Dating Advice from Mathematics

What's a good pairing? There are 2n people who each have a preference list (with no ties) of everyone. What's a good way of pairing them up? Maximize satisfaction Minimize dissatisfaction Maximize people who get first choice NO ROGUE COUPLES

Rogue Couples Two people who like each other more than their significant others. Unstable!!! Pairing is stable if it contains no rogue couples

No broken hearts possible? Is it always possible to have a stable pairing of 2n people? NO!

Medieval Times The sun orbits earth, and homosexuality was punishable by death. n boys and n girls. Boys only marry girls and vice versa. Is a stable pairing possible?

Traditional Marriage Algorithm Morning Every girl stands at her window Every guy serenades the next girl he hasn't crossed off Afternoon Girls tell all but one suitor “no” To the suitor she likes the most, she says “maybe” Evening Suitors that got rejected cross off that girl on their preference list. Ends when no boy gets told “no”

Traditional Marriage Algorithm Does it terminate? Yes! O(n2) Is it stable? Yes! Who's better off – boys or girls? Optimal match = gets highest ranked person for which there exists stable pairing Pessimal match = gets lowest ranked person for which there exists stable pairing

TMA is Boy Optimal! This means every boy gets his optimal match Proof by contradiction: Some boy b gets rejected by his optimal girl g in TMA Let t be earliest time this happens At time t, g preferred b* over b b* has not yet been rejected by his optimal girl because of (2) Thus b* likes g at least as much as his optimal There exists a stable pairing S in which b and g are married (def of optimal match) By (3) and (5), S is unstable. CONTRADICTION!

TMA is Girl Pessimal! Every girl gets her pessimal match Proof by contradiction: Let S be a stable pairing where girl g does worse than in TMA pairing T. In T, g is paired with b In S, g is paired with b* g likes b better than b* by assumption in (1) b likes g better than his mate in S (TDA is boy optimal) It follows from (4) and (5) that S is not stable.

Applications Proves that if you like someone, ASK THEM OUT, don't wait for them to serenade you. Used by online dating sites. Matching doctors to medical residencies (extended version) Matching law clerks and judges