Same Ballet Many Outcomes:

Tallying the Same Ballots under Different Procedures

Does it matter which election procedure is used?  If the outcome is the same regardless of which procedure is used, then no.  However, as we have seen, election outcomes often change when the method used to tally the votes changes.  In such a case, which election outcome represents the will of the people?  This phenomenon has been demonstrated in a few of the applications of different election procedures.  The following example demonstrates that the number of alternative election outcomes that may be possible from one set of voters’ preferences. 

Example:  Three Candidates and Four Election Outcomes
Suppose that 23 voters cast the following ballots by rank ordering candidates A, B, and C:

4
5
8
6
A
A
B
C
B
C
C
B
C
B
A
A

Even in the class of voting vectors, there are multiple election outcomes depending on which voting vectors are used.

  • If plurality rule is used to tally votes, a voter assigns one point to its top-ranked candidate and zero to the others, then A is the top candidate with nine votes, B comes in second place with 8 votes, and C loses with only six votes.

    This example demonstrates a worst-case scenario for voting vectors in which one set of voters’ ballots can yield four different outcomes.  Considering other election procedures may result in even more outcomes.  As each election outcome is backed by a procedure, an election outcome may be the result of which procedure is selected, as opposed to the outcome that the electorate desires.  Because there are so many election procedures, is there a way to eliminate some of the procedures – and, hence, their results.  Is there a best procedure?
  • If the Borda count is used to tally votes, a voter assigns two points to its top-ranked candidate, one point to its second-ranked candidate, and zero points to its bottom-ranked, then B finishes first with 20 points, A comes in second with 18 points, and C is in third again with 17 points.
  • If voters use a voting vector, the top-ranked candidate receives seven points, the second-ranked candidate receives four points, and the last-place candidate receives zero points, then B wins the election with 96 (= 8x7 + 10x4) points, C comes in second place with 94 (= 6x7 + 13x4) points, and A rounds out the field with 63 (= 9x7) points.
  • If anti-plurality is used to tally votes, a voter assigns the bottom-ranked candidate zero points and one point to the other candidates, then C receives 19 points, B receives 18 points, and A comes in last place with nine points.

Four different voting vectors were used and four different outcomes occurred!  There are three additional outcomes that have candidates tied.  Insights about how to construct such an example appear on the Mathematics page.

n
3
4
5
6
7
8
9
10
(n - 1) x (n - 1)!
4
18
96
600
4320
35280
322560
3265920

With as few as 10 candidates, once the ballots are cast, it is possible to have over three million different election outcomes that rank order the 10 candidates!