Class 7: Stable Matching

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MARKETS, MECHANISMS, MACHINES University of Virginia, Spring 2019 Class 7: Matching without Money 5 February 2019 cs4501/econ4559 Spring 2019 David Evans and Denis Nekipelov https://uvammm.github.io

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Plan Today: Resource allocation Stable matching algorithms Thursday: Kidney Exchange (focus of Project 3) 1

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How to allocate scarce resources? 2 How should we allocate scarce resources?

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3 Fish Market, Bangladesh

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Allocation Goals Maximize Utility Minimize Overhead Maximize Fairness / Minimize Repugnance 4

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Allocation Strategies 5 Decentralized Centralized

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Allocation Strategies 6 Decentralized Centralized Fish Market (Bangledesh)

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Centralized Match based on Decentralized Declared Preferences

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Formal Problem Definition Input: Output:

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Utility of a Match

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Utility of a Match ! "# , %& = the value of matching "# with %& 8 9 = { "; , %; , "< , %< , … , "> , %> } = @ !("B , %B ) > BD; Goal (?): find 9 that maximizes 8(9) Goal (?): find 9 that all individuals accept

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Stable Matching Problem Input: Output: Stable matching between E and F E, F, ≺H , ≺I A matching, 9 = { "; , %; , "< , %< , … , "> , %> } is a stable matching if there is no pair ("B , %J ) where %B ≺KL %J and "J ≺MN "B .

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Stable Matching Anna Elsa E F Hans Kristoff

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Stable Matching Anna Elsa E F Hans Kristoff Kristoff ≺OPPQ Hans Kristoff ≺RSTQ Hans Elsa ≺UQPT Anna Elsa ≺VWXTYZ[[ Anna

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Stable Matches? Anna Elsa Hans Kristoff Kristoff ≺OPPQ Hans Kristoff ≺RSTQ Hans Elsa ≺UQPT Anna Elsa ≺VWXTYZ[[ Anna {(Anna, Hans), (Elsa, Kristoff)} {(Anna, Kristoff), (Elsa, Hans)}

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Stable Matches? Anna Elsa Olaf Kristoff Olfa ≺OPPQ Kristoff Kristoff ≺RSTQ Olaf Anna ≺\SQ[ Elsa Elsa ≺VWXTYZ[[ Anna {(Anna, Olaf), (Elsa, Kristoff)} {(Anna, Kristoff), (Elsa, Olaf)}

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Students and Majors Avery: Computer Science, Economics, Music Blake: Music, Computer Science, Economics Corey: Economics, Music, Computer Science Computer Science: Blake, Corey, Avery Economics: Corey, Avery, Blake Music: Avery, Blake, Corey

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Gale- Shapley Algorithm Lloyd Shapley (1923-2016) accepting Nobel Prize (2012) (co-awarded to Alvin Roth)

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David Gale (1921 – 2008) Alvin Roth (Stanford) born 1951 Aaron Roth (U. Penn) Ben Roth (Harvard BS)

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A matching, 9 = { "; , %; , "< , %< , … , "> , %> } is a stable matching if there is no pair ("B , %J ) where %B ≺KL %J and "J ≺MN "B . Finding a Stable Matching

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Gale-Shapley Algorithm StableMatch E, F, ≺K , ≺M : 9 = { } # b"cdℎfg, hihch"jjk fblck foreach " ∈ E: "nQYop = ⊥, "rWZrZTst = 0 foreach % ∈ F: %vKwxy = ⊥ while ∃" ∈ E where "nQYop = ⊥ and "rWZrZTst < |F|: " ← pick(E) % = "ÅÇÉÑÖ "ÅÇÜÅÜÖÉá # highest − ranked F to whom " hasnãt already proposed if %vKwxy = ⊥: pair(", %) elseif %vKwxy ≺M ": # b prefers a to current match unpair(%vKwxy , %) pair(", %) "ÅÇÜÅÜÖÉá += 1

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Olfa ≺OPPQ Kristoff Kristoff ≺RSTQ Olaf Anna ≺\SQ[ Elsa Elsa ≺VWXTYZ[[ Anna Implementing Gale-Shapley

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Simplified Code (Non-Defensive) Do not write programs like this: assertions (especially) and comments are valuable (but don’t fit on slides)

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Cost of Gale-Shapley

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D. Gale and L. S. Shapley. College Admissions and the Stability of Marriage, The American Mathematical Monthly, 1962.

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Properties of a Matching Algorithm

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Proving Correctness For all valid inputs, gale_shapley returns a stable matching of its inputs. A matching, 9 = { "; , %; … , "> , %> } is a stable matching if there is no pair ("B , %J ) where %B ≺KL %J and "J ≺MN "B .

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Proving Optimality? For all valid inputs, gale_shapley returns a stable matching of its inputs. gale_shapley returns a matching that is optimal for the proposers (A).

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Is that fair? https://youtu.be/l_FWKLpoGBc?t=16

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National Resident Matching Program Medical residents in US, Canada, others 35,000 applicants

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New England Journal of Medicine, 1981

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Redesigned matching algorithm used in 1988

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Roth-Peranson Algorithm instability chaining: start with no matches loop through the proposers break matches when better found produces proposer-optimal stable match

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https://youtu.be/MMqThyNxbOw?t=23

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How can this possibly be legal?

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Anti-trust Exemption

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https://www.collegepravesh.com/cutoff/iit-bombay-cutoff-2018/ IIT Bombay

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Gale & Shapley’s Conclusion

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Who controls the match? Public schools in New York, Boston Singapore University Admissions Medical residents in US, Canada, others 35,000 applicants

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Who controls the match? Public schools in New York, Boston Singapore University Admissions Medical residents in US, Canada, others 35,000 applicants Use Trusted Third Party to run matching algorithm: - Receives all private rankings and keeps confidential - Produces correct result - uncorrupted

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Phase Time Initialization 2.07 hours Bidding 15.01 hours Total 17.08 hours Simulated 2016 US National Medical Residency Match: 35,476 prospective residents matching with 4836 programs with 30,750 total slots Secure Stable Matching Garbled Circuit Protocol secret input a secret input b r = f(a, b) r = f(a, b) Learns nothing else about b Learns nothing else about a

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Charge Scarce Resource Allocation is a core problem of economics Algorithms, Mechanism Design Project 3: Kidney Matching