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

Lecture Twenty-Six: Voting.

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

Voters Arrow Borda Condorcet

Alternatives chocolate strawberry vanilla

Rankings Arrow Arrow prefers strawberry to chocolate. chocolate strawberry

Rankings

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

Assumptions 1.Rankings are complete. Each voter has an opinion about each pair of alternatives. ?? ? ?? ?

Assumptions 2.Rankings are transitive.

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

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

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

Majority rule Two alternatives: Three alternatives?

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

Majority with 3 alternatives XYZYZXZXYXYZYZXZXY Majority XYXY YZYZ ZXZX Majority is not transitive

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

Are these preferences sensible? Che GuevaraJohn McCainBarack Obama

Single-peaked preferences CheCheneyObamaColbertMcCain Far left Far right

CheCheneyObamaColbertMcCain value Single-peaked: just one maximum

CheCheneyObamaColbertMcCain Y If for all candidates Y, then:

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

Who should win? CheCheneyObamaColbertMcCain # 1 st votes:(100 voters) Obama is the median alternative, i.e., the middle of the best alternatives.

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

Majority winner CheCheneyObamaColbertMcCain # 1 st votes:(100 voters) Obama is the median alternative, and the majority winner.

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

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

Computing Borda count Borda 321

Fact. Borda count always produces a complete transitive ranking.

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

YOU: Problem with Borda Then elect X (break ties alphabetically). Then elect Y ZYX YXZ XZY

Problem with Borda Borda

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

Reasonable voting schemes 1.Produce a complete transitive ranking. 2.Pareto Principle: If everyone prefers X to Y, rank X before Y. 3.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.

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.

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)

Trial A man stands accused of a horrible crime. The jury (voters).The verdict (alternatives).

Information aggregation Each voter receives a signal about the truth. Distortion flips the truth with probability p < 1/2 The unvarnished truth

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

Sequential voting? NO! Information cascades.

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.

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

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

Next time Epilogue.