Class 7: Stable Matching

Class 7: Stable Matching
https://uvammm.github.io/class7

Markets, Mechanisms, and Machines
University of Virginia
cs4501/econ4559 Spring 2019
David Evans and Denis Nekipelov
https://uvammm.github.io/

Transcript

1. 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
2. Plan Today: Resource allocation Stable matching algorithms Thursday: Kidney Exchange (focus of Project 3) 1
3. How to allocate scarce resources? 2 How should we allocate scarce resources?
4. 3 Fish Market, Bangladesh
5. Allocation Goals Maximize Utility Minimize Overhead Maximize Fairness / Minimize Repugnance 4
6. Allocation Strategies 5 Decentralized Centralized
7. Allocation Strategies 6 Decentralized Centralized Fish Market (Bangledesh)
8. Centralized Match based on Decentralized Declared Preferences
9. Formal Problem Definition Input: Output:
10. Utility of a Match
11. 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
12. 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 .
13. Stable Matching Anna Elsa E F Hans Kristoff
14. Stable Matching Anna Elsa E F Hans Kristoff Kristoff ≺OPPQ Hans Kristoff ≺RSTQ Hans Elsa ≺UQPT Anna Elsa ≺VWXTYZ[[ Anna
15. 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)}
16. 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)}
17. 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
18. Gale- Shapley Algorithm Lloyd Shapley (1923-2016) accepting Nobel Prize (2012) (co-awarded to Alvin Roth)
19. David Gale (1921 – 2008) Alvin Roth (Stanford) born 1951 Aaron Roth (U. Penn) Ben Roth (Harvard BS)
20. 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
21. 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
22. Olfa ≺OPPQ Kristoff Kristoff ≺RSTQ Olaf Anna ≺\SQ[ Elsa Elsa ≺VWXTYZ[[ Anna Implementing Gale-Shapley
23. Simplified Code (Non-Defensive) Do not write programs like this: assertions (especially) and comments are valuable (but don’t fit on slides)
26. Cost of Gale-Shapley
27. D. Gale and L. S. Shapley. College Admissions and the Stability of Marriage, The American Mathematical Monthly, 1962.
28. Properties of a Matching Algorithm
29. 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 .
30. 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).
31. Is that fair? https://youtu.be/l_FWKLpoGBc?t=16
32. National Resident Matching Program Medical residents in US, Canada, others 35,000 applicants
33. New England Journal of Medicine, 1981
34. Redesigned matching algorithm used in 1988
35. Roth-Peranson Algorithm instability chaining: start with no matches loop through the proposers break matches when better found produces proposer-optimal stable match
36. https://youtu.be/MMqThyNxbOw?t=23
37. How can this possibly be legal?
38. Anti-trust Exemption
44. https://www.collegepravesh.com/cutoff/iit-bombay-cutoff-2018/ IIT Bombay
45. Gale & Shapley’s Conclusion
46. Who controls the match? Public schools in New York, Boston Singapore University Admissions Medical residents in US, Canada, others 35,000 applicants
47. 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
48. 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
49. Charge Scarce Resource Allocation is a core problem of economics Algorithms, Mechanism Design Project 3: Kidney Matching