets Red people are one side of the market, black people are the other side. People on each side of the market have a number. Terence Johnson (ND) Two-Sided Matching December 1, 2017 16 / 47

Marriage Markets Red people are one side of the market, black people are the other side. People on each side of the market have a number. I will deal a set of preferences to each person on each side of the market over the people on the opposite side of the market. Whichever side is proposing can propose however it sees fit, in accordance with the rules of the Gale-Shapley algorithm; the deferred acceptance side can accept or reject however it sees fit Terence Johnson (ND) Two-Sided Matching December 1, 2017 16 / 47

Existence of stable matchings Theorem Every marriage market has at least one stable match, and the Gale-Shapley algorithm computes it. Terence Johnson (ND) Two-Sided Matching December 1, 2017 17 / 47

Existence of stable matchings Theorem Every marriage market has at least one stable match, and the Gale-Shapley algorithm computes it. Proof. Consider using the deferred acceptance algorithm. This procedure terminates in less than M W rounds, since men never revisit women who have previously rejected them, so it produces a matching µ. This match is stable: Suppose some man, A, prefers some woman, α, to his mate, µ(a). Then A must have proposed to α at some earlier stage and been rejected, since A ranks α higher than µ(a). Terence Johnson (ND) Two-Sided Matching December 1, 2017 17 / 47

Existence of stable matchings Theorem Every marriage market has at least one stable match, and the Gale-Shapley algorithm computes it. Proof. Consider using the deferred acceptance algorithm. This procedure terminates in less than M W rounds, since men never revisit women who have previously rejected them, so it produces a matching µ. This match is stable: Suppose some man, A, prefers some woman, α, to his mate, µ(a). Then A must have proposed to α at some earlier stage and been rejected, since A ranks α higher than µ(a). But then A must have been rejected in favor of someone that α liked better, because α always holds on to her most preferred proposal and A and α are not matched. Terence Johnson (ND) Two-Sided Matching December 1, 2017 17 / 47

Existence of stable matchings Theorem Every marriage market has at least one stable match, and the Gale-Shapley algorithm computes it. Proof. Consider using the deferred acceptance algorithm. This procedure terminates in less than M W rounds, since men never revisit women who have previously rejected them, so it produces a matching µ. This match is stable: Suppose some man, A, prefers some woman, α, to his mate, µ(a). Then A must have proposed to α at some earlier stage and been rejected, since A ranks α higher than µ(a). But then A must have been rejected in favor of someone that α liked better, because α always holds on to her most preferred proposal and A and α are not matched. Therefore, α prefers µ(α) to A, and the match is stable. Terence Johnson (ND) Two-Sided Matching December 1, 2017 17 / 47

Other matching algorithms Not all matching algorithms are so nice Terence Johnson (ND) Two-Sided Matching December 1, 2017 18 / 47

Other matching algorithms Not all matching algorithms are so nice Imagine taking a given proposed match. If it is stable, stop. Otherwise, find a blocking pair and make a new matching by marrying these people to each other. This is called a greedy algorithm, since it focuses on trying to improve the stability of the matching at each step without regard to broader consequences of the proposed changes Terence Johnson (ND) Two-Sided Matching December 1, 2017 18 / 47

Other matching algorithms Not all matching algorithms are so nice Imagine taking a given proposed match. If it is stable, stop. Otherwise, find a blocking pair and make a new matching by marrying these people to each other. This is called a greedy algorithm, since it focuses on trying to improve the stability of the matching at each step without regard to broader consequences of the proposed changes Is this algorithm stable? Terence Johnson (ND) Two-Sided Matching December 1, 2017 18 / 47

Other matching algorithms Recall the example with P(m 1 ) = (w 2, w 1, w 3 ) P(w 1 ) = (m 1, m 3, m 2 ) P(m 2 ) = (w 1, w 3, w 2 ) P(w 2 ) = (m 3, m 1, m 2 ) P(m 3 ) = (w 1, w 2, w 3 ) P(w 3 ) = (m 1, m 3, m 2 ) Terence Johnson (ND) Two-Sided Matching December 1, 2017 19 / 47

Other matching algorithms Recall the example with P(m 1 ) = (w 2, w 1, w 3 ) P(w 1 ) = (m 1, m 3, m 2 ) P(m 2 ) = (w 1, w 3, w 2 ) P(w 2 ) = (m 3, m 1, m 2 ) P(m 3 ) = (w 1, w 2, w 3 ) P(w 3 ) = (m 1, m 3, m 2 ) and consider the initial match ( w1 w µ 0 = 2 w 3 m 1 m 2 m 3 ). Terence Johnson (ND) Two-Sided Matching December 1, 2017 19 / 47

Other matching algorithms Recall the example with P(m 1 ) = (w 2, w 1, w 3 ) P(w 1 ) = (m 1, m 3, m 2 ) P(m 2 ) = (w 1, w 3, w 2 ) P(w 2 ) = (m 3, m 1, m 2 ) P(m 3 ) = (w 1, w 2, w 3 ) P(w 3 ) = (m 1, m 3, m 2 ) and consider the initial match ( w1 w µ 0 = 2 w 3 m 1 m 2 m 3 ). This match is unstable since (w 2, m 1 ) is a blocking pair. Let s implement that affair... Terence Johnson (ND) Two-Sided Matching December 1, 2017 19 / 47

Other matching algorithms... yielding ( w1 w µ 1 = 2 w 3 m 2 m 1 m 3 for which (w 2, m 3 ) is a blocking pair, yielding ( w1 w µ 2 = 2 w 3 m 2 m 3 m 1 for which (w 1, m 3 ) is a blocking pair, yielding ( w1 w µ 3 = 2 w 3 m 3 m 2 m 1 ), ), ), for which (w 1, m 1 ) is a blocking pair, yielding... Terence Johnson (ND) Two-Sided Matching December 1, 2017 20 / 47

Other matching algorithms... ( w1 w µ 0 = 2 w 3 m 1 m 2 m 3 which is where we started. ) Terence Johnson (ND) Two-Sided Matching December 1, 2017 21 / 47

Other matching algorithms... which is where we started. ( ) w1 w µ 0 = 2 w 3 m 1 m 2 m 3 Theorem The greedy algorithm need not converge to a stable match. Terence Johnson (ND) Two-Sided Matching December 1, 2017 21 / 47

Optimality What would it mean for a match to be optimal? Terence Johnson (ND) Two-Sided Matching December 1, 2017 22 / 47

Optimality What would it mean for a match to be optimal? In this world, there isn t a social welfare function N u i (x), i=1 where we can trade off each agent s welfare for those of the other agents, and even if we had such a thing, the outcome could easily be unstable (imagine an auction for partners where wealth and attractiveness are negatively correlated, leading to a Beauty and the [wealthy] Beast problem) Terence Johnson (ND) Two-Sided Matching December 1, 2017 22 / 47

Optimality What would it mean for a match to be optimal? In this world, there isn t a social welfare function N u i (x), i=1 where we can trade off each agent s welfare for those of the other agents, and even if we had such a thing, the outcome could easily be unstable (imagine an auction for partners where wealth and attractiveness are negatively correlated, leading to a Beauty and the [wealthy] Beast problem) But can we get a clear idea of society s preferences over the set of stable matches? Terence Johnson (ND) Two-Sided Matching December 1, 2017 22 / 47

Optimality Definition A match µ is M-preferred to µ if, for every man m, µ(m) m µ (m), and for some m, µ(m) m µ (m), and we write µ M µ. Terence Johnson (ND) Two-Sided Matching December 1, 2017 23 / 47

Optimality Definition A match µ is M-preferred to µ if, for every man m, µ(m) m µ (m), and for some m, µ(m) m µ (m), and we write µ M µ. A match µ is M-optimal if there does not exist a match µ such that µ M µ. Define W and W similarly for the women. Theorem If all preferences are strict, there is an M-optimal stable matching, and a W -optimal stable matching. The M-optimal stable match is computed when men propose in the Gale-Shapley algorithm, and likewise for the women. The strategy of the proof is to show that with the Gale-Shapley algorithm, if w rejects m at any point, there is no stable matching µ at which w and m are partners. Therefore, there is no stable match where a rejected man can get a more preferred partner, and this is then the best stable match for the men. Terence Johnson (ND) Two-Sided Matching December 1, 2017 23 / 47

Optimality Let P(m 1 ) = (w 1, w 2 ) P(m 2 ) = (w 2, w 1 ) P(w 1 ) = (m 2, m 1 ) P(w 2 ) = (m 1, m 2 ). Terence Johnson (ND) Two-Sided Matching December 1, 2017 24 / 47

Optimality Let P(m 1 ) = (w 1, w 2 ) P(m 2 ) = (w 2, w 1 ) P(w 1 ) = (m 2, m 1 ) P(w 2 ) = (m 1, m 2 ). Then there are two stable matches: ( ) ( w1 w µ 1 = 2 w1 w, µ m 1 m 2 = 2 2 m 2 m 1 ) Terence Johnson (ND) Two-Sided Matching December 1, 2017 24 / 47

Optimality Let P(m 1 ) = (w 1, w 2 ) P(m 2 ) = (w 2, w 1 ) P(w 1 ) = (m 2, m 1 ) P(w 2 ) = (m 1, m 2 ). Then there are two stable matches: ( ) ( w1 w µ 1 = 2 w1 w, µ m 1 m 2 = 2 2 m 2 m 1 ) but notice that the men obviously prefer µ 1 and the women obviously prefer µ 2. Terence Johnson (ND) Two-Sided Matching December 1, 2017 24 / 47

Optimality and Opposing Preferences Theorem When all sides have strict preferences, µ M µ if and only if µ W µ. Terence Johnson (ND) Two-Sided Matching December 1, 2017 25 / 47

Optimality and Opposing Preferences Theorem When all sides have strict preferences, µ M µ if and only if µ W µ. Proof. Let µ and µ be stable matchings such that µ M µ. Suppose that, by way of contradiction, µ W µ. Then there must be some woman w who strictly prefers µ to µ. Then w has a different partner at µ and µ, and so must her partner m = µ(w). But then m and w form a blocking pair for µ, since µ M µ by assumption and preferences are strict. But then µ is unstable, which is a contradiction. Therefore, it must be the case that µ W µ. Terence Johnson (ND) Two-Sided Matching December 1, 2017 25 / 47

Optimality and Opposing Preferences Theorem When all sides have strict preferences, µ M µ if and only if µ W µ. Proof. Let µ and µ be stable matchings such that µ M µ. Suppose that, by way of contradiction, µ W µ. Then there must be some woman w who strictly prefers µ to µ. Then w has a different partner at µ and µ, and so must her partner m = µ(w). But then m and w form a blocking pair for µ, since µ M µ by assumption and preferences are strict. But then µ is unstable, which is a contradiction. Therefore, it must be the case that µ W µ. Then any stable match preferred by the men is considered worse by the women, and vice versa: any gain to one side represents loss to the other. Terence Johnson (ND) Two-Sided Matching December 1, 2017 25 / 47

Optimality and Opposing Preferences Theorem When all sides have strict preferences, µ M µ if and only if µ W µ. Proof. Let µ and µ be stable matchings such that µ M µ. Suppose that, by way of contradiction, µ W µ. Then there must be some woman w who strictly prefers µ to µ. Then w has a different partner at µ and µ, and so must her partner m = µ(w). But then m and w form a blocking pair for µ, since µ M µ by assumption and preferences are strict. But then µ is unstable, which is a contradiction. Therefore, it must be the case that µ W µ. Then any stable match preferred by the men is considered worse by the women, and vice versa: any gain to one side represents loss to the other. This theorem has... implications. Terence Johnson (ND) Two-Sided Matching December 1, 2017 25 / 47

Equilibrium in the Gale-Shapley Algorithm The analysis above has no mention of equilibrium, so it is natural to wonder about the incentives of participants in the Gale-Shapley algorithm. Consider the game where players submit ordered lists, and a central authority uses the Gale-Shapley algorithm to compute a match (as is done in the NRMP). More formally, the game is Players simultaneously and non-cooperatively submit a list ˆP(m) or ˆP(w) to a matching authority The matching authority computes the outcomes of the Gale-Shapley algorithm with respect to the reported preferences, assigning a partner (or not) to each agent The agents true preferences are common knowledge Terence Johnson (ND) Two-Sided Matching December 1, 2017 26 / 47

Equilibrium in the Gale-Shapley Algorithm Let P(m 1 ) = (w 1, w 2 ) P(m 2 ) = (w 2, w 1 ) P(w 1 ) = (m 2, m 1 ) P(w 2 ) = (m 1, m 2 ). Terence Johnson (ND) Two-Sided Matching December 1, 2017 27 / 47

Equilibrium in the Gale-Shapley Algorithm Let P(m 1 ) = (w 1, w 2 ) P(m 2 ) = (w 2, w 1 ) P(w 1 ) = (m 2, m 1 ) P(w 2 ) = (m 1, m 2 ). Honesty in the GS algorithm yields the matching ( ) w1 w2 µ = m1 m2 Terence Johnson (ND) Two-Sided Matching December 1, 2017 27 / 47

Equilibrium in the Gale-Shapley Algorithm Let P(m 1 ) = (w 1, w 2 ) P(m 2 ) = (w 2, w 1 ) P(w 1 ) = (m 2, m 1 ) P(w 2 ) = (m 1, m 2 ). Honesty in the GS algorithm yields the matching ( ) w1 w2 µ = m1 m2 Can either of the women deviate in such a way that their preferred match is implemented? Terence Johnson (ND) Two-Sided Matching December 1, 2017 27 / 47

Equilibrium in the Gale-Shapley Algorithm Suppose agent w 1 submits the list P (w 1 ) = (m 1, w 1, m 2 ), stating that agent m 2 is unacceptable to her, while the others report honestly. Terence Johnson (ND) Two-Sided Matching December 1, 2017 28 / 47

Equilibrium in the Gale-Shapley Algorithm Suppose agent w 1 submits the list P (w 1 ) = (m 1, w 1, m 2 ), stating that agent m 2 is unacceptable to her, while the others report honestly. In the first round, m 1 and m 2 propose to w 1 and w 2, respectively. m 1 is rejected, and m 2 is conditionally matched to w 2 : ( ) m1 m µ 1 = 2 w 2 In the second round, m 1 proposes to w 2, who accepts, dumping m 1 : ( ) m1 m µ 1 = 2 w 2 Terence Johnson (ND) Two-Sided Matching December 1, 2017 28 / 47

Equilibrium in the Gale-Shapley Algorithm Suppose agent w 1 submits the list P (w 1 ) = (m 1, w 1, m 2 ), stating that agent m 2 is unacceptable to her, while the others report honestly. In the first round, m 1 and m 2 propose to w 1 and w 2, respectively. m 1 is rejected, and m 2 is conditionally matched to w 2 : ( ) m1 m µ 1 = 2 w 2 In the second round, m 1 proposes to w 2, who accepts, dumping m 1 : ( ) m1 m µ 1 = 2 w 2 Finally, m 2 proposes to w 1, who accepts, giving: ( ) m1 m µ 1 = 2 w 2 w 1 which is the preferred stable match for the women. Terence Johnson (ND) Two-Sided Matching December 1, 2017 28 / 47

Equilibrium in the Gale-Shapley Algorithm This shows agents on the conditionally accepting side may have incentives to misrepresent their preferences Terence Johnson (ND) Two-Sided Matching December 1, 2017 29 / 47

Equilibrium in the Gale-Shapley Algorithm This shows agents on the conditionally accepting side may have incentives to misrepresent their preferences Typically, this means cutting acceptable partners from the bottom of their lists to cause a rejection chain, thereby ending up with a better partner Computing a rejection chain is incredibly difficult, and requires knowing the preferences of all the agents in the market, which is unlikely: in large markets, it is extremely unlikely that one agent can perfectly execute a rejection chain of this type Terence Johnson (ND) Two-Sided Matching December 1, 2017 29 / 47

Equilibrium in the Gale-Shapley Algorithm What about incentives on the proposing side? Terence Johnson (ND) Two-Sided Matching December 1, 2017 30 / 47

Equilibrium in the Gale-Shapley Algorithm What about incentives on the proposing side? Theorem If the M-optimal version of the Gale-Shapley algorithm is used, it is a weakly dominant strategy for the men to report their preferences honestly. The intuition is that fixing the reports of the women if the men re-order or drop candidates from their lists, they can only miss out on the chance to get a particular partner, since women always hold on to their favorite proposal relative to their stated preferences. Terence Johnson (ND) Two-Sided Matching December 1, 2017 30 / 47

Private Information Suppose the preferences of the agents are not common knowledge, so it becomes a game of incomplete information. Suppose that an agent s preference P(m) or P(w) is private information. A direct revelation mechanism is one in which agents each make a report ˆP(m) or ˆP(w) to the matchmaker, who then uses some mechanism to compute a match from the reported preferences, ˆP. A mechanism implements honest reporting in weakly dominant strategies if, for whatever reports are made by the other players, each player finds it in his best interest to report his type honestly. Terence Johnson (ND) Two-Sided Matching December 1, 2017 31 / 47

Private Information Suppose the preferences of the agents are not common knowledge, so it becomes a game of incomplete information. Suppose that an agent s preference P(m) or P(w) is private information. A direct revelation mechanism is one in which agents each make a report ˆP(m) or ˆP(w) to the matchmaker, who then uses some mechanism to compute a match from the reported preferences, ˆP. A mechanism implements honest reporting in weakly dominant strategies if, for whatever reports are made by the other players, each player finds it in his best interest to report his type honestly. A mechanism implements honest reporting if, for each player i, when other players report their types honestly, it is a best response for i to do so also. Terence Johnson (ND) Two-Sided Matching December 1, 2017 31 / 47

Roth s Impossibility Theorem Theorem There does not exist a stable matching mechanism that implements honest reporting in weakly dominant strategies. Terence Johnson (ND) Two-Sided Matching December 1, 2017 32 / 47

Roth s Impossibility Theorem Theorem There does not exist a stable matching mechanism that implements honest reporting in weakly dominant strategies. Proof. (By contradiction) Let M = {m 1, m 2 } and W = {w 1, w 2 } with preferences P(m 1 ) = (w 1, w 2 ), P(m 2 ) = (w 2, w 1 ), P(w 1 ) = (m 2, m 1 ), P(w 2 ) = (w 1, w 2 ). Then there are exactly two stable matchings, µ = {(m 1, w 1 ), (m 2, w 2 )} and ν = {(m 1, w 2 ), (m 2, w 1 )}, so any stable mechanism must choose one of these matchings when P = ˆP is stated honestly. Terence Johnson (ND) Two-Sided Matching December 1, 2017 32 / 47

Roth s Impossibility Theorem Proof. Suppose the mechanism selects µ with strictly positive probability. Suppose, however, that w 2 changes her report to P (w 2 ) = (m 1, w 2 ). Then µ is no longer a stable match, because w 2 could always block by remaining single (if those were her true preferences). Therefore, if all other agents report honestly and w 2 submits P (w 2 ), the only stable match is ν, which must be chosen by any stable matching mechanism with probability 1. Therefore, w 2 has a profitable deviation. Terence Johnson (ND) Two-Sided Matching December 1, 2017 33 / 47

Roth s Impossibility Theorem Proof. Suppose the mechanism selects µ with strictly positive probability. Suppose, however, that w 2 changes her report to P (w 2 ) = (m 1, w 2 ). Then µ is no longer a stable match, because w 2 could always block by remaining single (if those were her true preferences). Therefore, if all other agents report honestly and w 2 submits P (w 2 ), the only stable match is ν, which must be chosen by any stable matching mechanism with probability 1. Therefore, w 2 has a profitable deviation. If µ was selected with zero probability, w 1 would have a similar profitable deviation. Terence Johnson (ND) Two-Sided Matching December 1, 2017 33 / 47

Summary of One-to-One Matching The M-optimal and W -optimal matches can be computed by using the Gale-Shapley algorithm Male and female preferences are opposed on the set of stable matches The proposing side always has a dominant strategy to report its preference list truthfully, but the conditionally accepting side does not There does not exist a stable matching mechanism, in general, that implements honest reporting when preferences over partners are private information Terence Johnson (ND) Two-Sided Matching December 1, 2017 34 / 47

Many-to-One Matching: Hospitals and Colleges To what extend can the marriage market results be extended to many-to-one matching models? Terence Johnson (ND) Two-Sided Matching December 1, 2017 35 / 47

Many-to-One Matching: Hospitals and Colleges To what extend can the marriage market results be extended to many-to-one matching models? There are two disjoint sets, students S = {s 1,..., s m } and colleges, C = {c 1,..., c m }. Each student has ordered, complete, transitive preferences over colleges, and each college has ordered, complete, transitive preferences over students. Each college j has a quota q cj, giving the maximum number of students it can admit. Terence Johnson (ND) Two-Sided Matching December 1, 2017 35 / 47

Many-to-One Matching: Hospitals and Colleges To what extend can the marriage market results be extended to many-to-one matching models? There are two disjoint sets, students S = {s 1,..., s m } and colleges, C = {c 1,..., c m }. Each student has ordered, complete, transitive preferences over colleges, and each college has ordered, complete, transitive preferences over students. Each college j has a quota q cj, giving the maximum number of students it can admit. A matching µ assigns students to college so that no student is matched to more than one college, and no college j has more than q cj students. Terence Johnson (ND) Two-Sided Matching December 1, 2017 35 / 47

The Many-to-One Deferred Acceptance Algorithm Consider breaking each college j into q cj copies of itself and constructing a one-to-one marriage market where each copy of college j has the same preferences, and the students preferences over copies is the same as the preferences over the original colleges. Theorem A matching is stable in the many-to-one market iff the matching is stable in the corresponding one-to-one market. So similar variations of the one-to-one results apply to the many-to-one market with responsive preferences. This is false, however, if the students have preferences over which students they attend school with (or which doctors have preferences over which doctors they do their residency with; namely their spouses) Terence Johnson (ND) Two-Sided Matching December 1, 2017 36 / 47

The Many-to-One Deferred Acceptance Algorithm For students 1 to 3, and for students 4 to 6, The college s preferences are s i : c 1 c 2 s i : c 2 c 1. but each college only has two seats. c 1 : s 1 s 4 s 5 s 3 s 2 s 6 c 1 : s 1 s 2 s 3 s 6 s 5 s 4 Terence Johnson (ND) Two-Sided Matching December 1, 2017 37 / 47

Assigning goods without money ( one-sided matching ) Instead of considering two-sided markets, consider the case where we want to distribute or re-allocate goods in a Pareto-improving way: There are a limited number of seats for students in classes, and the college wants to determine which students enroll in which courses In Israel, there were random allocations of families to government-owned houses to be fair... but ex post, many families were dissatisfied with their homes and preferred someone else s Terence Johnson (ND) Two-Sided Matching December 1, 2017 38 / 47

Assigning goods without money ( one-sided matching ) Instead of considering two-sided markets, consider the case where we want to distribute or re-allocate goods in a Pareto-improving way: There are a limited number of seats for students in classes, and the college wants to determine which students enroll in which courses In Israel, there were random allocations of families to government-owned houses to be fair... but ex post, many families were dissatisfied with their homes and preferred someone else s How can goods be assigned to consumers without money in Pareto-improving ways? Terence Johnson (ND) Two-Sided Matching December 1, 2017 38 / 47

Serial Random Dictatorship (SRD) Suppose there are agents i = 1, 2,..., N and goods k = 1, 2,..., M. Each agent has ordered, complete, transitive preferences over the goods. We want to assign the goods in a way that agents have a weakly dominant strategy to reveal their preferences honestly. Serial Random Dictatorship is the game where 1 Agents submit an ordered list 2 We pick agent an agent i at random, and allocate to i the remaining good that is ranked highest by i, and repeat until all goods are assigned or all agents have a good. Theorem Serial Random Dictatorship implements honest reporting in weakly dominant strategies. Terence Johnson (ND) Two-Sided Matching December 1, 2017 39 / 47

Serial Random Dictatorship (SRD) Suppose there are agents i = 1, 2,..., N and goods k = 1, 2,..., M. Each agent has ordered, complete, transitive preferences over the goods. We want to assign the goods in a way that agents have a weakly dominant strategy to reveal their preferences honestly. Serial Random Dictatorship is the game where 1 Agents submit an ordered list 2 We pick agent an agent i at random, and allocate to i the remaining good that is ranked highest by i, and repeat until all goods are assigned or all agents have a good. Theorem Serial Random Dictatorship implements honest reporting in weakly dominant strategies. If I omit something from my list, that might be the only item I could get; if I switch the order, I might get a worse item than I would have gotten if I were honest. Terence Johnson (ND) Two-Sided Matching December 1, 2017 39 / 47

Serial Random Dictatorship Goods 1 2 3 4 1 5 10 6 3 2 2 4 8 2 Agents 3 6 5 3 1 4 6 1 2 5 Terence Johnson (ND) Two-Sided Matching December 1, 2017 40 / 47

Serial Random Dictatorship Consider the following properties a game might have: Pareto Optimality: given the reported preferences, the game selects a Pareto optimal lottery, so that there is no lottery that makes all the agents better off and at least one agent strictly better off Symmetry: if agent i and i report the same preferences, they expect the same probabilities of getting each good Nash implementation: it is a Nash equilibrium to report preferences honestly Theorem There is no game that allocates goods to agents that satisfies Pareto optimality, symmetry, and implements truth-telling in Nash equilibrium for all profiles of agent preferences. Terence Johnson (ND) Two-Sided Matching December 1, 2017 41 / 47

Serial Random Dictatorship Consider the following properties a game might have: Pareto Optimality: given the reported preferences, the game selects a Pareto optimal lottery, so that there is no lottery that makes all the agents better off and at least one agent strictly better off Symmetry: if agent i and i report the same preferences, they expect the same probabilities of getting each good Nash implementation: it is a Nash equilibrium to report preferences honestly Theorem There is no game that allocates goods to agents that satisfies Pareto optimality, symmetry, and implements truth-telling in Nash equilibrium for all profiles of agent preferences. Proof is essentially by a complicated exercise in linear algebra. Terence Johnson (ND) Two-Sided Matching December 1, 2017 41 / 47