# Section 17.1: Voting

## 17.1 Outline

1. Dictatorship
2. Majority rule
1. vote
2. definition
3. symmetry
3. Plurality method
4. Borda count
1. definition
2. run-off election
5. Hare method
6. Pairwise comparison method
1. binary voting
2. definition
7. Tournament method
1. sequential voting
2. tournament method
3. tournament elimination method
8. Approval voting method
9. Summary

## 17.1 Essential Ideas

Majority Rule

By majority rule we mean voting to find an alternative which receives more than 50% of the vote. In other words, if the number of votes is n and n is even, then a majority is n/2 + 1. If the number of votes is odd, then a majority is (n + 1)/2.

Plurality Rule

By the plurality rule we mean the winner is the candidate with the highest number of votes. Each voter votes for one candidate. The candidate receiving the most votes is declared the winner.

Borda-Count Rule

Each voter ranks the candidates. If there are n candidates, then n points are assigned to the first choice for each voter, with n − 1 points for the next choice, and so on. The points for each candidate are added and if one has more votes, that candidate is declared the winner. Such a counting scheme is called a Borda count.

Hare-Count Rule

Each voter votes for one candidate. If a candidate receives a majority of the votes, that candidate is declared to be a first-round winner. If no candidate receives a majority of the votes, then the candidate(s) with the fewest number of first place votes is (are) eliminated.

Then each voter votes for one of the remaining candidates in the second round. If a candidate receives a majority, that candidate is declared to be the second-round winner.

If no candidate receives a majority of the votes, then eliminate the candidate(s) with the fewest number of first place votes.

Repeat this process until a candidate receives a majority of the votes. This voting procedure is called the Hare method or the plurality with a run-off method.

Pairwise-Comparison Rule

In the pairwise comparison method of voting, the voters rank the candidates. A series of comparisons in which each candidate is compared to each of the other candidates. If choice A is preferred to choice B, then A receives 1 point. If B is preferred to A, then V receives 1 point. If the candidates tie, each receives 1/2 point. The candidate with the most point is declared the winner.

Tournament Rule

Candidates are teamed head-to-head, with the winner of the first pairing facing a new component for the next election. This process continues until one candidate remains who is declared the winner.

Approval-Voting Rule

The approval voting method allows each voter to cast one vote for each candidate that meets with his or her approval. The candidate with the most votes is declared the winner.