Mathematics and Voting:

How to Vote

What could be easier than voting?  Cast a vote for your favorite candidate.  The candidate with the most votes wins. This is optimal for an election between two candidates.  But, for three or more candidates, most election procedures require voters to rank all the candidates from most-preferred to least-preferred, as opposed to voting for just a top-ranked candidate.  For elections with more than two candidates, voting for a single candidate may elect a candidate with less than a majority (i.e., greater than half) of all votes cast.  Knowing more information about voters’ preferences, e.g., taking into consideration voters’ second-place candidates, etc., can help determine a winner that better represents the will of the people.  As an example, suppose candidate A wins a three-candidate election with 34 first-place votes and 66 third-place votes.  The electorate may be enraged if candidate B receives 33 first-place votes and 67 second-place votes.

Suppose that a voter must decide between candidates A, B, and C.  Assuming that there is enough information for the voter to distinguish between the candidates so that there are no ties, then there are six ways to rank the three candidates:

1st     A A C C B B
2nd    B C A B C A
3rd    C B B A A C

A voter can rank n candidates in n! = n(n-1)(n-2) … (2)(1) ways. This follows because there are n choices for the top-ranked candidate.  Once this candidate has been chosen, then there are n-1 choices for the 2nd-ranked candidate.  Continuing with this logic, there are k choices for the candidate ranked in position n-k+1.  Finally, there is one way to rank the last-place candidate, as all of the other candidates have been ranked above it.  Because selecting the positions is independent, the number of rankings is the product:  n(n-1)(n-2) … (2)(1).