Generalizing Majority Rule

Two-candidate elections

Determining a winner for a two-candidate election is easy.  For an election between two candidates, A and B, there are two (= 2!) ways to rank the candidates:  A is preferred to B or B is preferred to A.  For an odd number of voters, one of the two candidates must be preferred to the other candidate by a majority of the voters.  Thus, there will be a candidate that wins a majority of the first-place votes. 

Ken May

In 1952, Kenneth May showed that majority rule is the only two-candidate election procedure in which

  • each voter is treated equally, that is, only the number of votes matters, not who casts the votes;
  • each candidate is treated equally, that is, only the number of votes that a candidate receives determines if he or she wins the election; and,
  • a candidate can never be harmed by receiving more votes, that is, if a candidate wins the election, then it would still win the election if some of the voters who had voted for the candidate’s opponent now voted for the candidate.

What happens if there are more than two candidates?

Elections with three or more candidates

Because May has shown that there is no difficulty in determining an election outcome in a two-candidate election, election procedures should reduce to majority rule when there are two candidates and an odd number of voters.  Indeed, this is the case for the election procedures described previously.

  • Plurality rule:  Recall that under plurality rule, a voter votes for the candidate he or she prefers most.  Because there are only two candidates and an odd number of voters, the candidate who receives the most votes has received a majority.
  • Voting vectors:  For two candidates, a voter assigns w1 points to its most-preferred candidate and w2 to its least-preferred candidate where w1w2.  Naturally, letting w1 = w2 fails to let voters distinguish between the two candidates; for a two-candidate election with k voters, w1 = w2 results in a tie in which each candidate receives kw1 points. Therefore, let w1 > w2.  Without loss of generality, assume that candidate A defeats candidate B by receiving more points.  If  m voters prefer A to B (each assigning w1 points to A and w2 points to B) and n voters prefer B to A (each assigning w1 points to B and w2 points to A), then A is the winner when mw1 + nw2 > nw1 + mw2 which reduces to

m(w1w2) > n(w1w2), or m > n,

because w1 > w2.   Hence, a majority m of the (m + n) voters prefer candidate A to candidate B; using a voting vector is equivalent to majority rule when there are two candidates and an odd number of voters.

  • Singular Transferable Vote/Instant Runoff:  Recall that an election under STV or instant runoff proceeds in a series of rounds.  But, for a two-candidate election with an odd number of voters, there can only be one round: the candidate with the least number of first-place votes is eliminated.  The candidate that receives the majority of the first-place votes remains and wins the election.
  • Approval Voting:  Recall that under approval voting, a voter indicates which candidates he or she approves.  A candidate receives one point for each voter that approves of the candidate.  A candidate receives no points for each voter that does not approve of the candidate.  For a single candidate election, the candidate(s) with the most points wins the election.  Naturally, approving of all candidates or disapproving of all candidates does not change the difference in the number of points the candidates receive.  If there are an odd number of voters and no voter approves or disapproves of both candidates, then approval voting is equivalent to majority rule: each voter gives one point to the candidate that he or she prefers and the candidate with a majority of the points wins the election.