arrow’s impossibility theorem -...

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