Second Choice Voting

Karl Sims


This page describes alternative voting systems for single-winner elections in which voters are given more than just a single choice vote. When implemented properly, second choice voting can have significant advantages over traditional voting:


The Problem with Traditional Voting

Below is a simple example that shows how a less popular third candidate can cause the candidate with the opposite policy to win when traditional single choice plurality voting is used. (Think Ralph Nader or Ross Perot.)

Consider a simplified scenario in which voters are electing a single candidate purely based on their preference on a single ideological dimension or policy. The voter popularity over a range of policy positions is drawn as a bell shaped curve, and candidates are arranged under the curve depending on their specific policy. If there are only two candidates, A and B, the one with the closest policy to what the majority wants (B) will win:

A 45%, B 55%
However, if a third candidate C is included, votes are taken away from the most similar candidate B, which can cause the opposite candidate A to win instead:
A 45%, B 40%, C 15%
This is not ideal, because B is actually still preferred over A by a majority of voters.

Sometimes voters may anticipate this effect, and avoid voting for C even if that would be their first choice. This is also unfortunate. It can strongly discourage third candidates and third parties, and could prevent desirable change in general. Second choice voting could help solve this problem.


Condorcet's Method

This method for second choice voting was originally proposed by the French mathematician Condorcet. For three candidates, ballots simply allow both a first choice and a second choice vote:

1st 2nd
[x] [  ]   The candidate you really want.
[  ] [x]   The candidate you prefer if your 1st choice doesn't win.
[  ] [  ]   The candidate you don't want.
When votes are collected, instead of just counting 1st choice votes, the winners between each pair of candidates are determined. In the example above, the voters with C as the 1st choice would probably have B as their 2nd choice, so:
If B ran alone against A: the winner would be B.
If B ran alone against C: the winner would be B.
If A ran alone against C: the winner would be A.
B is the winner of all of his pairwise matches, so B is declared the overall winner.

The voters' preferences are available for each of these one-on-one matches because of the 2nd choice votes. For a given pairwise match, if a voters 1st choice was not assigned to either candidate of that pair, the 2nd choice is used instead. When B is matched against A, any 1st choice votes for C are discarded and the 2nd choice votes of those voters are used instead, so B effectively reclaims the votes that would have been his if C had not run.


Multiple choices

This scheme can be extended for any number of candidates. Ideally the number of choices should be one less than the number of candidates, allowing voters the option of ranking all the candidates in order of preference. (Voters who do not have the patience to formulate more than their first few choices would just abstain from the additional choices.)

With traditional plurality voting, again the candidate (A) with the largest number of votes wins, even if most voters would actually prefer another candidate:

ABCDE
For multiple choice voting with any number of candidates, a grid can be constructed that shows who the winners would be if any one candidate ran against any other:
ABCDE
For each one-on-one match, each voter's vote is given to whichever candidate is ranked above the other in that voter's list of choices.

There should normally be one candidate that wins all of his pairwise competitions. In this case C is the winner. That is closest to what the majority actually prefers even though C might not have received the maximum number of 1st choice votes.


Of course in the real world, policy preferences are not one dimensional, and many other factors contribute to the fairness or unfairness of election results. But the only obvious disadvantage of second choice voting seems to be the slightly more involved technique for voting and counting the votes. In any case, for most measures of fairness, this method should always give an equal or more correct result than single choice voting.


Counting votes

There are several methods that could correctly calculate the grid of one-on-one election results as described above. Here is a technique that allows the results to be assembled and combined in a single pass rather than requiring multiple iterations through all voters' information.

First the ranked list from each voter is translated into a matrix of pairwise preferences. If a voter ranked 4 given candidates in order of A,B,C,D then her corresponding matrix would look like this:

A B C D
A   - 1 1 1   (A is preferred over all)
B   0 - 1 1   (B is preferred over C and D)
C   0 0 - 1   (C is preferred over D)
D   0 0 0 -   
Then, these matrices are simply summed for all voters to give the final result. For an example population of 10 voters, the matrix of sums might look something like this:
A B C D
A   - 3 3 4
B   7 - 4 6
C   7 6 - 8
D   6 4 2 -
This single matrix contains the results for all the one-on-one matches. For example row C shows that C defeats A by 7/10, defeats B by 6/10, and defeats D by 8/10.

In theory with Condorcet's method there is a chance that the "majority" can be in conflict with itself and create a cycle of preferences, such as: A defeats B, B defeats C, but C defeats A. This should not happen often in practice, but in case it does, another method is needed to select the winner from the subset of candidates in that cycle. This is an annoying side-effect of Condorset's Method that complicates any complete description of the procedure.


Instant Runoff Voting

Instant Runoff Voting or IRV is another voting system that gives voters more than just a single choice vote. It uses an iterative process to eliminate candidates with fewer 1st choice votes until one receives a majority. IRV has most of the same advantages over single choice plurality voting as Condorcet's method, and will probably give the same results as Condorcet's in most cases.

In the first example above where A, B, C receive 45%, 40%, and 15% of 1st choice votes, C would be eliminated after the first round. A second round of calculation would compare A and B alone in a runoff. For those voters that had C as their 1st choice, their 2nd choice votes are now used instead, so again B becomes the final winner.

In the second example above with five candidates, B would be eliminated after the first round with only 10% of 1st choice votes. Then D and E might be eliminated on the next two rounds, leaving A and C in the last runoff, and finally C as the winner.

It could be argued that IRV is not as ideal as Condorcet's Method. As an example, suppose there are three candidates: one centrist and two extremists. Even if the centrist is preferred over either extremist by a majority of voters, the centrist might not receive enough 1st choice votes to survive the first "runoff" round. With IRV voters can still be motivated to vote strategically rather than sincerely in some cases. The results of any runoff system can be sensitive to the order in which candidates are eliminated, but Condorcet's method effectively finds all possible runoff results simultaneously and avoids this effect.

On the other hand, there are some reasons that IRV may be more practical than Condorcet's Method:


More Information

See the Wikipedia page on Condorcets Method.

See also Accurate Democracy by Robert Loring for a discussion of the best voting methods and free software. This page by Bob Lanphier describes Condorcet's Method and provides software.

Here is also an Election Methods Resource site with information on many voting systems and a discussion of Ranked Pairs.

The Center for Voting and Democracy's site has information on Instant Runoff Voting initiatives.


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© 1999, Karl Sims. All rights reserved.