Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

Markets + Romance

Objective Study a simple model for a dating market Study stylized “dating ritual” Observe market’s predictions.

Love, marriage, & bipartite graphs the boys and girls Jake Elwood Curtis Ray Twiggy Claire Jill Holly Holly > Claire > Twiggy > Jill Claire > Jill > Twiggy > Holly Twiggy > Jill > Holly > Claire Holly > Claire > Twiggy > Jill Jake > Elwood > Curtis > Ray Jake > Curtis > Elwood > Ray Ray > Curtis > Elwood > Jake Ray > Jake > Elwood > Curtis

What do markets say on romance? Question A. Do “lasting” marriages always exist? Question B. Are they easy to find? Question C. Do conventional “dating rituals” result in lasting marriages?

What could keep marriages from lasting?

A pair (m,w) is a blocking pair for a matching if m prefers w to his match and w prefers m to her match. When is marriage stable? Jake Elwood Curtis Ray Twiggy Claire Jill Holly Holly > Claire > Twiggy > Jill Claire > Jill > Twiggy > Holly Twiggy > Jill > Holly > Claire Holly > Claire > Twiggy > Jill Jake > Elwood > Curtis > Ray Jake > Curtis > Elwood > Ray Ray > Curtis > Elwood > Jake Ray > Jake > Elwood > Curtis

When is marriage stable? A pair (m,w) is a blocking pair for a matching if m prefers w to his match and w prefers m to her mat c h. A matching is stable if there is no blocking pair.

The (stylized) courtship ritual Jake Elwood Curtis Ray Holly > Claire > Twiggy > Jill Claire > Jill > Twiggy > Holly Twiggy > Jill > Holly > Claire Holly > Claire > Twiggy > Jill Twiggy Claire Jill Holly Jake > Elwood > Curtis > Ray Jake > Curtis > Elwood > Ray Ray > Curtis > Elwood > Jake Ray > Jake > Elwood > Curtis

Facts about Courtship Ritual Fact 1. Courtship ritual terminates. Fact 2. Everyone marries. Fact 3. Resulting marriage is stable. Consequences: Answer A. Yes, stable matchings always exist. Answer B. Yes, stable matchings are easy to find. Answer C. Yes, the courtship ritual works.

Termination The courtship ritual terminates. 1.In each step, someone crosses off a name from their list. 2.Only a finite number of names, so ritual cannot run forever.

Everyone marries Everyone marries. Assume for contradiction: boy B isn’t married. 1.Then B has crossed everyone off his list. 2.So each girl was engaged at some point. 3.But, once a girl is engaged, always engaged. 4.So, all girls are married (to unique boy). 5.But number of girls equals number of boys. 6.So, B must be married. CONTRADICTION!

Stability The resulting matching is stable. Consider boy B and girl G that are not married to each other. 1.Suppose G was crossed off B’s list. Then G prefers husband to B, so won’t elope with B. 2.Suppose G is on B’s list. Then B didn’t propose to G yet, so B prefers wife to G, so won’t elope with G.

Facts about Courtship Ritual Fact 1. Courtship ritual terminates. Fact 2. Everyone marries. Fact 3. Resulting marriage is stable. Consequences: Answer A. Yes, stable matchings always exist. Answer B. Yes, stable matchings are easy to find. Answer C. Yes, the courtship ritual works.

Would you rather? Would you rather be a boy or a girl? (who are better off, the boys or the girls?)

Recall matching from ritual… Jake Elwood Curtis Ray Holly > Claire > Twiggy > Jill Claire > Jill > Twiggy > Holly Twiggy > Jill > Holly > Claire Holly > Claire > Twiggy > Jill Twiggy Claire Jill Holly Jake > Elwood > Curtis > Ray Jake > Curtis > Elwood > Ray Ray > Curtis > Elwood > Jake Ray > Jake > Elwood > Curtis

An alternate universe: girls propose Jake Elwood Curtis Ray Holly > Claire > Twiggy > Jill Claire > Jill > Twiggy > Holly Twiggy > Jill > Holly > Claire Holly > Claire > Twiggy > Jill Twiggy Claire Jill Holly Jake > Elwood > Curtis > Ray Jake > Curtis > Elwood > Ray Ray > Curtis > Elwood > Jake Ray > Jake > Elwood > Curtis

Conclusion: not unique Jake Elwood Curtis Ray Twiggy Claire Jill Holly Holly > Claire > Twiggy > Jill Claire > Jill > Twiggy > Holly Twiggy > Jill > Holly > Claire Holly > Claire > Twiggy > Jill Jake > Elwood > Curtis > Ray Jake > Curtis > Elwood > Ray Ray > Curtis > Elwood > Jake Ray > Jake > Elwood > Curtis

Stable spouses In general, there are many stable marriages. Boy B Girl G Girl G’ A person P is a stable spouse of a person P’ if P is married to P’ in some stable matching.

Boys are happier than girls In boys-propose ritual: 1.Boys marry their most preferred stable wife. 2. Girls get their least preferred stable husband! way

Assume for contradiction: some boy isn’t married to favorite stable spouse. 1.Let B be 1 st boy to lose his favorite stable wife G. 2.Then G must have had a proposal from a boy B’ she preferred to B. 3.Since B’ has not yet crossed off his favorite stable wife, B’ must love G more than any stable wife. 4.But then B’ and G will elope in marriage which matches B to G, contradicting stability of wife G.

Assume for contradiction: some girl G isn’t married to least favorite stable spouse. 1.Let M be the matching from boys-propose ritual where G is matched to B. 2.Let M’ be stable matching where G is matched to least favorite stable husband B’. 3.Then G prefers B to B’ by assumption. 4.But B’s favorite stable wife is G, So B and G prefer each other than spouses in M’. 5.But them M’ is not stable.

But of course … symmetry If girls propose, then they will get their favorite stable husbands.

Conclusions You’ll get a better match if you do the proposing!

Other applications of stable marriage Hospitals Medical Interns Schools Students Servers Requests

Next time Prediction markets.