Matching. Terence Johnson. April 17, University of Notre Dame. Terence Johnson (ND) Matching April 17, / 41

Matching Terence Johnson University of Notre Dame April 17, 2018 Terence Johnson (ND) Matching April 17, 2018 1 / 41

Markets without money What do you do when you can t use money to solve your problems? Terence Johnson (ND) Matching April 17, 2018 2 / 41

Markets without money What do you do when you can t use money to solve your problems? Matching: heterosexual men and women marrying in a small town, students matching to universities, workers to jobs where wages are rigid; 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) Matching April 17, 2018 2 / 41

Markets without money What do you do when you can t use money to solve your problems? Matching: heterosexual men and women marrying in a small town, students matching to universities, workers to jobs where wages are rigid; 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) Matching April 17, 2018 2 / 41

Markets without money When money can t be exchanged, what is the analog of the second price auction or Vickrey auction? 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. No one can deviate on their own: it s the overall suitableness of a match for everyone in the town that matters. We want a group equilibrium notion, not equilibrium agent-by-agent (one agent can have an affair only if someone else is willing to as well). Terence Johnson (ND) Matching April 17, 2018 3 / 41

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 uniquely interesting, or anything like that. Terence Johnson (ND) Matching April 17, 2018 4 / 41

Marriage Markets: model There are two disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w l } of women. Terence Johnson (ND) Matching April 17, 2018 5 / 41

Marriage Markets: model There are two disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w l } of women. Each man i has ordered, transitive, and complete preferences over the women, so that if w mi w, man i prefers woman w to woman w. If m i mi w, then m i prefers to be single or matched to himself rather than matched to w. The same definitions apply for the women, swapping the w s for m s and m s for w s. Terence Johnson (ND) Matching April 17, 2018 5 / 41

Marriage Markets: model There are two disjoint sets M = {m 1, m 2,..., m n } of men and W = {w 1, w 2,..., w l } of women. Each man i has ordered, transitive, and complete preferences over the women, so that if w mi w, man i prefers woman w to woman w. If m i mi w, then m i prefers to be single or matched to himself rather than matched to w. The same definitions apply for the women, swapping the w s for m s and m s for w s. A marriage market is the set of agents and their preferences. Notice, this is completely ordinal: no utility functions, no calculus Terence Johnson (ND) Matching April 17, 2018 5 / 41

Stability In a matching market, an outcome is stable if no two people want to cheat and have an affair. Terence Johnson (ND) Matching April 17, 2018 6 / 41

Stability In a matching market, an outcome is stable if no two people want to cheat and have an affair. 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) Matching April 17, 2018 6 / 41

Matchings Definition A matching µ is a one-to-one mapping from the set of men and women to the set of men and women so that each agent is matched at most once to someone else (or themselves), 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) Matching April 17, 2018 7 / 41

Matchings Definition A matching µ is a one-to-one mapping from the set of men and women to the set of men and women so that each agent is matched at most once to someone else (or themselves), 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) Matching April 17, 2018 7 / 41

Questions Does such a matching always exist? How do we find it through an auction-like mechanism? What is an optimal match? Is stability in conflict with optimality? What are the strategic incentives that matching mechanisms provide to participants? Terence Johnson (ND) Matching April 17, 2018 8 / 41

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 in one-sided matching. Terence Johnson (ND) Matching April 17, 2018 9 / 41

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 ( ) µ w1 w = 2 w 3 m 1 m 3 m 2 however, is stable. Terence Johnson (ND) Matching April 17, 2018 10 / 41

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine Terence Johnson (ND) Matching April 17, 2018 11 / 41

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine There are four students trying to match into two rooms, a, b, c, and d. Terence Johnson (ND) Matching April 17, 2018 11 / 41

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine There are four students trying to match into two 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) Matching April 17, 2018 11 / 41

Instability in one-sided markets There is not always a stable match in every kind of market you might imagine There are four students trying to match into two 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) Matching April 17, 2018 11 / 41

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) Matching April 17, 2018 12 / 41

The Gale-Shapley Algorithm Definition The Gale-Shapley algorithm is the procedure where Terence Johnson (ND) Matching April 17, 2018 13 / 41

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) Matching April 17, 2018 13 / 41

Marriage Markets On one side of the market, we have agents A, 2, 3, 4. On the other side of the market, we have agents 10, J, Q, K. Terence Johnson (ND) Matching April 17, 2018 14 / 41

Marriage Markets On one side of the market, we have agents A, 2, 3, 4. On the other side of the market, we have agents 10, J, Q, K. 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 (can t return to people you previously proposed to); the deferred acceptance side can accept or reject however it sees fit Terence Johnson (ND) Matching April 17, 2018 14 / 41

Existence of stable matchings Theorem Every marriage market has at least one stable match, and the Gale-Shapley algorithm computes it. Terence Johnson (ND) Matching April 17, 2018 15 / 41

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 this is the total number of proposals that could ever be made because men never revisit women who have previously rejected them. So GS 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) Matching April 17, 2018 15 / 41

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 this is the total number of proposals that could ever be made because men never revisit women who have previously rejected them. So GS 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) Matching April 17, 2018 15 / 41

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 this is the total number of proposals that could ever be made because men never revisit women who have previously rejected them. So GS 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) Matching April 17, 2018 15 / 41

Optimality What would it mean for a match to be optimal? Terence Johnson (ND) Matching April 17, 2018 16 / 41

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) Matching April 17, 2018 16 / 41

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) Matching April 17, 2018 16 / 41

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) Matching April 17, 2018 17 / 41

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. Terence Johnson (ND) Matching April 17, 2018 17 / 41

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. Terence Johnson (ND) Matching April 17, 2018 17 / 41

Optimality and Opposing Preferences Theorem When all sides have strict preferences, µ M µ if and only if µ W µ. Terence Johnson (ND) Matching April 17, 2018 18 / 41

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) Matching April 17, 2018 18 / 41

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) Matching April 17, 2018 18 / 41

Optimality and Opposing Preferences (w, µ(w)) is a blocking pair for µ : both sides can t both prefer one stable match µ to another µ Terence Johnson (ND) Matching April 17, 2018 19 / 41

Optimality and Opposing Preferences The previous result has a very stark consequence: Theorem If all preferences are strict, there is an M-optimal stable matching, and a W -optimal stable matching. The best stable match for the men is the worst stable match for the women, and vice versa. The M-optimal stable match is computed when men propose in the Gale-Shapley algorithm, and likewise for the women. Terence Johnson (ND) Matching April 17, 2018 20 / 41

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) Matching April 17, 2018 21 / 41

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) Matching April 17, 2018 21 / 41

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) Matching April 17, 2018 21 / 41

Other matching algorithms Not all matching algorithms are so nice Terence Johnson (ND) Matching April 17, 2018 22 / 41

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) Matching April 17, 2018 22 / 41

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) Matching April 17, 2018 22 / 41

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) Matching April 17, 2018 23 / 41

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) Matching April 17, 2018 23 / 41

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) Matching April 17, 2018 23 / 41

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) Matching April 17, 2018 24 / 41

Other matching algorithms... ( w1 w µ 0 = 2 w 3 m 1 m 2 m 3 which is where we started. ) Terence Johnson (ND) Matching April 17, 2018 25 / 41

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) Matching April 17, 2018 25 / 41

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) Matching April 17, 2018 26 / 41

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) Matching April 17, 2018 27 / 41

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) Matching April 17, 2018 27 / 41

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) Matching April 17, 2018 27 / 41

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) Matching April 17, 2018 28 / 41

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) Matching April 17, 2018 28 / 41

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) Matching April 17, 2018 28 / 41

Equilibrium in the Gale-Shapley Algorithm This shows agents on the conditionally accepting side may have incentives to misrepresent their preferences Terence Johnson (ND) Matching April 17, 2018 29 / 41

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) Matching April 17, 2018 29 / 41

Equilibrium in the Gale-Shapley Algorithm What about incentives on the proposing side? Terence Johnson (ND) Matching April 17, 2018 30 / 41

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) Matching April 17, 2018 30 / 41

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) Matching April 17, 2018 31 / 41

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) Matching April 17, 2018 31 / 41

Roth s Impossibility Theorem Theorem There does not exist a stable matching mechanism that implements honest reporting in weakly dominant strategies. Terence Johnson (ND) Matching April 17, 2018 32 / 41

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) Matching April 17, 2018 32 / 41

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) Matching April 17, 2018 33 / 41

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) Matching April 17, 2018 33 / 41

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) Matching April 17, 2018 34 / 41

Many-to-One Matching: Hospitals/Doctors and Colleges/Students To what extent can the marriage market results be generalized to many-to-one matching models? Terence Johnson (ND) Matching April 17, 2018 35 / 41

Many-to-One Matching: Hospitals/Doctors and Colleges/Students To what extent can the marriage market results be generalized to many-to-one matching models? There are two disjoint sets, students S = {s 1,..., s L } and colleges, C = {c 1,..., c K }. 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) Matching April 17, 2018 35 / 41

Many-to-One Matching: Hospitals/Doctors and Colleges/Students To what extent can the marriage market results be generalized to many-to-one matching models? There are two disjoint sets, students S = {s 1,..., s L } and colleges, C = {c 1,..., c K }. 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) Matching April 17, 2018 35 / 41

The Many-to-One Deferred Acceptance Algorithm Break each college j into q cj copies of itself and construct a one-to-one marriage market where each copy of college j has the same preferences, and the students preferences over copies are the same as the preferences over the original colleges. Theorem A matching is stable in the many-to-one market if and only if the matching is stable in the corresponding one-to-one market. Terence Johnson (ND) Matching April 17, 2018 36 / 41

The Many-to-One Deferred Acceptance Algorithm Break each college j into q cj copies of itself and construct a one-to-one marriage market where each copy of college j has the same preferences, and the students preferences over copies are the same as the preferences over the original colleges. Theorem A matching is stable in the many-to-one market if and only if the matching is stable in the corresponding one-to-one market. So all our one-to-one matching results apply to the many-to-one market, when we break the colleges/hospitals/employers/etc into copies of itself. Terence Johnson (ND) Matching April 17, 2018 36 / 41

The Many-to-One Deferred Acceptance Algorithm: faster implementation Using the idea of breaking the college or hospital into copies of itself is nice because it reduces the theory to the marriage market we ve already studied It is inconvenient because it makes the algorithm messier than it has to be: students propose to copies of the same school over and over before moving on to another school Instead, we can give each college a counter : when it conditionally accepts a student, the counter increases, and cannot exceed the school s quota. Terence Johnson (ND) Matching April 17, 2018 37 / 41

The Many-to-One Deferred Acceptance Algorithm: faster implementation Using the idea of breaking the college or hospital into copies of itself is nice because it reduces the theory to the marriage market we ve already studied It is inconvenient because it makes the algorithm messier than it has to be: students propose to copies of the same school over and over before moving on to another school Instead, we can give each college a counter : when it conditionally accepts a student, the counter increases, and cannot exceed the school s quota. If an additional student applies whom the college prefers to one of its existing students, it kicks out its least preferred student in favor of the new applicant. This significantly speeds things up (for doing calculations) Terence Johnson (ND) Matching April 17, 2018 37 / 41

The Many-to-One Deferred Acceptance Algorithm Suppose there are six students and two colleges. For students 1 to 3, s i : c 1 c 2 and for students 4 to 6, The college s preferences are 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) Matching April 17, 2018 38 / 41

Many-to-one matching: Package Preferences Welfare: whichever side gets to propose is, again, going to get its preferred stable match, which is the worst stable match with respect to the opposite side s preferences Strategy: the proposing side has a weakly dominant strategy to be honest, the accepting side might have a profitable deviation in cutting partners from the bottom of its list Terence Johnson (ND) Matching April 17, 2018 39 / 41

Many-to-one matching: Package Preferences Welfare: whichever side gets to propose is, again, going to get its preferred stable match, which is the worst stable match with respect to the opposite side s preferences Strategy: the proposing side has a weakly dominant strategy to be honest, the accepting side might have a profitable deviation in cutting partners from the bottom of its list Incomplete information: stable matching mechanisms that implement truth-telling in weakly dominant strategies do not exist Terence Johnson (ND) Matching April 17, 2018 39 / 41

Many-to-one matching: Package Preferences What if, instead of having a list of student preferences, a university had preferences over its entire class? What if, instead of having preferences over universities, students had preferences over the university and their peers? Suppose a college C has preferences over individual students and preferences over packages of students C. σ C is responsive if for every matching for C, µ(c), and an alternative µ = µ(c)\s s, µ C µ if and only if s C s. Then all our previous results work. Consider couples in the hopsital/doctor case: a hospital may want to hire a superstar at the cost of hiring a joker, if it s necessary to get the superstar Terence Johnson (ND) Matching April 17, 2018 40 / 41

Many-to-one matching: Package Preferences What if, instead of having a list of student preferences, a university had preferences over its entire class? What if, instead of having preferences over universities, students had preferences over the university and their peers? Suppose a college C has preferences over individual students and preferences over packages of students C. σ C is responsive if for every matching for C, µ(c), and an alternative µ = µ(c)\s s, µ C µ if and only if s C s. Then all our previous results work. Consider couples in the hopsital/doctor case: a hospital may want to hire a superstar at the cost of hiring a joker, if it s necessary to get the superstar This kind of change breaks the math underlying the one-to-one market in a fundamental way; there aren t good answers to this problem yet. Terence Johnson (ND) Matching April 17, 2018 40 / 41

How old are these ideas? Vickrey: 1996 Nobel prize, key paper written in 1961 Myerson/Maskin/Hurwicz: 2007 Nobel prize for mechanism design, key papers written in 1981 and 1977 Roth/Gale/Shapley: 2012 Nobel prize, key papers written in 1985, 1962 Terence Johnson (ND) Matching April 17, 2018 41 / 41