Regret-equality in Stable Marriage Frances Cooper Joint work with: - - PowerPoint PPT Presentation

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Regret-equality in Stable Marriage Frances Cooper Joint work with: - - PowerPoint PPT Presentation

Dec 23, 2023 155 likes •375 views

Regret-equality in Stable Marriage Frances Cooper Joint work with: Prof David Manlove 1 Outline Matching problems Fairness Finding fair stable matchings Experiments Future work Frances Cooper 2 Matching Problems

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SLIDE 1

Regret-equality in Stable Marriage

Frances Cooper

1

Joint work with: Prof David Manlove

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SLIDE 2

Frances Cooper

Outline

  • Matching problems
  • Fairness
  • Finding fair stable matchings
  • Experiments
  • Future work

2

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SLIDE 3

Frances Cooper

Matching Problems

3

  • Assign one set of entities

to another set of entities

  • Based on preferences and

capacities

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SLIDE 4

Frances Cooper

m4, m3, m2, m1 m2, m1, m4, m3 w1, w2, w3, w4

m1 m3 m4 m2

Men Women

Stable Marriage

4

w1 w3 w4 w2

w4, w3, w2, w1 w2, w1, w4, w3 w3, w4, w1, w2 m3, m4, m1, m2 m1, m2, m3, m4

Rank Cost: cU(M) = 10, cW(M) = 10 Degree: dU(M) = 4, dW(M) = 4 Blocking pair

A stable matching is a matching with no blocking pairs

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SLIDE 5

Frances Cooper

Stable Marriage

5

  • A stable matching is a matching with no blocking

pairs

  • Many stable matchings per instance
  • We can find a stable matching in linear time using the

man-oriented or woman-oriented Gale-Shapley

  • Algorithm. O(m) time where m is total length of

preference lists

  • Man-oriented Gale-Shapley Algorithm: finds a man-
  • ptimal (woman-pessimal) stable matching (and vice

versa)

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SLIDE 6

Frances Cooper

Fairness

6

  • Want to find a stable

matching that provides some kind of equality between men and women

  • Several different

fairness measures

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SLIDE 7

Frances Cooper

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Fairness measures

Minimises the maximum Minimises the difference Minimises the sum

Among all stable matchings, find the stable matching that… Cost: cU(M), cW(M) Degree: dU(M), dW(M)

Balanced stable matching Sex-equal stable matching Egalitarian stable matching Minimum-regret stable matching * Regret-equal stable matching * Min-regret sum stable matching NP-hard NP-hard Poly Poly ? ?

balanced score sex-equal score egalitarian cost degree regret-equal score regret sum score

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SLIDE 8

Frances Cooper

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Fairness measures (degree based)

Min-regret

Degree: 3 Regret-equality score: 0 Min-regret sum score: 6

m1: w1, w2, w3, w4 m4: w4, w3, w2, w1 m2: w2, w1, w4, w3 m3: w3, w4, w1, w2

w1: m4, m3, m2, m1 w3: m2, m1, m4, m3 w2: m3, m4, m2, m1 w4: m1, m2, m3, m4

m1: w1, w2, w3, w4 m4: w4, w3, w2, w1 m2: w2, w1, w4, w3 m3: w3, w4, w1, w2

w1: m4, m3, m2, m1 w3: m2, m1, m4, m3 w2: m3, m4, m2, m1 w4: m1, m2, m3, m4

m1: w1, w2, w3, w4 m4: w4, w3, w2, w1 m2: w2, w1, w4, w3 m3: w3, w4, w1, w2

w1: m4, m3, m2, m1 w3: m2, m1, m4, m3 w2: m3, m4, m2, m1 w4: m1, m2, m3, m4

Degree: 3 Regret-equality score: 1 Min-regret sum score: 5 Degree: 4 Regret-equality score: 3 Min-regret sum score: 5

10 stable matchings for this instance Min-regret Min-regret sum & Min-regret sum & Regret-equal

Over all stable matchings: Minimum degree = 3 Minimum regret-equality score = 0 Minimum regret sum score = 5

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SLIDE 9

Frances Cooper

Finding a Regret-Equal Stable Matching

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Frances Cooper

Rotations

  • Rotation - series of man-woman pairs that take us from
  • ne stable matching to another when permuted

10

R1

  • O(n2) algorithm to find all rotations
  • Rotations form a structure to allow enumeration of all

stable matchings. All rotation makes some men worse off and some women better off

M1 m1 m2 m3 m4 w2 w1 w4 w3 R1 m1 m4 w2 w3 M2 m1 m2 m3 m4 w3 w1 w4 w2

  • Can only eliminate exposed rotations

R2 m1 m2 w1 w2

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SLIDE 11

Frances Cooper

Algorithm

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  • 1. Find the man-optimal stable matching M0
  • Each man has their best partner in any stable matching.

Say dU(M0) = 2 and dW(M0) = 5 d(M0) = (2, 5)

  • Then, a regret equal stable matching must exist within

the following degrees pairs:

why are these the only possible degrees?

  • M0 has a r-e score of 3
  • men can only get worse
  • women can only get better

(2, 5) (2, 4) (2, 3) (2, 2) (2, 1) (3, 5) (3, 4) (3, 3) (3, 2) (3, 1) (4, 5) (4, 4) (4, 3) (4, 2) (5, 5) (5, 4) (5, 3) (6, 5) (6, 4) (7, 5)

r-e score: 3 r-e score: 2 r-e score: 1 r-e score: 0 r-e score: 1 r-e score: 2

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SLIDE 12

Frances Cooper

Algorithm

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  • 2. If dU(M0) >= dW(M0) then exit with M0
  • 3. For each man m and for each column c:
  • 1. rotate m down to c (if possible)
  • 2. rotate women down column c who have worst rank

(2, 5) (2, 4) (2, 3) (2, 2) (2, 1) (3, 5) (3, 4) (3, 3) (3, 2) (3, 1) (4, 5) (4, 4) (4, 3) (4, 2) (5, 5) (5, 4) (5, 3) (6, 5) (6, 4) (7, 5)

r-e score: 3 r-e score: 2 r-e score: 1 r-e score: 0 r-e score: 1 r-e score: 2

  • Stop iterating women up the

column when dU(M) >= dW(M)

  • Save the best matching as you

go

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SLIDE 13

Frances Cooper

Time complexity

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  • Find man-optimal stable matching & all rotations O(n2)
  • For each man O(n)
  • For each column O(2 * |dU(M0) - dW(M0)|) = O(c)
  • Rotate man down and women down O(n2)

Total O(n3c)

2 * man-optimal difference

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SLIDE 14

Frances Cooper

Experiments

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Frances Cooper

Methodology

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  • Performance of the Regret-equal Algorithm compared to an Enumeration algorithm

(exponential in worst case)

  • Instances size {10, 20, …, 100, 200, …, 1000}, complete preference lists, 500

instance per size.

  • looked at properties over several types of optimal stable matching (balanced, sex-

equal, egalitarian, minimum regret, regret-equal, min-regret sum)

  • Java, Python, Bash, GNU parallel
  • Correctness
  • all matchings found were stable
  • Regret-equality scores matched
  • CPLEX up to size n = 50 for the enumeration algorithm
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SLIDE 16

Frances Cooper

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Mean time (ms)

100 1000 10000 100000

n

1 2 3 4 5 6 7 8 9 1

Enumeration Algorithm Regret-equal Algorithm

Time taken

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SLIDE 17

Frances Cooper

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Mean regret-equality score

30 60 90

n

1 2 3 4 5 6 7 8 9 1

Balanced Sex-equal Egalitarian Minimum regret Regret-equal Min-regret sum

Regret-equality score for different optimal matchings

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SLIDE 18

Frances Cooper

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Mean sex-equal score

2000 4000 6000 8000 10000 12000 14000

n

1 2 3 4 5 6 7 8 9 1

Balanced Sex-equal Egalitarian Minimum regret Regret-equal Min-regret sum Regret-Equal Algorithm

Sex-equal score for different

  • ptimal matchings
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SLIDE 19

Frances Cooper

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Mean number of stable matchings

1 10 100 B a l a n c e d S e x

  • e

q u a l E g a l i t a r i a n M i n i m u m r e g r e t R e g r e t

  • e

q u a l M i n

  • r

e g r e t s u m

100 400 700 1000

Frequency of different optimal stable matchings

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Frances Cooper

Future Work

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  • Improving the O(n3c) Regret-equal Algorithm, where c = |dU(M0)
  • dW(M0)|
  • Grouping women - e.g. women are workers and men are jobs to

assign to workers.

  • Woman optimal stable matching would naturally satisfy

‘balanced’, ‘min-regret', ‘egalitarian’ and ‘min-regret sum’ criteria

  • Can find a ‘regret-equal’ stable matching in O(n4) time
  • Open problem for ‘sex-equality’ -> grouped-women-

equality

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SLIDE 21

Frances Cooper

Thank you

Summary

  • Matching problems
  • Fairness
  • Finding fair stable matchings
  • Experiments
  • Future work: finding improved

algorithms

21

f.cooper.1@research.gla.ac.uk http://fmcooper.github.io

EPSRC Doctoral Training Account