Contemporary Mathematics Topic 23 Voting Objections Voting Objections

Contemporary Mathematics Topic 23: Voting Objections

Voting Objections › Is there a truly fair voting system? › In this section, we will look at criteria that can be used to evaluate voting systems.

Majority Criterion › If a candidate receives a majority of the first-place votes, then that candidate should be the winner.

Example 1: › We will use this example throughout, so refer back to it. › Check if the majority criteria is violated in plurality voting, plurality with elimination, Borda count, and comparison methods. #pairwise of Votes 60 25 15 First Place R D R Second Place D I D Third Place R I R

Majority Criterion › The Borda Count method can violate it. The other methods do not.

Head-to-Head Condorcet Criterion › If a candidate is favored when compared head-tohead with every other candidate, then that candidate should be the winner. › Use the table in Example 1 to test the head-to-head Condorcet criterion for plurality voting, Borda count, plurality with elimination, and pairwise comparison methods.

Head-to-Head Condorcet Criterion › The pairwise comparison method never violates this criterion. The rest have the possibility to do so.

Monotonicity Criterion › If a candidate is the winner of a first nonbinding election and then GAINS additional support without losing any of the original support, then the candidate should be the winner of the second election. › We won’t practice an example here for this one, but know that the plurality with elimination method has the potential for violating the monotonicity criterion.

Irrelevant Alternatives Criterion › In this criterion, if a candidate is the winner of an election, and in a second election one or more of the losing candidates is removed, then the winner of the first election should be the winner of the second election. › In Example 1, remove I from the mix, giving his votes to D. Run the different voting methods. Does anything change? › ALL of the methods have the potential of violating this criterion.

Arrow’s Impossibility Theorem › Of the four criteria we talked about, there is NO voting method in existence (or one that could ever be made) that will simultaneously satisfy all four of the voting criterion. › In other words, Arrow’s Impossibility Theorem says that we can NEVER find a voting system that does everything we want. › There is no 100% truly fair voting system. Period.