voting method: tie-breaking in IRV:
I Like IRVing
donald at mich.com
Wed May 9 08:31:32 PDT 2001
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Hi Derek, you wrote:
"My group is voting on names for conferences rooms.
We have 18 nominations, and about as many voters."
In any election in which we are going to elect more than one
candidate, or room names as in your case, we need to first decide which
type of election it is to be, that is, a majority type election or a
proportional type election.
If you want a majority of your voters to approve of each room name,
that is a majority type election and you should use the Multi-Single-Seat
method, which is a number of single-seat elections conducted in one field
of candidates using IRVing. See text below:
A proportional type election is when each proportional part of the
voters selects one room name. For example: if you have five conference
rooms and fifteen voters, it would only take three votes for a name to be a
winner. This is called STV, which I assume you already know.
I would say that your election of room names should be a majority type
election, because all the rooms are to represent all the people.
Besides, otherwise three voters might just give a silly name to one of
your conference rooms.
Now, the question of ties. Whenever we have an election with only a
few voters, we are going to have a problem with ties. The tie solutions of
Low Series and Lower Choices (see below) are best suited for larger
elections, but they can be used on smaller elections, more so if you will
be using my suggested Multi-Single-Seat method.
You did not say how many rooms there are to be named, but if it is
five, the Multi-Single-Seat method will be counting the first five choices
of each voter as first votes, which yields seventy-five votes plus lower
choices for us to count, all this from fifteen voters.
Hopefully, this larger number of votes and choices will seperate the
best names from the less popular names and inturn give us not only less
ties but ties that can be solved.
Nominations: With five rooms, you should have at least ten nominations. I
would say eighteen is not too many, but you could rule only one nomination
per voter.
If you are only naming two rooms, then eighteen nominations are too
many. In this case, you need some more rules of nomination in order to
reduce the list of names. You need to restrict the fifteen voters to only
one nomination or second per person. This will reduce the nominations down
to seven. You can reduce the nominations down to five by ruling only one
nomination or second or third per person. This is not unfair because if a
name cannot muster the support of one or two other persons then it is going
to fail in the election, better it fail in the nominations. Rules of
nomination can act like a primary. See text on nominations below.
Regards, Donald
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Multi-Single-Seat Elections:
Multi-Single-Seat means that more than one Single-Seat Election is being
conducted at the same time in one field of candidates.<P>
<B>Counting More Than One Set of Choices:</B> When only one judge is
to be elected, IRVing(IRV) is to be used. When more than one is being
elected we will still use IRVing but with a difference. That difference is
that instead of only counting the first choice of each ballot, we count a
number of choices equal to the number of judges to be elected. We add all
these choices together and then drop the lowest candidate. And instead of
working the candidates down to only one we will work them down to the
number of judges to be elected. Each judge in order to win must get a
majority - because each judge will be sitting as a single person and must
serve the entire community like any single office holder.<P>
If five judges are to be elected then each voter has five votes and is
allowed to select up to five candidates as his first choices - plus the
voter is allowed to select additional choices to act as run-off selections
if needed. The first five choices on each ballot are considered to be equal
and therefore receive one vote each. The math will work as follows: The
first tally of the candidates is obtained by adding the first five choices
of all the ballots together. At this point we drop the lowest candidate
beyond the fifth candidate and transfer his votes to the next choices on
the ballots. Again the first five choices are added together to yield us
the second tally of the candidates. We continue dropping the lowest
candidate and adding the first five choices until we only have five
candidates left. If these five all have a majority then all five are
elected. A majority being more than fifty percent of the number of electors
that voted in the race for these judges.<P>
It is possible to end up with less than five with a majority. It is
tempting to declare the top five as winners but I say that only the ones
with a majority each are elected - the others did not receive a majority.
This is important because many times in the election of judges the number
of candidates is equal to only the number of seats. These candidates should
not be guaranteed winning an election because of no opposition. The people
have the right to not vote for a candidate - and that candidate has no
right to expect to be a winner without having received a majority in the
election.<P>
Some judge positions have been elected for hundreds of years and most
likely these positions will be elected for hundreds of years into the
future. If that is to be so then I hold that the math I have presented here
is the best math to elect these judges.<P>
This is not a proportional representation type election. It is one
single person election being done five times in one field of candidates.
The exception to this would be if a panel of judges was being elected to
serve as one body. Then they are to be elected using the rules of
multi-seat elections, as one five seat election.<P>
The above system can also be used for any election of persons that
serve the same function but work separately - marshalls are an example.
(01-16-97)<P>
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Tie solutions:
<B>The Low Series Solution:</B><BR>
Ties will be rare, more so in larger election, but when we do get one the
first step is for us to check and see if the tie is part of a Low Series of
candidates. If so, then we are allowed to drop the entire Low Series of
candidates, that inturn will solve the tie.<BR>
We have a Low Series when we have two or more of the lowest candidates
with a total vote count that is less than the one candidate ahead of the
Low Series. If we do have a Low Series we are allowed to drop the entire
Low Series of candidates because none of those candidates have any
mathematical possibility of getting ahead of any candidate that is in front
of the Low Series. The tie becomes moot when the entire Low Series is
dropped.<BR>
Example please:
<PRE>
150A 140B 140C 120D 40H 30I 20J 20K
</PRE>
We have two ties in this group. Candidates B and C are tied, but we do
not need to do anything about that tie at this time. The transferring of
votes from eliminated candidates will most likely solve that tie.<BR>
Also tied are candidates J and K, the two lowest. We need to do
something about this tie because we need to know which candidate to drop, J
or K, but please notice that we have a Low Series. The 110 sum of the votes
of H, I, J, and K is less than the 120 votes of D. These lowest four make
up a Low Series and we can drop all four at one time. This means that we do
not have to deal with the tie of J and K. So, the first thing to do when
we are faced with a tie between the two lowest is to check and see if we
have a Low Series. If so, we have a quick solution. Keep in mind that a Low
Series could be longer than the number of tied candidates. <P>
<B>The Lower Choices Solution:</B><BR>
We can find a difference in the lower choices between the lowest
candidates who happen to be tied, a difference that we can use to solve the
tie. It would be best to consider only the lower choices of the other
candidates, other than the tied candidates. To use the lower choices of
the tied candidates will violate the Golden Rule of Preferences which
states that lower preferences are not to help nor harm earlier
preferences.<BR>
In using the lower choices of the other candidates, we do not want to
use all the lower choices at the same time. We only consider one part of
them at a time so that we have a back up in case the tie is not broken by
the first part of the choices we use.<BR>
We should have an order defined before the election. The following is
the order that I suggest, but a different order would also be acceptable as
long as it was agreed upon before the election.<BR>
Suggested Order: The first steps will be to use the next choices of
each of the other candidates, one by one, starting with the first candidate
ahead of the tied candidates. As soon as we have a difference in the lower
choices of one of these other candidates, the tie is solved.<BR>
If tie has not been broken by one of these steps, then the next steps
are to use the second next choices of each of the other candidates, one by
one, starting with the first candidate ahead of the tied candidates.<BR>
The tie most likely will be solved somewhere in one of these steps.<BR>
Suppose a four candidate election as follows: 35 A, 25 B, 20 C, 20 D:
We have a tie between C and D. We first look to the next choices of the
votes of candidate B. Whichever of the candidates, C or D, that has the
least number of next choices is the candidate that is eliminated. In the
rare event C and D are still tied, then we look to the next choices of the
votes of candidate A.<BR>
This solution does not violate the Golden Rule of Preferences. A lower
choice of candidate A or B is being used to help keep the same lower choice
in contention as a candidate, which is the intent of the voter when the
choice was made.<P>
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Nominations:
It would be helpful if the number of candidates could be reduced
without harming the rights of anyone. This can be done by setting some
rules of nomination. (1) A candidate must be nominated by another council
or board member. If a member wishes to be considered for the position that
member should start with at least enough support to get nominated. (2) The
nominator can only nominate one candidate that accepts the nomination. The
nominator must make a commitment to only one candidate. (3) The nominator
cannot be a candidate and a candidate cannot be a nominator. The nominator
cannot run against the person he nominated - again he must be committed to
the person he nominated. And each candidate must be committed to
themselves as candidates. These rules would limit the nominations to a
maximum of five for a ten member council or board. These rules still allow
any member to be nominated with only the low support of one other
member.<P>
We can further improve on this by requiring a second nomination. The
person who is the second would also come under the same rules of
nomination. Using a second would limit the nominations to a maximum of
three for a ten member council or board.
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----------- Original Letter -----------
From: "Truesdale, Derek" <Derek.Truesdale at a...>
To: "'donald at mich.com'" <donald at mich.com>
Subject: voting method: tie-breaking in IRV
Date: Fri, 4 May 2001 17:02:38 -0700
Hi Donald,
My group is voting on names for conference rooms.
We have 18 nominations, and about as many voters.
Our secretary was going to have everyone just give their #1 pick, but this
seemed silly to me, with so many different choices.
In the past, I've been in elections where IRV balloting (or as we usually
called it, preferential balloting) was used. I like this a lot better than
other balloting methods. For instance, an alternate suggestion is to rank
the 18 candidates, 1-18. Add up the rankings, and whichever room has the
lowest rank wins. Unfortunately, with this method, it's advantageous to
know which candidates are the genuine contenders. Then you want to put the
contender you like 1st, and the contenders you like less at the bottom. You
want to put the non-contenders at the top, right after #1. and, in the end,
your ballot doesn't represent your true preferences, but rather how savvy
you are of the election system. This seems wrong.
Anyway, I told my group that I'd be doing the election using preferential
balloting. I told them the basic mechanics, and that there was no advantage
to ranking contenders above or below the non-contenders.
Certainly, choosing conference room names is not the most important thing in
the world, but it is pointing out to me some of the difficulties of the IRV.
I was hoping you could share your thoughts.
Now, I'm facing some questions about tie-breaks. With 18 nominations, and
15 ballots, there are a lot of choices that get one vote after the first
round, and I need to drop one of them. Tie breaks continue to occur
throughout the rounds.
Your voting suggestion seems somewhat inadequate.
The Low Series solution is quite reasonable. Of course, to be consistent, I
feel I may as well just go ahead and break the tie, knowing that it will be
irrelevant anyway.
When there is not a low series, there are problems.
Let's say I get the following ballots:
AEDCB
ADBCE
ACDBE
BDCAE
CBDAE
DBAEC
EBCAD
According to your method, I would first eliminate B, right? B, C, D, and E
are tied for last, so I look at the 2nd round votes of A, and find that E,
D, and C are represented. B loses. A goes on to win. Clearly, however, B
was a strong candidate. If I had eliminated C, D, or E first, B would have
gone on to win. This is tough to reconcile.
Furthermore, the order that the A voters have chosen effects whether their
candidate A gets elected. This is a case, where, even if they prefer
candidate B to candidate E, they should rank E higher because E is less of a
threat. This is also a problem.
Thirdly, it suddenly matters how far apart the distance is in the A voters'
rankings. It is not sufficient for the voter to think that putting in some
off the wall candidate high up is irrelevant. Because the tie break is done
based on the next choice down, if you have an F as your second choice
instead of a more reasonably candidate, the more reasonable candidate is
less likely to get elected.
In general, there is a problem in that we don't know whose votes will be
redistributed later. You suggest using the next-to-last candidate's votes
to determine who should win, probably under the assumption that these are
the votes most likely to be redistributed. but there's nothing that
guarantees that, and it could be that the person who ranked right above the
next to last will be the ones getting redistributed and should have the most
weight in a tie breaker.
---
I don't know how to solve the problems. In general, I dislike the idea of
going item by item down a list, because this means that throwing a
non-contender high in your list could devalue your vote. Instead, I think
there needs to be a comparison of which candidate was ranked above the
others, not just which was ranked 2nd. or 3rd.
For instance, in my above example, I might argue that in the tied votes,
B outranked C 3 times
B outranked D 3 times
B outranked E 3 times
C outranked B 1 time
C outranked D 2 times
C outranked E 2 times
D outranked B 1 time
D outranked C 2 times
D outranked E 3 times
E outranked B 1 time
E outranked C 2 times
E outranked D 1 time
I'd probably then score E at the bottom of the list. regardless of where
any As or Fs or Gs come into play.
In a two way tie, this is not useful, since the result would wind up with a
tie.
but then, when using the other ballots to break the tie, I would still check
out the number of times A outranked B and vice-versa, rather than just going
down the list.
However, this still does not solve the problems of whose ballots to use to
break the tie, nor does it solve the problem that voters should put their
competitors at the bottom of their list, so that they will not win a tie
break.
-----------------
Lastly, I might observe that IRV is not perfect.
If 100 voters vote 100 different people as their first choice, and everyone
votes a 101st candidate as the 2nd choice, it might seem that the person
listed as the 2nd choice on all the ballots should win. Yet, this person is
the first one eliminated.
I don't think there's much that can be done about that.
Any thoughts on how I should resolve my tie-break problems?
Derek.
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Regards, Donald Davison - Host of New Democracy, www.mich.com/~donald
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| Q U O T A T I O N |
| "Democracy is a beautiful thing, |
| except that part about letting just any old yokel vote." |
| - Age 10 |
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