# Voting and Election Theory

## Preference

Voters express their preference in ballots. Ballots ask voter to rank the candidates by preference in a variety of voting methods.

Vocab:

- Preference Ballot: Ballot where the voters rank the candidates based on preference

- Linear Ballot: A ballot where ties aren't allowed

- Transitive: If a voter prefers candidate A over B and candidate B over C, then it will prefer A over C.

## Plurality Method

The Plurality Method is one of the voting methods. The candidate with the most 1st places votes wins with this method.

There are criterions that are in place to make sure whether or not each voting method is fair.

The majority criterion says that the winner of an election should win because they had the majority of the votes which is more than half.

The plurality method can follow the majority criterion when there are 2 candidates but three or more may not ensure that it is fair.

Also, the plurality method only looks at 1st place votes which can be a bad decisions based on other votes. For example, if there are 3 candidates, A, B, and C. Candidate A can have the most 1st place votes but also the most last place votes while B and C have an equal amount of votes that are 2nd and 3rd place. Since A has the most 1st place votes it automatically even though it has the most last place votes.

The criterion that this violates is the Condorcet Criterion. The Condorcet criterion states that the winning candidate should win in head to head comparisons against the other candidates.

Vocab:

- Insincere Voting: When the person you want to win doesn't look like they will win then you change your vote

## Borda Count Method

On a preference ballot each vote is assigned points. The number of points are determined by number of candidates and it goes down from that number to 1 point. For example, if there are 3 candidates, 1st place gets 3 points, 2nd place gets 2 points, and last place gets 1 point. To find the number of votes, you multiply the candidate's rank by the number of voters put them as that rank. Once the points are distributed you add up the number of points from each candidate's place.

The Borda Count Method also violates some fairness criterions. It can violate the majority criterion because even though a person may have the most votes all together they might not have majority of first place votes. Once the majority criterion is violated so is the Condorcet criterion.

## Plurality with Elimination Method

In this method you would do the same thing you do in the original Plurality method. But, the person with the least amount of 1st place votes gets eliminated. Then after the elimination you cross out the eliminated candidate's name and recount the votes. This process continues until there is 1 candidate left with majority of the first place votes.

This method violates the Monotonicity criterion. The monotonicity criterion states that if a candidate is the winner of an election then in the reelection the candidate should remain the winner.

## Method of Pairwise Comparisons

This method was created to satisfy the Condorcet criterion. This method is like round robin tournaments, each candidate goes up against the other candidates. Each candidate gets a tally mark if they win their head to head matchup. Head to head matchups are won based upon which candidate has the most votes. The candidate with the most tallies wins.

The fairness criterion that this method violates is the Independence of Irrelevant Alternatives Criterion. This criterion states that if a candidate wins an election, if a candidate withdraws or gets disqualified they should still be the winner. With this method that is not the case. Also, there are times where in this method there is a tie for the number of tally a candidate gets. In order to pick a winner you have to use one of the other methods. The only problem with that is each method will give you a different winner.

## Extended Ranking Methods

Each of the methods have an extended version to them. Once you go through the original methods, you go through the process again to rank the other places after 1st.

## Recursive Ranking Methods

Just like extended ranking method this method requires extra steps after the original method is complete. Instead of just going down to rank the rest of the candidates after the winner, you will eliminate the winner from the preference ballot and make a whole new ballot without the winner. Then that process will continue until every candidate is ranked.