A paradox of ranked choice voting

A paradox of ranked choice voting

In a multi-choice election, ranked choice voting offers participants the ability to give a detailed account of their preferences.  However, there are different ways to evaluate ranked voting results, which will provide different outcomes.  These different ways of counting ranked votes are equally fair, assuming that each participant’s ballot is counted by the same method.  But in an election in which certain options or candidates are polarizing (tending to be at the top of a large group of voters’ rankings and at the bottom of the rankings of another large group) and other options are compromise choices, tending to be somewhere in the middle of most voters’ rankings, the way in which these rankings are weighted in determining the outcome will effect the result of the election.

Consider the following case, in which four voters are choosing among four options (or candidates) A, B, C and D.  Each voter has made a different choice from among the 24 possible ways to rank these four choices.  Here are their votes:

	1	2	3	4
1st	A	A	B	C
2nd	B	C	D	D
3rd	D	D	C	B
4th	C	B	A	A

In this case, A is the polarizing option, D is the compromise option, and B and C have garnered responses that are evenly distributed among the rankings.   If we assign 3 points to each 1^st place ranking, 2 points to each 2^nd place ranking, 1 point to each 3^rd place ranking and 0 points to 4^th place rankings, we find that all four options are tied, with 6 points each.

If, however, we give extra weight (4 or more points) to each first ranking, the polarizing candidate will win the election.  Conversely, if we weigh 4^th place rankings more heavily, subtracting 1 or more points for each 4^th place vote while keeping the other rankings at 3, 2, and 1 points respectively, the compromise option will prevail.

More generally, in choosing among 4 options, only the relative weighting of the 1^st and 4^th rankings will affect the outcome.  (Mathematically inclined readers can take a moment to confirm for themselves that this is correct., and then consider what happens when the range of options increases.) In our example above, there is no way to weight the rankings such that candidates B and C will beat both A and D. 

While we tend to think of voting as an expression of preferences, of selecting a favorite among available options, in practice our choices are often more crucially about avoiding the worst options.  How we decide to weight the results of ranked voting may be as important as the voting itself for the process of establishing and maintaining consensus.