Topics in Mathematics Midterm 1

Independence of Irrelevant Alternatives Fairness Criterion

if a non-winning candidate drops from an election, then this should not change the orginal election results

Did a non-winning candidate drop out of the election? (IIA FC)

No = No violation

Yes = Find the winner of the election after the specified candidate dropped out. Are the results the same as the results you had when no candidate dropped out? If yes, then no violation; if no, then a violation occurs.

Monotonicity Fairness Criterion

if votes on ballots are changed in favor of the winner (and only the winner), then this should not change the original election results

Were the ballots changed in favor of the winning candidate and only the winning candidate? (Monotonicity FC)

If no, then no violation.

Yes = Find the winner of the election after changing the ballots. Are the results the same as those you had before changing the ballots? If yes, then no violation; if no, then a violation occurs.

Majority Fairness Criterion

if a candidate is the Majority Candidate (meaning the candidate has more than half of all the first place votes), then that candidate should win the election

is there a Majority Candidate? (Majority FC)

If no, then no violation.

Yes = Is the Majority Candidate the winner of the election according to the voting method applied? If yes, no violation; if no, a violation occurs.

Condorcet Fairness Criterion

if a candidate is a Condorcet Candidate (meaning the candidate won all head-to-head matches they are actually in), then that candidate should win the election

Is there a Condorcet Candidate? (Condorcet FC)

If no, then no violation.

Yes = Is the Condorcet Candidate the winner of the election according to the voting method applied? If yes, then no violation; if yes, a violation occurs. If yes, a violation occurs.

Plurality Voting Method

only look at first choice votes and whoever has the most wins

Method of Pairwise Comparisons

have to write out all the head-to-head matches and use a point system to choose overall winner

Instant Runoff Voting Method

every round, the candidate with the fewest first choice votes is eliminated; the winner of the last round wins the whole election

Borda Count Method

1 point for last place vote, 2 points for second place vote, etc. and tally all points

Strategic Voting

a change in ballots does effect who wins, and the new winner is better according to the voters who changed their ballotsThis occurs when voters choose a less preferred candidate to prevent an undesirable candidate from winning.

Fact

Plurality, Pairwise, and Instant Runoff never violate Majority FC, but Borda CAN violatemajority fairness criteria.

Fact

only one of the four methods can violate Monotonicity: Instant Runoff

Player

people or groups voting on a motion; denoted P1, P2, etc.

Weight

each player controls a certain amount of votes; denoted W1, W2, etc.

Quota

minimum number of votes needed to pass a motion; cannot be larger than the sum of the weights (if everyone votes in favor, the motion should pass); must be greater than half of the total weight to ensure a majority

Dummy

a player who has no impact on the outcome of the motion

Dictator

player who controls the outcome of a motion by themselves; if there is a dictator, everyone else is a dummy, and dictators always have veto power

Veto Power

player who is needed to pass a motion (sum of all other weights is less than 2)

Winning Coalition

group of players with more than q votes who can pass a motion

Critical Player

someone in a winning coalition that cannot be removed; if they are removed, the coalition is no longer able to pass the motion

Banzhaf Power Distribution Procedure

1. List all winning coalitions using {…} notation

2. In each winning coalition, identify the critical players (if any)

3. For each player, count the number of times they are critical

4. Count the total number of “instances of criticality”

5. Each player’s Banzhaf Power Index = number of times they are critical / number of instances of criticality

Shapley-Shubik Power Distribution

based on “pivotal” players in “sequential coalitions”

Sequential Coalition

ordered list of all players; players join coalition in the specified order, at some point the coalition will be winning

Pivotal Player

player whose act of joining first makes a coalition a winning coalition

Shapley-Shubik Procedure

1. List all sequential coalitions for n players, there are n! of them (3 players = 3! = 3 × 2 × 1 = 6 sequential coalitions)

2. In each sequential coalition, identify the one and only pivotal player

3. Count the number of times each player is pivotal

4. Count the number of instances of pivotality; this should be n! again

5. Shapley-Shubik power index for each player is: number of times player is pivotal / total number of instances of pivotality

Power

describe voting system, then measure power with Banzhaf or Shapley-Shubik

Ex. {ABCD}, {ABC}, {ABD}, {ACD}, {BCD}, {AD}

Who is critical in {ABD}?

A