Two-Sided Matching Terence Johnson University of Notre Dame September 1, 2011 Terence Johnson (ND) Two-Sided Matching September 1, 2011 1 / 37
One-to-One Matching: Gale-Shapley (1962) 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. If agents are never indifferent between two potential partners, then preferences are strict. 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 September 1, 2011 2 / 37
Matchings Definition A matching µ is a one-to-one correspondence from the set M W onto itself that is idempotent (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 September 1, 2011 3 / 37
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 µ 2 = ( ) m1 m 2 w 2 w 1 So there is no reason to expect the set of stable matchings to be unique. Terence Johnson (ND) Two-Sided Matching September 1, 2011 4 / 37
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 September 1, 2011 5 / 37
Stable Matches and the Core Recall that the set of core allocations are those where no coalition of agents can improve or block them. The payoffs associated with a Vickrey auction are in the core (when there is only one good for sale or bidders consider the goods to be substitutes) The set of Walrasian equilibria is in the core for sufficiently large economies. Terence Johnson (ND) Two-Sided Matching September 1, 2011 6 / 37
Stable Matches and the Core Recall that the set of core allocations are those where no coalition of agents can improve or block them. The payoffs associated with a Vickrey auction are in the core (when there is only one good for sale or bidders consider the goods to be substitutes) The set of Walrasian equilibria is in the core for sufficiently large economies. Stable matches are important because they correspond to these Walrasian equilibria or Vickrey prices for matching games: Theorem The set of stable matchings is the core of a marriage market. Terence Johnson (ND) Two-Sided Matching September 1, 2011 6 / 37
Uses of Matching Theory How should children be assigned to public schools, students to universities, doctors to residency programs,...? How should organ exchange programs be organized, like the market for kidneys? If there are many families trying to get houses and a price system cannot be used, what market designs can improve welfare? Terence Johnson (ND) Two-Sided Matching September 1, 2011 7 / 37
The Matching Research Agenda The matching market research agenda can be summed up in the following questions: When do stable matchings exist? What games, markets, or algorithms compute matches that are stable? What is the mathematical structure of the set of stable matchings? When do there exist non-cooperative games that implement the same outcomes as a stable matching algorithm? Terence Johnson (ND) Two-Sided Matching September 1, 2011 8 / 37
The Gale-Shapley Algorithm Definition The Gale-Shapley algorithm is the procedure where Terence Johnson (ND) Two-Sided Matching September 1, 2011 9 / 37
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. Terence Johnson (ND) Two-Sided Matching September 1, 2011 9 / 37
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 September 1, 2011 9 / 37
Example 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 ) Terence Johnson (ND) Two-Sided Matching September 1, 2011 10 / 37
Example 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 ) In the first round, m 1 proposes to w 2, while m 2 and m 3 propose to w 1. w 2 conditionally accepts m 1 while w 1 chooses m 3 over m 2, giving ( ) w1 w µ 1 = 2 w 3 m 3 m 1 w 3 Terence Johnson (ND) Two-Sided Matching September 1, 2011 10 / 37
Example 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 ) In the first round, m 1 proposes to w 2, while m 2 and m 3 propose to w 1. w 2 conditionally accepts m 1 while w 1 chooses m 3 over m 2, giving ( ) w1 w µ 1 = 2 w 3 m 3 m 1 w 3 In the second round, the only bachelor m 2 proposes to w 3. She prefers him to remaining single, so they match, and no single men remain: ( ) w1 w µ 2 = 2 w 3 m 3 m 1 m 2 Terence Johnson (ND) Two-Sided Matching September 1, 2011 10 / 37
Existence Theorem The set of stable matchings is non-empty. 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. Terence Johnson (ND) Two-Sided Matching September 1, 2011 11 / 37
Existence Theorem The set of stable matchings is non-empty. 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 September 1, 2011 11 / 37
Existence Theorem The set of stable matchings is non-empty. 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 September 1, 2011 11 / 37
Existence Theorem The set of stable matchings is non-empty. 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. This is basically the original proof of Gale and Shapley. This case is special: It does not extend to markets with one side or more than two sides, for example. Terence Johnson (ND) Two-Sided Matching September 1, 2011 11 / 37
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 similarly for the women. Terence Johnson (ND) Two-Sided Matching September 1, 2011 12 / 37
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 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 can be computed by allowing the men to 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 September 1, 2011 12 / 37
Opposing Preferences Theorem When all sides have strict preferences, if µ M µ iff 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. Terence Johnson (ND) Two-Sided Matching September 1, 2011 13 / 37
Opposing Preferences Theorem When all sides have strict preferences, if µ M µ iff 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). Terence Johnson (ND) Two-Sided Matching September 1, 2011 13 / 37
Opposing Preferences Theorem When all sides have strict preferences, if µ M µ iff 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). 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, µ > W µ. Terence Johnson (ND) Two-Sided Matching September 1, 2011 13 / 37
The Structure of Stable Matchings Suppose we ask all the men to point to the woman they most prefer who they are matched to at any stable matching. At the M-optimal stable match, they obviously point to their current partner, and no two men point to the same woman. Terence Johnson (ND) Two-Sided Matching September 1, 2011 14 / 37
The Structure of Stable Matchings Suppose we ask all the men to point to the woman they most prefer who they are matched to at any stable matching. At the M-optimal stable match, they obviously point to their current partner, and no two men point to the same woman. Suppose for two matches, µ and µ, we do the same, asking each man m to point to whichever of µ(m) or µ (m) he prefers. It turns out that if µ and µ are stable, again, no two men point to the same woman. (There is no obvious reason to expect this to be true) Terence Johnson (ND) Two-Sided Matching September 1, 2011 14 / 37
The Structure of Stable Matchings Suppose we ask all the men to point to the woman they most prefer who they are matched to at any stable matching. At the M-optimal stable match, they obviously point to their current partner, and no two men point to the same woman. Suppose for two matches, µ and µ, we do the same, asking each man m to point to whichever of µ(m) or µ (m) he prefers. It turns out that if µ and µ are stable, again, no two men point to the same woman. (There is no obvious reason to expect this to be true) Suppose preferences are strict. Define the function on M W M W λ = µ M µ as λ(m) = µ(m) if µ(m) > m µ (m) and λ(m) = µ (m) otherwise, and λ(w) = µ(w) if µ(w) < w µ (w) and λ(w) = µ (w) otherwise. Define µ M µ similarly, but instead of giving each man his preferred partner of µ(m) and µ (m), give him his less preferred partner. Terence Johnson (ND) Two-Sided Matching September 1, 2011 14 / 37
The Structure of Stable Matchings Theorem When all preferences are strict, if µ and µ are stable matchings, then µ M µ and µ M µ are stable matchings. Terence Johnson (ND) Two-Sided Matching September 1, 2011 15 / 37
The Structure of Stable Matchings Theorem When all preferences are strict, if µ and µ are stable matchings, then µ M µ and µ M µ are stable matchings. Definition Let X be a partially ordered set. The join of two elements x, x X is x x = sup{z X : z x, z x}. The meet of two elements x, x X is x x = inf{z X : z x, z x}. A set X is a lattice if it is a partially ordered set, where the meet and join of any two elements are also in X. Terence Johnson (ND) Two-Sided Matching September 1, 2011 15 / 37
The Structure of Stable Matchings Theorem When all preferences are strict, if µ and µ are stable matchings, then µ M µ and µ M µ are stable matchings. Definition Let X be a partially ordered set. The join of two elements x, x X is x x = sup{z X : z x, z x}. The meet of two elements x, x X is x x = inf{z X : z x, z x}. A set X is a lattice if it is a partially ordered set, where the meet and join of any two elements are also in X. Theorem The set of stable matchings is a lattice, partially ordered by M, with meet M and join M, and likewise for W, W, and W. This way of thinking about matching is exploited in Hatfield-Milgrom through Tarski s fixed point theorem: Every isotone function on a lattice has a fixed point. Terence Johnson (ND) Two-Sided Matching September 1, 2011 15 / 37
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. Terence Johnson (ND) Two-Sided Matching September 1, 2011 16 / 37
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 static 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 September 1, 2011 16 / 37
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 ), and P(w 2 ) = (m 1, m 2 ). 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 September 1, 2011 17 / 37
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 ), and P(w 2 ) = (m 1, m 2 ). 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 Terence Johnson (ND) Two-Sided Matching September 1, 2011 17 / 37
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 ), and P(w 2 ) = (m 1, m 2 ). 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 September 1, 2011 17 / 37
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 ), and P(w 2 ) = (m 1, m 2 ). 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 Terence Johnson (ND) Two-Sided Matching September 1, 2011 17 / 37
Equilibrium in the Gale-Shapley Algorithm However, the following is true: 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 September 1, 2011 18 / 37
Equilibrium in the Gale-Shapley Algorithm However, the following is true: 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. But, Theorem When any stable mechanism is applied to a marriage market in which preferences are strict and there is more than one stable matching, then at least one agent can profitably misrepresent his or her preferences if the others tell the truth. Terence Johnson (ND) Two-Sided Matching September 1, 2011 18 / 37
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 as a Bayesian Nash equilibrium 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 September 1, 2011 19 / 37
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 ). Terence Johnson (ND) Two-Sided Matching September 1, 2011 20 / 37
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. Suppose the mechanism selects µ with strictly positive probability. Terence Johnson (ND) Two-Sided Matching September 1, 2011 20 / 37
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. 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). Terence Johnson (ND) Two-Sided Matching September 1, 2011 20 / 37
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. 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 September 1, 2011 20 / 37
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. 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 September 1, 2011 20 / 37
Achieving Dominant Strategy Implementation The mechanism in which Agents submit reports of their preferences over partners The matchmaker either randomly draws an agent and matches him or her to his or her favorite mate who is currently unmatched is called serial random dictatorship. Terence Johnson (ND) Two-Sided Matching September 1, 2011 21 / 37
Achieving Dominant Strategy Implementation The mechanism in which Agents submit reports of their preferences over partners The matchmaker either randomly draws an agent and matches him or her to his or her favorite mate who is currently unmatched is called serial random dictatorship. Theorem Serial random dictatorship is the only mechanism that implements honest reporting in weakly dominant strategies for both sides. Terence Johnson (ND) Two-Sided Matching September 1, 2011 21 / 37
Summary of One-to-One Matching The M-optimal and W -optimal matches can be computed by using the Gale-Shapley algorithm The set of stable matches is a lattice, partially ordered by M and W 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 September 1, 2011 22 / 37
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 has a quota q Ci, giving the maximum number of students it can admit. A matching µ is a function from the set C S into the set of all subsets of C S such that µ(s) 1, µ(c i ) q Ci, µ(s) = C iff s µ(c). Terence Johnson (ND) Two-Sided Matching September 1, 2011 23 / 37
Many-to-One Matching: Preferences To deal with externalities between students for a college, the following definition is introduced: Definition The preference relation P # (C) over sets of students is responsive to the preferences P(C) over individual students if, whenever µ (C) = (µ(c) {s k })\{s }, for s µ(c) and s k / µ(c), then C prefers µ (C) to µ(c) iff C prefers s k to s. In words, Preferences are responsive if, whenever we swap s k for s at match µ, college C prefers the swap only if s k is preferred to s. This basically assumes that whatever externalities exist between students never change the college s preference ordering, conditional on the students the college already has. Terence Johnson (ND) Two-Sided Matching September 1, 2011 24 / 37
The Many-to-One Deferred Acceptance Algorithm Consider breaking each college into q Ci marriage market. copies of itself and constructing a Terence Johnson (ND) Two-Sided Matching September 1, 2011 25 / 37
The Many-to-One Deferred Acceptance Algorithm Consider breaking each college into q Ci copies of itself and constructing a marriage market. Theorem If preferences are responsive, a matching is stable in the many-to-one market iff the matching is stable in the corresponding marriage 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 September 1, 2011 25 / 37
Many-to-One Matching with Transfers One reason this literature is popular is that money is not necessary. What if transfers are allowed back into the environment? There are i = 1, 2,..., m workers and j = 1, 2,..., n firms. Each firm can hire unlimited workers, but each worker can work at only one firm. Workers are indifferent about the other workers at the firm. Worker i s payoff at the j firm with salary s ij is u i (j, s ij ). Firm j s payoff of hiring the set of workers C j is π j (C j, s j ) = y j (C j ) i C j s ij, where y j (C j ) are its gross profits. Let worker i s reservation wage at firm j be u i (j, σ ij ) = u i (0, 0). Suppose that y j (C {i}) y j (C) σ ij 0 and y j ( ) = 0. Terence Johnson (ND) Two-Sided Matching September 1, 2011 26 / 37
Many-to-One Matching with Transfers Kelso and Crawford (1982) introduce a gross substitutes condition that is revisited in Hatfield and Milgrom (2002): Let M j (s j ) = max C π j (C, s j ) be the profit-maximizing package of workers at salary vector s j. Let s j and s j be two different vectors of salaries. Let T j (C j ) = {i : i C j and s ij = s ij ; i.e., the set of chosen workers paid the same at the two salary vectors. Then workers are gross substitutes, if, for every firm j, if C j M j (s j ) and s j s j, then there exists C j M j ( s j ) such that T j (C j ) C j. This says that workers are gross substitutes if increases in other worker s salaries never causes a firm to withdraw an offer from a worker whose salary has not risen. Additively separable production technology, for example, would satisfy this restriction. Terence Johnson (ND) Two-Sided Matching September 1, 2011 27 / 37
The Salary Adjustment Process R1: Firms face a set of salaries s ij (0) = σ ij. Each firm makes an initial offer to all the workers. Terence Johnson (ND) Two-Sided Matching September 1, 2011 28 / 37
The Salary Adjustment Process R1: Firms face a set of salaries s ij (0) = σ ij. Each firm makes an initial offer to all the workers. R2: Each successive round, each firm makes offers to the members of one of its favorite sets of workers, given the schedule of prices s j (t). Any offer made by firm j in round t 1 that was not rejected must be repeated in round t. Terence Johnson (ND) Two-Sided Matching September 1, 2011 28 / 37
The Salary Adjustment Process R1: Firms face a set of salaries s ij (0) = σ ij. Each firm makes an initial offer to all the workers. R2: Each successive round, each firm makes offers to the members of one of its favorite sets of workers, given the schedule of prices s j (t). Any offer made by firm j in round t 1 that was not rejected must be repeated in round t. R3: Each worker who receives one or more offers rejects all but his or her favorite, which he or she tentatively accepts. Terence Johnson (ND) Two-Sided Matching September 1, 2011 28 / 37
The Salary Adjustment Process R1: Firms face a set of salaries s ij (0) = σ ij. Each firm makes an initial offer to all the workers. R2: Each successive round, each firm makes offers to the members of one of its favorite sets of workers, given the schedule of prices s j (t). Any offer made by firm j in round t 1 that was not rejected must be repeated in round t. R3: Each worker who receives one or more offers rejects all but his or her favorite, which he or she tentatively accepts. R4: Offers not rejected in previous periods remain in force. So if worker i rejected an offer from firm j at t 1, s ij (t) = s ij (t 1) + δ, where δ is some small price increment. Otherwise, s ij (t) = s ij (t 1). Terence Johnson (ND) Two-Sided Matching September 1, 2011 28 / 37
The Salary Adjustment Process R1: Firms face a set of salaries s ij (0) = σ ij. Each firm makes an initial offer to all the workers. R2: Each successive round, each firm makes offers to the members of one of its favorite sets of workers, given the schedule of prices s j (t). Any offer made by firm j in round t 1 that was not rejected must be repeated in round t. R3: Each worker who receives one or more offers rejects all but his or her favorite, which he or she tentatively accepts. R4: Offers not rejected in previous periods remain in force. So if worker i rejected an offer from firm j at t 1, s ij (t) = s ij (t 1) + δ, where δ is some small price increment. Otherwise, s ij (t) = s ij (t 1). R5: The process stops when no rejections are issued in some period. Terence Johnson (ND) Two-Sided Matching September 1, 2011 28 / 37
Adjustment Process Outcomes Theorem The salary adjustment process converges in finite time to a core allocation (of a suitably discretized market, to deal with the discrete increments of the price process). So Kelso-Crawford (1982) has elements of auctions as well as matching. But it is missing the lattice-theoretic structure of the set of stable matches, and the strategic incentives are not at all clear (These issues are revisited in Hatfield-Milgrom (2002)). Terence Johnson (ND) Two-Sided Matching September 1, 2011 29 / 37
My Research: Implementation and Intermediaries Suppose agents have private signals about their characteristics as a partner, and transfers can be paid to an intermediary matchmaker. What parts of this theory survive? Terence Johnson (ND) Two-Sided Matching September 1, 2011 30 / 37
My Research: Implementation and Intermediaries Suppose agents have private signals about their characteristics as a partner, and transfers can be paid to an intermediary matchmaker. What parts of this theory survive? It is not an ex post equilibrium to report information honestly in serial random dictatorship, or for either side in the Gale-Shapley algorithm. In a generalization of the Vickrey-Clarke-Groves mechanism to matching environments with transfers, honest reporting is never a dominant strategy. Terence Johnson (ND) Two-Sided Matching September 1, 2011 30 / 37
My Research: Implementation and Intermediaries Suppose agents have private signals about their characteristics as a partner, and transfers can be paid to an intermediary matchmaker. What parts of this theory survive? It is not an ex post equilibrium to report information honestly in serial random dictatorship, or for either side in the Gale-Shapley algorithm. In a generalization of the Vickrey-Clarke-Groves mechanism to matching environments with transfers, honest reporting is never a dominant strategy. For one-to-one matching markets where agents have private information about their quality and quality is supermodular, sufficient conditions are constructed for a direct revelation mechanism to implement efficient and stable matching. The conditions are much stricter for a profit maximizing matchmaker; namely, the agents may prefer a positive assortative match, while the matchmaker prefers a negative assortative match. Terence Johnson (ND) Two-Sided Matching September 1, 2011 30 / 37
My Research: Implementation and Intermediaries Suppose agents have private signals about their characteristics as a partner, and transfers can be paid to an intermediary matchmaker. What parts of this theory survive? It is not an ex post equilibrium to report information honestly in serial random dictatorship, or for either side in the Gale-Shapley algorithm. In a generalization of the Vickrey-Clarke-Groves mechanism to matching environments with transfers, honest reporting is never a dominant strategy. For one-to-one matching markets where agents have private information about their quality and quality is supermodular, sufficient conditions are constructed for a direct revelation mechanism to implement efficient and stable matching. The conditions are much stricter for a profit maximizing matchmaker; namely, the agents may prefer a positive assortative match, while the matchmaker prefers a negative assortative match. For complementary matching problems, I investigate when dynamic auctions work for two-sided and many-sided matching, and when efficient or stable matching can be achieved as an ex post equilibrium. Terence Johnson (ND) Two-Sided Matching September 1, 2011 30 / 37
General Open Questions in the Matching Literature Externalities between partners in many-to-one matching: Married doctors Strategic Incentives and Incomplete Information: What are the incentive properties of different mechanisms under incomplete information? How can markets be engineered to balance the release of information vs. the tendency to misrepresent one s characteristics to obtain an advantage? One-, Three- and k-sided matching What is the empirical content of matching models, and how can they be taken to data? (There are several papers now that use structural models to estimate preferences and compare real outcomes to the estimated counterfactual under the Gale-Shapley algorithm) Terence Johnson (ND) Two-Sided Matching September 1, 2011 31 / 37
References Gale, David and Lloyd Shapley. College Admissions and the Stability of Marriage. 1962, American Mathematical Monthly, Vol 69, p. 9-15 Roth, Alvin and Marilda Sotomayor. Two-Sided Matching: A study in game-theoretic modeling and analysis, 1990, Cambridge: Cambridge University Press Roth, Alvin and Marilda Sotomayor. Two-sided Matching, in The Handbook of Game Theory with Economic Applications, 1992, Vol 1, p. 485-541. (Free download from Science Direct) Crawford, Vincent and Alexander Kelso. Job Matching, Coalition Formation, and Gross Substitutes. Econometrica, 1982, Vol 50, No 6, p. 1483-1504 Roth, Alvin. The Economics of Matching: Stability and Incentives. Mathematics of Operations Research, 1982, Vol 7, No 4, p. 617-628 Muriel Niederle and Alvin Roth have useful web pages on matching research Terence Johnson (ND) Two-Sided Matching September 1, 2011 32 / 37
Paper 1: State of the Art Matching Theory Hatield and Milgrom. Matching with Contracts, The American Economic Review, 2005, Vol. 95, No. 4, p. 913-935 Hatfield-Milgrom extend the idea of a deferred acceptance algorithm or the M / M construction to a lattice that includes wages. This unifies the Gale-Shapley-Roth literature with the Kelso-Crawford work. Terence Johnson (ND) Two-Sided Matching September 1, 2011 33 / 37
Paper 2: Deferred Acceptance with Incomplete Information Niederle, Muriel and Leeat Yariv. Decentralized Matching with Aligned Preferences. Working Paper, Stanford University. Niederle and Yariv study a matching market where agents know their preferences over partners, but not the preferences of anyone else. They study when a decentralized version of the Gale-Shapley algorithm achieves a stable outcome. Many of their results have the flavor, Provided that agents are sufficiently patient,... Terence Johnson (ND) Two-Sided Matching September 1, 2011 34 / 37
Paper 3: Matching over Time and Search Shimer, Robert and Lones Smith. Assortative Matching and Search. Econometrica, 2003, Vol. 68, No. 2, p. 343-369 Suppose agents meet randomly and have to decide whether to match or not. If they match, each period there is an exogenous chance their partnership is broken, and they have to begin dating again. Shimer-Smith studied the acceptance/rejection behavior of agents in a nice framework, and this has lead to many follow-up papers. Terence Johnson (ND) Two-Sided Matching September 1, 2011 35 / 37
Paper 4: Networks of Buyers and Sellers Kranton, Rachel and Deborah Minehart. A Theory of Buyer-Seller Networks, The American Economic Review, 2001, Vol. 91, No. 3, p. 485-508 One way of looking at a matching is that it is a bi-partite graph. If the matching algorithm is replaced by a network formation game where buyers are matched to sellers who then compete in markets, what do the equilibria look like? Terence Johnson (ND) Two-Sided Matching September 1, 2011 36 / 37
I ll post a copy of the presentation and the Latex/Beamer code at nd.edu/ tjohns20/matching.html Terence Johnson (ND) Two-Sided Matching September 1, 2011 37 / 37