# [EM] Fw: borda count

Steve Eppley seppley at alumni.caltech.edu
Sun Nov 7 15:04:02 PST 2004

```Hi,

Paul K wrote:
> James Gilmour wrote:
>> Steve E wrote:
>>-snip-
>>> But I accept Paul's point.  There might be some decision,
>>> somewhere, where Borda would be a good voting method.
>>
>> No matter how you manipulate the points allocated to
>> successive preferences, it will, I think, always be
>> possible for the Borda winner to be a candidate other
>> than the one candidate who secured an absolute majority
>> of the first preferences.

That depends on what's meant by the "Borda" method.  In the
narrow sense, the way I use the term, Borda is a linear
scoring rule, and given that meaning James is correct
(assuming there are more than 2 candidates).  But there
are other scoring rules.  In general, a scoring rule
for n candidates is defined by a sequence of n numbers
S1, S2, S3, ..., Sn such that the following condition
holds:
S1 >= S2 >= S3 >= ... >= Sn
For each ballot in which candidate x is ranked in the ith
position (where 1st position is the top of the ballot and
nth is the bottom), x scores Si points.  The candidate
that scores the most points wins.

If James used the term Borda to refer to any scoring rule,
which is a possible interpretation of his premise about
"no matter how" the points are allocated to successive
preferences, then his supposition is wrong.  Plurality
rule is a scoring rule where S1 = 1 and S2=S3=...=Sn=0,
and it always elects a candidate ranked first by an
absolute majority, if such a candidate exists.  Any scoring
rule such that S1 >= 2 * (S2 + S3 + ... + Sn) has that
property. (So do the scoring rules where n = 2 and
S1 > S2.)

(Since the number of candidates is often not known before
the voting method is selected, a "scoring rule voting
method" is actually a set of scoring rules, one rule
for each feasible integer n>1.)

>> How then can Borda be "a good voting method"?

I don't want to be misinterpreted as advocating Borda.
I'm just commenting on the "first choice of a majority"
criterion that James mentioned.

I believe that when I mentioned my majority rule heuristic
a month or so ago, I wrote that "the larger the number of
voters who prefer x over y, the more likely it is that
x is better for society than y."  It's just a heuristic;
it's possible the majority is wrong from time to time.
Particularly in a close election, which is why it doesn't
make sense to me to spend a \$billion trying to make sure
every last vote is counted.

I neglected when I mentioned that heuristic to append the
caviat "all else being equal."  There are several ways in
which things can be unequal:  Some choices might be harder
to undo than others, so perhaps those should require a high
level of support before being chosen.  Some voters may be
less competent or less socially responsible than others,
and thus their votes might reasonably be discounted (as
we typically do to children and felons).  Some voters may
have more intense preferences, and in such cases it would
not be unreasonable to defer to an intense minority over a
nearly indifferent majority, if there were a way to measure
sincere preference intensities. (Similarly, some choices
might considerably increase the well-being of a minority
at small cost to a majority.)  Majority preferences can
cycle, in which case there is additional evidence that
should reduce confidence in the (smaller) majority.

Of the several ways things can be unequal, the one that
seems most relevant to the question of whether Borda can
ever be a good method is the one about intensities (or
utilities).  Borda purports to glean information about
voters' intensities from their orders of preference.
For example, it assumes a voter's preference for her
top choice over her second choice is much weaker than
her preference for her top choice over her bottom choice,
and it assumes every voter has similar intensities.
I don't see why either of these assumptions is reasonable,
unless we also know something about the voters and the
set of candidates and the strategic incentives induced
by the voting method.  We know Borda's strategic incentives
are horrible, but it's possible that some things about the
voters & alternatives could be known that would suggest
Borda would be a good method in some cases.  If the voters
were the "honest men" that the Duc de Borda acknowledged
his method required, and if the alternatives were a
well-distributed sample from the space of potential
alternatives, that might be such a case. (It might also
be a case where some "cardinal ratings" voting method
would be even better than Borda.)

One final response to James' question:  The case that
concerned him, where some choice is ranked 1st by an
absolute majority but is not elected by Borda, might
be unlikely in some scenarios, for instance if the
voters are a diverse group, and if so, other criteria
might be relatively more important then.

> This is why I was careful to distinguish between a voting
> method and an election method (or system). I would never
> use Borda to elect a government, BUT... when there was
> no clear majority and my purpose is to quantify a
> "consensus", a Borda count is a quick and efficient
> way to do that.

If the method first tests to see whether there is
a "clear" majority, and elects the majority choice
in that case, then it's not Borda.

> For example, if I ask 65 sportswriters to rank 117 sports
> teams in a league, I might use some form of Borda as an
> alternative to just averaging the ordinal rankings they
> provide (which has worse problems, especially since most
> won't bother to do it right after the 25 teams they know
> something about). It is useful in cases like this because
> the objective is not just to pick which team is number 1,
> where just 33 of 65 "votes" would suffice, the objective
> is to order all 117 teams.
-snip-

Some of the 65 sportswriters may have their own objectives,
such as having a particular team--their alma mater?--finish
on top.  Indeed, a few years ago there was a scandal in a
ranking of college football teams.  I don't recall if the
voters were sportswriters, coaches, or a mix.  The voting
method was some form of Borda.  The team that finished
second was ranked near the bottom by two of the voters,
and it would have finished first if those two had voted
sincerely.  The scandal set off a scramble to find a
better way to generate a ranking of the teams.

--Steve

```