arrow’s impossibility theorem -...
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Arrow’s Impossibility TheoremLecture 11Section 1.6
Robb T. Koether
Hampden-Sydney College
Wed, Sep 13, 2017
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 1 / 23
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 2 / 23
Outline
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 3 / 23
The Majority Criterion
Definition (Majority Criterion)The Majority Criterion says that if a candidate has a majority offirst-place votes, then that candidate should be the winner. (There mayor may not be a majority candidate.)
The Borda count method and Coombs’ method may violate theMajority Criterion.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 4 / 23
The Majority Criterion
Example (The Majority Criterion – Borda Count Method)
No. of Votes 8 6 5 31st A A B B2nd B B C C3rd C D D A4th D C A D
Does any candidate have a majority?Who wins by the Borda Count Method?
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 5 / 23
The Majority Criterion
Example (The Majority Criterion – Coomb’s Method)
No. of Votes 8 6 5 21st D A A B2nd B B C C3rd C D D A4th A C B D
Does any candidate have a majority?Who wins by Coomb’s Method?
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 6 / 23
The Majority Criterion
The Borda count method and Coombs’ method may violate theMajority Criterion.The other methods (I’m pretty sure) do not violate it.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 7 / 23
Outline
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 8 / 23
The Condorcet Criterion
Definition (Condorcet Winner)The Condorcet winner is a candidate who beat every other candidatein pairwise comparisons. (There may or may not be a Condorcetwinner.)
Definition (Condorcet Criterion)The Condorcet Criterion says that if there is a Condorcet winner, thenthat candidate should be the winner (by whatever method used).
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 9 / 23
The Condorcet Criterion
Example (The Condorcet Criterion – Borda Count Method)
No. of Votes 8 6 5 31st A A B B2nd B B C C3rd C D D A3rd D C A D
Is there a Condorcet winner?Who wins by the Borda Count Method?
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 10 / 23
The Condorcet Criterion
The plurality method, Borda count method,plurality-with-elimination method, and Coombs’ method mayviolate the Condorcet Criterion.The other method (I’m pretty sure) does not violate it.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 11 / 23
Outline
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 12 / 23
The Monotonicity Criterion
Definition (Monotonicity Criterion)The Monotonicity Criterion says that if candidate X is the winner, thenX would still be the winner if a voter had placed X higher in his ranking.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 13 / 23
The Monotonicity Criterion
Example (The Monotonicity Criterion – Plurality-with-EliminationMethod)
10 8 7 41st B A C C2nd C B A B3rd A C B A
Who is the winner by the Plurality-with-Elimination Method?Suppose that the last four voters decided to rank B over C. Nowwho is the winner?
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 14 / 23
The Monotonicity Criterion
The plurality-with-elimination Method may violate the MonotonicityCriterion.The other methods (I’m pretty sure) do not violate it.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 15 / 23
Outline
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 16 / 23
The Independence-of-Irrelevant-Alternatives Criterion
Definition (Independence-of-Irrelevant-Alternatives Criterion)The independence-of-irrelevant-alternatives criterion (IIA) says that ifcandidate X is the winner, then X would still be the winner if one ormore of the losing candidates had not been in the race.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 17 / 23
The Independence-of-Irrelevant-Alternatives Criterion
Example (The Independence-of-Irrelevant-Alternatives Criterion– Plurality-with-Elimination Method)
10 8 7 41st B A C C2nd C B A B3rd A C B A
Who is the winner by the Plurality-with-Elimination Method?Suppose that candidate C drops out. Now who is the winner?
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 18 / 23
The Independence-of-Irrelevant-Alternatives Criterion
The plurality method and the plurality-with-elimination methodmay violate the IIA Criterion.The other methods (I’m pretty sure) do not violate it.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 19 / 23
Outline
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 20 / 23
Arrow’s Impossibility Theorem
Theorem (Arrow’s Impossibility Theorem)If there are at least 3 candidates, then there is no voting method thatcannot violate any of the four desired properties (Majority, Condorcet,Monotonicity, Independence of Irrelevant Alternatives).
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 21 / 23
Outline
1 The Majority Criterion
2 The Condorcet Criterion
3 The Monotonicity Criterion
4 The Independence-of-Irrelevant-Alternatives Criterion
5 Arrow’s Impossibility Theorem
6 Assignment
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 22 / 23
Assignment
AssignmentChapter 1 Exercises 51, 52, 53, 54, 55, 56.
Robb T. Koether (Hampden-Sydney College) Arrow’s Impossibility Theorem Wed, Sep 13, 2017 23 / 23