Mathmatics and Politics

Algorithim

a step-by-step mathematical process used to decide an outcome

Proof

A formal argument used to justify a theorm/proposition/lennen

Counter-Example

Something that proves a statement as false

Function

A rule which assigns every element in the domain to exactly one element in the codomain

Domain

Input set

Codomain/Range

Output Set

Social Choice Functions

functions whose domain is a profile and their range is the canidates

Electorate

Set of voters

Slate

Set of canidates

Ballot

A column of candidates in preference order from top to bottom

Profile

A collection of ballots for an election

Simple Majority SCF

The 2 candidate SCF where the winner has more than 50% of first place votes or the electon results in a tie

Super Majority SCF

The election winner is determined by a certain parameter, if no one achieves this parameter, then the election is a tie

Status Quo Method

There is a base SCF, if a tie is produced than the winner is automatically the status quo canidate

Bloc Voting SCF

essentially US Electoral College, voters are split into blocs, election is determined based on simple majority in these blocs.

Monarchy Method

1 Canidate is the monarch and they win

Dictator Method

One voter is the dictator, and their selected canidate wins

Parity Method

If one canidate gets an even number of votes they win, if both or neither canidates gets even number of votes, than election result is a tie

All-ties Method

End result of the election is always a tie

Deciseveness

A scf is decisive if there is never a tie (monarchy, dictatorship, status quo)

Not Decisive

an election is not desive if it can end in a tie

Anominity

an scf is anonymous if swapping the ballots of two voters never changes the election result

Neutrality

An election is neutral if it treats all candidates equally

Monotonicity

a scf is monotonic if adding a vote to the winner doesnt change the result of the election

Near Decissiveness

a scf is nearly decisive if the only tie results from both canidates earning the exact same number of votes

May’s Theorm

In a 2 canidate election, the only voting method which is anonymous, monotonic, neutral a nearly decisive is simple majority.