Dating Scenario

1. Dating Scenario • There are n boys and n girls • Each girl has her own ranked preference list of all the boys • Each boy has his own ranked preference list of all the girls • The lists have no ties • Question: How to pair them off?

2. Stable Marriage • Would like a pairing that is stable • No pairs (g1,b1) and (g2,b2) exist such that g1 prefers b2 over b1 and b2 prefers g1 over g2 • The pair (g1,b2) will be referred to as a rogue couple • A stable marriage is a pairing that allows no possibility of rogue couples

3. Example

4. Stable Marriage Problem • Given preference lists, determine a stable pairing • A more fundamental question: • Does a stable pairing always exist??

5. Instructive Variant: Bisexual case • Consider 4 participants A, B, C, and D • A: B C D • B: C A D • C: A B D • D: does not matter • Any pairing is unstable: • say (A,B) and (C,D) • C prefers B over D and B prefers C over A

6. Existence of Stable Pairing • For the heterosexual case, stable pairing always exists! • Will show that a simple algorithm – the traditional marriage algorithm (TM) – always yields a stable marriage

7. TM Algorithm (Overview) • The algorithm proceeds in rounds. • In each round: • Each boy proposes to one girl • Each girl rejects all but one boy • At the end, a stable pairing will result

8. TM Algorithm • In each round: • Each boy proposes to his most-preferred girl, who has not rejected him • Each girl considers the proposals in the round; rejects all but the most-preferred proposal • Repeat until no rejections occur • Each girl marries the sole proposer in last round • We have a stable marriage!

9. Properties of TM • Improvement: Once a girl has a suitor, she will always have a suitor; furthermore, the ranking of the suitor will never decrease • Matching: No boy can be rejected by all girls • Termination: TM always terminates in at most rounds • Stability: The final pairing is stable

10. Improvement • Suppose a girl G has at least one proposer in round i. • Suppose B is the proposer not rejected by G in round i • G has a suitor B • B will repeatedly propose to G until G rejects B and takes a better suitor (or marries B) • Thus, once G has a suitor, she always has a suitor • Ranking of the suitor never decreases

11. Matching • Suppose boy B is rejected by all girls • Since B has proposed to all girls, each of them has had a suitor at some point • This implies they all have suitors now (by the improvement lemma) • But this is a contradiction, since we have n girls and n-1 suitors (since B is not one)

12. Termination • A boy cannot rejected by a girl twice • There are a total of possible rejections • Each round – until the end – there is at least one rejection • So at most rounds before TM terminates

13. Stability • Suppose the resultant pairing is not stable • (G1, B1) and (G2, B2) are couples • B1 prefers G2 over G1 and G2 prefers B1 over B2 • Can this happen? • B1 must have proposed to G2 before G1 • G2 rejected B1 for some other boy B3 who was higher on G2’s list • The final suitor for G2, B2, is at least as high on G2’s list as B3 • So G2 prefers B2 over B1

14. Who Fares Better in TM? • Boys: Start from the top of their list • Girls: The ranking of their suitor progressively increases • In final solution: • Each boy is paired with the highest ranked girl on his list that he can conceivably get in a stable world • Each girl is paired with the lowest ranked boy on her list that she can conceivably get in a stable world

15. Stable Marriage in Practice • Matching Residents to Hospitals: • Since WWII, medical school graduates have been matched to hospitals (as residents) using a stable marriage algorithm • College Admissions