Social Networks 101

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

2. Lecture Twenty-Six: Voting.

3. Group decision-making Netflix recommendation system A jury’s verdict Ranking of search results US presidential elections Ranking of college football teams

4. Voters Condorcet Arrow Borda

5. Alternatives vanilla strawberry chocolate

6. Rankings strawberry chocolate Arrow Arrow prefers strawberry to chocolate.

7. Rankings

8. Problem Given: Set of voters, set of alternatives, rankings Output: Global ranking of alternatives

9. Assumptions • Rankings are complete. Each voter has an opinion about each pair of alternatives. ? ? ? ? ? ?

10. Assumptions • Rankings are transitive. &

11. In other words, … Assumptions imply rankings are complete rank-ordered lists.

12. Question How can we combine individual rankings to produce a group ranking of the alternatives?

13. Voting schemes Consensus Dictatorship Which scheme works best? … always produces a valid ranking … is not subject to manipulation … makes the “right” decision Majority Electoral college Borda count

14. Majority rule Two alternatives: Three alternatives?

15. Majority rule For every pair of alternatives X and Y, Rule. Rank X above Y if more voters rank X Y than Y X

16. Majority with 3 alternatives X Y Z Y Z X Z X Y X Y 1. Majority Y Z 2. 3. Z X Majority is not transitive.

17. Condorcet triple 1. 2. X Y Z Y Z X Z X Y 3. For any winner, there is another winner a majority of voters prefer: e.g., If X wins, 2 and 3 would prefer Z.

18. Are these preferences sensible? Che Guevara John McCain BarackObama

19. Single-peaked preferences Far left Far right Che Colbert Obama McCain Cheney

20. value Che Colbert Obama McCain Cheney Single-peaked: just one maximum

21. Che Colbert Obama McCain Cheney If for all candidates Y, Y then: ?

22. Result Majority works for single-peaked preferences! (it always outputs a transitive ranking)

23. Who should win? Che Colbert Obama McCain Cheney # 1st votes: 7 43 35 13 2 (100 voters) Obama is the median alternative, i.e., the middle of the best alternatives.

24. Majority winner Che Colbert Obama McCain Cheney # 1st votes: 7 43 35 13 2 (100 voters) All these voters prefer Obama to McCain or Cheney. All these voters prefer Obama to Che or Colbert. Obama beats McCain and Cheney in Majority Rule. Obama beats Colbert and Che in Majority Rule. = 85 voters > ½ the voters (since Obama is median alternative). ½ voters < 50 voters = (since Obama is median alternative).

25. Majority winner Che Colbert Obama McCain Cheney # 1st votes: 7 43 35 13 2 (100 voters) Obama is the median alternative, and the majority winner.

26. Fact. For single-peaked preferences, majority winner = median alternative!

27. Borda count Rule. Assign a score to each alternative based on rank, output rank-by-score. (break ties alphabetically)

28. Computing Borda count 2 1 0 Borda 3 2 1 2 1 0

29. Fact. Borda count always produces a complete transitive ranking.

30. Voting in Borda count Experiment: YOU: Z Y X Y X Z X Z Y 2. 3. If you elect: Z: 2 pts. Y: 1 pt. X: 0 pts.

31. Problem with Borda Then elect X (break ties alphabetically). Then elect Y. YOU: Z Y X 2. Y X Z 3. X Z Y

32. Problem with Borda 0 3 2 1 0 3 2 1 Borda 0 5 4 3 4 3 2

33. Problem with Borda Highest-ranked alternative can change depending on how individuals rank low-ranked alternatives.

34. Reasonable voting schemes • Produce a complete transitive ranking. • Pareto Principle: If everyone prefers X to Y, rank X before Y. • Independence of Irrelevant Alternatives: For any three alternatives X, Y, and Z, group ranking of X and Y does not depend on how individuals rank Z.

35. Arrow’s impossibility result Only reasonable voting scheme is dictatorship. (or, if you prefer, there is no reasonable voting scheme) Pick an arbitrary voter and output her ranking.

36. But, … Information aggregation: There is a “correct” alternative but each voter has a noisy signal. Stability to noise: Each voter’s vote is subjected to some random noise. Majority rules (and dictators drool)

37. Trial A man stands accused of a horrible crime. The jury (voters). The verdict (alternatives). Guilty Not Guilty

38. Information aggregation Each voter receives a signal about the truth. Not Guilty Signal The unvarnished truth Distortion flips the truth with probability p < 1/2

39. Voting to aggregate information How should the jury vote? Majority rule aggregates information. How should this be implemented?

40. Sequential voting? NO! Information cascades.

41. Simultaneous voting? Problem: Sincerity. Suppose it is really bad for society to convict an innocent man. If even one jury member receives an innocent signal, the man should be acquitted.

42. Simultaneous voting A jury of three with two alternatives: Alternative A is better if any voter sees A. A B Doesn’t matter. I better vote A! A B A A B B Scenario 1: Scenario 2: Scenario 3:

43. Simultaneous voting For majority: Sincere voting is not a Nash equilibrium. You should always vote as if you’re pivotal.

44. Next time Epilogue.