Stable Matching Examples Ron K Cytron http www

Stable Matching Examples Ron K. Cytron http: //www. cs. wustl. edu/~cytron/ Examples drawn from Knuths’ lecture in French: Marriages stables et leurs relations avec d’autres problemes combinatoires 24 November 2008

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Stable Matching Problem CSE 131

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Stable Matching Problem CSE 131

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Bb Stable Matching Problem CSE 131

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Bb Cc Stable Matching Problem CSE 131

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Bb Cc Dd Stable Matching Problem CSE 131

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Bb Cc Dd is not stable Do you see why? Stable Matching Problem CSE 131

Can’t we just guess? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Bb Cc Dd is not stable Ab elope Stable Matching Problem CSE 131

Let infidelity derive a solution? #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Aa Bb Cc Dd is not stable Ab elope Stable Matching Problem CSE 131

Let infidelity derive a solution? Let Ab elope and…. . #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Ab Ba Cc Dd is not stable Cb elope Stable Matching Problem CSE 131

Let infidelity derive a solution? Let Cb elope and…. . #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Ac Ba Cb Dd is not stable Dc elope Stable Matching Problem CSE 131

Let infidelity derive a solution? Let Cb elope and…. . #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Ad Ba Cb Dc is stable! How can we show this? Stable Matching Problem CSE 131

Proof of no infidelity #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Ad Ba Cb Dc is stable! Stable Matching Problem CSE 131

Proof of no infidelity #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Ad Ba Cb Dc is stable! Algorithm? Stable Matching Problem CSE 131

Your ideas? • Let infidelity rule and see what happens • Put top choices together and see • Compute average of X y y X and do something with that score • Favor one side with its preference, ignore the other side • Make smaller instances and try that • Start with worst matches and try to improve Stable Matching Problem CSE 131

Iterate until stable? Stable Matching Problem CSE 131

Iterate until stable? A b a c a A C B B -- -- -- b C A B C a b c c -- -- -- Ab elope Stable Matching Problem CSE 131

Iterate until stable? A b a c a A C B B -- -- -- b C A B C a b c c -- -- -- Ab elope Cb elope Stable Matching Problem CSE 131

Iterate until stable? A b a c a A C B B -- -- -- b C A B C a b c c -- -- -- Ab elope Ca elope Stable Matching Problem CSE 131

Iterate until stable? A b a c a A C B B -- -- -- b C A B C a b c c -- -- -- Ab elope Ca elope Aa elope – back to start Stable Matching Problem CSE 131

Try everything? • How many possibilities are there? Stable Matching Problem CSE 131

How many possible stable matchings? 1 2 3 4 … n-1 n 1 2 3 4 n-1 n 2 1 4 3 … … … … 1 2 3 4 n n-1 … … … … n-1 n Stable Matching Problem CSE 131

How many possible stable matchings? 1 2 3 4 … n-1 n 1 2 3 4 n-1 n 2 1 4 3 … … … … 1 2 3 4 n n-1 … … … … n-1 n Left group can decide to approach distinct people from the right group Stable Matching Problem CSE 131

How many possible stable matchings? 1 2 3 4 … n-1 n 1 2 3 4 n-1 n 2 1 4 3 … … … … 1 2 3 4 n n-1 … … … … n-1 n Each adjacent pair will decide between their first and second choices. For example, 1 and 2 …. Stable Matching Problem CSE 131

How many possible stable matchings? 1 2 3 4 … n-1 n 1 2 3 4 n-1 n 2 1 4 3 … … … … 1 2 3 4 n n-1 … … … … n-1 n Go with their first choices 1 and 2 are stable from the left Or……. Stable Matching Problem CSE 131

How many possible stable matchings? 1 2 3 4 … n-1 n 1 2 3 4 n-1 n 2 1 4 3 … … … … 1 2 3 4 n n-1 … … … … n-1 n Go with their second choices 1 and 2 are stable from the right Stable Matching Problem CSE 131

How many possible stable matchings? 1 2 3 4 … n-1 n 1 2 3 4 n-1 n 2 1 4 3 … … … … 1 2 3 4 n n-1 … … … … n-1 n n/2 groups can go 1 of 2 ways here Exponential number of possibilties: 2 n/2 Stable Matching Problem CSE 131

The algorithm Stable Matching Problem CSE 131

Algorithm applied to first example #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D Stable Matching Problem CSE 131

Algorithm applied to first example #1 #2 #3 #4 A c b d a a A B D C B b a c d b C A D B C b d a c c C B D A D c a d b d B A C D C and D are unbreakable Little a could try for big A, but big A won’t go for it Little d could try for big B, but big B won’t go for it Stable! Stable Matching Problem CSE 131

Desirable properties • Termination • Correctness – Everybody paired up – Stable arrangement • Fairness – Is it fair? – If not, who suffers? Stable Matching Problem CSE 131

Fairness #1 #2 #3 #4 A a b c d a B C D A B b c d a b C D A B C c d a b c D A B C D d a b c d A B C D Stable Matching Problem CSE 131

Fairness #1 #2 #3 #4 A a b c d a B C D A B b c d a b C D A B C c d a b c D A B C D d a b c d A B C D Big letters have the advantage. Hospitals hosting residents. Stable Matching Problem CSE 131

How about picking roommates? • Like stable matching, but instead of two populations – ABC… – abc… • We have just one population: – A, B, C, … • Example: – A, B, C, D – Make any one pairing (such as AB) and the other paring (CD) is determined Stable Matching Problem CSE 131

Roommates #1 #2 #3 A C B D B A C D C B A D D A B C We can cover all possibilities by trying A with C, then B, then D Let’s see what happens Stable Matching Problem CSE 131