Voting

The Plurality Method

Each voter votes for one candidate, and the candidate with the most votes wins. The winning candidate does not have to have a majority of the votes.

Limitation: alternative choices are not considered

The Borda Count Method

If there are 𝑛 candidates or issues in an election, each voter ranks the candidates or issues by giving 𝑛 points to the voter’s first choice, 𝑛 − 1 points to the voter’s second choice, and so on, with the voter’s least favorite choice receiving 1 point. The candidate or issue that receives the most total points is the winner.
Plurality with Elimination Method

Pairwise Comparison Voting Method

✓ The “head-to-head” method.
✓ Each candidate is compared one-on-one with each of the other candidates.
✓ A candidate receives 1 point for a win, 0.5 point for a tie and 0 point for a loss.
✓ The candidate with the greatest number of points wins the election.

Majority Criterion

The candidate who receives a majority of first-place votes is the winner.

Monotonicity Criterion

If candidate A wins an election, then candidate A will also win the election if the only change in the voters’ preferences is that supporters of a different candidate change their votes to support candidate A.

Condorcet Criterion

A candidate who wins all possible head to head matchups should win an election when all candidates appear on the ballot.

Independence of Irrelevant Alternatives

If a candidate wins an election, the winner should remain the winner in any recount in which losing candidates withdraw from the race.

Arrow’s Impossibility Theorem

There is no voting method involving three or more choices that satisfies the fairness criteria.

Coalition

is a set of voters each of whom votes the same way, either for or against a resolution.

Winning coalition

Is a set of voters the sum of whose votes is greater than or equal to the quota.

Losing coalition

Is set of voters the sum of whose votes is less than the quota.

Critical Voter

A voter who leaves a winning coalition and thereby turns it into a losing coalition.

Dictator

A voter who has a weight that is greater than or equal to the quota.

Dummy

A voter who is never a critical voter has no power and is referred.

𝟐𝒏 − 𝟏

The number of possible coalitions of 𝑛 voters

Banzhaf Power Index

Determines the power of a voter in a weighted voting system.
𝐵𝑃𝐼 (𝑣) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑣𝑜𝑡𝑒𝑟 𝑣 𝑖𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑜𝑡𝑒𝑟 / 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎𝑛𝑦 𝑣𝑜𝑡𝑒𝑟 𝑖𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑜𝑡𝑒𝑟