Math and Politics-Chapter 1
Terms
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- social choice function
- decision procedure or method that is used to render the result of an election
- domain
- input from one set, set of all possible preferences of the voters (ex: ballots)
- range
- output chosen from another set A wins, B wins, or tie (declares winner)
- plurality
- candidate who gets more votes than any other candidate
- majority
- candidate who gets more than half the votes
- quota
- minimum number of votes (q) that a candidate needs to win
- supermajority method/quota method
- any candidate who gets a fraction p or more of all the votes
- status quo
- if you win the base method over the challenger, you are the winner for the status quo method as well
- partition method
- every voter is in exactly one block
- monarchy method
- one of the candidates is a monarch, the candidate wins no matter how anybody votes
- dictatorship method
- one of the voters is a dictator, whoever the dictator prefers is the winner
- parity method
- a candidate who gets an even number of votes wins, if both candidates get an odd number of votes or if both candidates get an even number of votes, it is a tie
- all-ties method
- the election is a tie no matter how the electorate votes
- criteria
- conditions, rules, or properties that we can require our election methods to satisfy
- anonymity criterion
- treats all voters equally
- neutrality criterion
- treats both candidates equally
- monotonicity criterion
- whenever one or more voters change their votes from one candidate to the other and the other candidate was the winner of the election before the change, that candidate must remain the winner after the change
- decisiveness criterion
- always chooses a winner
- near-decisiveness criterion
- can only be a tie if the candidates receive exactly the same number of votes
- proof
- persuasive logical argument that shows that the method satisfies the given criterion, behaves in a certain way, in all possible situations
- counterexample
- no answer that begs for an explanation
- May's Theorem
- simple majority method is the only method (only one that is anonymous, monotone, and neutral)
- corollary
- it is impossible for a voting method with two candidates to satisfy anonymity, neutrality, monotonicity, and decisiveness