Borda Count discussions
From: Paul McClintock [mailto:paulmc@verizon.net]
Sent: Wednesday, January 30, 2008 4:32 AM
Subject: Borda count
John,
(a) If the Borda count is specified in bylaws, do the bylaws need to specify the details, such as how many points the first place choice receives (n, or n-1?), and whether n is the number of nominees, or the number of names receiving a vote (which could include write-in names, or some nominees may not receive any votes)? Do you have bylaw wording for the Borda count?
(b) Assuming candidates A, B and C, and
11 voters prefer A>B>C
10 voters prefer B>C>A
Then the n-1 Borda count for A is 22, for B is 31, and for C is 10, so B wins, whereas A is the majority winner.
The n Borda count for A is 43, for B is 52, and for C is 31, so B wins, whereas A is the majority winner.
The Nauru Borda count (1 pt for first choice, 1/2 pt for 2nd, 1/n pt for nth) for A is 14.333, for B is 15.5, and for C is 8.333, so B wins, whereas A is the majority winner.
I'd think that this is so contrary a result to what folks would expect (B winning instead of A), that some Borda count implementations would utilize Borda count only when a majority winner is not obtained from first-choice votes. What do you recommend, and what have you observed in any implementations you know of?
(c) Does the n Borda count ever produce a different winner from the n-1 Borda count? What is the original method, and the history of the other? Which does Saari analyze? (I'm guessing the original is "n" and Saari version is "n-1").
(d) My references for the Borda count are:
http://en.wikipedia.org/wiki/Borda_count
http://www.ctl.ua.edu/math103/Voting/borda.htm
http://www.math.harvard.edu/~bruff/voting/borda.html
Do you have others you recommend?
(See my coverage in http://paulmcclintock.com/edu/preferential-voting.htm.)
Thanks,
Paul McClintock
From: Stackpole John D [mailto:jstackpo@alum.mit.edu]
Sent: Wednesday, January 30, 2008 5:17 AM
To: Paul McClintock
Subject: Re: Borda count
Good questions...
First let me send you your homework - two documents attached.
One (the big one) just came out in the AIP's Parliamentary Journal
(January 2008) and the smaller one last April.
Watch for your e-mail - when I take care of a couple of pressing
chores I'll respond to those questions.
Enjoy!
From: Stackpole John D [mailto:jstackpo@alum.mit.edu]
Sent: Wednesday, January 30, 2008 3:59 PM
To: Paul McClintock
Cc: John D Stackpole
Subject: Re: Borda count
Various replies and comments, probably unnecessary now that you have
done your homework and seen all your questions already answered....
n vs. n-1? Makes no difference what point count (or weights) you use
as long as they are equally spaced - linear. As long as the
difference between neighboring weights is equal any set of numbers
will do. For a 3 way choice (3,2,1) is just as good as (2,1,0) or
(1,0,-1) or any other triple. The Borda scores will change of course,
but the ranking in the outcome will not. Fractions work fine, too.
In a 5-choice you can use (1, 3/4, 1/2, 1/4, 0) just as well.
I think the "n-1" choice predominates because it is easier to work
with in theory - the bottom ranked choice gets a Zero so there is one
less quantity to keep in equations and calculations.
Majority winner not selected by Borda? Yup. You anticipated the last
bit of that long paper where I suggested just your notion of falling
back on Borda if no majority winner shows up. But it is a compromise.
Lets look at your example, however - is A "really" people's choice?
(I am lifting this discussion from a new book by Saari, not yet
published.) I think you will agree that B is clearly preferred over
C. C is bottom ranked by (just about) half the voters and middle-
ranked by the others. On the other hand, B is TOP ranked by (almost)
half and middle ranked by the rest. Clearly B is collectively more
favored than C. A is "better" than C for the same reasons.
What about A vs. B? Well, B is top ranked by (almost) half and middle
ranked by the rest. A is also top ranked by (almost) half but BOTTOM
ranked by the others. Sounds like B has a broader base of approval
overall when you take into account the full range of the voters
preferences. And that is what the Borda count does - looks at the
full range of the peoples preferences, not just the first choice. The
latter is what the "majority" vote-for-one election system does.
I used "almost" in that little discussion because there were more of
your A>B>C voters than B>C>A ones. But not many more. As the A>B>C
option gains more adherents (Obama wins more primaries?) at the
expense of B>C>A the Borda count will shift, of course, and when there
are enough shifts, the Borda and majority will agree. The crossover
point is about when A gets 2/3 of all the votes cast.
Never heard of the Nauru system - it amounts to a different set of
(non-linear) weights. As soon as you depart from Borda's "equally
spaced" requirement your options are infinite. Saari discusses some
of the consequences and shows many bad things can happen, far "worse"
that the majority and Borda not agreeing on the "winner". Pick your
weights carefully and ANYBODY can win.
John
John D Stackpole, CPP, PRP
OEO, Parliamentary Services
11 Battersea Lane
Fort Washington, MD 20744-7203
jstackpo@alum.mit.edu
Voice: 301.292.9479
Fax: 301.292.9527
From: Stackpole John D [mailto:jstackpo@alum.mit.edu]
Sent: Wednesday, January 30, 2008 4:34 PM
To: Paul McClintock
Cc: John D Stackpole
Subject: Re: Borda count
And here is a little game to play with your set of 21 voters...
Ask them to vote AGAINST the person they LEAST like and tabulate the
count of hate-votes each person gets.
Then, of course, declare the person with the fewest hate-votes as the
winner.
You might be surprised.
I think this is sometimes called the Texas Ballot.
John