voting methods examples of voting methods (other than majority rules) –plurality –borda count...

Voting Methods

• Examples of Voting Methods (other than majority rules)

– Plurality– Borda Count– Hare System– Sequential Pairwise– Approval Voting

Examples of Voting Methods

Consider the following preference schedule for 9 voters considering 4 candidates A, B, C and D.

Number of Voters (9 total)

Rank 3 1 1 1 1 1 1

1st A A B B C C D

2nd D B C C B D C

3rd B C D A D B B

4th C D A D A A A

(a) Who wins by the plurality method?


Rank 3 1 1 1 1 1 1

1st A A B B C C D

2nd D B C C B D C

3rd B C D A D B B

4th C D A D A A A

With the plurality method, we consider only each voter’s 1st place ranking.

A voter can only vote for his or her first choice.

(a) Who wins by the plurality method?


Rank 3 1 1 1 1 1 1

1st A A B B C C D

We consider only each voter’s 1st place ranking.

Because A has the most first place votes, A is the winner by the plurality method.

(b) Who wins by the Borda Count method of voting?

• Note that A has 13 points, B has 15 points, C has 13 points, and D has 13 total points. • Because B has the most total points, B is the winner by the Borda Count method.


Rank 3 1 1 1 1 1 1

1st

(3points)

A

(9)

A

(3)

B

(3)

B

(3)

C

(3)

C

(3)

D

(3)

2nd

(2points)

D

(6)

B

(2)

C

(2)

C

(2)

B

(2)

D

(2)

C

(2)

3rd

(1 point)

B

(3)

C

(1)

D

(1)

A

(1)

D

(1)

B

(1)

B

(1)

4th

(0 points)

C

(0)

D

(0)

A

(0)

D

(0)

A

(0)

A

(0)

A

(0)

(c) Who wins by the Hare System voting method?

• First, we must eliminate candidate D because the least number of voters rank D as a first choice.


Rank 3 1 1 1 1 1 1

1st A A B B C C D

2nd D B C C B D C

3rd B C D A D B B

4th C D A D A A A



Rank 3 1 1 1 1 1 1

1st A A B B C C D

2nd D B C C B D C

3rd B C D A D B B

4th C D A D A A A

We cross out every instance of candidate D in the preference lists of all voters.



Rank 3 1 1 1 1 1 1

1st A A B B C C

2nd B C C B C

3rd B C A B B

4th C A A A A

Once candidate D is removed as an option, we assume individual voter’s preferences remained unchanged with respect to the other candidates, and so we move candidates up into any open spaces.



Rank 3 1 1 1 1 1 1

1st A A B B C C C

2nd B B C C B B B

3rd C C A A A A A

4th

Once candidate D is removed as an option, we assume individual voter’s preferences remained unchanged with respect to the other candidates, and so we move candidates up into any open spaces.



Rank 4 2 3

1st A B C

2nd B C B

3rd C A A

4th (none)

There are now some repetitions, so we can re-organize the table.

(c) Who wins by the Hare System Method of Voting?


Rank 4 2 3

1st A B C

2nd B C B

3rd C A A

• Now, we begin again, by identifying any candidates which have the least number ofvoters ranking them in first place.

• Remember there could be a tie for least, for example if candidate B and C were both ranked first by 2 voters. If that had been the case then both B and C would be eliminated and A would be declared the winner.

• In this example, we eliminate candidate B because B tops only 2 voters’ preferencelists.

(c) Who wins by the Hare System Voting Method?


Rank 4 2 3

1st A B C

2nd B C B

3rd C A A

• Now, we begin again, by identifying any candidates which have the least number ofvoters ranking them in first place.

• Remember there could be a tie for least, for example if candidate B and C were both ranked first by 2 voters. If that had been the case then both B and C would be eliminated and A would be declared the winner.

• In this example, we eliminate candidate B because B tops only 2 voters’ preferencelists.



Rank 4 2 3

1st A C

2nd C

3rd C A A

Now we have eliminated candidate B from contention. Again, we assume voter preferences will remain unchanged with respect to the other candidates and we will move the remaining candidates up into any open spaces.



Rank 4 2 3

1st A C C

2nd C A A

3rd

Now we have eliminated candidate B from contention. Again, we assume voter preferences will remain unchanged with respect to the other candidates and we will move the remaining candidates up into any open spaces.



Rank 4 5

1st A C

2nd C A

Now we re-organize the table because of the repetitions in the preference lists. At this point, A is eliminated because A tops the least number of preference lists (4) compared to C (with 5 lists). Because C is the only candidate remaining after the sequence of eliminations, C is declared the winner.

Note that if both A and C where ranked first by the same number of voters (for example if both lists represented 5 voters) we would declare the two remaining candidates tied.

(d) Who wins by the sequential pairwise voting method?

• We must determine the order in which we will compare candidates head-to-head. This order is called the agenda. Suppose we will compare them in alphabetical order: A, B, C, then D.

• With the voter preferences listed below, we begin by comparing candidates A and B.

• We consider all voter lists and note that B ranks above A on 5 lists whereas A ranks over B on only 4 lists. Therefore, B wins over A by a vote of 5 to 4.

• Because B wins over A, based on the agenda, we now compare B with C.


Rank 3 1 1 1 1 1 1

1st A A B B C C D

2nd D B C C B D C

3rd B C D A D B B

4th C D A D A A A

(d) Who wins by the sequential pairwise voting method?

• Comparing B and C head-to-head, we recognize that B ranks above C for 6 out of 9 voters. That is, B beats C by a vote of 6 to 3.

• Following the agenda A, B, C, and D, we now compare B with D.• We find that D is preferred to B among voters by a ratio of 5 to 4 and because we

have reached the last candidate, D is declared the winner by sequential pairwise voting.


Rank 3 1 1 1 1 1 1

1st A A B B C C D

2nd D B C C B D C

3rd B C D A D B B

4th C D A D A A A

Is this at all disconcerting? Different voting methods producing different results?

• Candidate A prefers plurality• Candidate B prefers Borda Count• Candidate C prefers the Hare System• Candidate D prefers sequential pairwise voting (but with

agenda A,B,C,D)

Which voting method is the best? How do we decide?

How about approval voting?

An example of Approval Voting

• When voters use approval voting they cast votes only for candidates they find acceptable.

• This means we can imagine voters will “draw a line” between acceptable and unacceptable candidates as they vote.

• Voters need not decide on a preference list and rank all of the candidates, however we may continue using preference lists without loss of generality. In other words, it does no harm to suppose the voters actually do rank the candidates and then choose to separate them into two categories: acceptable and unacceptable.

• For our examples of approval voting we will continue using preference lists and actually draw a line to represent the point of division between acceptable and unacceptable candidates for each voter.

An example of Approval Voting

• Consider the preference list shown with lines added to indicate an “approval line”• Voters will cast approval votes for candidates above the approval line and will

not cast approval votes for candidates below that line.


33 33 34

1st A B C

2nd B A A

3rd C C B

Notice the lines indicate 33 voters approve of both candidates A and B, 33 votersApprove of only candidate B and 34 candidates approve of only candidate C.

If the election were held with these preferences using approval voting, the winnerwould be candidate B who would get 66 approval votes. Note that A gets 33 approval votes and C gets 34 approval votes.

Which Method is Best?

• Is approval voting the answer ?– Is this the best method of voting ?

• We saw 4 different voting methods each give us a different winner for the same election.

• There are two problems we must face:– First, there is the problem that different voting methods can

produce different winners and we must decide which method to use.

– Second, each of our voting methods have problems in and of themselves. Even approval voting has its share of problems.

voting methods examples of voting methods (other than majority rules) –plurality –borda count...

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