Explaining the Impossible: Kenneth Arrow s Nobel Prize Winning Theorem on Elections

Transcription

1 Explaining the Impossible: Kenneth Arrow s Nobel Prize Winning Theorem on Elections Dr. Rick Klima Appalachian State University Boone, North Carolina U.S. Presidential Vote Totals, 2000 Candidate Bush Gore Nader Others Abstained Popular Votes 50,456,002 50,999,897 2,882,955 1,066,246 (a lot) Electoral Votes

2 U.S. Presidential Vote Totals, 1876 Candidate Hayes Tilden Others Popular Votes 4,034,311 4,288,546 90,244 Electoral Votes Majority Criterion If there is a candidate in an election who is preferred as the winner by a majority (more than half) of the voters, then that candidate should be the winner of the election.

3 Associated Press Top 20 Football Poll November 25, 1968 Team First Place Votes Borda Points 1. Ohio State USC Penn State Georgia Kansas California Gubernatorial Recall Election, candidates were on the ballot.

4 California Gubernatorial Recall Election, candidates were on the ballot. What is the smallest percentage of the votes that Arnold Schwarzenegger could have received and won the election? (Less than 1%) What is the largest percentage of voters who could have preferred Arnold the least among the 135 candidates in order for him to have won? (More than 99%) Minnesota Gubernatorial Election, 1998 Percentage of Voters Rank 35% 28% 20% 17% 1 Coleman Humphrey Ventura Ventura 2 Humphrey Coleman Coleman Humphrey 3 Ventura Ventura Humphrey Coleman

5 Condorcet Loser Criterion If there is a candidate in an election who would lose a head-to-head contest against each of his or her opponents, then that candidate should not be the winner of the election. Condorcet Winner Criterion If there is a candidate in an election who would win a head-to-head contest against each of his or her opponents, then that candidate should be the winner of the election.

6 IOC Voting for Site of 2004 Olympics City Athens Buenos Aires Cape Town Rome Stockholm Votes City Buenos Aires Cape Town Abstained Votes City Athens Cape Town Rome Stockholm Votes City Athens Cape Town Rome Votes City Votes Athens 66 Rome 41 Women s Figure Skating Judging 2002 Olympics Position First Second Third Before Final Competitor Skated Michelle Kwan Sarah Hughes After Final Competitor Skated Sarah Hughes Irina Slutskaya Michelle Kwan

7 Independence of Irrelevant Alternatives Criterion Suppose candidate A finishes ahead of candidate B in an election. If it is decided that one of the other candidates will be removed from the ballots, and the ballots (with the one candidate removed) will be reevaluated, then A should still finish ahead of B. I began to get the idea that maybe there was no voting method that would satisfy all the conditions that I regarded as reasonable. It was at this point that I set out to prove it. And it actually turned out to be a matter of only a few days work.

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9 Arrow s Actual Impossibility Theorem For an election with more than two candidates, it is impossible for a voting system to satisfy all of the following five conditions: Independence of Irrelevant Alternatives (IIA) Monotonicity - Changes to the ballots cast in an election that favor only one candidate should never cause that candidate to drop in the final ranking of the candidates. Universality - Voting systems should never restrict how the voters in an election are allowed to rank the candidates. Citizen Sovereignty - Every final ranking of the candidates in an election should be possible through some combination of personal rankings by the voters. Nondictatorship - There should never be a voter in an election for whom the voter s personal ranking of the candidates is always identical to the final ranking of the candidates.

10 Arrow s Theorem The Proof To Prove: For an election with more than two candidates, a voting system that satisfies IIA, monotonicity, universality, and citizen sovereignty must be a dictatorship. Unanimity (The Pareto Condition) - If all of the voters in an election prefer candidate A over candidate B, then A should finish ahead of B in the election. Lemma 1: For an election with more than two candidates, a voting system that satisfies IIA, monotonicity, and citizen sovereignty must satisfy unanimity. Arrow s Theorem The Proof Lemma 1: For an election with more than two candidates, a voting system that satisfies IIA, monotonicity, and citizen sovereignty must satisfy unanimity. Sketch of Proof: Let S 1 be a collection of preferences in which all of the voters prefer candidate A over candidate B. Citizen sovereignty implies that there must be a collection S 2 of preferences for which A finishes ahead of B. Create S 3 from S 2 by moving A ahead of B in all of the voter preferences. Monotonicity implies that A will finish ahead of B given S 3. IIA then implies that A will finish ahead of B given S 1.!

11 Arrow s Theorem The Proof Lemma 2: For an election with more than two candidates and a voting system that satisfies IIA, universality, and unanimity, if every voter ranks candidate B alone in first or last, then B must finish alone in first or last. Sketch of Proof: Suppose B does not finish alone in first or last. Find candidates A and C for which A finishes ahead of or tied with B, and C finishes behind or tied with B. Note then that A must finish ahead of or tied with C. If every voter moves C above A on their ballot, but makes no other changes, then unanimity implies that C must finish ahead of A. But IIA also implies that A must still finish ahead of or tied with C.! Arrow s Theorem The Proof Theorem: For an election with more than two candidates, a voting system that satisfies IIA, universality, and unanimity must be a dictatorship. Sketch of Proof: Call the voters v 1, v 2,, v n (in some order), and consider the case when every voter ranks candidate B alone in last. Then unanimity implies B must finish alone in last. Consider the voters one-by-one (in order) moving B on their ballot from alone in last to alone in first. Unanimity implies there must be a voter v j for whom when v j moves B to first, for the first time B moves to finishing first. We claim that this pivotal voter v j is a dictator.

12 Arrow s Theorem The Proof Sketch of Proof (Continued): Rank v 1!!! v j-1 v j v j+1!!! First B!!! B??!!!? M M M M M M Last?!!!? B B!!! B Table 1: B finishes last Rank v 1!!! v j-1 v j v j+1!!! First B!!! B B?!!!? M M M M M M Last?!!!?? B!!! B Table 2: B finishes first v n v n Arrow s Theorem The Proof Sketch of Proof (Continued): Consider candidates A and C, neither being B, and let S 1 be a collection of voter preferences for which v j prefers A over C. Create a collection S 2 of voter preferences by making the following changes to S 1, none of which by IIA can affect how A and C would finish relative to each other: v 1,, v j-1 move B to being alone in first v j moves B to between A and C v j+1,, v n move B to being alone in last Then A relates to B in S 2 as in Table 1 (where B finishes last), and B relates to C in S 2 as in Table 2 (where B finishes first). By IIA, A must finish ahead of C given S 2, and thus also by IIA, A must finish ahead of C given S 1.

13 Arrow s Theorem The Proof Sketch of Proof (Continued): Consider candidates A and B, and suppose that for some other candidate C the pivotal voter is v i. Then v i completely controls how every pair of candidates not including C would finish relative to each other. But v j can clearly affect how A and B would finish relative to each other. Thus v i = v j.! Voting is a basic tool of every democracy. We vote to choose the name for a pet dog, a textbook, a department chair, a U.S. Senator, the president of the United States. But does the election outcome capture what the voters really want? Not necessarily.

14 Intensity of Binary Independence Criterion Suppose candidate A finishes ahead of candidate B in an election. If the ballots are changed, but the number of candidates ranked between A and B is not changed on any of the ballots, and the ballots are reevaluated, then A should still finish ahead of B. Saari s Non-Impossibility Theorem For an election with more than two candidates, it is possible for a voting system to satisfy all of the following five conditions: Intensity of Binary Independence Monotonicity Universality Citizen Sovereignty Nondictatorship And the Borda count is one such voting system! But wait can t the Borda count violate the majority criterion?

15 California Gubernatorial Recall Election, 2003 (Hypothetical) Rank 1 51% of voters Schwarzenegger 49% of voters Leo Gallagher 2 Leo Gallagher Gary Coleman M M M 135 Gary Coleman Schwarzenegger