Chapter 1- The Mathematics of Voting: Paradoxes of Democracy

Single choice ballot

"pick the candidate you like best and keep the rest of your opinions to yourself"

partial ranking

a want for a broader outcome than just the winner. Like considering who got 2nd and 3rd place.

full ranking

a want to rank all of the candidates (1st, 2nd, 3rd,...last)

The Monotonicity criterion

If candidate X is the winner, then X would still be the winner had a voter ranked X higher in his preference ballot.

The independent-of-irrelevent-alternatives criterion

If candidate X is the winner, then X would still be the winner had one or more of the irrelevant candidates (losing candidates) had not been in the race.

Arrow's fairness criterea

The criterion for the ideal voting method

The Condorcet criterion

a Condorcet candidate (a candidate that beats the other candidates in pairwise comparisons) should always be the winner

condorcet candidate

a candidate preferred by a majority of the voters when the candidates are compared in head to head comparisons.

The Majority criterion

a majority candidate (the candidate with a majority first place votes) should always be the winner

plurality with elimination method

after every round, the candidate with the least amount of first place votes gets eliminated. Those votes are then transferred to the available candidate one place below the eliminated candidate.

plurality method of voting

all that matters is how many first place votes a candidate gets. It is the simplest voting method.

borda count method of voting

each place on a ballot is assigned points. In an election with N candidates, last place is given 1 point, second-to-last is given 2 points. The number of points given to 1st place is worth N amount of candidates.

method of pairwise comparisons

for each comparison between candidates, the winner gets 1 point, the loser gets 0 points, and if there is a tie, each candidate gets 1/2 a point. The one with the most points gets 1st place Number of candidates= N => Number of pairwise comparisons= N(N-2)/2

Pairwise comparisons

given any two candidates (X and Y) we can count how many voters rank X over Y and how many rank Y over X.

preference schedule

summarizes all of the elements that constitute the input to an election: the candidates, voters, and ballots.

Preference ballot

the voter is asked to rank all of the candidates in order of preference

Truncated preference ballot

the voter is asked to rank some, but not all, of the candidates