Some Perspective on the Single Seat Election Dispute
donald at mich.com
Fri Feb 7 02:15:35 PST 1997
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - February 07 1997
Greetings to Election Methods List,
I posted the following study material on the Elections Reform list - I now
post it here.
Something needs to be said about the other methods of electing single seat
positions. There is a dispute between the Other Methods and Preference
Run-off (also known as Instant Run-off). This dispute needs to be put into
In single seat elections there are two schools - the Preference Run-off
School and the Other Methods School. The differences between these two
schools can be put into the form of questions.
Should other preferences be considered in selecting the winner?
Is a preference the same as a vote?
Is it proper to drop the last candidate?
These questions make up the heart of the dispute. If you are going to
pick between the two schools you must resolve these questions in your mind.
I have resolved these questions in my mind and I pick the Preference
Run-off school. Allow me to present my reasons and answers to these
Should other preferences be considered in selecting the winner?
Preference Run-off uses the other preferences in two ways: When a candidate
is dropped the votes of this candidate are not dropped - the votes are
salvaged by transferring them to the next preferences. The second way in
which preferences are used is in the event of a tie between candidates. The
next preferences of the tied candidates are used to solve the tie - the
candidate with the lowest number of next preferences is dropped. Using
other preferences to solve ties is acceptable because the candidates are at
a equal point together on a level playing field - they have equal
mathematical consideration to the same number of possible preference votes.
This is the extent of the use of the other preferences by Preference
Run-off. The Other Methods school uses the other preferences more
extensively. They have many methods for you to consider - I list some
below. All these methods have been devised to include the use of other
preferences to determine a winner.
Condorcet using rated, not ranked, ballots
Smith-Condorcet using rated, not ranked, ballots
The Greater Majority is an Other Method that I devised - I do not advocate
it - I do not advocate any of these methods. I devised the Greater Majority
method as a method for any person who believes that more than one set of
preferences should be used to decide a winner. The math is simple: Each
voter makes a series of preferences. The first two preferences of all the
voters are added together. The low candidate is dropped. At this point some
preferences from the third set will now fill into the first two sets - more
than only the first two sets of preferences is used in this method. Again
we add the first two preferences and again we drop the low candidate. We
continue this routine until we have one candidate left - which is the
Instead of dropping the low candidate of the first set of preferences, this
method drops the low candidate of the combined first two sets of
preferences. This method considers the strength a candidate may have in
both sets together. The Greater Majority is a method that is honest in
considering some of the other preferences. If a person feels that other
preferences should be considered this method says - Fine! How about
considering the first two sets of preferences as being completely equal?
When we do consider other preferences we must face the possibility of one
candidate having a greater majority in two preferences than the majority
another candidate may have in the first set of preferences. In other words
- this method will not always elect the candidate that may have received a
majority on the tally of the first preferences. The Greater Majority
Method is true to itself - it will elect the candidate that has the
greatest support in the first two sets of selections. Its policy is
enforced even if there is a majority candidate in the first set of
preferences. Any person that decides other preferences are to be considered
must accept the fact that the majority candidate of the first set of
preferences may lose - a person cannot have it both ways. You either
consider other preferences or you do not consider other preferences - and
if you consider them you must consider them always.
Because the Greater Majority method uses two sets of preferences its
results are based on twice as many votes - two hundred percent of the
votes. Therefore the majoity requirement for this method is one hundred
percent plus one vote of the total votes cast in the election. So you see,
fifty-one percent of the votes in only the first set is not enough to win -
you have to think about this - and while you are thinking I am going to
give you another point to think about.
There is an outer limit to the use of preferences. The Outer Limit would be
reached if you were to regard all preferences as being equal and used them
all - added together. When the Outer Limit is reached all candidates will
be tied - each candidate will have received one hundred percent of the
votes cast in the election. For example: Suppose a single seat election
with four candidates - every voter makes four selections. The majority
requirement would be two hundred percent plus one vote. The results of this
election would reach the Outer Limit. Each of the four candidates would
receive exactly one hundred percent of the votes cast - these results are
absured. Other Methods do not go all the way to the Outer Limit but they do
go part way towards the Outer Limit - the Other Methods are partially
You may notice that in this Greater Majority method some other preferences
are regarded as votes - this is true in all the Other Methods. The Other
Methods school regard preferences as votes or parts of votes - and that
takes us into the next question.
Is a preference the same as a vote?
The position of the Run-off school is that preferences are not votes. Each
voter only has one vote and that vote belongs to the candidate that is the
first preference of the voter. That vote must stay with that candidate and
cannot be taken away as long as that candidate is a contender. If the time
comes in which that candidate is last and no longer a contender, the
candidate is dropped but the vote is not dropped - the vote is salvaged and
transferred to the next preference of the voter. The Other Methods School's
position is that every preference is a vote and can be added to the votes
of other preferences per whichever other method is being used. The Other
Methods people are claiming that every preference is the same as the first
preference, and that all preferences should carry the same weight. This is
not true. WHEN A VOTER SELECTS ONE CANDIDATE AS HIS FIRST PREFERENCE THAT
MEANS THAT THE VOTER PREFERS THAT CANDIDATE MORE THAN ANY OTHER CANDIDATE.
The other preferences are only to be used to salvage the voter's vote in
the event the first preference is dropped - and in cases of ties between
In politics there is the saying: One person - one vote. That saying applies
here - but the reasons are more than politics - there is also a
mathematical reason. When preferences are added together as votes there are
two mathematical effects that will be produced. One: The majority
requirement will be increased. Two: The percentage differences between the
candidates will be reduced. The results of these two effects will be that
it will be harder and harder to have a candidate with a majority. Now it is
possible to construct an example in which we do get a majority winner when
we add preferences, but on the average the higher percent candidates will
go down and the lower percent candidates will go up and all candidates will
stay under a majority.
Is it proper to drop the last candidate?
The Run-off School's position is that after each election the last
candidate is no longer a contender and shall be dropped. The Other Methods
School's position is that after the general election all candidates are to
be considered as being contenders and some way should be found and used so
as to meet this end. The Other Methods people think it is wrong for Run-off
Method to drop the last candidate - but many of these other methods use
pairing, the practice of conducting a run-off between every possible pair
of candidates and they see no conflict of policy when they drop four
candidates at one time in each of fifteen pairings in a six candidate race.
And on the average these four dropped candidates will have together a
majority of the votes.
Before the election all candidates have equal mathematical consideration to
all the votes - a level playing field - how many each receives will depend
on the voters - they will decide which candidate is going to be last. After
the election the candidates are no longer equal. How much consideration are
the voters of the last place candidate entitled to at this point? The
Run-off Method allows these voters to salvage their votes and place them on
another preference. Some of these voters will be making "lesser of two
evils" decisions, but most people are not opposed to their second
preferences. And consider this - these voters will be in the position of
deciding the winner of the contest - they will be the "King Makers" so to
speak. What Run-off does not give these voters is a chance for their
dropped candidate to win. In a single seat election we are only going to
have one winner - we must have a system that reduces the field down to one.
The Preference Run-off method is already engineered and works fine and part
of it is the policy of dropping the last candidate.
Another point that you should be aware of: The people who advocate some
Other Method for you to consider will be showing you examples like the
following 49AB 10B 41CB - only three vote combinations for three
candidates. It should be pointed out that these examples will not be
realistic examples. With three candidates there are fifteen possible
combinations. In a real election the voters will be using all combinations.
All the voters that voted for a certain candidate are not going to march
lock step and select the same candidate as their second preference. When
examples become real the Condorcet winning candidate will be the same as
the Preference Run-off winning candidate. Condorcet will only produce a
different winner in unrealistic examples of election results - results that
will never happen in a real election in the real world. These people do
this because they are living in a world of their own.
It is one thing to talk about protecting the rights of all candidates but
it is another thing to produce the mathematics that will do it. So far the
Other Methods people do not have anything. What they have is not
acceptable. They need to come up with a method that does not add
preferences together - or if they must add preferences they must put all
candidates on a level playing field. The current Other Methods favor the
lower candidates and disfavor the higher candidates. Mathematically these
current Other Methods try to even up all the candidates after the first set
of preferences. The lower candidates have a better chance to gain more
votes in the other preferences beyond the first set. This mathematical bias
will exists in any method that uses other preferences to decide an
election. This bias does not exist in Preference Run-off. If and when the
Other Methods people do come up with a new method there is a way to check
and see if it is better. I have constucted The Davison Standard that I use
to check these Other Methods - new or old.
The Davison Standard
This standard uses ratios to remove the mathematical bias. In the math of
this standard the sum of the next preferences of each candidate is subject
to a ratio. The ratio is different for each candidate. What the ratios do
is to give all candidates the same number of chances to gain votes from the
next set of preferences - the ratios level the playing field as far as math
is concerned. The ratio is the votes of the lead candidate of the first
preferencs divided by the difference between the total votes less the votes
of a candidate in the first column of preferences. The equation is:
Test of C is equal to C(0ne) plus C(two) times -------------------------
Total Votes less C(one)
C(one) is the votes of candidate C from the first column of preferences.
C(two) is the votes of candidate C from the second column of preferences.
Lead votes(one) is the votes of the leading candidate in the first column.
All candidates are tested using this math. The low candidate is dropped and
his votes are transferred. Then the testing is done again for the remaining
candidates. All values may have changed - we must use any new values. We
continue testing until we have one remaining candidate. This is the
standard to compare the other methods. If some new Other Method produces
the same final candidate then we can say that this new Other Method meets
the Davison Standard. But - Until the Other Method people have something
good on the drawing board, Preference Run-off is the method for us to use
in single seat elections.
- - - - - - An Example of the Davison Standard - - - - - - - - - - - - - -
On February 04 I took one of DEMOREP's two examples and compared it to the
DEMOREP's second example: (Example expanded by Donald)
>A slightly less extreme example-- (one) (two)
> 14 CE 14 C 14 E
> 6 CD 6 C 6 D
> 45 DC 45 D 45 C
> 29 EC 29 E 29 C
> 6 ED 6 E 6 D
>With Approval Voting, C gets 94, D gets 57, E gets 49, C wins.
>With head to head, D beats C, 51 to 49 and D beats E 51 to 49. D wins.
Donald writes: I will use this example to show the math of the Davison Standard.
Each candidate will be tested using the following equation:
Test of candidate A is equal to A(one) plus A(two) times --------------------
total votes less A(one)
Test of candidate C = 20 + 74 X 45/(100-20) = 61.625
Test of candidate D = 45 + 12 X 45/(100-45) = 54.818
Test of candidate E = 35 + 14 X 45/(100-35) = 44.692
Candidate E received the lowest test amount and is dropped.
The new vote tally is as follows:
6 C 6 D
45 D 45 C
----- We now have a candidate with a majority.
100 Candidate D is the winner with 51/100
In this example the Head to Head method meets the Davison Standard.
Head to Head also elects the same candidate as Preference Run-off -
maybe because this second example is less exteme than the first example. As
examples become more real we can expect the Other Methods examples will
elect the Preference Run-off candidates.
Donald Eric Davison of New Democracy at http://www.mich.com/~donald
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