on heuristics for two-sided...

Intro Representation G-S Challenges ABM GA Conclusion

On Heuristics for Two-Sided MatchingRevisiting the Stable Marriage Problem as a Multiobjective

Problem

Steven O. Kimbrough Ann Kuo

University of Pennsylvania

GECCO, Portland, OR. 2010-7-10, Saturday, 14:25–14:50,Room: Salon D, 14:00–15:40

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Acknowledgements

Robert Axtell, Rob Kurzban

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Outline

1 Intro

2 Representation

3 G-S

4 Challenges

5 ABM

6 GA

7 ConclusionEnd Notes

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Markets and two-sided matching

(Material drawn largely from [Kimbrough and Kuo, 2010].)

Two-sided matching problems.Sides: X and Y . Problem: form pairs, one member fromeach side.Model for a market: buyers and sellers, etc.Distributed versus centralized markets.Market failures and the move to centralization.Labor markets: interns and hospitals. Admissions:students and schools.Widely practiced in the USA.

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Markets and two-sided matching

Agents and strategic considerations.Assignment versus matching.Stability.Equity.Social welfare.

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Representative representation

Sides X and Y . Pairs (x , y), x ∈ X , y ∈ Y . Match array Mis size |X | × |Y |.

M =

0 0 0 10 1 0 01 0 0 00 0 1 0

or

y1 y2 y3 y4

x1 0 0 0 1x2 0 1 0 0x3 1 0 0 0x4 0 0 1 0

This is the Simple Marriage Matching problem.

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Simple Marriage Matching problem

n men, n women, all of whom want to be paired up with amember of the complementary gender.This is very stylized. Lots of variations possible (andconsidered in the literature).More generally on matching, many elaborations possibleand encountered in practice.

“Two-body” problem for physicians, "n-body" for students.Admissions problem: more physicians than hospitals, morestudents than schools.Various ad hoc, special considerations.

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Simple Marriage Matching problem

Focus of this paper: the very elementary, stylized SimpleMarriage Matching problem.Note:

1 It is prototypical, representative in important ways.2 It is most favorable to the standard approach, which we are

challenging, or rather seeking to augment.Our points are only strengthened for less stylized, morerepresentative models.

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The Gale-Shapley deferred acceptance algorithm

The algorithm (informally):

1 Do until there are no unaccepted men:1 Each currently unaccepted man proposes to his most

preferred woman, provided she hasn’t already rejected him.2 Each woman with more than one proposal rejects all but

her most preferred proposer, who becomes perforcecurrently accepted by that woman.

2 Stop. The matching is determined by the currentlyaccepted proposals.

Alternate version: the women propose, the men dispose.

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The Gale-Shapley deferred acceptance algorithm

Stable and unstable matches. The equilibrium concept formatching.Key properties of GS/DAA.

For special cases of matching (including the SimpleMarriage Matching problem), guaranteed to terminate(quickly) and to find a stable match. (And for otherproblems, it isn’t, doesn’t.)Deterministic. It can only find two distinct stable matches(one if there is only one).Asymmetric. Proposers get optimal match, disposers getpessimal match.Blind to considerations other than stability, e.g., fairness,social welfare.

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Can we do better?

Key property of matching problems: may be exponentiallymany stable matches.Can we find stable matches that are better on fairnessand/or on social welfare?What about ‘nearly-stable’ matches?Must rely on heuristics.

Larger question (comes later): Can we design high-qualitycentralized markets that will work well in practice?

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Matching with a Simple ABM

Model of a market.Agents epistemically powerful; see the entire field.Agents randomly get the floor and make a feasible swapthat is most favorable to them.

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ABM Results: n = 40. Transaction Cost = 0 (†Mean=3,Max=124)

1st Qu Median 3rd QuInit. # unstable pairs 314 340 367Final # unstable pairs† 0 0 0InitialSocialWelfareSum 1570 1638 1709Final SocialWelfareSum 476 499 529Initial Equity 492 532 572Final Equity 200 224 245SwapCount 1610 5731 22389InitialSumXScores 769 819 869Final SumXScores 217 246 280InitialSumYScores 772 820 869Final SumYScores 224 253 293Number of Runs 100Number of Replications 100TransactionCost 0

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ABM Results: n = 40

Points arising:1 The initial match, which is randomly generated, is normally

a very poor one. It is not stable, and it fares poorly onsocial welfare, equity, and individual scores.

2 In driving towards a stable solution, however myopically,the market process reliably improves the scores on socialwelfare and equity, as well as the individual scores. Here isan invisible hand at work with broadly sanguine effects.

3 There is, however, a large churning cost. The mediannumber of swaps is 5731 to achieve stability in the society.Divided by 40, that’s just over 573 breakups experiencedon average by each agent (5731×4/40). What a cruelsystem that would break so many hearts.

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ABM Results: n = 100

1st Qu Median 3rd QuInit. # unstable pairs 1709 1808 1908Fin. # unstable pairs 0 0 0Init. # unstbl pairs NTC 2061 2165 2270Fin. # unstbl pairs NTC 28 32 37InitialSocialWelfareSum 9830 10104 10376Final SocialWelfareSum 1984 2048 2113Initial Equity 3173 3329 3492Final Equity 878 934 994SwapCount 655 1049 1901InitialSumXScores 4859 5050 5242Final SumXScores 941 1020 1106InitialSumYScores 4858 5055 5249Final SumYScores 939 1018 1103Number of Runs 100Number of Replications 100TransactionCost 5

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ABM Results: n = 100

Points arising:1 The behavior when n = 100, with and without transaction

costs, is broadly similar to cases with n = 40 and smaller.That is, the market actions by myopic agents generallyimprove both individual and social welfare, from a randomstart.

2 Equilibrium, in the form of stability, is not achieved unlesswe factor in substantial transaction costs.

3 Even when equilibrium is attained, it is remarkably costly interms of the number of matches made and abandoned.

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Implemented a GA solver

Solutions are permutations, search in permutation space.Every permutation is feasible.Used rather conventional approach (see paper for details).Reporting results on randomly-generated test problems.First: 25, 20×20 problems.

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Comparisons with GS/DAA: GA. Points arising:

1 In these 25 test problems, the GA is able to find very manystable matches that are Pareto-superior to the GS/DAAsolutions. By Pareto-superior, I mean stable and strictlybetter than the GS/DAA solutions on both social welfareand equity.

2 The GA and the ABM do about equally well at the task offinding solutions that are Pareto-superior to those ofGS/DAA.

3 These points are illustrated vividly in Figure 1, which is fortest case 7.

4 We also used the GA to find and collect ‘one-away’solutions, that is, matches with only one unstable pair ofcouples. When we relax the requirement for strict stability,many more attractive options appear. See Figure 2 as anexample.

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Comparisons with GS/DAA

9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11 11.24

4.5

5

5.5

6

6.5

avgSum

avgAbsDiff

../Data/20x20x25/PrefConfig20x207.txt

Figure: Plot of alternate stable solutions found for test case 7.GS/DAA in red ∗, GA in ◦, agent model in 4.

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One-away solutions. 40×40.

11.4 11.6 11.8 12 12.2 12.4 12.6 12.84.5

5

5.5

6

6.5

7

avgSum

avgAbsDiff

../Data/40x40x80/PrefConfig40x406.txt

Figure: One-away solutions compared to GS/DAA solutions, 40x40,case 6. GS/DAA in red ∗.

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Discussion and Conclusion

1 Two-sided matching models apply widely.2 The standard approach . . . Gale-Shapley deferred

acceptance algorithm. . . . important virtues, includingcomputational tractability. Operates without regard tosocial welfare and in contravention to equity.

3 Many stable matches in a two-sided matching problem.GS/DAA can only find two of these, which two have veryspecial optimality and pessimality properties.

4 Metaheuristics can find stable matches that GS/DAAcannot find and that are superior with regard to equityand/or social welfare.

5 ‘one-away’ matches with attractive social welfare andequity profiles.

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Next steps include

Exploration with real data.(Have ideas and plans. Also actively seeking data. . . )Further exploration with simulated data, especially variousforms of ‘correlation’ (e.g., everyone agrees on who theprettiest girl is).Looking at ‘mating markets’. Consider multidimensionalmodels.Investigate strategic misrepresentation.Consider: much broader application of centralized markets.The real-world as just one run of a simulation.Long term: Can we design high-quality centralized marketsthat will work well in practice?

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Thank-you for your attention

Questions?

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End Notes

Kimbrough, S. O. and Kuo, A. (2010).On heuristics for two-sided matching: Revisiting the stablemarriage problem as a multiobjective problem.In Proceedings of the Genetic and EvolutionaryComputation Conference (GECCO-2010). ACM.

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End Notes

$Id: marriage-matching-beamer.tex 1640 2010-07-05 10:43:54Z sok $

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