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

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Lecture Twenty-Six: Voting.

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

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Voters Arrow Borda Condorcet

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Alternatives chocolate strawberry vanilla

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Rankings Arrow Arrow prefers strawberry to chocolate. chocolate strawberry

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Rankings

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

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

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Assumptions 2.Rankings are transitive.

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

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

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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

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Majority rule Two alternatives: Three alternatives?

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

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

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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

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Are these preferences sensible? Che GuevaraJohn McCainBarack Obama

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Single-peaked preferences CheCheneyObamaColbertMcCain Far left Far right

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CheCheneyObamaColbertMcCain value Single-peaked: just one maximum

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CheCheneyObamaColbertMcCain Y If for all candidates Y, then:

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

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

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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.

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

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

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

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Computing Borda count Borda 321

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Fact. Borda count always produces a complete transitive ranking.

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

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

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Problem with Borda Borda

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

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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.

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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.

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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)

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

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

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

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Sequential voting? NO! Information cascades.

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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.

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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!

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

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Next time Epilogue.

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