Date: Nov 11, 2000 9:04 PM
Author: Guy Brandenburg
Subject: Borda count

Because of the tremendous interest by my students in the recent
presidential election, I decided on Thursday to scrap the
previously-planned lesson and instead to do a lesson on voting theory.
We compared :

* plurality voting (our current system, where the person with the
greatest number of votes, even if only around 27%, wins -- and that
happens to be the percentage tallied by the winner in my school board

* runoff voting, where one takes the top two vote-getters and holds a
runoff election between them;

* repeated runoff voting, where one takes the lowest vote-getter and
eliminates him or her, and does a runoff between the remainder,
repeating if necessary until a majority is reached;

* Borda counting, where the voters give a numerical preference to the
candidates, and an algorithm of some sort is used to decide how many
points to give to each voter's first choice, second choice, third
choice, and so on.

(I mentioned that some countries have instant-runoff voting. Students
complained about the Electoral College and all of the irregularities in
this year's election, but that was not the main focus of the lesson.)

I made up an example with four candidates, and the following preferences
among the following groups of voters. The candidates were Julia, Kate,
Larry, and Max. From memory, I think I had the preferences as follows:

# of voters first choice second choice third choice fourth choice
6 Julia Kate Larry Max
8 Max Kate Larry Julia
5 Larry Kate Julia Max
3 Kate Julia Larry Max

(obviously there are 24 ways of arranging preferences for 4 candidates,
but I was trying to keep it simple.)

This means that for 6 voters, Julia is the first choice, but they think
that Max is the worst; and so on...

Max wins the plurality vote, even though 14 of the voters think that Max
is the worst.

Julia wins a runoff between the 2 top vote getters by 14 to 8, a
majority, because she gets the votes from those who wanted Larry or Kate
as their first choice.

In a repeated runoff count, Kate is eliminated from the first round, and
those votes go to Julia. In the second round, Julia has 6+3=9 votes, Max
has 8, and Larry has 5, so he is eliminated, and his votes again go to
Julia, who wins again in the third round, 14 to 8 against Max.

In a borda count, we award 3 points to 1st choice, 2 to 2nd choice, 3 to
3rd choice, and 0 to 4th choice.
So Julia gets 6*3 + 8*0 + 5*1 + 3*2 = 29 points.
And Max gets 24 + 0 + 0+0=24.
And Larry gets 6+8+15+3=32.
And Kate gets 12+16+15+9=52, and wins by a lot.

Notice that Kate was either the first or second choice of all of the
voters. I wonder how McCain would have done if we used a Borda count?

My students generally concluded that the plurality method we use is the
worst of the four methods I mentioned, and that the Borda count is the
best at actually expressing mathematically the will of the people.

No, I did not go into Condorcet voting. I am not of the opinion that it
is a good way of doing voting. There are too many cases where voters
will prefer A to B, and B to C, and C to A. Thus we become irrational
because we are intransitive.

Lani Guinier was blackballed because she proposed a slight variation on
the Borda count: a voter could cast all of his or her points for 1
candidate, i.e. calling one candidate your first AND second AND third
AND fourth AND fifth AND sixth choice if one wants. To me, that minor
variation is not all that important.

Certainly a Borda count could be handled by voting machines. With a
well-designed national or state-wide ballot, we could have instructions
that made more sense than what they had to face in Palm Beach County,
FL. And clearly there are voting machines that will simply not permit a
voter to attempt to cast a vote that does not follow the rules, hence
forcing the voter to re-do his or her ballot until it makes sense.

But it'll never happen. Too bad.

Guy Brandenburg

My souces for this lesson were, more or less, the COMAP book For All
Practical purposes, which is an interesting source of ideas; columns I
had read a long time ago in Scientific American by Martin Gardner and I
suspect others on Kenneth Arrow's apparent proof that the only type of
election procedure that actually follows a few apparently simple axioms
is a dictatorship; and a much more recent article on voting theory by a
person whose name I cannot recall right now because I forgot to bookmark
it and instead simply printed out and took to school. He contradicts
Arrow. I will attempt to find this citation on Monday.