Approval Voting and Broda Count are not Neutral

[New Democracy]

Note: Lorrie Faith Cranor posted her dissertation titled: Declared-Strategy
Voting at http://dworkin.wustl.edu/~lorracks/dsv/diss/book.html
      On December 31 1996 I sent her an eMail dealing with my comments on
the neutrality of Apporval Voting and Broda Count.

Dear Doctor Lorrie Cranor,

     I am going through your paper on the Web. In the "Vote Aggregation
Methods" section you have checked Approval Voting and Broda as being
neutral. I take the position that they are not neutral. You define
neutrality as: "A voting system is neutral if the system does not favor any
alternative."

     Apporval Voting and Broda favor the lower candidates and disfavor the
higher candidates - from the first tally. Any election method that adds
selections will try to even up the results.

     On the first set of selections the candidates may have differences
from one to fifty percent but as other sets of selections or parts of other
sets are added, all candidates will approach an average value - average
being one hundred percent divided by the number of candidates in the race.

     An example can show this mathematical bias. This example will have
five candidates and twenty voters. Five candidates mean that the average
per candidate will be twenty percent. On the first selection I have a range
that goes from five to fifty percent. As each set of selections is added
all candidates will drift towards twenty percent.

                                         B R O D A     C O U N T
Example:     Same example expanded:      X5    X4   X3   X2   X1
2 ABCDE      2A   2B   2C   2D   2E      10A   8B   6C   4D   2E
2 BCEDA      2B   2C   2E   2D   2A      10B   8C   6E   4D   2A
1 EABCD      1E   1A   1B   1C   1D       5E   4A   3B   2C   1D
2 ABDEC      2A   2B   2D   2E   2C      10A   8B   6D   4E   2C
2 BDCAE      2B   2D   2C   2A   2E      10B   8D   6C   4A   2E
2 AECDB      2A   2E   2C   2D   2B      10A   8E   6C   4D   2B
2 AEDBC      2A   2E   2D   2B   2C      10A   8E   6D   4B   2C
2 ADBCE      2A   2D   2B   2C   2E      10A   8D   6B   4C   2E
3 CEDAB      3C   3E   3D   3A   3B      15C  12E   9D   6A   3B
2 DAEBC      2D   2A   2E   2B   2C      10D   8A   6E   4B   2C


Plurality will give us these results: 10A     4B     3C     2D     1E
Yielding these percentages: >          50%    20%    15%    10%     5%

Approval Voting - adding two sets     13A     8B     5C     6D     8E
Yields these percentages: >          32.5%  20.0%  12.5%  15.0%  20.0%

Approval Voting - adding three sets   13A    11B    11C    13D    12E
Yields these percentages: >          21.7%  18.3%  18.3%  21.7%  20.0%

Approval Voting - adding four sets    18A    15B    14C    19D    14E
Yields these percentages: >          22.5%  18.75% 17.5%  23.75% 17.5%

Apporval Voting - adding five sets    20A    20B    20C    20D    20E
Yields these percentages: >           20%    20%    20%    20%    20%

     The final limit is reached when all possible selections are added
together. When that limit is reached all candidates will have received
exactly the same votes and the same average value percent - twenty percent
in this example of five candidates.

     Now - I know these methods do not go all the way to the limit - but
any distance along the path towards this limit will produce partial
influence towards reducing the difference between the candidates.

     Another point to take note of is that when more selections are added
into the math of a method the harder it is to get a winner with a majority.


     Next I will use the same example and do the Broda Count - which is
similar to adding three sets of selections.

The Broda Count gives us the following results:

                            74 A    58 B    53 C    60 D    55 E
Yielding these percentages: 24.7%   19.3%   17.7%   20.0%   18.3%

     All candidates have drifted closer to twenty percent - the average
value. Also note that no candidate has a majority.

     This is only one example but you or anyone can make as many examples
as they care to make - the average over many examples will show results
that will be similar to the example above. When we add selections we tend
to even out the differences between the candidates.

     Otherwise your paper is the best source about different election
methods that I have been able to find on the internet. I was referred to
your web site because I was asking for information on the Coombs' Method.
Your site is the only site I have found that even mentions Coombs.

Yours sincerely,

Donald Eric Davison