The
Case Against Borda
by James Green-Armytage
Thesis:
I argue that the Borda count is an arbitrary, chaotic, and unreliable vote-processing rule.
General and single-winner arguments against Borda
Totally irrelevant candidates altering the result
Independence of clones criterion
Differential voting power and favorite betrayal
Multiple-winner arguments against Borda
GENERAL AND SINGLE-WINNER ARGUMENTS AGAINST BORDA
•
Totally
irrelevant candidates altering the result: One weird thing about Borda is
that the resulting winner can change if another candidate is added to the
election, even if absolutely nobody ranks the additional candidate at all on
their ballot.
For
example, let's say that there is an election with 100 voters for one open seat,
and the ballots are cast as follows:
12 voters: 1.A
40 voters: 1.A, 2.B
30 voters: 1.B, 2.A
18 voters: 1.B
Let's say
that A and B are the only two candidates in the race. If so, then first place
would be worth 2 points, and second place (or no place at all) would be worth 1
point. The point totals would then be calculated as follows:
A: (12x2)+(40x2)+(30x1)+(18x1) = 152
B: (12x1)+(40x1)+(30x2)+(18x2) = 148
So A wins,
which seems reasonable enough. However, let's imagine that there is a third
candidate in the race, C, who is not ranked on anyone's ballot at all. In this
case, since there are three candidates, a first place vote is now worth 3
points, a second place vote is now worth 2 points, and a last place vote is
worth one point. The ballots cast are exactly the same, but the point totals
have changed:
A: (12x3)+(40x3)+(30x2)+(18x1) = 234
B: (12x1)+(40x2)+(30x3)+(18x3) = 236
C: (12x1)+(40x1)+(30x1)+(18x1) = 100
All of a
sudden B wins? This is just an introduction to how weird Borda is.
•
Independence
of clones criterion: There is a criterion for voting methods known as
independence of clones, which Borda fails spectacularly. This criterion
essentially says that it should be neither an advantage or a disadvantage to a
given viewpoint if it is represented by a group of similar candidates rather
than one single candidate.
It turns
out that given the Borda count, it is a huge advantage for a viewpoint to be
represented by a large number of similar candidates. The more similar
candidates are run, the larger the advantage becomes, and the more other
viewpoints can be crowded out.
For
example, let’s say that there is 1 position to be filled, 100 voters, and 2
candidates, candidates A and B. 60 people vote for A first and B second. 40
people vote for B first and A second. Of course A should win, right? Right, and
A does win.
But what if
B gets his three friends C, D, and E to run? Let’s say that the 60 people who
voted A first still vote A first, and the 40 people who vote A last still vote
A last. B’s friends are more similar to B than they are to A, but most people
prefer B over his friends. For example, you might have a result that looks like
this:
30 voters: 1.A, 2.B, 3.C, 4.D, 5.E
15 voters: 1.A, 2.C, 3.B, 4.D, 5.E
15 voters: 1.A, 2.C, 3.D, 4.B, 5.E
35 voters: 1.B, 2.C, 3.D, 4.E, 5.A
5 voters: 1.C, 2.B, 3.D, 4.E, 5.A
A’s total
is (30x5)+(15x5)+(15x5)+(35x1)+(5x1) = 340.
B’s total
is (30x4)+(15x3)+(15x2)+(35x5)+(5x4) = 390.
C’s total
is (30x3)+(15x4)+(15x4)+(35x4)+(5x5) = 375.
D and E’s
totals are less than these.
B is now in
first place, followed by C in second place, and A (who is preferred over all
other candidates by a 60-40 majority) is in third place. This is an absurd
result. (The gap between B and A will be greater if the preference of B over C,
D, and E is more unanimous. If the preferences between these four friends are
very evenly mixed, then A might win.) This is a fatal problem in
multiple-winner cases, as well as in single-winner cases like this one.
(Candidate
A still wins this example using Condorcet’s method, which will be described
along with the single transferable vote method in the next section, after I
have finished making my case against Borda.)
•
Mutual
majority criterion: In the process of the above example, Borda has also
failed the “mutual majority criterion,” for single winner voting methods, which
states that if a majority of the voters always rank a given set of candidates
above every other candidates, then the winning candidate should come from that
set. In the above example, a majority ranks A above all other candidates. Since
A does not win using Borda, Borda fails the mutual majority criterion.
Is Borda
somehow a superior alternative to majority rule? Definitely not. Rather, it is
a method which lists between majority rule and total arbitrariness. If any
majority is determined and well-organized enough, rest assured that they can
determine the outcome by themselves, just as with any other reasonable
single-winner methods that don’t require supermajorities or unanimity to settle
on a decision (and which must therefore risk deadlock). For example, if all 60
voters who prefer A in the above example voted for A as their first choice and
left all the other candidates off the ballot, then A would win sure enough. The
difference between Borda and majority rule isn’t that Borda somehow leaves
greater room for minority input. The difference is that Borda will sometimes
ignore a majority preference out of sheer chaos and disorganization.
•
Incentives
for truncation: Borda is riddled with unusually strong strategic
incentives, that is, incentives for voters to list the candidates in an order
different from their sincere preferences, or to neglect to express a preference
that one might have between some of the candidates.
The
specific strategic incentives involved in Borda depend on what version of Borda
is being used. One factor is whether truncated ballots are allowed. If they are allowed, there are very strong
incentives towards truncation. Consider the following example, with 100 voters,
3 candidates (A, B, and C), and 1 position to be filled. First I will give the
sincere preferences of the voters:
30 voters: 1.A, 2.B, 3.C
25 voters: 1.B, 2.A, 3.C
25 voters: 1.C, 2.A, 3.B
20 voters: 1.C, 2.B, 3.A
Notice that
a majority of voters (55) prefer either A or B over C, while a minority prefer
C. Also, a majority (55 voters) prefers
A over B. If the voters vote according to their sincere preferences, then the
point totals are as follows:
A: (30x3)+(25x2)+(25x2)+(20x1) = 210
B: (30x2)+(25x3)+(25x1)+(20x2) = 200
C: (30x1)+(25x1)+(25x3)+(20x3) = 190
A has the
highest total and wins. (A also wins with Condorcet.) So far so good. But what
if the 25 voters who prefer B most of all decide that in voting A in second
place, they are adding a point to B’s closest rival, and so decide to truncate
their ballots after listing B?
30 voters: 1.A, 2.B, 3.C
25 voters: 1.B
25 voters: 1.C, 2.A, 3.B
20 voters: 1.C, 2.B, 3.A
A: (30x3)+(25x1)+(25x2)+(20x1) = 185
B: (30x2)+(25x3)+(25x1)+(20x2) = 200
C: (30x1)+(25x1)+(25x3)+(20x3) = 190
If the B
voters truncate as above, B wins, stealing a well-deserved victory from A. What
if, on the other hand, the A voters believe that B is A’s closest rival, and
decide that they shouldn’t give B an extra point?
30 voters: 1.A
25 voters: 1.B, 2.A, 3.C
25 voters: 1.C, 2.A, 3.B
20 voters: 1.C, 2.B, 3.A
A: (30x3)+(25x2)+(25x2)+(20x1) = 210
B: (30x1)+(25x3)+(25x1)+(20x2) = 170
C: (30x1)+(25x1)+(25x3)+(20x3) = 190
If the A
voters truncate, but the B voters don’t, then A wins, as above. (A still wins
with Condorcet in this example.) But what if both the A voters and the B voters
truncate?
30 voters: 1.A
25 voters: 1.B
25 voters: 1.C, 2.A, 3.B
20 voters: 1.C, 2.B, 3.A
A: (30x3)+(25x1)+(25x2)+(20x1) = 185
B: (30x1)+(25x3)+(25x1)+(20x2) = 170
C: (30x1)+(25x1)+(25x3)+(20x3) = 190
In this
case, C wins. (C would win with Condorcet as well, but the strategic incentive
which pulled the result to this point would not have existed in the first
place.) Thus the strong strategic incentives towards truncation, pulling on
multiple groups of voters at once, can end up with the election of a candidate
who would have lost in a one-on-one election with every other candidate in the
race. Or, truncation strategies can be successful and end up electing a winner
who doesn’t deserve to win in light of the sincere preferences, such as B in
the above example.
•
Differential
voting power and compromising: When there are more than two
candidates, Borda gives voters differential voting power on deciding a contest
between any two of those candidates. This in turn leads to a fairly strong
incentive for a strategy known as the “compromising strategy.”
For
example, let’s say that there is a close race between two front-runners A and
B, and there is a third candidate C with a smaller share of the vote. There is
one position open, and there are 100 voters. Let’s say that the sincere
preferences look something like this:
40 voters: 1.A, 2.B, 3.C
13 voters: 1.C, 2.A, 3.B
47 voters: 1.B, 2.A, 3.C
(A is a
clear Condorcet winner.) If most of the voters realize ahead of time that C is
unlikely to win, then they will realize that the primary contest is between A
and B. Thus, the incentive for both ABC and BAC voters to truncate is extremely
strong. Given a 53-47 split, if 53 vote 1.A, 2.B, 3.C and 47 vote 1.B only,
then B will win over A, 247 points to 206. The gap would of course be larger if
it was the ABC voters who truncated, since they are after all a majority. So,
let’s assume that all of the voters follow this incentive, as it regards the
contest between A and B. The result looks like this:
40 voters: 1.A
13 voters: 1.C, 2.A
47 voters: 1.B
A’s point total: (40x3)+(13x2)+(47x1) = 193
B’s point total: (40x1)+(13x1)+(47x3) = 194
C’s point total: (40x1)+(13x3)+(47x1) = 126
B has
beaten A by one point. But wait, why does B win if a majority of voters (53
versus 47) prefer A to B? (A is still a Condorcet winner.) The reason is that
13 of those 53 voters have less voting power than the other 87 voters with
respect to the contest between A and B. The reason they have less voting power
is that they have ranked their sincere favorite, C, before A. This does not
necessarily mean that they value A less than the other 40 voters do, or than
the 47 voters value B; it could simply mean that they have a special respect
for candidate C. Candidate C is analogous to a third party candidate in the
American political system: a sincere favorite who is unlikely to win.
However,
the pragmatic thing for the CAB voters to do in this example (as in many cases
in American politics under the current plurality system) is to abandon their
sincere favorite so that they have their full voting power to weigh in favor of
A over B. If they do this, the result is as follows:
40 voters: 1.A
13 voters: 1.A, 2.C
47 voters: 1.B
A’s point total: (40x3)+(13x3)+(47x1) = 206
B’s point total: (40x1)+(13x1)+(47x3) = 194
C’s point total: (40x1)+(13x2)+(47x1) = 113
A, the
rightful winner, has been elected, but at the cost of voters abandoning their
sincere favorite. This kind of thing is not only a problem for symbolic
purposes; it is also a problem because of the risk of people abandoning a
candidate that is viewed as being a long shot, but who actually could have won
given sincere votes. Of course, the fact that they could have won is never
discovered. The dilemma between supporting a sincere favorite and a compromise
candidate cannot always be resolved, and often leaves the voter in a state of
uncertainty.
MULTIPLE-WINNER ARGUMENTS AGAINST BORDA
•
Disproportionality:
Aside from the problems with the Borda count already mentioned, the principle
problem regarding multiple-winner elections in particular is that
multiple-winner Borda is not a method of proportional representation.
What is
proportional representation? While in a single-winner election it is possible
to completely ignore the will of a given minority, the goal of proportional
representation is to provide representation for all segments of the electorate
such that the representation of a group in the set of elected options is in
proportion to the relative size of the group within the electorate.
For
example, if there is a set of voters who constitute 30% of the electorate,
there is no guarantee that they will have any input in determining the outcome
of a single winner election. However, in a proportional election filling 100
seats in a council, they should in theory be able to determine how 30 of those
seats are filled.
Many
countries around the world use some form proportional representation as their
primary system of proportional representation, including most of the nations
that are widely considered to be advanced democracies. Notable exceptions are
the U.S., U.K., and Canada. The U.K. and Canada have made many inroads toward
the use of proportional representation, and the U.S. is exceptional in its
ignorance of the issue.
Proportional
representation is the most effective way to go beyond majority rule in
representative government, and to provide voices for a wider range of
viewpoints.
Let me show
a couple examples which illustrate the fact that multiple-winner Borda is NOT a
system of proportional representation. In the first example, there are 3 seats
open, 5 candidates, and 100 voters. I will imagine that there is a distinct
group of 63 voters who represent a majority viewpoint, and a group of 37 voters
who represent a minority viewpoint. Candidates A, B, and C appeal to the
majority viewpoint, while candidate D appeals to the minority viewpoint. The
minority voters maximize their chances of winning by truncating their ballots
after D. Here are the ballots:
24 voters: 1.A, 2.B, 3.C, 4.D
21 voters: 1.B, 2.C, 3.A, 4.D
18 voters: 1.C, 2.A, 3.B, 4.D
37 voters: 1.D
A’s point total: (24x4)+(21x2)+(18x3)+(37x1) = 229
B’s point total: (24x3)+(21x4)+(18x2)+(37x1) = 229
C’s point total: (24x2)+(21x3)+(18x4)+(37x1) = 220
D’s point total: (21x1)+(21x1)+(21x1)+(37x4) = 211
If there
are three seats to be decided, and one distinct group which accounts for at
least one third of the electorate (100÷3 = 33.33, rounded up to 34), then any
decent proportional representation method should allocate one seat to that
group. That is, if all members of that group rank a given set of candidates
over all other candidates, then at least one candidate from that set should be
elected. (Single transferable vote would award seats to A, B, and D.)
Here is one
more example, with 7 seats open, 8 candidates, and 100 voters. Candidates A
through G are sympathetic to the majority viewpoint held by 70 voters, and
candidate H represents the views of a 30 voter minority. Again, the minority
voters truncate their ballots after voting for candidate H. Here are the
ballots:
11 voters: 1.A, 2.B, 3.C, 4.D, 5.E, 6.F, 7.G, 8.H
10 voters: 1.B, 2.C, 3.D, 4.E, 5.F, 6.G, 7.A, 8.H
10 voters: 1.C, 2.D, 3.E, 4.F, 5.G, 6.A, 7.B, 8.H
10 voters: 1.D, 2.E, 3.F, 4.G, 5.A, 6.B, 7.C, 8.H
10 voters: 1.E, 2.F, 3.G, 4.A, 5.B, 6.C, 7.D, 8.H
10 voters: 1.F, 2.G, 3.A, 4.B, 5.C, 6.D, 7.E, 8.H
9 voters: 1.G, 2.A, 3.B, 4.C, 5.D, 6.E, 7.F, 8.H
30 voters: 1.H
A’s vote total:
(11x8)+(9x7)+(10x6)+(10x5)+(10x4)+(10x3)+(10x2)+(30x1) = 381
B’s vote total:
(10x8)+(11x7)+(9x6)+(10x5)+(10x4)+(10x3)+(10x2)+(30x1) = 381
C’s vote total:
(10x8)+(10x7)+(11x6)+(9x5)+(10x4)+(10x3)+(10x2)+(30x1) = 381
D’s vote total: (10x8)+(10x7)+(10x6)+(11x5)+(9x4)+(10x3)+(10x2)+(30x1)
= 381
E’s vote total:
(10x8)+(10x7)+(10x6)+(10x5)+(11x4)+(9x3)+(10x2)+(30x1) = 381
F’s vote total:
(10x8)+(10x7)+(10x6)+(10x5)+(10x4)+(11x3)+(9x2)+(30x1) = 381
G’s vote total: (9x8)+(10x7)+(10x6)+(10x5)+(10x4)+(10x3)+(9x2)+(30x1)
= 374
H’s vote total:
(10x1)+(10x1)+(10x1)+(10x1)+(10x1)+(10x1)+(10x1)+(30x8) = 310
The 7 seats
are filled by candidates A, B, C, D, E, F, and G. (Single transferable vote
would pick candidates A, B, C, D, E, F, and H.) Again, despite the fact that
30% of the voters ranked H first, H failed to gain a seat. Given 7 seats and
100 votes, any decent proportional method should award a seat to any candidate
ranked above all other candidates by 15 or more voters. (100÷7 = 14.29, rounded
up to a whole number.)
Admittedly, multi-winner Borda could possibly be classified as a sort of semi-proportional system, as it is probably more proportional than methods such as at-large plurality. However, why settle for a tenuous semi-proportionality when fully proportional systems are available? Also, the proportionality of Borda can be undermined factors such as the clone effect, and other problems described above.
I advise that the Borda count should not be used for real elections, especially contentious elections.