Doesn’t preferential voting favour middle of the road candidates?

It first bears asking what does it mean to “favour” a candidate? The implication is that a preferential ballot confers an unfair benefit to such candidates, somehow putting a thumb on the scale to fudge the outcome on their behalf.

I take the notion of “favouring” or “advantaging” in this sense to mean:

  1. the outcome for a given candidate differs from the “correct” result; and
  2. the outcome is successful for the given candidate.

But it’s hard to address this question in any meaningful way without settling on what, in fact, constitutes a “correct” result, without also, in fact, begging the question.

If we take the FPTP result as the benchmark for “correctness,” for example, then anything that produces a different result than FPTP is a divergence from the “correct” result, and thus, by definition, favours any non-FPTP winner.

The underlying position here, however, is that the “correct” result is the result that best represents the will of the majority, this being the candidate who is most-preferred by that majority, or in other words, the Condorcet winner if such exists. By this token, any time the Condorcet winner exists and, in fact, wins, there is no advantage, and any time the Condorcet winner exists but fails to win, he or she is disadvantaged.

Where you start is where you end-up.

1. Proposition

But let’s look at the proposition, anyway: the typical middle-of-the-road scenario is that given a field of, say, a centrist candidate, a rightist candidate, and a leftist candidate – for those voters for whom the centrist is their first choice, their second choices would be expected to be more or less split between the rightist and the leftist, depending upon which side of center the given voter identifies; and when their first choice is either the leftist or the rightist, then, as the argument goes, their second choice is more likely the centrist in both cases.

The proposition, then, in this context, is that a preferential ballot gives an advantage to the centrist candidate.

Let us consider this from the standpoint of FPTP, being the status quo, IRV, being a widely used preferential method, and any method that determines a Condorcet winner, in recognition of the Condorcet approach advocated here:

  1. Let there be L leftist, C centrist, and R rightist first-preference voters.
  2. The total number of ballots, the whole pie, then, is: L + C + R.
  3. Now, let us assume that the second-preference:

    1. For either a leftist or rightist first-preference voter, will always be centrist;
    2. For centrist first-preference voters, will split exactly evenly between the leftist and the rightist.

2. Tally Sheet

For our Condorcet analysis, with these assumptions, we will get a tally sheet as follows:

Candidate
Pairs
B more-
preferred
than A
No-
Preference
A more-
preferred
than B
Total
A: leftist
vs
B: centrist
R + C
0
L
L + C + R
A: leftist
vs B: rightist
R + C/2
0
L + C/2
L + C + R
A: centrist
vs B: rightist
R
0
L + C
L + C + R

3. Voting Outcomes

With respect to our centrist candidate, there is symmetry regarding the leftist and rightist. Without loss of generality let us take R >= L, and from the above tally we get:

Majority
Candidate
Minority
Candidate
Majority
Vote
Minority
Vote
centrist
leftist
R + C
L
rightist
leftist
R + C/2
L + C/2
rightist
centrist
R
L + C

R is a majority, which means it’s more than half the pie, so the other two together make up the rest of the pie, and must be less than R, thus: R > L + C

The rightist wins all pairings in which he or she occurs, and thus is the Condorcet winner.

Majority
Candidate
Minority
Candidate
Majority
Vote
Minority
Vote
centrist
leftist
R + C
L
rightist
leftist
R + C/2
L + C/2
centrist
rightist
L + C
R

Where none of L, C, or R is a majority, any two must total more than the other, i.e.: if each is less than half the pie, the other two together must account for the majority of it.

In particular, whether or not C is a majority: L + C > R, and R + C > L;

The centrist wins all pairings in which he or she occurs, and thus is the Condorcet winner.

If L = R, we get rightistleftist, which would create a last-place tie, but has no effect on the centrist as the Condorcet winner.

(We can get the L > R case by exchanging R vs. L and leftist vs. rightist in the above; it makes no difference to our consideration of centrist outcomes.)

4. Analysis

Now, considering an election involving the above candidates:

  1. In any case where we get a first-preference majority (Case 1, and Case 2 centrist-majority variant), the first-preference majority candidate is the Condorcet winner in all cases, and this candidate always wins whether we use FPTP, IRV, or any Condorcet method; no advantage to anyone.
  2. Where we don’t have a first-preference majority (Case 2, non-majority variant), we have either a first-preference plurality, or two or more candidates are tied for first-preference first place. Regardless, as shown, it follows directly that the centrist is the Condorcet winner. Who wins the election?

    1. By FPTP, with a first-preference first-place tie there is no winner; disadvantage to the centrist. With a first-preference plurality, the plurality candidate wins: if this is the centrist, no advantage to anyone; otherwise disadvantage to the centrist.
    2. By IRV, if there happens also to be a first-preference last-place tie, there is ambiguity about who to eliminate, but, in any event: if we eliminate the centrist either another candidate wins, or there is a tie, both of which disadvantage the centrist; and if we don’t eliminate the centrist, the centrist wins, with no advantage to anyone.
    3. By any Condorcet method, the centrist wins; no advantage to anyone.

5. Conclusion

All said, then, with the given first- and second-preference assumptions, which most strongly support the original proposition, the proposition fails: while there is demonstrated disadvantage to the centrist in some non-Condorcet votes, there is no advantage whatsoever to the centrist due to either preferential approach, whether IRV, or any Condorcet method.

This conclusion rests of course on our stated criteria of “favouring,” which, as noted, do beg the question, as they themselves rest on our fundamental thesis that the Condorcet winner is the “correct” outcome. Where you start is where you end-up. But it’s hard to see that a standard based upon, say, FPTP, would be more appropriate, or more reasonable.

It is true, in any event, that in these scenarios the centrist wins by a preferential vote in some cases where he or she would lose with FPTP. But we should expect differences – differences are the very point of the exercise; if there were, in fact, no differences in outcomes between the two systems there’d be little point in changing. Mere difference does not imply either advantage or disadvantage.

The danger of second-guessing voter preferences in this simplistic way is further revealed in a bit more BC history:

“For the 1952 provincial election, the Liberal-Conservative provincial coalition government switched the electoral system from first past the post to the Alternative Vote. The coalition was nervous about the growing popularity of the socialist Co-operative Commonwealth Federation (forerunner of the NDP). With the expectation that Conservative voters would list the Liberals as their second choice and vice versa, the two parties believed they’d garner enough votes between them to stay in power.” — BC Social Credit Party

Here, as stated, was an expectation that the new preferential-voting system would confer advantage to middle-of-the-road candidates, thwarting a perceived threat from an upstart, non-middle-of-the-road, rival. Instead, it opened the door on the “other” side of the field to a completely different, evidently under-appreciated, one:

“… much to the Socreds’ own surprise, the party garnered enough second preference votes to become the largest party in the legislature with 19 seats, one more than the CCF, while the Liberals and Conservatives were practically wiped out….” — BC Social Credit Party

Presuming upon a simplistic model of voter preferences is perilous indeed. The assumptions themselves are fundamentally flawed: the political landscape is not trivially one-dimensional, ranging in a straight line from the left through the center to the right. It is multi-dimensional, and, depending on the actual parties and their leaders and platforms, as well as the sundry issues and positions that don’t fit a uniform one-dimensional paradigm, voters’ second and subsequent choices are far more complex and unpredictable than a simplistic left-center-right scenario would suppose.

But let’s now look at the question in a different way: let us go back to what “middle of the road” means in this context.

Voting is about making collective decisions about things that it makes sense to decide collectively. For such questions, when a democratic decision of a group is required, the democratic ideal is that “the will of the majority” should prevail. Once we have determined what that will is, when a choice does indeed have such broad appeal it won’t seem “extreme,” or “radical” it will, pretty much by definition, be considered “middle of the road.”

Arguably, the will of the majority is the “middle of the road.” In this context, if it is indeed true that preferential voting favours a “middle of the road” result – this is not an indictment of it, but rather the point of it.

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