Voting & Social Choice - Mathematics of Decision Making

Majority Rule

candidate receiving a majority (over 50%) of the votes cast is elected

Condorcet’s Method

beat every other candidate in a one on one competition using majority rule. The winner of each one-on-one race will have over 50% of votes.

Condorcet’s Voting Paradox

paradox in which with three or more candidates, there are elections in which Condorcet yields no winners

Plurality Method

candidate with the most first place votes on preference list ballots is the winner. Not necessarily a majority, and doesn’t satisfy the Condorcet Winner Criterion

Borda Count Method

assign point value to each position and tally up total points to determine winner. Doesn’t satisfy IIA.

Independence of Irrelevant Alternatives

if a new election is held in which every voter maintains their previous relative ranking of the winner and non-winner, but the outcome changes, this is not satisfied

Sequential Pairwise Voting

given a particular agenda, it pits the first candidate against the second candidate in a one-on-one contest. The winner goes on to confront the third candidate on the agenda, and the loser is eliminated.

Hare System

winner is determined by repeatedly deleting candidates that are at least preferred (at the top of the fewest preference list) fewest first place votes. Doesn’t satisfy monotonicity.

Plurality Runoff

there is a runoff between two candidates receiving the most first-place votes. Runoff is between those tied for the most first place votes or the lone candidate with the most first place votes alone with those tied for the second most first place votes.

Approval Voting

each voter may vote for as many candidates as they wish. Candidate with highest number of approval votes wins.

Preference List Ballots

ranking candidates

Manipulable Methods

Hare system, plurality runoff, sequential pairwise, plurality (group),

n!

number of possible preference list ballots

n!

number of possible preference list ballots

Condorcet Winner Criterion

for every possible sequence of preference list ballots either there is no Condorcet winner or the voting system produces the same winner as Condorcet’s method. Plurality Voting does NOT satisfy this criterion.

Voting Manipulability

rearranging choices to achieve better outcomes

Insincere Ballot

voting weakness where voters can achieve their preferred result by submitting a ballot that misrepresents their actual preferences

Weighted Voting

each participant has voting weights where a quota is specified, does not treat all voters equally, required to meet or exceed the quota, includes veto power, dummy voter, dictator

Dictator Voter

a participant in a voting system who has all the power

Veto Power

a participant in a voting system whose vote is required to pass any measure

Dummy Voter

a participant in a voting system who has no power.

[5: 5, 2, 1]

Dictator power

[5: 4, 2, 1]

Dummy voter

[6: 4, 2, 1]

Veto power

Formula for SSPI

number of times a voter is pivotal / total number of permutations

Formula for Banzhaf Power Index

number of times voters are critical / sum of all critical voters

• pivotal voter

• voting permutations (n!)

• no binary

• fraction of voting permutations where voter is pivotal

Shapley-Shubick Power Index

Banzhaf Power Index

• voting combo’s (2^n)

• use binary to list voting combinations

• calculate extra votes, yes weights, and critical voters

• critical voters should be greater than extra votes to count