Chapter 12 sec 1

Voting Methods

Def. Each person votes for his or her favorite candidate. The candidate receiving the most votes is declared the winner.

Plurality Method

Plurality method is the simplest way to determine the outcome of an election;

Many state and local elections use the plurality method because it is easy to determine the winner.

The students have organize a union in order to improve their salaries and working conditions. A group of 33 students, have just had and election to choose their president.

Using the plurality method, who won the election?

Example

Ann 10

Ben 9

Mike 11

Zach 3

Example

Mike won, for he had the most votes.

Mike won the election even though 22/33 = 66.7% of the group voted against him.

Permits the voter to “fine-tune” his or her vote in the sense that the voter can designate not only a first choice but also a second choice, a third choice, and so on.

Borda Count Method

In the previous election, we could have used the Borda Method by specifying that on the voter’s ballot the first choice would be 4 points, the second choice 3 points, third choice 2 points, and the fourth choice 1 point.

Borda

Def.If there are k candidates in an

election, each voter ranks all candidates on the ballot. Then the first choice is given k pts, the second choice is given k-1 pts, the third choice is given k-2 pts, and so on. The candidate who receives the most total pts wins the election.

ExamplePreferenc

e6 7 5 3 9 3

1st M A M A B Z2nd A M Z Z A A3rd B B B B Z M4th Z Z A M M BThe top numbers are the number of people

that voted in the particular preference ballot. The first column means that there were 6 people choose Mike first, Ann second, Ben third, and Zach fourth.

To determine the winner we calculate the ballots.

Ann = 10x4+18x3+0x2+5x1 = 99 ptsBen = 9x4+0x3+21x2+3x1 = 81 pts Mike = 11x4+7x3+3x2+12x1 = 83 ptsZach = 3x4+8x3+9x2+13x1 = 67 pts

We notice that Ann wins the election.

Many polls use the Borda to rank sports teams.

Try to calculate the winner

A WINS WITH 78 POINTS

Preference 8 7 5 71st C D C A2nd A A B D3rd B B D B4th D C A C

Def. Each voter votes for on candidate. A

candidate receiving a majority of votes is declared the winner. If no candidate receives a majority of votes, then the candidate(or candidates) with the fewest votes is dropped from the ballot and a new election is held. This process is continues until a candidate receives a majority of votes.

Plurality-with-Elimination Method

Viewing the first place, Zach has the fewest votes so he is eliminated. The remaining candidates move up.

Preference

6 7 5 3 9 3

1st M A M A B Z2nd A M Z Z A A3rd B B B B Z M4th Z Z A M M B

By looking at the 3rd column and the last column they are the same, therefore we can combine them.

Preference

6 7 5 3 9 3

1st M A M A B A2nd A M B B A M3rd B B A M M B

Viewing the first row Ben has the fewest votes, therefore he is eliminated.

Preference

6 10 5 3 9

1st M A M A B2nd A M B B A3rd B B A M M

Combine the same columns together.

Ann wins with 22 votes and Mike has 11 votes.

Preference

6 10 5 3 9

1st M A M A A2nd A M A M M

Try this

C wins

Preference 8 9 5 4 21st C E B A A

2nd A D C D C

3rd B B E B B

4th E C A C E

5th D A D E D

Def.Voters first rank all candidates. If A

and B are a pair of candidates, we count how many voters prefer A to B and vise versa. Whichever candidate is preferred the most receives 1 pt. If A and B are tied, then each receives ½ pt. Do this comparison, assigning pts., for each pair of candidates. Candidates with the most pts wins.

Pairwise Comparison Method

You are a owner of a restaurant and want to add to its menu. You did a survey in which the customers were asked to rank their preferences for (B)urritos, (S)ushi, & (H)amburger.

Example

We must compare :a) S with H, b) S with B, and c) H

with B

Preference

2,108

864

1,156

1,461

1,587

1,080

1st S S H H B B

2nd H B S B S H

3rd B H B S H S

A) S with HComparing S with H, we will ignore all

references to B4, 559 prefer S over H3, 697 prefer H over STherefore we award S, 1 pt.

Preference

2,108

864

1,156

1,461

1,587

1,080

1st S S H H B B

2nd H B S B S H

3rd B H B S H S

Compare S with B4, 128 prefer S over B4, 128 prefer B over STherefore S and B are tied, so each

receive ½ pt each.

Preference

2,108

864

1,156

1,461

1,587

1,080

1st S S H H B B

2nd H B S B S H

3rd B H B S H S

Compare H and B4, 725 prefer H over B3, 531 prefer B over HTherefore award 1 point to H

Preference

2,108

864

1,156

1,461

1,587

1,080

1st S S H H B B

2nd H B S B S H

3rd B H B S H S

S has 1½ ptH has 1 ptB has ½ pt

Therefore the winner is S.

Who is the winner?

Use the pairwise comparison to determine the winner. Also list how many pts each item gets.

H has 2 pts. B has 1 pt. S has 0 pts.

Try this Preferenc

e985 86

41,156

1,021

1,187

1,080

1st S S H H B B

2nd H B S B S H

3rd B H B S H S