I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win the election

I A voting method satisfies the majority criterion if a candidate with a majority of first-preference votes will win the election

I A voting method satisfies the public enemy criterion is a candidate with a majority of last-preference votes cannot win the election

I Condorcet criterion:

I majority criterion:

I public enemy criterion:

Review

Fairness criteria for voting methods: I A voting method satisfies the majority criterion if a candidate with a majority of first-preference votes will win the election

I A voting method satisfies the public enemy criterion is a candidate with a majority of last-preference votes cannot win the election

I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win the election I majority criterion:

I public enemy criterion:

Review

Fairness criteria for voting methods: I Condorcet criterion:

I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the majority criterion if a candidate with a majority of first-preference votes will win the election

I A voting method satisfies the public enemy criterion is a candidate with a majority of last-preference votes cannot win the election

I majority criterion:

I public enemy criterion:

Review

Fairness criteria for voting methods: I Condorcet criterion:

I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win the election I A voting method satisfies the public enemy criterion is a candidate with a majority of last-preference votes cannot win the election

I public enemy criterion:

Review

Fairness criteria for voting methods: I Condorcet criterion:

I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win the election I majority criterion:

I A voting method satisfies the majority criterion if a candidate with a majority of first-preference votes will win the election Review

Fairness criteria for voting methods: I Condorcet criterion:

I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win the election I majority criterion:

I A voting method satisfies the majority criterion if a candidate with a majority of first-preference votes will win the election I public enemy criterion:

I A voting method satisfies the public enemy criterion is a candidate with a majority of last-preference votes cannot win the election I no

I A

I A

I C

I B

I B I Is there a majority winner?

I Is there a public enemy?

I Who is the plurality winner?

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner? I no

I A

I A

I C

I B

I Is there a majority winner?

I Is there a public enemy?

I Who is the plurality winner?

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I A

I A

I C

I B

I no I Is there a public enemy?

I Who is the plurality winner?

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner? I A

I A

I C

I B

I Is there a public enemy?

I Who is the plurality winner?

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I A

I C

I B

I A I Who is the plurality winner?

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy? I A

I C

I B

I Who is the plurality winner?

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I C

I B

I A I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I Who is the plurality winner? I C

I B

I Who is the instant runoff winner (10 to win)?

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I Who is the plurality winner?

I A I B

I C I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I Who is the plurality winner?

I A I Who is the instant runoff winner (10 to win)? I B

I Is there Borda winner?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I Who is the plurality winner?

I A I Who is the instant runoff winner (10 to win)?

I C I B

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I Who is the plurality winner?

I A I Who is the instant runoff winner (10 to win)?

I C I Is there Borda winner? Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Is there a Condorcet winner?

I B I Is there a majority winner?

I no I Is there a public enemy?

I A I Who is the plurality winner?

I A I Who is the instant runoff winner (10 to win)?

I C I Is there Borda winner?

I B Summary

Condorcet Majority Public Enemy Plurality no yes no Instant Runoff no yes yes Borda no no yes I Rio de Janeiro is eliminated in the first round I Tokyo wins

I Candidates are Rio de Janeiro, Madrid, and Tokyo

I Poll yields the following preferences: Number of Voters 7 8 10 4 1st choice MRTM 2nd choice RTMT 3rd choice TMRR

I If the election were held right now, who would win?

Example

I Instant runoffs are being used to determine the host city for the 2016 Olympics I Rio de Janeiro is eliminated in the first round I Tokyo wins

I Poll yields the following preferences: Number of Voters 7 8 10 4 1st choice MRTM 2nd choice RTMT 3rd choice TMRR

I If the election were held right now, who would win?

Example

I Instant runoffs are being used to determine the host city for the 2016 Olympics

I Candidates are Rio de Janeiro, Madrid, and Tokyo I Rio de Janeiro is eliminated in the first round I Tokyo wins

I If the election were held right now, who would win?

Example

I Instant runoffs are being used to determine the host city for the 2016 Olympics

I Candidates are Rio de Janeiro, Madrid, and Tokyo

I Poll yields the following preferences: Number of Voters 7 8 10 4 1st choice MRTM 2nd choice RTMT 3rd choice TMRR I Rio de Janeiro is eliminated in the first round I Tokyo wins

Example

I Instant runoffs are being used to determine the host city for the 2016 Olympics

I Candidates are Rio de Janeiro, Madrid, and Tokyo

I Poll yields the following preferences: Number of Voters 7 8 10 4 1st choice MRTM 2nd choice RTMT 3rd choice TMRR

I If the election were held right now, who would win? I Tokyo wins

Example

I Instant runoffs are being used to determine the host city for the 2016 Olympics

I Candidates are Rio de Janeiro, Madrid, and Tokyo

I Poll yields the following preferences: Number of Voters 7 8 10 4 1st choice MRTM 2nd choice RTMT 3rd choice TMRR

I If the election were held right now, who would win?

I Rio de Janeiro is eliminated in the first round Example

I Instant runoffs are being used to determine the host city for the 2016 Olympics

I Candidates are Rio de Janeiro, Madrid, and Tokyo

I Poll yields the following preferences: Number of Voters 7 8 10 4 1st choice MRTM 2nd choice RTMT 3rd choice TMRR

I If the election were held right now, who would win?

I Rio de Janeiro is eliminated in the first round I Tokyo wins I Madrid is eliminated in the first round I Rio de Janeiro wins!?!

I The change looks like it only benefits Tokyo (the presumed winner) I Who wins?

I So Tokyo getting more first round votes caused them to lose

Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR

Example

I Suppose that the contingent of 4 decides to help Tokyo: I Madrid is eliminated in the first round I Rio de Janeiro wins!?!

I The change looks like it only benefits Tokyo (the presumed winner) I Who wins?

I So Tokyo getting more first round votes caused them to lose

Example

I Suppose that the contingent of 4 decides to help Tokyo: Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR I Madrid is eliminated in the first round I Rio de Janeiro wins!?!

I Who wins?

I So Tokyo getting more first round votes caused them to lose

Example

I Suppose that the contingent of 4 decides to help Tokyo: Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR

I The change looks like it only benefits Tokyo (the presumed winner) I Madrid is eliminated in the first round I Rio de Janeiro wins!?!

I So Tokyo getting more first round votes caused them to lose

Example

I Suppose that the contingent of 4 decides to help Tokyo: Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR

I The change looks like it only benefits Tokyo (the presumed winner) I Who wins? I Rio de Janeiro wins!?!

I So Tokyo getting more first round votes caused them to lose

Example

I Suppose that the contingent of 4 decides to help Tokyo: Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR

I The change looks like it only benefits Tokyo (the presumed winner) I Who wins?

I Madrid is eliminated in the first round I So Tokyo getting more first round votes caused them to lose

Example

I Suppose that the contingent of 4 decides to help Tokyo: Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR

I The change looks like it only benefits Tokyo (the presumed winner) I Who wins?

I Madrid is eliminated in the first round I Rio de Janeiro wins!?! Example

I Suppose that the contingent of 4 decides to help Tokyo: Number of Voters 7 8 10 4 1st choice MRT —MT 2nd choice RTM —TM 3rd choice TMRR

I The change looks like it only benefits Tokyo (the presumed winner) I Who wins?

I Madrid is eliminated in the first round I Rio de Janeiro wins!?!

I So Tokyo getting more first round votes caused them to lose I Yes

I Yes

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner

I Instant runoffs do not satisfy the monotonicity criterion I Does plurality voting satisfy the monotonicity criterion?

I Does the Borda method satisfy the monotonicity criterion?

The Monotonicity Criterion

I The monotonicity criterion: I Yes

I Yes

I Instant runoffs do not satisfy the monotonicity criterion I Does plurality voting satisfy the monotonicity criterion?

I Does the Borda method satisfy the monotonicity criterion?

The Monotonicity Criterion

I The monotonicity criterion:

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner I Yes

I Yes

I Does plurality voting satisfy the monotonicity criterion?

I Does the Borda method satisfy the monotonicity criterion?

The Monotonicity Criterion

I The monotonicity criterion:

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner

I Instant runoffs do not satisfy the monotonicity criterion I Yes

I Yes I Does the Borda method satisfy the monotonicity criterion?

The Monotonicity Criterion

I The monotonicity criterion:

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner

I Instant runoffs do not satisfy the monotonicity criterion I Does plurality voting satisfy the monotonicity criterion? I Yes

I Does the Borda method satisfy the monotonicity criterion?

The Monotonicity Criterion

I The monotonicity criterion:

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner

I Instant runoffs do not satisfy the monotonicity criterion I Does plurality voting satisfy the monotonicity criterion?

I Yes I Yes

The Monotonicity Criterion

I The monotonicity criterion:

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner

I Instant runoffs do not satisfy the monotonicity criterion I Does plurality voting satisfy the monotonicity criterion?

I Yes I Does the Borda method satisfy the monotonicity criterion? The Monotonicity Criterion

I The monotonicity criterion:

I A voting method satisfies the monotonicity criterion if the winner of an election cannot lose a repeat election if preferences are altered only to the benefit of the original winner

I Instant runoffs do not satisfy the monotonicity criterion I Does plurality voting satisfy the monotonicity criterion?

I Yes I Does the Borda method satisfy the monotonicity criterion?

I Yes Summary

Condorcet Majority Public Enemy Monotonicity Plurality no yes no yes Instant Runoff no yes yes no Borda no no yes yes I A

I A

I A I Who is the instant runoff winner (11 to win)?

I Who is the Borda winner?

One Last Criterion

Number of Voters 8 6 4 3 1st choice ABCD 2nd choice DCDB 3rd choice BAAC 4th choice CDBA

I Who is the plurality winner? I A

I A

I Who is the instant runoff winner (11 to win)?

I Who is the Borda winner?

One Last Criterion

Number of Voters 8 6 4 3 1st choice ABCD 2nd choice DCDB 3rd choice BAAC 4th choice CDBA

I Who is the plurality winner?

I A I A

I A I Who is the Borda winner?

One Last Criterion

Number of Voters 8 6 4 3 1st choice ABCD 2nd choice DCDB 3rd choice BAAC 4th choice CDBA

I Who is the plurality winner?

I A I Who is the instant runoff winner (11 to win)? I A

I Who is the Borda winner?

One Last Criterion

Number of Voters 8 6 4 3 1st choice ABCD 2nd choice DCDB 3rd choice BAAC 4th choice CDBA

I Who is the plurality winner?

I A I Who is the instant runoff winner (11 to win)?

I A I A

One Last Criterion

Number of Voters 8 6 4 3 1st choice ABCD 2nd choice DCDB 3rd choice BAAC 4th choice CDBA

I Who is the plurality winner?

I A I Who is the instant runoff winner (11 to win)?

I A I Who is the Borda winner? One Last Criterion

Number of Voters 8 6 4 3 1st choice ABCD 2nd choice DCDB 3rd choice BAAC 4th choice CDBA

I Who is the plurality winner?

I A I Who is the instant runoff winner (11 to win)?

I A I Who is the Borda winner?

I A I C

I C

I C I Who is the instant runoff winner (11 to win)?

I Who is the Borda winner?

One Last Criterion

Now suppose that B drops out: Number of Voters 8 6 4 3 1st choice A XBCCD 2nd choice D XCAD XBC 3rd choice XBC XAD A XCA XC XD XB XA

I Who is the plurality winner? I C

I C

I Who is the instant runoff winner (11 to win)?

I Who is the Borda winner?

One Last Criterion

Now suppose that B drops out: Number of Voters 8 6 4 3 1st choice A XBCCD 2nd choice D XCAD XBC 3rd choice XBC XAD A XCA XC XD XB XA

I Who is the plurality winner?

I C I C

I C I Who is the Borda winner?

One Last Criterion

Now suppose that B drops out: Number of Voters 8 6 4 3 1st choice A XBCCD 2nd choice D XCAD XBC 3rd choice XBC XAD A XCA XC XD XB XA

I Who is the plurality winner?

I C I Who is the instant runoff winner (11 to win)? I C

I Who is the Borda winner?

One Last Criterion

Now suppose that B drops out: Number of Voters 8 6 4 3 1st choice A XBCCD 2nd choice D XCAD XBC 3rd choice XBC XAD A XCA XC XD XB XA

I Who is the plurality winner?

I C I Who is the instant runoff winner (11 to win)?

I C I C

One Last Criterion

Now suppose that B drops out: Number of Voters 8 6 4 3 1st choice A XBCCD 2nd choice D XCAD XBC 3rd choice XBC XAD A XCA XC XD XB XA

I Who is the plurality winner?

I C I Who is the instant runoff winner (11 to win)?

I C I Who is the Borda winner? One Last Criterion

Now suppose that B drops out: Number of Voters 8 6 4 3 1st choice A XBCCD 2nd choice D XCAD XBC 3rd choice XBC XAD A XCA XC XD XB XA

I Who is the plurality winner?

I C I Who is the instant runoff winner (11 to win)?

I C I Who is the Borda winner?

I C I A voting method satisfies the I.I.A. criterion if the winner of an election would still win if other candidates were disqualified

I Plurality, instant runoffs, and the Borda method do not satisfy I.I.A.

Independence of Irrelevant Alternatives

I The independence of irrelevant alternatives criterion: I Plurality, instant runoffs, and the Borda method do not satisfy I.I.A.

Independence of Irrelevant Alternatives

I The independence of irrelevant alternatives criterion:

I A voting method satisfies the I.I.A. criterion if the winner of an election would still win if other candidates were disqualified Independence of Irrelevant Alternatives

I The independence of irrelevant alternatives criterion:

I A voting method satisfies the I.I.A. criterion if the winner of an election would still win if other candidates were disqualified

I Plurality, instant runoffs, and the Borda method do not satisfy I.I.A. Summary

Cond. Maj. P.E. Mono. I.I.A. Plurality no yes no yes no Instant Runoff no yes yes no no Borda no no yes yes no I Candidate earns one point for each candidate that they beat

I Candidate with the most points wins

Method of Pairwise Comparisons (Condorcet Method)

Method of pairwise comparisons

I Compare each pair of candidates I Candidate with the most points wins

Method of Pairwise Comparisons (Condorcet Method)

Method of pairwise comparisons

I Compare each pair of candidates

I Candidate earns one point for each candidate that they beat Method of Pairwise Comparisons (Condorcet Method)

Method of pairwise comparisons

I Compare each pair of candidates

I Candidate earns one point for each candidate that they beat

I Candidate with the most points wins I yes

I yes

I yes

I no

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I Does it satisfy the public enemy criterion?

I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons? I yes

I yes

I yes

I no

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I Does it satisfy the public enemy criterion?

I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B I yes

I yes

I yes

I no

I Does it satisfy the majority criterion?

I Does it satisfy the public enemy criterion?

I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I yes

I yes

I no

I yes I Does it satisfy the public enemy criterion?

I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion? I yes

I yes

I no

I Does it satisfy the public enemy criterion?

I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I yes

I no

I yes I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I Does it satisfy the public enemy criterion? I yes

I no

I Does it satisfy the monotonicity criterion?

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I Does it satisfy the public enemy criterion?

I yes I no

I yes I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I Does it satisfy the public enemy criterion?

I yes I Does it satisfy the monotonicity criterion? I no

I Does it satisfy the I.I.A. criterion?

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I Does it satisfy the public enemy criterion?

I yes I Does it satisfy the monotonicity criterion?

I yes I no

Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I Does it satisfy the public enemy criterion?

I yes I Does it satisfy the monotonicity criterion?

I yes I Does it satisfy the I.I.A. criterion? Example Number of Voters 7 5 6 1st choice ABC 2nd choice BCB 3rd choice CAA

I Who wins via the method of pairwise comparisons?

I B

I The method of pairwise comparisons satisfies the Condorcet criterion I Does it satisfy the majority criterion?

I yes I Does it satisfy the public enemy criterion?

I yes I Does it satisfy the monotonicity criterion?

I yes I Does it satisfy the I.I.A. criterion?

I no I Nobody (rock, paper, scissors)

n2−n I If there are n candidates, you have to do 2 comparisons I For other methods, you have to do ∼ n computations

I Who wins via the method of pairwise comparisons?

I Complicated/inefficient:

I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB

Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Nobody (rock, paper, scissors)

I Who wins via the method of pairwise comparisons?

n2−n I If there are n candidates, you have to do 2 comparisons I For other methods, you have to do ∼ n computations I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB

Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Complicated/inefficient: I Nobody (rock, paper, scissors)

I Who wins via the method of pairwise comparisons?

I For other methods, you have to do ∼ n computations I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB

Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Complicated/inefficient: n2−n I If there are n candidates, you have to do 2 comparisons I Nobody (rock, paper, scissors)

I Who wins via the method of pairwise comparisons?

I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB

Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Complicated/inefficient: n2−n I If there are n candidates, you have to do 2 comparisons I For other methods, you have to do ∼ n computations I Nobody (rock, paper, scissors)

I Who wins via the method of pairwise comparisons?

Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Complicated/inefficient: n2−n I If there are n candidates, you have to do 2 comparisons I For other methods, you have to do ∼ n computations I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB I Nobody (rock, paper, scissors)

Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Complicated/inefficient: n2−n I If there are n candidates, you have to do 2 comparisons I For other methods, you have to do ∼ n computations I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB

I Who wins via the method of pairwise comparisons? Method of Pairwise Comparisons

Problems with the method of pairwise comparisons? I Complicated/inefficient: n2−n I If there are n candidates, you have to do 2 comparisons I For other methods, you have to do ∼ n computations I There might not be winner: Number of Voters 7 5 6 1st choice ABC 2nd choice BCA 3rd choice CAB

I Who wins via the method of pairwise comparisons?

I Nobody (rock, paper, scissors) Summary

Cond. Maj. P.E. Mono. I.I.A. Plurality no yes no yes no Instant Runoff no yes yes no no Borda no no yes yes no Pairwise Comparisons yes yes yes yes no I Condorcet criterion

I majority criterion

I monotonicity criterion

I I.I.A. criterion

Arrow’s Impossibility Theorem There is no voting method that satisfies:

Another Method?

I Is there a method that satisfies all of these criteria? Another Method?

I Is there a method that satisfies all of these criteria?

Arrow’s Impossibility Theorem There is no voting method that satisfies:

I Condorcet criterion

I majority criterion

I monotonicity criterion

I I.I.A. criterion