[EM] Fw: borda count

[Steve Eppley]

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