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. 1. 2. 3.
XYZYZXZXYXYZYZXZXY For any winner, there is another winner a majority of voters prefer: e.g., If X wins, 2 and 3 would prefer Z. 1. 2. 3. 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 74335132# 1 st votes:(100 voters) Obama is the median alternative, i.e., the middle of the best alternatives.
Majority winner CheCheneyObamaColbertMcCain 74335132# 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 74335132# 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)
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?