Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com
Transcript of Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com
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Dynamic Matching —Part 1
Leeat Yariv
Yale University
February 21, 2016
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Two-sided One-to-one Matching Markets
I Two classes of agents: men and women, workers and firms,rabbis and congregations, patients and organ donors, etc.
I An agent from each class cares about whom they match withfrom the other class
I Today:I Static matching for pedestriansI Why care about dynamic matching?I Some approaches to dynamic matching
I Wednesday: A particular framework for studying dynamicmatching mechanisms
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Two-sided One-to-one Matching Markets
I Two classes of agents: men and women, workers and firms,rabbis and congregations, patients and organ donors, etc.
I An agent from each class cares about whom they match withfrom the other class
I Today:I Static matching for pedestriansI Why care about dynamic matching?I Some approaches to dynamic matching
I Wednesday: A particular framework for studying dynamicmatching mechanisms
![Page 4: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/4.jpg)
Static Matching on a Shoestring
Participants:
I A set of Squares S , with a typical firm s ∈ SI A set of Rounds R, with a typical round r ∈ RI Squares and rounds can stand for men and women, workersand firms, patients and organ donors, etc.
Preferences:
I Each square s has a complete ranking �s over R ∪ {s}I Each round r has a complete ranking �r over S ∪ {r}I Today: ignore indifferences (strict preferences)I A round r is acceptable to square s if r �s s, andunacceptable otherwise; Similarly for acceptable squares
![Page 5: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/5.jpg)
Static Matching on a Shoestring
Participants:
I A set of Squares S , with a typical firm s ∈ SI A set of Rounds R, with a typical round r ∈ RI Squares and rounds can stand for men and women, workersand firms, patients and organ donors, etc.
Preferences:
I Each square s has a complete ranking �s over R ∪ {s}I Each round r has a complete ranking �r over S ∪ {r}I Today: ignore indifferences (strict preferences)I A round r is acceptable to square s if r �s s, andunacceptable otherwise;
Similarly for acceptable squares
![Page 6: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/6.jpg)
Static Matching on a Shoestring
Participants:
I A set of Squares S , with a typical firm s ∈ SI A set of Rounds R, with a typical round r ∈ RI Squares and rounds can stand for men and women, workersand firms, patients and organ donors, etc.
Preferences:
I Each square s has a complete ranking �s over R ∪ {s}I Each round r has a complete ranking �r over S ∪ {r}I Today: ignore indifferences (strict preferences)I A round r is acceptable to square s if r �s s, andunacceptable otherwise; Similarly for acceptable squares
![Page 7: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/7.jpg)
Matchings and Stable Matchings
I A matching is a one-to-one mapping µ : S ∪ R −→ S ∪ Rsatisfying:
I µ is of order 2: µ2(x) = x for all xI µ(s) ∈ R ∪ {s} for all s ∈ SI µ(r) ∈ S ∪ {r} for all r ∈ R
I A matching µ is stable if:I It is individually rational : If µ(s) = r , then s �r r and r �s sI There are no blocking pairs, pairs (s ′, r ′) such thatr ′ �s ′ µ(s ′) and s ′ �r ′ µ(r ′)
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Matchings and Stable Matchings
I A matching is a one-to-one mapping µ : S ∪ R −→ S ∪ Rsatisfying:
I µ is of order 2: µ2(x) = x for all xI µ(s) ∈ R ∪ {s} for all s ∈ SI µ(r) ∈ S ∪ {r} for all r ∈ R
I A matching µ is stable if:I It is individually rational : If µ(s) = r , then s �r r and r �s sI There are no blocking pairs, pairs (s ′, r ′) such thatr ′ �s ′ µ(s ′) and s ′ �r ′ µ(r ′)
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Stable Matchings —Example 1
Suppose S = {s1, s2}, R = {r1, r2}, and
s1 : r1 � r2 � s1 r1 : s2 � s1 � r1s2 : r2 � r1 � s2 r2 : s1 � s2 � r2
I µ(s1) = r1, µ(s2) = r2 stable (squares are happy)I µ(s1) = r2), µ(s2) = r1 stable (rounds are happy)
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Stable Matchings —Example 1
Suppose S = {s1, s2}, R = {r1, r2}, and
s1 : r1 � r2 � s1 r1 : s2 � s1 � r1s2 : r2 � r1 � s2 r2 : s1 � s2 � r2
I µ(s1) = r1, µ(s2) = r2 stable (squares are happy)
I µ(s1) = r2), µ(s2) = r1 stable (rounds are happy)
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Stable Matchings —Example 1
Suppose S = {s1, s2}, R = {r1, r2}, and
s1 : r1 � r2 � s1 r1 : s2 � s1 � r1s2 : r2 � r1 � s2 r2 : s1 � s2 � r2
I µ(s1) = r1, µ(s2) = r2 stable (squares are happy)I µ(s1) = r2), µ(s2) = r1 stable (rounds are happy)
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Stable Matchings —Example 2
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
)stable
µ̃ =
(s1 s2 s3r2 r1 r3
)unstable
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Stable Matchings —Example 2
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
)stable
µ̃ =
(s1 s2 s3r2 r1 r3
)unstable
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Stable Matchings —Example 2
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
)stable
µ̃ =
(s1 s2 s3r2 r1 r3
)unstable
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Stable Matchings —Example 2
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
)stable
µ̃ =
(s1 s2 s3r2 r1 r3
)unstable
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Stable Matchings —Example 2
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
)stable
µ̃ =
(s1 s2 s3r2 r1 r3
)unstable
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Stable Matchings —Example 2
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
)stable
µ̃ =
(s1 s2 s3r2 r1 r3
)unstable
![Page 18: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/18.jpg)
An Economist Goes to a Bar (Example 3)
Becker (1992), “A Theory of Marriage”
I Society as a cocktail party with rational-minded daterssearching for the most desirable viable partners
I Each individual has an observable “pizzazz”parameterI Everyone agrees that greater pizzazz is more desirableI Unique stable matching: positive assortative matching, menand women of similar desirability partner with one another.
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An Economist Goes to a Bar (Example 3)
Becker (1992), “A Theory of Marriage”
I Society as a cocktail party with rational-minded daterssearching for the most desirable viable partners
I Each individual has an observable “pizzazz”parameterI Everyone agrees that greater pizzazz is more desirableI Unique stable matching: positive assortative matching, menand women of similar desirability partner with one another.
![Page 20: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/20.jpg)
An Economist Goes to a Bar (Example 3)
Becker (1992), “A Theory of Marriage”
I Society as a cocktail party with rational-minded daterssearching for the most desirable viable partners
I Each individual has an observable “pizzazz”parameterI Everyone agrees that greater pizzazz is more desirable
I Unique stable matching: positive assortative matching, menand women of similar desirability partner with one another.
![Page 21: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/21.jpg)
An Economist Goes to a Bar (Example 3)
Becker (1992), “A Theory of Marriage”
I Society as a cocktail party with rational-minded daterssearching for the most desirable viable partners
I Each individual has an observable “pizzazz”parameterI Everyone agrees that greater pizzazz is more desirableI Unique stable matching: positive assortative matching, menand women of similar desirability partner with one another.
![Page 22: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/22.jpg)
Existence and Features of Stable Matchings
Theorem (Gale and Shapley, 1962)There exists a stable matching in any one-to-one matching market.
Proof: By construction, using the Deferred Acceptance Algorithm.
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Existence and Features of Stable Matchings
Theorem (Gale and Shapley, 1962)There exists a stable matching in any one-to-one matching market.
Proof: By construction, using the Deferred Acceptance Algorithm.
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The Deferred Acceptance Algorithm
I Each square proposes to highest round on her list
I Rounds make a “tentative match”based on their preferredoffer, and reject other offers, or all if none are acceptable
I Each rejected square removes round from her list, and makesa new offer
I Continue until no more rejections or offers, at which pointimplement tentative matches
I This is the square-proposing deferred acceptance algorithm;Could analogously define the round-proposing version
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The Deferred Acceptance Algorithm
I Each square proposes to highest round on her listI Rounds make a “tentative match”based on their preferredoffer, and reject other offers, or all if none are acceptable
I Each rejected square removes round from her list, and makesa new offer
I Continue until no more rejections or offers, at which pointimplement tentative matches
I This is the square-proposing deferred acceptance algorithm;Could analogously define the round-proposing version
![Page 26: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/26.jpg)
The Deferred Acceptance Algorithm
I Each square proposes to highest round on her listI Rounds make a “tentative match”based on their preferredoffer, and reject other offers, or all if none are acceptable
I Each rejected square removes round from her list, and makesa new offer
I Continue until no more rejections or offers, at which pointimplement tentative matches
I This is the square-proposing deferred acceptance algorithm;Could analogously define the round-proposing version
![Page 27: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/27.jpg)
The Deferred Acceptance Algorithm
I Each square proposes to highest round on her listI Rounds make a “tentative match”based on their preferredoffer, and reject other offers, or all if none are acceptable
I Each rejected square removes round from her list, and makesa new offer
I Continue until no more rejections or offers, at which pointimplement tentative matches
I This is the square-proposing deferred acceptance algorithm;Could analogously define the round-proposing version
![Page 28: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/28.jpg)
The Deferred Acceptance Algorithm
I Each square proposes to highest round on her listI Rounds make a “tentative match”based on their preferredoffer, and reject other offers, or all if none are acceptable
I Each rejected square removes round from her list, and makesa new offer
I Continue until no more rejections or offers, at which pointimplement tentative matches
I This is the square-proposing deferred acceptance algorithm;Could analogously define the round-proposing version
![Page 29: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/29.jpg)
The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of rounds
I Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejectedI At that point, r ′ preferred his tentative match to sI As algorithm goes forward, r ′ can only do betterI So r ′ prefers her final match to sI Therefore, there are NO BLOCKING PAIRS
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The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of roundsI Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejectedI At that point, r ′ preferred his tentative match to sI As algorithm goes forward, r ′ can only do betterI So r ′ prefers her final match to sI Therefore, there are NO BLOCKING PAIRS
![Page 31: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/31.jpg)
The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of roundsI Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejected
I At that point, r ′ preferred his tentative match to sI As algorithm goes forward, r ′ can only do betterI So r ′ prefers her final match to sI Therefore, there are NO BLOCKING PAIRS
![Page 32: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/32.jpg)
The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of roundsI Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejectedI At that point, r ′ preferred his tentative match to s
I As algorithm goes forward, r ′ can only do betterI So r ′ prefers her final match to sI Therefore, there are NO BLOCKING PAIRS
![Page 33: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/33.jpg)
The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of roundsI Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejectedI At that point, r ′ preferred his tentative match to sI As algorithm goes forward, r ′ can only do better
I So r ′ prefers her final match to sI Therefore, there are NO BLOCKING PAIRS
![Page 34: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/34.jpg)
The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of roundsI Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejectedI At that point, r ′ preferred his tentative match to sI As algorithm goes forward, r ′ can only do betterI So r ′ prefers her final match to s
I Therefore, there are NO BLOCKING PAIRS
![Page 35: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/35.jpg)
The Deferred Acceptance Algorithm
Proof of the Gale-Shapley(1962) Theorem:
I Algorithm must end in a finite number of roundsI Suppose s, r are matched, but s prefers r ′
I At some point, s proposed to r ′ and was rejectedI At that point, r ′ preferred his tentative match to sI As algorithm goes forward, r ′ can only do betterI So r ′ prefers her final match to sI Therefore, there are NO BLOCKING PAIRS
![Page 36: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/36.jpg)
Features of Stable Matchings
I There is a square-optimal stable matching, which is theround-pessimal stable matching and vice versa
I The square-proposing stable matching implements thesquare-optimal stable matching
I In fact, the set of stable matchings has a lattice structureI Stable matchings are Pareto effi cient (but not vice versa, seeExample 2)
![Page 37: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/37.jpg)
Features of Stable Matchings
I There is a square-optimal stable matching, which is theround-pessimal stable matching and vice versa
I The square-proposing stable matching implements thesquare-optimal stable matching
I In fact, the set of stable matchings has a lattice structureI Stable matchings are Pareto effi cient (but not vice versa, seeExample 2)
![Page 38: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/38.jpg)
Features of Stable Matchings
I There is a square-optimal stable matching, which is theround-pessimal stable matching and vice versa
I The square-proposing stable matching implements thesquare-optimal stable matching
I In fact, the set of stable matchings has a lattice structure
I Stable matchings are Pareto effi cient (but not vice versa, seeExample 2)
![Page 39: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/39.jpg)
Features of Stable Matchings
I There is a square-optimal stable matching, which is theround-pessimal stable matching and vice versa
I The square-proposing stable matching implements thesquare-optimal stable matching
I In fact, the set of stable matchings has a lattice structureI Stable matchings are Pareto effi cient (but not vice versa, seeExample 2)
![Page 40: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/40.jpg)
Stability in the Real World
In 1984, Al Roth discovered that the centralized clearinghouse usedto match newly-minted doctors with residencies is theGale-Shapley algorithm (discovered independently by doctors)!
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Stability in the Real World
From Roth (2002), McKinney, Niederle, and Roth (2003), and Lee(2011)
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“Folk Theorem”about the Success of Stable Mechanisms
I Suppose a centralized clearinghouse does not implement astable matching
I Then there would be some participants who would prefer tobe unmatched, or form blocking pairs
I They will try to undo the match (severing connections,poaching rounds from other squares)
I So ultimately, a stable matching would be implemented, butmay entail some effi ciency loss if there are any frictions
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“Folk Theorem”about the Success of Stable Mechanisms
I Suppose a centralized clearinghouse does not implement astable matching
I Then there would be some participants who would prefer tobe unmatched, or form blocking pairs
I They will try to undo the match (severing connections,poaching rounds from other squares)
I So ultimately, a stable matching would be implemented, butmay entail some effi ciency loss if there are any frictions
![Page 44: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/44.jpg)
“Folk Theorem”about the Success of Stable Mechanisms
I Suppose a centralized clearinghouse does not implement astable matching
I Then there would be some participants who would prefer tobe unmatched, or form blocking pairs
I They will try to undo the match (severing connections,poaching rounds from other squares)
I So ultimately, a stable matching would be implemented, butmay entail some effi ciency loss if there are any frictions
![Page 45: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/45.jpg)
“Folk Theorem”about the Success of Stable Mechanisms
I Suppose a centralized clearinghouse does not implement astable matching
I Then there would be some participants who would prefer tobe unmatched, or form blocking pairs
I They will try to undo the match (severing connections,poaching rounds from other squares)
I So ultimately, a stable matching would be implemented, butmay entail some effi ciency loss if there are any frictions
![Page 46: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/46.jpg)
“Folk Theorem”about the Success of Stable Mechanisms
I Suppose a centralized clearinghouse does not implement astable matching
I Then there would be some participants who would prefer tobe unmatched, or form blocking pairs
I They will try to undo the match (severing connections,poaching rounds from other squares)
I So ultimately, a stable matching would be implemented, butmay entail some effi ciency loss if there are any frictions
![Page 47: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/47.jpg)
A Note on Stable Matchings in the Lab
I Decentralized Matching (Echenique and Yariv, 2015):I Stable matchings are often implementedI The median stable matching has a strong drawI Cardinal presentation of preferences matters a lot
I Centralized Matching a-la DA (Echenique, Wilson, andYariv, 2015):
I Stable matchings are implemented around 50% of the timeI Many markets culminate in the receiver-optimal stablematching
I Market features matter (cardinal presentation, number ofstable matchings)
![Page 48: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/48.jpg)
A Note on Stable Matchings in the Lab
I Decentralized Matching (Echenique and Yariv, 2015):I Stable matchings are often implementedI The median stable matching has a strong drawI Cardinal presentation of preferences matters a lot
I Centralized Matching a-la DA (Echenique, Wilson, andYariv, 2015):
I Stable matchings are implemented around 50% of the timeI Many markets culminate in the receiver-optimal stablematching
I Market features matter (cardinal presentation, number ofstable matchings)
![Page 49: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/49.jpg)
A Note on Mechanism Design in Static Settings
I No stable mechanism is incentive compatible
I Unless harsh restrictions on preferences (say, if there is alwaysa unique stable matching)
I When there are frictions (say, incomplete information) all hellbreaks loose
I Stable matchings are not necessarily utilitarian effi cient. Forinstance:
r1 r2s1 5 4s2 3 1
I µ(si ) = ri unique stable matching, but µ(si ) = r3−i is effi cientI (but asymptotically, not necessarily an issue, Lee and Yariv,2015)
![Page 50: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/50.jpg)
A Note on Mechanism Design in Static Settings
I No stable mechanism is incentive compatibleI Unless harsh restrictions on preferences (say, if there is alwaysa unique stable matching)
I When there are frictions (say, incomplete information) all hellbreaks loose
I Stable matchings are not necessarily utilitarian effi cient. Forinstance:
r1 r2s1 5 4s2 3 1
I µ(si ) = ri unique stable matching, but µ(si ) = r3−i is effi cientI (but asymptotically, not necessarily an issue, Lee and Yariv,2015)
![Page 51: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/51.jpg)
A Note on Mechanism Design in Static Settings
I No stable mechanism is incentive compatibleI Unless harsh restrictions on preferences (say, if there is alwaysa unique stable matching)
I When there are frictions (say, incomplete information) all hellbreaks loose
I Stable matchings are not necessarily utilitarian effi cient. Forinstance:
r1 r2s1 5 4s2 3 1
I µ(si ) = ri unique stable matching, but µ(si ) = r3−i is effi cientI (but asymptotically, not necessarily an issue, Lee and Yariv,2015)
![Page 52: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/52.jpg)
A Note on Mechanism Design in Static Settings
I No stable mechanism is incentive compatibleI Unless harsh restrictions on preferences (say, if there is alwaysa unique stable matching)
I When there are frictions (say, incomplete information) all hellbreaks loose
I Stable matchings are not necessarily utilitarian effi cient. Forinstance:
r1 r2s1 5 4s2 3 1
I µ(si ) = ri unique stable matching, but µ(si ) = r3−i is effi cient
I (but asymptotically, not necessarily an issue, Lee and Yariv,2015)
![Page 53: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/53.jpg)
A Note on Mechanism Design in Static Settings
I No stable mechanism is incentive compatibleI Unless harsh restrictions on preferences (say, if there is alwaysa unique stable matching)
I When there are frictions (say, incomplete information) all hellbreaks loose
I Stable matchings are not necessarily utilitarian effi cient. Forinstance:
r1 r2s1 5 4s2 3 1
I µ(si ) = ri unique stable matching, but µ(si ) = r3−i is effi cientI (but asymptotically, not necessarily an issue, Lee and Yariv,2015)
![Page 54: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/54.jpg)
A Note on Transfers
I Can define stability with transfers
I I consider here markets without transfers:I Transfers are “repugnant” (Roth, 2007) and illegal in manymarkets: cannot buy babies or organs in the U.S.
I Even in labor markets, tailored wages are often notimplemented:
I Salaries for new medical residents exhibit low variance acrossthe market and across years, and are uniform across specialtieswithin a hospital
I Hall and Krueger (2012): A large fraction of jobs in the U.S.have posted wages
![Page 55: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/55.jpg)
A Note on Transfers
I Can define stability with transfersI I consider here markets without transfers:
I Transfers are “repugnant” (Roth, 2007) and illegal in manymarkets: cannot buy babies or organs in the U.S.
I Even in labor markets, tailored wages are often notimplemented:
I Salaries for new medical residents exhibit low variance acrossthe market and across years, and are uniform across specialtieswithin a hospital
I Hall and Krueger (2012): A large fraction of jobs in the U.S.have posted wages
![Page 56: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/56.jpg)
Why Think about Dynamic Matching?
I Recall the “folk theorem”— If we do not implement a stablematching, the market will ultimately converge to a stablematching, possibly inducing ineffi ciencies
I When would such a decentralized, and plausibly dynamic,process yield stable outcomes?
I Many matching processes are inherently dynamic:I Fixed set of participants, but dynamic interactions: many labormarkets
I Changing sets of participants over time
![Page 57: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/57.jpg)
Why Think about Dynamic Matching?
I Recall the “folk theorem”— If we do not implement a stablematching, the market will ultimately converge to a stablematching, possibly inducing ineffi ciencies
I When would such a decentralized, and plausibly dynamic,process yield stable outcomes?
I Many matching processes are inherently dynamic:I Fixed set of participants, but dynamic interactions: many labormarkets
I Changing sets of participants over time
![Page 58: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/58.jpg)
Changing Sets of Participants over Time
I Child Adoption:I About 1.6 million, or 2.5%, of children in the U.S. are adoptedI One adoption facilitator —11 new potential parents and 13newly relinquished children each month
I Labor Markets:I U.S. Bureau of Labor Statistics —approximately 5 million newjob openings and slightly fewer than 5 million newlyunemployed workers each month this year
I Organ Donation:I Organ Donation and Transplantation Statistics —a new patientadded to the kidney transplant list every 14 minutes and about3000 patients added each month
I In 2013, about a third of ˜17, 000 kidney transplants involvedliving donors
![Page 59: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/59.jpg)
Changing Sets of Participants over Time
I Child Adoption:I About 1.6 million, or 2.5%, of children in the U.S. are adoptedI One adoption facilitator —11 new potential parents and 13newly relinquished children each month
I Labor Markets:I U.S. Bureau of Labor Statistics —approximately 5 million newjob openings and slightly fewer than 5 million newlyunemployed workers each month this year
I Organ Donation:I Organ Donation and Transplantation Statistics —a new patientadded to the kidney transplant list every 14 minutes and about3000 patients added each month
I In 2013, about a third of ˜17, 000 kidney transplants involvedliving donors
![Page 60: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/60.jpg)
Changing Sets of Participants over Time
I Child Adoption:I About 1.6 million, or 2.5%, of children in the U.S. are adoptedI One adoption facilitator —11 new potential parents and 13newly relinquished children each month
I Labor Markets:I U.S. Bureau of Labor Statistics —approximately 5 million newjob openings and slightly fewer than 5 million newlyunemployed workers each month this year
I Organ Donation:I Organ Donation and Transplantation Statistics —a new patientadded to the kidney transplant list every 14 minutes and about3000 patients added each month
I In 2013, about a third of ˜17, 000 kidney transplants involvedliving donors
![Page 61: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/61.jpg)
Modeling Dynamic Matching
I Fixed participants, dynamic interactions
I Dynamic participation and interactions
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Dynamic Interactions with Fixed Participants
I Definition of dynamic stability —Doval (2015)I Some references: Blum, Roth, and Rothblum (1997),Haeringer and Wooders (2009), Diamantoudi, Miyagawa, andXue (2007), Alcade (1996), Alcalde, Pérez-Castrillo, andRomero-Medina (1998), Alcalde and Romero-Medina (2000),Niederle and Yariv (2009)
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Dynamic Market Game
I t = 0 : squares and rounds enter the market
I t = 1, 2, ... : two stages as follows
Stage 1: Squares simultaneously decide whether and towhom to make an offer. Unmatched square canhave at most one offer out
Stage 2: Each round r who has received an offer from scan accept, reject, or hold the offer
I Once an offer is accepted, round r is matched to square sirreversibly
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Payoffs
I Square s matched to round r at time t → payoffs δtuSsr andδtuRsr , where δ ≤ 1 is the market discount factor. Unmatchedagents receive 0
I To ease getting stable matching: focus on high δ
![Page 65: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/65.jpg)
Payoffs
I Square s matched to round r at time t → payoffs δtuSsr andδtuRsr , where δ ≤ 1 is the market discount factor. Unmatchedagents receive 0
I To ease getting stable matching: focus on high δ
![Page 66: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/66.jpg)
Market Monitoring
I Squares and rounds observe receival, rejection, and deferralonly of own offers. When an offer is accepted, the wholemarket is informed of the match. Similarly, when there ismarket exit
I Equilibrium notion: Nash equilibrium (Bayesian NashEquilibrium if we introduce uncertainty about preferences)
![Page 67: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/67.jpg)
Market Monitoring
I Squares and rounds observe receival, rejection, and deferralonly of own offers. When an offer is accepted, the wholemarket is informed of the match. Similarly, when there ismarket exit
I Equilibrium notion: Nash equilibrium (Bayesian NashEquilibrium if we introduce uncertainty about preferences)
![Page 68: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/68.jpg)
Stable Matchings as Equilibrium Outcomes
Proposition: When the underlying market has a unique stablematching, there exists a Nash equilibrium in strategies that are notweakly dominated that generates the unique stable matching.
Intuition:
I t = 1 : each square s makes offer to µ(s)I t = 1 : each round r accepts square µ(r) or more preferred, orexits if no offers
But there can be other (unstable) equilibrium outcomes...
![Page 69: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/69.jpg)
Stable Matchings as Equilibrium Outcomes
Proposition: When the underlying market has a unique stablematching, there exists a Nash equilibrium in strategies that are notweakly dominated that generates the unique stable matching.
Intuition:
I t = 1 : each square s makes offer to µ(s)I t = 1 : each round r accepts square µ(r) or more preferred, orexits if no offers
But there can be other (unstable) equilibrium outcomes...
![Page 70: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/70.jpg)
Stable Matchings as Equilibrium Outcomes
Proposition: When the underlying market has a unique stablematching, there exists a Nash equilibrium in strategies that are notweakly dominated that generates the unique stable matching.
Intuition:
I t = 1 : each square s makes offer to µ(s)
I t = 1 : each round r accepts square µ(r) or more preferred, orexits if no offers
But there can be other (unstable) equilibrium outcomes...
![Page 71: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/71.jpg)
Stable Matchings as Equilibrium Outcomes
Proposition: When the underlying market has a unique stablematching, there exists a Nash equilibrium in strategies that are notweakly dominated that generates the unique stable matching.
Intuition:
I t = 1 : each square s makes offer to µ(s)I t = 1 : each round r accepts square µ(r) or more preferred, orexits if no offers
But there can be other (unstable) equilibrium outcomes...
![Page 72: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/72.jpg)
Stable Matchings as Equilibrium Outcomes
Proposition: When the underlying market has a unique stablematching, there exists a Nash equilibrium in strategies that are notweakly dominated that generates the unique stable matching.
Intuition:
I t = 1 : each square s makes offer to µ(s)I t = 1 : each round r accepts square µ(r) or more preferred, orexits if no offers
But there can be other (unstable) equilibrium outcomes...
![Page 73: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/73.jpg)
Example 2 (Continued)
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
), µ̃ =
(s1 s2 s3r2 r1 r3
)
µ unique stable matching, can implement µ̃
![Page 74: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/74.jpg)
Example 2 (Continued)
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
), µ̃ =
(s1 s2 s3r2 r1 r3
)
µ unique stable matching, can implement µ̃
![Page 75: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/75.jpg)
Example 2 (Continued)
Suppose S = {s1, s2, s3}, R = {r1, r2, r3}, all acceptable, and
s1 : r2 � r1 � r3s2 : r1 � r2 � r3s3 : r1 � r2 � r3
r1 : s1 � s3 � s2r2 : s2 � s1 � s3r3 : s1 � s3 � s2
µ =
(s1 s2 s3r1 r2 r3
), µ̃ =
(s1 s2 s3r2 r1 r3
)
µ unique stable matching, can implement µ̃
![Page 76: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/76.jpg)
Multiplicity (Continued)
In “sub-market”without (s3, r3), multiple stable matchings(Example 1...):
s1 : r2 � r1s2 : r1 � r2
,r1 : s1 � s2r2 : s2 � s1
µ =
(s1 s2 s3r1 s2 r3
), µ̃ =
(s1 s2 s3r2 r1 r3
)
Stage 1 : s3 and r3 match, Stage 2: follow µ̃
µ̃ induces the square-preferred stable matching in stage 2
![Page 77: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/77.jpg)
Multiplicity (Continued)
In “sub-market”without (s3, r3), multiple stable matchings(Example 1...):
s1 : r2 � r1s2 : r1 � r2
,r1 : s1 � s2r2 : s2 � s1
µ =
(s1 s2 s3r1 s2 r3
), µ̃ =
(s1 s2 s3r2 r1 r3
)Stage 1 : s3 and r3 match, Stage 2: follow µ̃
µ̃ induces the square-preferred stable matching in stage 2
![Page 78: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/78.jpg)
Multiplicity (Continued)
I With enough “alignment”of preferences across squares androunds, no such examples
I In general, dynamic mechanisms with irreversible matchings atcertain periods are prone to these issues
![Page 79: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/79.jpg)
Frictions Make Things Even Worse (Hand-waving)
I Suppose squares and rounds know their own utilities frommatching with others, but have incomplete information abouteveryone else’s preferences
I Incomplete information is likely in large markets (but notmuch is known, even in the static world)
I To make things easy, suppose we know that in eachrealization of preferences/utilities, there is a unique stablematching, can we implement it?
I First guess: Yes, squares and rounds can simply emulate thedeferred acceptance strategies
![Page 80: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/80.jpg)
Frictions Make Things Even Worse (Hand-waving)
I Suppose squares and rounds know their own utilities frommatching with others, but have incomplete information abouteveryone else’s preferences
I Incomplete information is likely in large markets (but notmuch is known, even in the static world)
I To make things easy, suppose we know that in eachrealization of preferences/utilities, there is a unique stablematching, can we implement it?
I First guess: Yes, squares and rounds can simply emulate thedeferred acceptance strategies
![Page 81: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/81.jpg)
Frictions Make Things Even Worse (Hand-waving)
I Suppose squares and rounds know their own utilities frommatching with others, but have incomplete information abouteveryone else’s preferences
I Incomplete information is likely in large markets (but notmuch is known, even in the static world)
I To make things easy, suppose we know that in eachrealization of preferences/utilities, there is a unique stablematching, can we implement it?
I First guess: Yes, squares and rounds can simply emulate thedeferred acceptance strategies
![Page 82: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/82.jpg)
Frictions Make Things Even Worse (Hand-waving)
I Suppose squares and rounds know their own utilities frommatching with others, but have incomplete information abouteveryone else’s preferences
I Incomplete information is likely in large markets (but notmuch is known, even in the static world)
I To make things easy, suppose we know that in eachrealization of preferences/utilities, there is a unique stablematching, can we implement it?
I First guess: Yes, squares and rounds can simply emulate thedeferred acceptance strategies
![Page 83: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/83.jpg)
Simply Use Gale-Shapley?
Suppose there is some discounting (frictions)
Example: complete information
U1 =4 13 2
I s2 knows r2 will accept an offer immediatelyI s2 will not make an offer to r1
![Page 84: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/84.jpg)
Simply Use Gale-Shapley?
Suppose there is some discounting (frictions)
Example: complete information
U1 =4 13 2
I s2 knows r2 will accept an offer immediatelyI s2 will not make an offer to r1
![Page 85: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/85.jpg)
Simply Use Gale-Shapley?
Suppose there is some discounting (frictions)
Example: complete information
U1 =4 13 2
I s2 knows r2 will accept an offer immediatelyI s2 will not make an offer to r1
![Page 86: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/86.jpg)
Simply Use Gale-Shapley?
Suppose there is some discounting (frictions)
Example: complete information
U1 =4 13 2
I s2 knows r2 will accept an offer immediatelyI s2 will not make an offer to r1
![Page 87: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/87.jpg)
Simply Use Gale-Shapley?
I Generally not an equilibriumI Agents can deduce that some on their preference list areinconceivable partners —will prefer to pass over
I Can we just modify deferred-acceptance strategies to skip overinconceivable partners?
I Yes, but still tricky in equilibrium when there is incompleteinformation (timing of actions is informative andmanipulable...)
I Getting a stable matching implemented in equilibrium requiresharsh restrictions on preferences
![Page 88: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/88.jpg)
Simply Use Gale-Shapley?
I Generally not an equilibriumI Agents can deduce that some on their preference list areinconceivable partners —will prefer to pass over
I Can we just modify deferred-acceptance strategies to skip overinconceivable partners?
I Yes, but still tricky in equilibrium when there is incompleteinformation (timing of actions is informative andmanipulable...)
I Getting a stable matching implemented in equilibrium requiresharsh restrictions on preferences
![Page 89: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/89.jpg)
Simply Use Gale-Shapley?
I Generally not an equilibriumI Agents can deduce that some on their preference list areinconceivable partners —will prefer to pass over
I Can we just modify deferred-acceptance strategies to skip overinconceivable partners?
I Yes, but still tricky in equilibrium when there is incompleteinformation (timing of actions is informative andmanipulable...)
I Getting a stable matching implemented in equilibrium requiresharsh restrictions on preferences
![Page 90: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/90.jpg)
Simply Use Gale-Shapley?
I Generally not an equilibriumI Agents can deduce that some on their preference list areinconceivable partners —will prefer to pass over
I Can we just modify deferred-acceptance strategies to skip overinconceivable partners?
I Yes, but still tricky in equilibrium when there is incompleteinformation (timing of actions is informative andmanipulable...)
I Getting a stable matching implemented in equilibrium requiresharsh restrictions on preferences
![Page 91: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/91.jpg)
Search Frictions and Matching
I Original references: Burdett and Coles (1997), Eeckhout(1999), Shimer and Smith (2000)
I Distribution of squares and rounds in each period stationaryI Squares and rounds have “pizzazz”parameters on theirforeheads (Example 3)
I Squares and rounds are paired randomly in each period anddecide whether to accept one another and leave, or continuesearching, with common discount factor
I Each participant solves an option value problemI Recall Becker’s prediction of assortative matching
![Page 92: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/92.jpg)
Search Frictions and Matching
I Original references: Burdett and Coles (1997), Eeckhout(1999), Shimer and Smith (2000)
I Distribution of squares and rounds in each period stationaryI Squares and rounds have “pizzazz”parameters on theirforeheads (Example 3)
I Squares and rounds are paired randomly in each period anddecide whether to accept one another and leave, or continuesearching, with common discount factor
I Each participant solves an option value problemI Recall Becker’s prediction of assortative matching
![Page 93: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/93.jpg)
Search Frictions and Matching
I Original references: Burdett and Coles (1997), Eeckhout(1999), Shimer and Smith (2000)
I Distribution of squares and rounds in each period stationaryI Squares and rounds have “pizzazz”parameters on theirforeheads (Example 3)
I Squares and rounds are paired randomly in each period anddecide whether to accept one another and leave, or continuesearching, with common discount factor
I Each participant solves an option value problemI Recall Becker’s prediction of assortative matching
![Page 94: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/94.jpg)
Search Frictions and Matching
I Original references: Burdett and Coles (1997), Eeckhout(1999), Shimer and Smith (2000)
I Distribution of squares and rounds in each period stationaryI Squares and rounds have “pizzazz”parameters on theirforeheads (Example 3)
I Squares and rounds are paired randomly in each period anddecide whether to accept one another and leave, or continuesearching, with common discount factor
I Each participant solves an option value problem
I Recall Becker’s prediction of assortative matching
![Page 95: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/95.jpg)
Search Frictions and Matching
I Original references: Burdett and Coles (1997), Eeckhout(1999), Shimer and Smith (2000)
I Distribution of squares and rounds in each period stationaryI Squares and rounds have “pizzazz”parameters on theirforeheads (Example 3)
I Squares and rounds are paired randomly in each period anddecide whether to accept one another and leave, or continuesearching, with common discount factor
I Each participant solves an option value problemI Recall Becker’s prediction of assortative matching
![Page 96: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/96.jpg)
Search Frictions and Matching
I Each square of pizzazz Ps is willing to match immediatelywith rounds of pizzazz of at least P∗r (s)
I Each round of pizzazz Pr is willing to match immediately withsquares of pizzazz of at least P∗s (r)
![Page 97: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/97.jpg)
Search Frictions and Matching
![Page 98: Dynamic Matching ŒPart 1 - cpb-us-w2.wpmucdn.com](https://reader033.fdocuments.net/reader033/viewer/2022051813/62833bb14886a703286dd3c8/html5/thumbnails/98.jpg)
Search Frictions and Matching
I In general, “almost” stable/assortative matchingsI Achieve assortative matching as discount factor → 0
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THE END