Plurality voting as preferential voting with incomplete information

In this post I will compare two voting systems, plurality voting (aka simple majority) and preferential voting (Condorcet method), under some general assumptions including equal voter intent. First I will consider an abstract example where voter preference information is incomplete. Then I will present a concrete example with complete preference information.

Consider a simple example of preferential voting, defined by the following

1) Single winner

2) Three candidates, A, B and C

3) The number of first choice votes for A is a, for B is b, for C is c

4) a > b and a > c

We have no further information as to voter preferences beyond the first choice. We model this as equal preference as to the to other candidates. If we describe a ballot as

(first choice, second choice, third choice)

then we have that

a ballots contain (A, B/C)

b ballots contain (B, A/C)

c ballots contain (C, A/B)

where for example B/C represents a lack of preference between B and C.

We will now determine the Condorcet winner for this vote. According to wikipedia, the Condorcet winner is

the candidate whom voters prefer to each other candidate, when compared to them one at a time

Calculating the winner is a matter of iterating over all the pairwise combinations and determining each net pairwise preference. There are six possible combinations, which reduce to three due to symmetry. The net pairwise preference is therefore

A vs B: a – b + 0 + 0 = a – b

A vs C: a – c + 0 + 0 = a – c

B vs C: b – c + 0 + 0 = b – c

Where the zeroes are due to the fact that there are no preferences beyond the first choice. These net results represent the net preference of a candidate over another one, given all votes. A positive net value in A vs B means that A is preferred to B by a majority of voters. A negative net value means the opposite. These values can be visualized in a preference matrix

 A

 B

 C

 A

 a-b

 a-c

 B

 b-a

 –

 b-c

 C

 c-a

 c-b

 –

 

We have that a > b and a > c, and the relationship between b and c is unspecified. This yields

A

B

C

A

 WIN

 WIN

B

 LOSS

 ?

C

 LOSS

 ?

 

So the  Condorcet winner is A, irrespective of the contents marked with ‘?’, and the exact values of a, b and c.

But recall that a > b and a > c. This implies that under plurality voting (where the ballots only record one choice) the same voter intent would also yield A as the winner.

In conclusion, preferential voting reduces to plurality voting when there is insufficient information as to voter’s preferences[1]. Or in other words, plurality voting is an approximation to preferential voting, because preference information necessary to produce the correct[2] result is unavailable.

Let’s see an example where this information is available and how it produces different results for the two voting systems. Say we have voters with the following preferences

10 voters with preference (A, B, C)

8 voters with preference (B, C, A)

6 voters with preference (C, B, A)

Under plurality voting, this would result in A winning the election, as A would have more votes than B or C. Now let’s calculate the Condorcet winner:

A vs B: 10 – 8 – 6 = -4

A vs C: 10 – 8 – 6 = -4

B vs C: 10 + 8 – 6 = 12

The matrix is

 A

 B

 C

 A

 -4

 -4

 B

 4

 –

 12

 C

 4

 -12

 –

 

which yields

 A

 B

 C

 A

 LOSS

 LOSS

 B

 WIN

 –

 WIN

 C

 WIN

 LOSS

 –

So the Condorcet winner is B, while in plurality voting it was A. The interpretation is that the extra information available allowed a better choice to be made, improving upon the approximation of plurality voting.


[1] Under the given assumptions, of course.

[2] Note that when using better or more correct I am assuming that the Condorcet winner, that is the result of all the pairwise comparisons, is fundamentally better than a simple majority. This assumption seems intuitively correct, a candidate that is pairwise preferred to all other candidates should be the winner.

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