The Mathematics of Voting

Democracy is the worst form of government except [for] all those other forms that have been tried from time to time.

-- Winston Churchill

Some Criteria for a Fair Election

  1. Majority Criterion

    If there is a candidate or alternative that is the first choice of a majority of the voters, then that candidate or alternative should be the winner of the election.

  2. Condorcet Criterion

    If there is a candidate or alternative that wins in a one-to-one comparison between it and any other alternative, then that candidate or alternative should be the winner of the election.

  3. Monotonicity Criterion

    If X is the winner of an election, and in a reelection, all the voters who change their preferences do so in a way that is favorable only to X, then X should be the winner of the election.

  4. Independence of Irrelevant Alternatives Criterion

    If X is the winner of an election, and one or more of the other candidates or alternatives are removed and the ballots recounted, then X should be the winner of the election.

Given the following preference schedule, find the winner under the different methods.

18 12 10 9 4 2
A B C D E E
D E B C B C
E D E E D D
C C D B C B
B A A A A A

  1. Plurality winner

    A is the plurality winner because A has 18 first place votes as opposed to 12 for B, 10 for C, 9 for D and 6 for E.

  2. Pairwise Comparison

    A - B;--> B wins
    A - C;--> C wins
    A - D;--> D wins
    A - E;--> E wins
    B - C;--> C wins
    B - D;--> D wins
    B - E;--> E wins
    C - D;--> D wins
    C - E;--> E wins
    D - E;--> E wins

    This means:
    A has 0 wins, B has 1 win, C has 2 wins, D has 3 wins, and E has 4 wins. Therefore E (with the most number of wins) is the winner. In this particular case E is also a Condorcet winner. Because he has won every head-to-head (one-to-one) contest. In general this is not the case with the pairwise comparison

  3. Borda Count Winner

    A 5(18) + 4(0) + 3(0) + 2(0) + 1(37) = 127
    B 5(12) + 4(14) + 3(0) + 2(11) + 1(18) = 156
    C 5(10) + 4(11) + 3(0) + 2(34) + 1(0) = 162
    D 5(9) + 4(18) + 3(18) + 2(10) + 1(0) = 191
    E 5(6) + 4(12) + 3(37) + 2(0) + 1(0) = 189

    This means D is the Borda Count winner.

  4. Runoff (Two Candidates) winner

    E, D, and C (the three lowest first vote getters) get eliminated.

    18 12 10 9 4 2
    A B C D E E
    D E B C B C
    E D E E D D
    C C D B C B
    B A A A A A

    This leaves us with

    18 37
    A B
    B A

    Thus B is the Runoff winner.

  5. Plurality-With-Elimination

If this is the case, which one of the above is a fair method of election?