Social Networks 101

1. Social Networks 101 Prof. Jason Hartline and Prof. Nicole Immorlica

2. Markets+Romance

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

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

5. Whatdo 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?

6. What could keep marriages from lasting?

7. When is marriage stable? Twiggy Jake Jake > Elwood > Curtis > Ray Holly > Claire > Twiggy > Jill Claire Elwood Jake > Curtis > Elwood > Ray Claire > Jill > Twiggy > Holly Jill Curtis “blocking pair” Ray > Curtis > Elwood > Jake Twiggy > Jill > Holly > Claire 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. Holly Ray Holly > Claire > Twiggy > Jill Ray > Jake > Elwood > Curtis

8. When is marriage stable? “blocking pair” “stability” 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. A matching is stable if there is no blocking pair.

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

10. 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.

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

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

13. Stability Fact 3. The resulting matching is stable. Why: • Consider boy B and girl G that are not married to each other. • Suppose G was crossed off B’s list. Then G prefers husband to B, so won’t elope with B. • 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.

14. 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.

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

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

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

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

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

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

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

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

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

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

25. Other applications of stable marriage National Residency Matching Program (NRMP). School Choice (Boston and New York). Medical Interns Students Hospitals Schools Akamai Servers Requests

26. Next time Prediction markets.