[EM] FWD - Re: Multiple Winner Elections
Advance Copy
donald at mich.com
Mon Feb 26 06:48:37 PST 2001
------------ Forwarded Letter ------------
>From: "Moe St. EverGreen" <evergreen at lovemail.com>
>To: "Advance Copy" <donald at mich.com>
>Subject: Re: Reply to: [EM] Multiple Winner Elections
>Date: Sat, 24 Feb 2001 16:12:29 -0500
>Although I understand that we would not want to use
>approval voting for multiple candidate situations,
>I do not see how you can say IRV is better
>than approval voting in a single winner situation.
Don: I have the reverse problem. I do not see how anyone can say Approval
Voting is better than Irving in a single winner situation.
Do you have a most preferred choice when you vote in a single seat
election? If so, you are like me, but Approval Voting will not allow us to
have a top choice, unless of course we only make one choice, but then if
our top choice loses we will have no second choice to become our top
choice. Besides, is it still Approval Voting if people only make one
choice?
If we make two selections, our second selection will help some other
candidate defeat our top choice. Most people, once they are aware of this
situation will refuse to make any more selections than their top choice.
I like Irving - Approval is too much trouble.
>In STV, it seems that the affiliation doesn't matter?
>in that case, I would not want unaffiliated person to have
>greater significance.
>I know that in some proportional systems proposed, affiliation
>matters, and that affiliation can mean unaffiliated person
>have a significantly less chance of winning which makes
>the system unfair, which is why I mentioned that.
Don: Affiliation does matter to the people and the results may be uneven
but it is not unfair. The people are not required to give every candidate
the same number of votes. The people are allowed to be uneven. That is what
an election is all about, get use to it.
When the people vote as a body, we expect them to give more votes to
some candidates than to others, we expect them to be `unfair'. An election
is not a `feel good' exercise in which every candidate receives the same
gold star on the forehead. The people are doing what they are supposed to
do. Don't try to offset their voting.
Besides, better reasons can be presented on why people should vote for
affiliated candidates than for unaffiliated candidates. It is not wrong if
the people act on these reasons, they are merely sticking to the business
of voting. The people are more informed about their own affiliations than
they are about the candidates. It is best that they vote for the candidate
that is affiliated with their affiliation.
>Could you send me a pointer to a clear description
>of an STV system exactly equivalent that that which you describe?
Don: You may wish to see what the Center for Voting and Democracy has to
say about STV on their web site at: www.fairvote.org
>Right now our committee is tending towards using Approval Voting
>and Cumulative Voting, as they are both simple to define and implement,
>and both better than mere plurality.
Don: If your committee does go with Approval Voting for single seat
elections, you can do us on the Election Methods list a big favor by being
so kind as to supply us with returns. We would not need all the ballots,
just the vote combinations. For example, a four candidate race could have
15 possible combinations as follows: A, B, C, D, AB, AC, AD, BC, BD, CD,
ABC, ABD, ACD, BCD, and ABCD. We on the EM list have discussed Approval
Voting without any reality. Results from a real election in the real world
will greatly help resolve this dispute.
For this favor from you I will favor you with a tip on how to vote in
an Approval election. If your candidate is a possible winner, instruct your
faction to only vote for your candidate, not to make any other selections.
Votes from your faction will only help your candidate, but there will be
votes in the other factions that will help your candidate. This difference
will give your candidate an edge, maybe a large enough edge to win.
>We have a few multiple winner positions of two to three people, a couple
>of single winner positions, and one large (30 to 600 person ) committee.
>- Moe.
Don: Thank you for writing and I'll be looking for the results of your
first Approval Voting election.
Regards, Donald Davison - Host of New Democracy, www.mich.com/~donald
+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+
| 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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Subject: [EM] 'The two Pij definitions are different'-reformatted
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Hello everyone,
My 25 Feb posting on Pij definitions, prepared in so-called 'rich text', was
judged by the latest EM-digest to be in an irreproducible format. In hopes
of getting a posting which WILL reproduce in an EM-digest, the present
posting restates the earlier one, but in (what I believe to be) 'plain text'
format.
Joe Weinstein
Long Beach CA USA
THE TWO Pij DEFINITIONS ARE DIFFERENT
PREAMBLE AND ABSTRACT.
Fellow EM-listers: Most of us on this list, like me, have other lives - both
at work and elsewhere - and often lack the time to prepare intelligent
postings - even when we’ve got new ideas or insights to share, or would like
to rebut ill-considered criticisms of our prior postings. Even when others’
postings get really silly or confusing, time constraints keep most of us
helpless spectators. That’s been my situation for the past six weeks.
Most frustrating has been Mike and Richard’s recent furious exchange as to
whether two different definitions of Pij - a probability value defined for
an impending election - really must amount to the same thing. Would-be
conclusions have been obfuscated by various irrelevant asides, including
dubious pronouncements and basic queries about math and logic conventions
and about probabilities.
Unfortunately, I lack time to answer the queries (but thanks to Anthony
Simmons for a good show on the logic part) - despite the fact that I did do
my math Ph.D. dissertation on logic, and now for a living (as an applied
statistician) am always computing probabilities. But here in this posting I
will do my best to set the record straight on the original Pij question.
Namely, I give an example and use it to illustrate conclusively and derive
numerically that the two Pij definitions can refer to distinctly DIFFERENT
probability values - which thereby yield different ‘strategic voting’.
I first review, restate and give names to the two definitions at issue.
Next, I give an illustrative (if in some respects fanciful) example of an
impending election. Then I derive and calculate the numerical probability
values. Finally, I use the example’s situation and numbers to illustrate why
(despite both authors’ confusing asides)
Richard’s main argument is correct and Mike’s main counter is erroneous.
DEFINITIONS AT ISSUE.
The two definitions are reviewed in a 17 Feb. posting by Richard,which
begins by noting that earlier Mike wrote:
> >> > Pij is the probability that i & j will be the 2 frontrunners.
> >
> >
> >Slight correction/clarification: The precise meaning of Pij is usually
> >taken to mean the probability, given a tie exists for first place, that
> >i and j will be involved in that tie.
>
>Different wording, same thing. If i & j are the 2 highest vote-getters,
>and there's a tie for 1st place, then it's between them. If they
>aren't the 2 highest vote-getters, and there's a tie for 1st place, then
>it isn't between them.
Richard then comments:
"They aren't the same. It's easy to think they'd be proportional, but keep
in mind Bayes' Theorem for conditional probabilities..."
My side comment:
Indeed, as we shall see, the two definitions of Pij aren’t the same at all.
However - as we shall also see - in order to understand and explain the
difference, it is unnecessarily arcane to appeal to such devices as
conditional probability or Bayes' Theorem. Richard’s extended comment may
have been prompted by the following two facts, true but unneeded here: (1)
one way the two definitions of Pij differ is that the first-defined
probability is ‘unconditional’ but the second is ‘conditional’; and (2) in
some situations (not really here) Bayes' Theorem aids computations with
conditional probabilities.
Let’s restate and briefly name the two probability definitions (or
essentially similar ones) at issue. Let i and j be any two candidates in an
impending election.
(1) ‘lead’: Pij = probability that i and j will be the two leading
candidates (i.e. each will get more votes than any of the candidates other
than i and j).
(2) ‘tie’: Pij = probability, given that precisely two candidates tie for
first place, that i and j will be those candidates.
In some postings, the ‘lead’ definition was ascribed to Mike; and the ‘tie’
definition was ascribed to Bart Ingles (who in turn ascribed it to a
published reference).
Note that, for outcomes of a given election, we are discussing PRIOR
probability values, NOT POST-facto values. After the fact, an election has
but one actual outcome and its data, so post-facto outcome probabilities
must be 0 or 1; but an impending election has a set of possible outcomes,
each with its data - so prior probabilities need not all be 0 or 1.
EXAMPLE.
This example has two versions, one for lone-mark (‘plurality’) voting and
one for unrestricted pass-fail (‘approval’) voting.
In the state of Catatonia, in order to avoid a steep fine, each potential
voter appears at the local polling place by 9 AM on Election Day, and
remains there until 9:15 AM. Prior to Election Day, each voter has received
one ballot for each race, and may mark it before entering the polling place.
No marking of ballots is allowed within the polling place. Upon exit, each
voter may - but need not - cast any or all of the voter’s ballots received
for the various races.
There are two political parties: the Hots and the Colds. Cold voters always
cast their ballots. In order to decide whether to cast ballots, all Hots
listen to the result - broadcast in all polling places at 9:05 AM - of the
latest State Lottery random drawing of six digits (000000 through 999999).
If the drawn number is 900000 or more, all Hots cast all their ballots; if
not, they cast no ballots.
Hot and Cold voters of both genders are very partisan. On each ballot
received, each such voter will mark only suitable candidates, i.e.
candidates of the same party and gender; conversely, so long as at least one
suitable candidate exists, each such voter will mark at least one. Further,
for every married couple (and Catatonia allows marriage only between
opposite genders), if either spouse is Hot, then both are Hot; and if either
is Cold, then both are Cold.
The electorate comprises a grand total of T voters, split as follows:
60% of all voters are married Hots.
34% of all voters are married Colds.
4% of all voters are single female Colds.
2% of all voters are single Independents.
The example’s specific election contest is for state governor, between four
candidates: #1 is a female Hot; #2 is a male Hot; #3 is a female Cold; and
#4 is a male Cold.
PROBABILITY VALUES.
It is easy to see that the partisan (i.e., Hot or Cold) 98% of the
electorate will produce just two outcomes. These outcomes differ only
according as the Hots do or do not cast ballots.
Apart from the Independent voters’ ballots, candidates #3 and #4 will always
get respectively 21% x T and 17% x T votes.
Suppose - as will occur with probability 10% - the Hots do cast ballots.
Then, candidates #1 and #2 are the leaders - each with at least 30% x T
votes. (Meanwhile, # 3 and #4 each have at most 23% x T votes). Depending on
how the Independents vote, #1 and #2 may tie for first place.
Suppose - as will occur with probability 90% - the Hots do not cast ballots.
Then candidates #3 and #4 are the leaders and cannot tie: #3 has at least
21% x T votes and #4 has at most 19% x T votes. (Meanwhile #1 and #2 each
have at most 2% x T votes).
Using the ‘lead’ definition: P12 = 10% and P34 = 90%. That is, probability
is 10% that the two Hot candidates #1 and #2 will be the leaders, and is 90%
that the Cold candidates #3 and #4 will be the leaders.
However, using the ‘tie’ definition, P12=100% and P34=0%. That is,
considering just all cases with a first-place tie: in all of these cases the
tie is between Hot candidates #1 and #2.
INTERPRETATIONS AND CONSEQUENCES.
The above example and numbers illustrate that each of the defined
probabilities may be interpreted simply as a fraction (valued between 0 and
1 inclusive). Thus,with the ‘lead’ definition, Pij is just the fraction of
all possible outcomes which have candidates #i and #j as the leaders. With
the ‘tie’ definition, Pij is the fraction -
of those outcomes which have a first-place tie - where the first-place tie
is between candidates #i and #j.
The two fractions have different denominators. For the ‘lead’ definition,
the denominator counts all possible outcomes, whereas the ‘tie’ definition’s
denominator counts only certain of the outcomes - namely those which meet
the extra condition of having a first-place tie. For this reason, usual
terminology calls the ‘lead’
probability Pij ‘unconditional’ and the ‘tie’ probability Pij ‘conditional’.
What do the two definitions’ Pij values tell us about possible ‘strategic’
voting on the part of the only voters - namely the Independents - who are
open to such voting? Clearly, the only way an Independent voter can have any
winner-determining effect is to break a first-place tie (among all other
voters). Here, this situation can possibly occur only if her (or his) ballot
is marked for exactly one of the two candidates - namely the Hot candidates
#1 and #2 - who could possibly be in a first-place tie. The ‘tie’
probability values (P12=100% and Pij=0% for all other choices of i and j
with i<j) directly reflect this reality; but the ‘lead’ probability values
do not.
CORRECT AND INCORRECT ARGUMENTS.
A 20 February posting by Richard quoted and critiqued the crux of Mike’s
argument that necessarily the lead and tie definitions must yield the same
Pij value. Richard wrote:
"MIKE OSSIPOFF wrote:
...
>But when I showed you why those 2 probabilities can't be different,
>I asked you which part of my argument you disagree with. You didn't
>answer ...
You didn't show they can't be different. Instead you wrote this:
>1. We agree that if i & j are the 2 top vote-getters in the election,
>then, if there's a tie for 1st place, it will be between them.
>
>2. We agree that if i & j are not the 2 biggest vote-getters in the
>election, then, if there is a tie for 1st place, it will not be between
>them.
>
>3. That means that either both of the following 2 statements are true,
>or neither of them are true:
>
>a) i & j are the 2 biggest vote-getters in the election.
>b) If there's a tie for 1st place, it will be between i & j.
>
>4. Since, after the election, those 2 statements are either both true
>or both false, then the probability, at a time before the election
>that a) will be true after the election must be the same as the
>probability,
>at that same time before the election, that b) will be
>true after the election."
As Richard goes on to comment, Mike’s argument is subtly erroneous. Namely,
despite the truth of Mike’s statements 1 and 2, statement 3 in fact can in
some cases be false: it does not follow from 1 and 2.
Detailed reasons for this failure of derivation are worth noting. Statement
1 asserts that ‘if a) then b)’. Statement 2 is intended to assert that ‘if
not-a), then not-b)’: then statement 3 WOULD follow from statements 1 and 2.
However, what statement 2) actually asserts is in effect that ‘if not-a)
then not-c)’, where c) is not quite
equivalent to b). Namely, b) asserts that ‘if there is a first-place tie,
then #1 and #2 tie for first place’, whereas c) asserts that ‘there is a
first-place tie and (moreover) #1 and #2 tie for first place’ - or, in
effect, simply that ‘#1 and #2 tie for first place’.
In fact, contra 3, statement b) can be true and meanwhile statement a) can
be false.
For instance, to simplify and concretize Richard’s argument, take our
example with i=#1 and j=#2. For ALL our election outcomes, #1 and #2 are the
only possible first-place-tied candidates, so for ALL outcomes statement b)
is true: IF there is a first-place tie (at all), THEN #1 and #2 tie for
first place. On the other hand, for the 90% of our example’s outcomes where
Hots do not cast ballots, statement a) is false, because the leaders are not
#1 and #2.
To be sure, some people may be confused by the idea that, for our example,
b) is always true (i.e. is true for all possible election outcomes). Well,
b) is a conditional statement of the form ‘if X then Y’. Here X (‘there is a
first-place tie’) is sometimes true and sometimes false, but anyhow Y (‘#1
and #2 tie for first place’) happens always to be true whenever X is true -
which is all that is required for truth of a statement of form ‘if X then
Y’.
[As Anthony Simmons has noted, this approach to truth of if-then statements
not only very well describes ordinary usage but also is a precise convention
of mathematics and logic.]
THAT’S ALL!! THANKS FOR YOUR HEED!!
Joe Weinstein
Long Beach CA 90807 USA
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Subject: [EM] I hate the Cold voters!
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There's no disagreeing with an example. Yes, if the 2 Pij are always
the same, then the 2000 Presidential election was legitimate & valid.
I'd taken it for granted that which 2 candidates are at the front of
the pack wasn't related to whether the front 2 candidates' totals were close
enough for a tie, and so I believed that the probability of i & j
being frontrunners when there's a tie for 1st place would be the same
as their probability of being frontrunners.
But the example shows that that isn't necessarily so. And the example
makes it look obvious. If 1 candidate has a durable
lead over another, maybe they can be frontrunners, but they can't be
in a tie.
In fairness to myself, I asked for an example demonstrating this early
in the discussion. I didn't look for one myself, because I was completely
convinced that it was impossible.
Mike Ossipoff
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