## Re: [RangeVoting] Criteria for selection of an electoral method.

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• Let me add my own definition of simplicity: 1) voters can easily understand how to use the ballots 2) voters, given internal preferences as to feelings toward
Message 1 of 24 , Dec 11, 2005
Let me add my own definition of simplicity:

1) voters can easily understand how to use the ballots
2) voters, given internal preferences as to feelings toward each candidates,
can easily convert this information to the terms needed to fill out the
ballots.
3) Voters need no knowledge of anything other than the candidates themselves
to vote the most effectively. They should not need to understand the voting
method nor know anything about which candidates are most likely to win
4) There is no conflict of interest between expressing their internal
preferences, and voting in the most effective way toward the best results as
described by those preferences.

As to 3 and 4, these also make a big difference on the "consensus"
criterion. Its pretty hard to say that something reaches a consensus unless
you can predict how people (with given preferences) will vote, using any
mathematical model. Unfortunately, if you have a system where voters must
be well informed (about more than the candidates) to vote effectively, or
worse yet, where they may be conflicted between strategy and sincerity, all
predictions go out the window.

BTW, I believe approval fails 2 and 3 but passes 4, while everything but
approval and condorcet fail 2, 3 and 4. All of them can pass 1 with a good
ballot design.

On 12/11/05, rob brown <rob@...> wrote:
>
> Hi Carolyn
>
> I prefer Approval to Range on the simplicity factor, but not to Condorcet
> methods.
>
> At first you might think it Approval is simpler, and you are right that a
> typical 12 year old might have the easiest time with Approval compared to
> Condorcet.
>
> The problem is that by the time they reach 13 or 14, they start to see how
> complex it really is. "Like" vs."Dislike" sounds simple, but quickly
> becomes obvious that it is a relative term. Relative to the other
> candidates? Well, if you want to vote effectively, it's really "relative to
> who you think might win".
>
> Which means that to effectively fill out an Approval ballot, you must not
> only understand strategy, but have a good knowledge of who is likely to
> win.....in other words those who have the best knowledge of the polls will
>
> With Condorcet, on the other hand, it just asks you to rank them relative
> to each other, and knowledge of the polls or of strategy is not going to buy
> you anything (well, only in rare elections, and if you have incredibly
> accurate knowledge of the polls, and if you are really really smart, it is
> going to make a small difference to vote with anything other than utter
> sincerity).
>
> To vote effectively with Range, you need to know everything you need to
> know with Approval, plus have the additional knowlede that it is inadvisable
> to vote at anything but the extremes, i.e., you should treat it as
> Approval.
>
> -rob
>
> On 12/11/05, Carolyn <gusrabson@...> wrote:
> >
> > Of course if we are going to choose an electoral method we need to
> > have some criteria. We hope that we can find criteria that are
> > non-contradictory. Here are two criteria that I consider absolutely
> > necessary:
> >
> > 1. Simplicity. An average 12 year old child should be able to fill out
> > a ballot with no additional instruction.
> > 2. Consensus. Given two electoral methods the better one is the one
> > whose results are accepted by the most people.
> >
> > Now I have got to admit that I am an advocate of approval voting and
> > these criteria will favor approval voting. But I favor approval voting
> > because I accept these criteria rather than vice versa. My second
> > criterion does not consider the possibility that there can be degrees
> > of disapproval. I admit this � but I feel that in view of the sketchy
> > and distorted information available to the electorate any fine-tuning
> > of degrees of disapproval is spurious.
> >
> > Does anybody know which of Arrow's conditions Approval Voting fails to
> > satisfy? What about Range Voting?
>
>

[Non-text portions of this message have been removed]
• ... --rank-order ballots appear to lead to more voter errors causing more invalidated ballots than approval, range, or plurality.
Message 2 of 24 , Dec 11, 2005
--- In RangeVoting@yahoogroups.com, rob brown <rob@k...> wrote:
>
> Let me add my own definition of simplicity:
>
> 1) voters can easily understand how to use the ballots

--rank-order ballots appear to lead to more voter errors causing more invalidated
ballots than approval, range, or plurality.
http://math.temple.edu/~wds/crv/SPRates.html

> 3) Voters need no knowledge of anything other than the candidates themselves
> to vote the most effectively. They should not need to understand the voting
> method nor know anything about which candidates are most likely to win

> 4) There is no conflict of interest between expressing their internal
> preferences, and voting in the most effective way toward the best results as
> described by those preferences.

--this would be great. However, it is not achievable by any voting method,
due to Gibbard's theorem. Also, "not understanding the voting method"
Furthermore, if voters truly treat the voting method as a "black box:"
then they certaintly will strategically exaggerate and cause the "DH3 pathology"
http://math.temple.edu/~wds/crv/DH3.html
so in some sense you can immediately see your (3) and (4) are
inherently incompatible.

wds
• ... This is crucial. Many people studying election methods don t agree with this, apparently, so it is quite natural and inevitable that they do not agree on
Message 3 of 24 , Dec 11, 2005
At 11:07 AM 12/11/2005, Carolyn wrote:

>2. Consensus. Given two electoral methods the better one is the one
>whose results are accepted by the most people.

This is crucial. Many people studying election
methods don't agree with this, apparently, so it
is quite natural and inevitable that they do not
agree on the best election methods.

>Now I have got to admit that I am an advocate of approval voting and
>these criteria will favor approval voting. But I favor approval voting
>because I accept these criteria rather than vice versa. My second
>criterion does not consider the possibility that there can be degrees
>of disapproval. I admit this  but I feel that in view of the sketchy
>and distorted information available to the electorate any fine-tuning
>of degrees of disapproval is spurious.

One method of dealing with the problem is to
directly measure approval. That is, a *question*
is presented to the voters: Shall So-And-So be elected to the Office?

I'd suggest that to be elected, at the very
minimum, a majority of voters should assent to
this question. But the goal should be much
higher. I know one organization, very large and
very successful, that requires delegates to be
elected with a two-thirds approval. If they can't
get that within a certain process and time, they
select the delegate by lot from among the top two
candidates. What this does is to create a
delegate assembly that represents the consensus
(at least a 2/3 consensus) of the organization,
and where it does not, there is a rough form of
proportional representation that is created.
• ... There is no disagreement that Range Voting gives more information about the strength of third parties. But Warren has adopted a very strange standard for
Message 4 of 24 , Dec 11, 2005
At 01:04 PM 12/11/2005, warren_d_smith31 wrote:
>If you consider that argument too ivory-tower/abstract, then consider
>the fact that in our exit poll study, range voting and approval
>voting gave VERY
>different results. Specifically, range voting gave FAR HIGHER scores
>experimentally to all third-party candidates, than does approval voting.
>This is called the "nursery effect." It is due to extra voter honesty which
>is not permitted by AV. It is discussed here:
>
><http://math.temple.edu/~wds/crv/NurseryEffect.html>http://math.temple.edu/~wds/crv/NurseryEffect.html

about the strength of third parties. But Warren has adopted a very
strange standard for determining the effect of Range Votes. If a
candidate gets very few votes in a standard plurality election, no
overvoting allowed, the candidate may well get more votes in
Approval. What happens in Range?

Warren takes a Range score, notes that it is higher numerically than
an Approval vote, and thus claims that this is the "nursery effect."
Yet his examples, if you look at the data, are, for example:

Badnarik, Plurality 0.32%, Approval 0.6%, Range 9%.

What Warren has never answered is the charge that the Range numbers,
quite simply, are apples to the Approval and Plurality numbers'
oranges. Approval and Plurality numbers show a preference; we can
assume that 0.6 percent of the voters either preferred Badnarik or
would have accepted Badnarik as a reasonable outcome.

Nothing like this is true of Range 9%. If *all* voters voted 9% for
Badnarik, and I suspect that if you look at the ballots, most voters
voted close to 100% for candidates they preferred, but certainly much
higher than 50%, a 9% range rating is *lousy*. Unless you analyze the
ballot numbers themselves, you cannot derive *any* support for
Badnarik from 9% Range. It could mean that voters entirely range
between utter hatred and serious disapproval of Badnarik. How is this
suppose to create a "nursery effect"?

Individual ballot analysis, however, could come up with better
information. If 9% of the voters actually *preferred* Badnarik, that
would be a very strong showing!

Contrary to what Warren asserts, it can't be said that Badnarik, and
the other candidates, did significantly "better" with Range than with
Approval. The Approval result is far more solid.

>This effect has the very important consequence that third parties (who are the
>voting reform) will want RV far more than they want AV, because
>they do not like getting far fewer votes. And that means you can hope
>to get a unified dedicated group of 100,000 troops trying to get RV
>in the USA.
>But you cannot hope that for AV. And that means only voting systems
>satisfying the Nursery effect can hope for adoption in the USA.

Warren claims this. Given that it is extremely easy to see through
this claim -- I'm just talking about the alleged Nursery Effect as
having been proven by Warren's study -- he's off on a windmill-tilt.

But my purpose in writing here is to point out something that may
have been missed. In every state there are not only the third
parties, but also often one of the *major* parties is in the
minority. And, *in that state*, that party is being hurt by the
present system, as it bleeds votes to third parties.

Approval, indeed, does resolve this problem with minimum fuss.

Warren's argument holds no weight at all unless he can convince the
third parties that it is in their interest to promote Range, and even
that is small stuff compared to convincing, in a state, the minority
party among the top two. Approval has a basic equity argument in its
favor that might allow multiple paths to implementation. Essentially,
there is no coherent reason for *not* allowing overvoting. And if you
allow overvoting, you have Approval. All it takes is removing a
sentence or two from the Election Code. Count the votes as they are.
No changes in ballots or election machines (which are all capable of
counting multiple votes, they have to be, because there are elections
with more than one winner, it is all a matter of how one sets the
options for a particular election).

Range is essentially an improved form of Approval. Any voter can vote
Range as if it were Approval; and many will, under present
conditions. But Range allows an additional option: voting an
intermediate vote. Further, it thus incorporates in the ballot far
intelligent elections. Approval stuffs complex relationships into a
binary value, thus distorting the outcome. Range has the potential
for overcoming this. Approval could pave the way.
• ... The difference between these two views is illusory. Nonacceptance of the results is quite likely to be highly correlated with unhappiness! Further, we have
Message 5 of 24 , Dec 11, 2005
At 01:36 PM 12/11/2005, warren_d_smith31 wrote:
> > > 2. Consensus. Given two electoral methods the better one is the one
> > > whose results are accepted by the most people.
>
>My personal view on this is that what actually matters is:
>the best voting system delivers winners which yield the highest average
>utility for society. I.e. it is about making the world better off and
>increasing human happiness the most.
>
>I don't care how many people "accept the results." If nobody accepted
>the results and we got the most human happiness, that'd be fine with me
>(although that scenario would not actually happen).

The difference between these two views is illusory. Nonacceptance of
the results is quite likely to be highly correlated with unhappiness!

Further, we have made the discovery that attempting to decide what is
good for people, over their dead bodies, so to speak, doesn't work
well at all. Democracy is about people deciding themselves what is
good for them. Sometimes they will be wrong.

Got a better idea?

I think Range results would be widely accepted, *if* it is properly
implemented. So that is what we must work on, how to implement it,
how to get from here to there.

>Now computer simulations can address precisely this, surprisingly directly.
>That is what Bayesian regret is all about.
>
><http://math.temple.edu/~wds/crv/BayRegDum.html>http://math.temple.edu/~wds/crv/BayRegDum.html
>Range Voting really shines in terms of Bayesian Regret, doing better than
>every other voting system simulated so far.
>
>
>----
>
>Also about Range versus Approval -
>AV is simpler, but
>1) does not lead to the nursery effect and hence may still yield
>2-party domination,

toss that reason, Warren. Nobody believes it but you. Unless you can
explain to me the basis on which you compared Range results with
Approval results, i.e., assuming that a 9% Range vote was "better"
than, say, a 2% Approval Vote.

A 9% Range Vote could represent widespread rejection of a candidate!
-- as long as people rejecting the candidate rated the candidate as,
say, 9%, i.e., little better than a warm body. With standard Range
strategy, as some recommend, 9% might mean "really bad, but not as
bad as X, whom I rated as zero."

To translate Range votes to Approval votes, you need to define an
Approval cutoff. It might be 50%. In which case Range 9% might be
equivalent to Approval 0 or some very small number. Do you think,
Warren, that leaders of political parties are going to be deceived by
this legerdemain? Look, I'm on your side, and you have not come close
to convincing me on this one. I think Range is great, I think you
could even prove that mathematically, and maybe you have, but Nursery
Effect? Approval does just as well and maybe better. It makes it
possible for third parties to establish voter loyalty. That's about
all that is needed, as far as third parties are going to see.

There are other arguments for why Range is better. But Approval could
be a good stepping-stone to Range.
• ... --this is true, I do not deny it. Indeed, if acceptance is regarded as a continuum (varying degrees of acceptance and not) then range voting arguably is
Message 6 of 24 , Dec 11, 2005
> >I don't care how many people "accept the results." If nobody accepted
> >the results and we got the most human happiness, that'd be fine with me
> >(although that scenario would not actually happen).
>
> The difference between these two views is illusory. Nonacceptance of
> the results is quite likely to be highly correlated with unhappiness!

--this is true, I do not deny it.
Indeed, if "acceptance" is regarded as a continuum (varying degrees of acceptance and
not) then range voting arguably is the best system by this criterion - i.e.
the most people accept the winner.

--Second, the claim range voting is forcing people to accept what they do not
want, is pretty silly for this same reason.
• It s been proposed by Mr. Brown that Range Voting should be analyzed such that the winner has the highest median rating. It s an interesting proposal. This
Message 7 of 24 , Dec 11, 2005
It's been proposed by Mr. Brown that Range Voting should be analyzed
such that the winner has the highest median rating.

It's an interesting proposal. This would mean that a majority of the
population rated that candidate at or higher than the winning rating,
which is, of course, higher than that of any other candidate.

This automatically discounts outliers, exaggerated ratings, without
specifically disqualifying them. They do still have an effect, just a
greatly reduced one. Which is fair. This would reduce the incentive
to vote strategically in Range, since you have the same effect on the
median if you meet or exceed it by a notch or you go all the way to
the maximum rating, and the same argument applies to the minimum.

The median really does guarantee majority approval of the winner,
something the mean does not do. It does this by creating an effective
definition of Approval that is quite reasonable: the overall Approval
rating is the rating of the winner! You could say that this would be
Approval strategy on steroids, Approval Voting without any need to
know how the other candidates are doing....

I like it. Very much. In fact, why didn't we think of this before?
Has the idea been raised and discussed? I do miss a good deal of what goes on!

Thanks, Mr. Brown!

I thought at first that it might be harder to count. Yes, a little
harder. But not *really* harder. The same totals (for each rating,
for each candidate) would be recorded that we conceive of using for
mean Range, but instead of adding them up at the precinct level, they
would simply be passed on and analyzed at the end. You end up with a
bar graph for each candidate, showing the number of voters who rated
the candidate at each rating. This, by the way, would be a good idea
in any case, since it detects strategic voting; or exaggerated
voting, which amounts to the same thing, only the deception may be
internal. I.e., Bush is really the antichrist....

Come to think of it.... :-)

(My political preferences aren't relevant to election reform, or I
would say a lot more!)
• ... Okay, the first problem. If Range has limited granularity, the probability of ties with a simple calculation of median is relatively large. The
Message 8 of 24 , Dec 11, 2005
At 06:51 PM 12/11/2005, Abd ul-Rahman Lomax wrote:
>I like it. Very much. In fact, why didn't we think of this before?
>Has the idea been raised and discussed? I do miss a good deal of what goes on!

Okay, the first problem. If Range has limited granularity, the
probability of ties with a simple calculation of median is relatively
large. The determination of the median would need to be adjusted to
compensate for the normal fact that there would be unequal numbers
above and below any actual rating. I don't think this would be
difficult, to derive a fractional rating from integral ones.

But it does whack the simplicity of the idea. Three-digit Range would
make ties pretty unlikely. Indeed, the way Range works might make
ties unlikely anyway. But we'll have to face the problem if we want
to promote median Range. Three-digit Range makes for a costlier
ballot, though there are tricks that can be used.
• ... No, I did not. I said If you are voting on numbers, for chrissake use median not average But then went on to say that that doesn t solve the problem
Message 9 of 24 , Dec 11, 2005
On 12/11/05, Abd ul-Rahman Lomax <abd@...> wrote:
>
> It's been proposed by Mr. Brown that Range Voting should be analyzed
> such that the winner has the highest median rating.

No, I did not.

I said "If you are voting on numbers, for chrissake use median not average"

But then went on to say that that doesn't solve the problem here, because we
are not voting on numbers, we are voting on candidates. In the article I
linked I demonstrated that median is way to vote for numbers (again:
http://karmatics.com/voting/moose-example.html ), and I *do* think the
article is relevant here, especially with regard to your quote of "vote for
the scores you want them to have".

However I never suggested that using median with Range to produce scores
would fix or even improve it for voting on candidates.

There are analogous ways to fix the problem, however. The DSV system that I
"reinvented" would do it. Condorcet would as well. Neither use median in
any direct way, but they would result in almost the exact same effect as
using median when voting on numbers such as a budget . For instance, in the
moose lodge "voting on a budget" example, you could let anyone nominate
"candidates" (which would simply be numerical values for the budget), people
would rank them based on how close to their preferred amount they are, then
elect them with any Condorcet method of your liking. Or people could "rate"
them 1 to 10, again based on proximity to their preferred amount, and use
the DSV system I suggested that converted them into "maximally strategic"
Approval ballots, then elected them that way. Either one would elect a
candidate (i.e. a budget amount) very close to the median, assuming people
rank them rationally.

Range would not. It would produce erratic results if significant numbers of
people voted strategically, parties would form, and then it would stabilize
on a two party duopoly with all the negatives of that. Just like plurality
in the moose lodge example.

I'll skip addressing the rest of your post because it was clearly written
with a vast misunderstanding of my words.

[Non-text portions of this message have been removed]
• ... to ... absolutely ... fill out ... one ... mathematical proof... ... I don t like the word vague. I have given a pretty clear prescription for how to
Message 10 of 24 , Dec 11, 2005
I wrote:

> > Of course if we are going to choose an electoral method we need
to
> > have some criteria. We hope that we can find criteria that are
> > non-contradictory. Here are two criteria that I consider
absolutely
> > necessary:
> >
> > 1. Simplicity. An average 12 year old child should be able to
fill out
> > a ballot with no additional instruction.
> > 2. Consensus. Given two electoral methods the better one is the
one
> > whose results are accepted by the most people.

Warren responded:
> --well, your two criteria are a bit vague, i.e not susceptible to
mathematical proof...
> although aside from that I like them...

I don't like the word vague.
I have given a pretty clear prescription for how to measure them.
You can measure simplicity by testing a group of students.
You can measure consensus by a post election poll.
I admit these are not the kinds of conditions mathematicians usually
work with -- but this is not really mathematics.

I wrote:
> > I feel that in view of the sketchy
> > and distorted information available to the electorate any fine-
tuning
> > of degrees of disapproval is spurious.

Warren's reply follows -- I don't agree with it. It seems by
averaging a great many false or meaningless numbers you will get a
false or meaningless number.

> --I disagree. You can for example average 1000000 numbers with
large amounts of
> noise pre-added to each, and still get an accurate result. But if
you say "oh, there
> is tremendous noise in these numbers so it is silly to ask each
number to be typed in
> accurately, let's just ask for 1 bit of each number" then you can
> get an extemely innaccurate result.
>
> If you consider that argument too ivory-tower/abstract, then
consider
> the fact that in our exit poll study, range voting and approval
voting gave VERY
> different results. Specifically, range voting gave FAR HIGHER
scores
> experimentally to all third-party candidates, than does approval
voting.
> This is called the "nursery effect." It is due to extra voter
honesty which
> is not permitted by AV. It is discussed here:
> http://math.temple.edu/~wds/crv/NurseryEffect.html
> This effect has the very important consequence that third parties
(who are the
> voting reform) will want RV far more than they want AV, because
> they do not like getting far fewer votes. And that means you can
hope
> to get a unified dedicated group of 100,000 troops trying to get
RV in the USA.
> But you cannot hope that for AV. And that means only voting
systems
> satisfying the Nursery effect can hope for adoption in the USA.
> And that means if you actually want to get voting reform as
opposed to just
> talking about it, you only have one choice at present, since only
range voting is
> known to engender the Nursery effect.
>
> > Does anybody know which of Arrow's conditions Approval Voting
fails to
> > satisfy? What about Range Voting?
>
> --Arrow's theorem is simply not applicable to either AV or RV
because
> Arrow only applies to voting systems with rank-orders as votes.
>
> If Range is converted to rank-order votes, then it satisfies all
of Arrow
> conditions except one, described last post, which I claim is
actually undesirable.
>
• I wrote: Consensus. Given two electoral methods the better one is the one whose results are accepted by the most people. Warren Replied: My personal view on
Message 11 of 24 , Dec 11, 2005
I wrote:
Consensus. Given two electoral methods the better one is the one whose results are accepted by the most people.

Warren Replied: My personal view on this is that what actually matters is: the best voting system delivers winners which yield the highest average utility for society. I.e. it is about making the world better off and increasing human happiness the most.
I don't care how many people "accept the results." If nobody accepted the results and we got the most human happiness, that'd be fine with me.

I strongly disagree: In the first place. as you would say, it is a little vague.

In the second place who is going to decide what would make the world better off and what would increase my happiness the most? I wouldn't trust anybody who has the power to make these decisions to decide what would make me happy.

In the third place experience shows that when roughly half the electorate feels unrepresented there is hell to pay. Either the president gets impeached, or there is a civil war, or at best the government is deadlocked and nothing can be done.

[Non-text portions of this message have been removed]
• ... --not necessarily so. For example, suppose the voters deliver a candidate s true score, which is, say, 0.4, except that they deliver 0.4+noise, where
Message 12 of 24 , Dec 12, 2005
> By averaging a great many false or meaningless numbers you will get a
> false or meaningless number.

--not necessarily so. For example, suppose the voters
deliver a candidate's "true" score, which is, say, 0.4, except
that they deliver 0.4+noise, where noise is, say, X-Y where
both X and Y are independent random uniform
numbers in [0, 0.4] representing two perturbing effects.

Let us assume the same thing happens for two other candidates whose "true"
scores are 0.5 and 0.6.

Well, that was just about total garbage in, right?
But if everybody behaves like this, then, with ten million voters, it is extremely likely
that the average scores we get out, will be 0.4, 0.5, and 0.6 within errors
of 0.001.

Now, instead suppose the voters take their proposed range vote (0.4+noise)
and round it off to 0 or 1, whichever is nearer, then submit that "approval" vote.
Then what happens? The scores we get out, with high probability, will be
0.2812, 0.5, and 0.7188 within errors of 0.002.

The point is that allowing the voters to deliver range votes reconstructed the
truth to high accuracy despite the very large noise.
But forcing the voters to round off to approval-style votes
introduced biases which caused the less-popular candidates to get
fewer, and the more-popular candidates to get more, votes than they
"deserved." These biases can in fact be comparable to or far larger than
the original perturbations you were worried about.
This is one example of the "nursery effect" in action -
third party candidates get far fewer votes with approval voting than
with range voting.

More generally, the entire idea of voting, medical studies, etc
is to combine opinions/data from many sources, weird and
misinformed and strange though they may be, to get a hopefully sensible result.
If those many inputs are each pre-distorted, then you can get an insane
combined result, whereas without the pre-distortion it can be possible
to get extremely sensible results. Indeed, if, say, we were doing a medical study
counting lymphocytes in 1000s of patients, or something, and you decided
to just round off the counts to "zero" or "high" on all patients, since after all,
these counts are fairly imprecise and many diseases other than
the one we are studying could be affecting the counts, etc -,
then you would probably be accused of scientific misconduct or incompetence
for willingly throwing away your precious data. It would be an outrage
for anybody to do that, and the thrown-away data could easily make
the difference between spotting a promising AIDS drug, say, versus not.

But oddly, when it comes to choosing politicians, we see the opposite thinking:
"the data is cruddy, so let's begin operations by throwing it away."
I find this quite odd.

I could give more examples from everyday experience.
The Olympic judges for gymnasts, divers, etc use (essentially) range voting.
Not approval, not borda, not plurality, and not condorcet.
And they use an 0-100 scale (well, 0-10 by 0.1 increments.)

Why? I suggest to you, that if they used those other systems to choose
the gold medalists, then everybody could instantly see the system was insane.
When quality matters commercially (because of TV revenue), they adopt a good voting
system. When it is our very lives on the line (actual politics), they don't.
Quite odd, don't you think?

The same for internet movie-rating and recipe-rating services
like allrecipes.com. They use range voting. Why? Why not just approval voting?
After all, surely anybody's opinion of a movie is just balderdash (I mean, I've
actually met people who liked - nay even RECOMMENDED - "Good Will Hunting"
and we know they are just completely nuts), so let's round it off to 0 or 1. It'll be simpler.

But, they don't do that, because everybody knows that would be ridiculous
and nobody would go for that movie-rating web site. Again, when it is
about capitalism and competition for money, they go for a quality system.
Now of course, these votes for movies are manipulable by strategic voting.
And of course it is economically-irrational to go to the trouble of voting on a movie and
then waste it by not maxing out your vote. But, amazingly, a lot of voters are
honest, and that is generally perceived as obviously leading to better movie-rating
results.

But hey, feel free to disagree. Start your own movie-rating site based on "approve"
or "not" voting. Advertise it as "more fair" because nobody
will be able to overwhelm honest voters by being strategic.
See how well your wonderful new movie-rating site does.
• ... accepted by the most people. ... system delivers winners which yield the highest average utility for society. I.e. it is about making the world better off
Message 13 of 24 , Dec 12, 2005
--- In RangeVoting@yahoogroups.com, "Gus" <gusrabson@c...> wrote:
>
>
>
> I wrote:
> Consensus. Given two electoral methods the better one is the one whose results are
accepted by the most people.
>
>
> Warren Replied: My personal view on this is that what actually matters is: the best voting
system delivers winners which yield the highest average utility for society. I.e. it is about
making the world better off and increasing human happiness the most.
> I don't care how many people "accept the results." If nobody accepted the results and we
got the most human happiness, that'd be fine with me.
>
>
> I strongly disagree: In the first place. as you would say, it is a little vague.

--first of all, it is NOT vague, because "Bayesian Regret" has
a precise mathematical defintion. Check the hyperlinks I give.
In this case, that hyperlink was
http://math.temple.edu/~wds/crv/BayRegDum.html
Second of all, we were not really in disagreement here.
I am really offering a more refined version of your own idea.
As has been observed, the two ideas in practice are highly correlated.

> In the second place who is going to decide what would make the world better off and
what would increase my happiness the most? I wouldn't trust anybody who has the power
to make these decisions to decide what would make me happy.

--you are under the misimpression that I am somehow deciding that for people
and imposing my decision on them. To the contrary. They are deciding,
they are voting. I am merely sitting back and watching my
computer simulate millions of elections with lots of different voting systems,
then asking the simulated "people" inside the computer how happy they are about
the election result, then observing that range voting delivers the highest
average happiness. I am not imposing anything. I am observing the facts.

Now before you tell me your further self-generated misimpressions
(I've been this route before so I am quite used to the ones they generate),
let me tell you that this observed fact (Range voting has superior Bayesian Regret)
is NOT dependent on having "honest" voters, it also works for
strategic voters. It also works for "misinformed" voters.
It does NOT depend on voters "downweighting" themselves. (Mine
didn't do that.) It works for a large number of voters. It works
for a small number of voters. It works for a large number of candidates.
It works for a small number of candidates.
It works if the voter's happiness for each candidate is random, or based
on 1 "issue" or based on 2 "issues" or based on an infinite
set of issues. I've tried a lot of computer experiments as you can see.

> In the third place experience shows that when roughly half the electorate feels
unrepresented there is hell to pay. Either the president gets impeached, or there is a civil
war, or at best the government is deadlocked and nothing can be done.

--to some extent this is false (e.g. Bush - no civil war, no deadlock,
no impeach -- bad government, I grant you) and to some extent it is unavoidable.
It also would be sort of unknown. E.g. in the much cited scenario where
60% vote A=6, B=5 while 40% win the day with A=0, B=10,
(is this a good thing or a bad thing? I say good) nobody would actually know
that had happened because votes are secret. All we would know is the
total - B wins heavily. Would there then be hell to pay? Perhaps.
Perhaps there should be. For example, a hypothetical
vote to free the slaves with slaves allowed to vote, might work in exctly this kind of
manner. Freeing the slaves then would cause considerable unrest.
But I think it would be the right decision. Only range voting allows this
possibility - i.e. that if enough voters choose to be honest, then a societally
best, but condorcet-lost, decision can be made.
That in my view is a clear advantage of RV over other systems.
• ... Mr. Brown, while it is possible I m off on a wild goose chase, it is also possible that my misunderstanding of your words was due to an insight that I
Message 14 of 24 , Dec 12, 2005
At 07:56 PM 12/11/2005, rob brown wrote:
>I'll skip addressing the rest of your post because it was clearly written
>with a vast misunderstanding of my words.

Mr. Brown, while it is possible I'm off on a wild goose chase, it is
also possible that my "misunderstanding" of your words was due to an
insight that I recognized that I thought you were expressing.

If it wasn't your idea, fine. Then I suppose it's mine. Or to whom
would we ascribe it?

I've thought of at least one problem, of a solution, and Warren seems
to think he sees others. I'll be looking at Warren's objection.
• On 12/11/05, Abd ul-Rahman Lomax wrote: ... I agree that comparing Plurality, Approval and Range numbers is like comparing plums, apples
Message 15 of 24 , Dec 12, 2005
On 12/11/05, Abd ul-Rahman Lomax <abd@...> wrote:
...
> Warren takes a Range score, notes that it is higher numerically than
> an Approval vote, and thus claims that this is the "nursery effect."
> Yet his examples, if you look at the data, are, for example:
>
> Badnarik, Plurality 0.32%, Approval 0.6%, Range 9%.
>
> What Warren has never answered is the charge that the Range numbers,
> quite simply, are apples to the Approval and Plurality numbers'
> oranges. Approval and Plurality numbers show a preference; we can
> assume that 0.6 percent of the voters either preferred Badnarik or
> would have accepted Badnarik as a reasonable outcome.

I agree that comparing Plurality, Approval and Range numbers is like
comparing plums, apples and rhubarb. They are all figures of merit,
and "more is better" in each case, but they can't be directly compared
because the measurements are done differently.

What you CAN do is compare the vote results of different candidates.
For example, Badnarik with Bush (the top vote-getter under Plurality):

Badnarik got 1/158th as many votes as Bush. With such tiny vote
results, Badnarik is "obviously" an obscure, fringe candidate. Why
should the press give him and his party any coverage? Why should the
average person who is not a political junkie bother to find out about
Badnarik, what he and his party stand for, etc.? Why should anyone
waste campaign contributions on him? There's no way he could possibly
win!

Now, we can do the same with Approval votes. Let's compare with
Kerry, since Kerry got the most Approval votes in Warren's data set:

Badnarik got 1/102 as many votes as Kerry. He still looks like an
obscure, fringe candidate. (But Nader comes out of obscurity with 21%
predicted that kind of result for Nader, _many_ people would likely

Finally, we do the same thing with the Range scores:

Badnarik 9%, Kerry 55% average Range scores.

but it's an order of magnitude better than how he fared under Approval
or Plurality.

People make those candidate-to-candidate comparisons all the time.
Otherwise, what would be the point of taking and publishing polls?
So, I claim that taking the ratio of the voting results for different
candidates under a given voting method, as I have done above, is a
reasonable thing to do. We could call that ratio (a given candidate's
poll or vote results / winner's poll or vote results) the "prominence"
of that candidate. Candidates with higher prominence in a campaign
have gotten and will continue to get more attention from the public
than candidates with low prominence.

WE are students of voting methods and voting reformers. What we do
all the time on these discussion groups is compare different voting
methods in just about every way imaginable. I claim that it is quite
reasonable to compare the _prominence_ of a given candidate or
candidates among different voting methods. For example:

Badnarik's prominence: Plurality 1/158, Approval 1/102, Range 1/6.
Nader's prominence: Plurality 1/133, Approval 1/6, Range 1/2.2 .
Cobb's prominence: Plurality 1/507, Approval 1/30, Range 1/11.

Warren's point, which I agree with, is that _all_ the alternative
candidates stand to gain more prominence under Range Voting than under
Approval.

> Nothing like this is true of Range 9%. If *all* voters voted 9% for
> Badnarik, and I suspect that if you look at the ballots, most voters
> voted close to 100% for candidates they preferred, but certainly much
> higher than 50%, a 9% range rating is *lousy*. Unless you analyze the
> ballot numbers themselves, you cannot derive *any* support for
> Badnarik from 9% Range. It could mean that voters entirely range
> between utter hatred and serious disapproval of Badnarik. How is this
> suppose to create a "nursery effect"?
>
> Individual ballot analysis, however, could come up with better
> information. If 9% of the voters actually *preferred* Badnarik, that
> would be a very strong showing!

I don't think the average voter is going to do such a careful analysis
I don't thing some voter, Joe, would look at Badnarik's 0.6% Approval
(say it's a poll result) and decide to find out more about Badnarik,
but in a parallel universe using Range Voting, the same Joe would look
at Badnarik's 9% Range score and decide NOT to find out more, because
9% is supposedly a lousy score.

> Contrary to what Warren asserts, it can't be said that Badnarik, and
> the other candidates, did significantly "better" with Range than with
> Approval. The Approval result is far more solid.

They do better in terms of prominence, which is what matters to the
alternative candidates and their supporters.

> >This effect has the very important consequence that third parties (who are the
> >voting reform) will want RV far more than they want AV, because
> >they do not like getting far fewer votes.

Warren, I think we should talk about Range "scores" or "results", not
"votes". I think applying the word "vote" to a range score is
confusing, because we usually think of votes as binary - either you
vote for someone or you don't.

I think the way to describe the difference between voting methods
without defining "prominence" is to say something like "Range Voting
allows alternative candidates to get much better vote results
(compared to the winner's results) than other voting methods." Or,
"My exit poll study showed that Badnarik would have gotten a Range
score of 9, compared to a score of 40 for Bush. Under Plurality,
Badnarik got 0.38% of the vote, and Bush gor 50.3%. Thus, under Range
Voting, it seems that Badnarik would have gotten MUCH better, more
impressive results relative to Bush, than he did in the actual 2004
Plurality election."

Cheers,
- Jan

> > And that means you can hope
> >to get a unified dedicated group of 100,000 troops trying to get RV
> >in the USA.
> >But you cannot hope that for AV. And that means only voting systems
> >satisfying the Nursery effect can hope for adoption in the USA.
>
> Warren claims this. Given that it is extremely easy to see through
> this claim -- I'm just talking about the alleged Nursery Effect as
> having been proven by Warren's study -- he's off on a windmill-tilt.
>
> But my purpose in writing here is to point out something that may
> have been missed. In every state there are not only the third
> parties, but also often one of the *major* parties is in the
> minority. And, *in that state*, that party is being hurt by the
> present system, as it bleeds votes to third parties.
>
> Approval, indeed, does resolve this problem with minimum fuss.
>
> Warren's argument holds no weight at all unless he can convince the
> third parties that it is in their interest to promote Range, and even
> that is small stuff compared to convincing, in a state, the minority
> party among the top two. Approval has a basic equity argument in its
> favor that might allow multiple paths to implementation. Essentially,
> there is no coherent reason for *not* allowing overvoting. And if you
> allow overvoting, you have Approval. All it takes is removing a
> sentence or two from the Election Code. Count the votes as they are.
> No changes in ballots or election machines (which are all capable of
> counting multiple votes, they have to be, because there are elections
> with more than one winner, it is all a matter of how one sets the
> options for a particular election).
>
> Range is essentially an improved form of Approval. Any voter can vote
> Range as if it were Approval; and many will, under present
> conditions. But Range allows an additional option: voting an
> intermediate vote. Further, it thus incorporates in the ballot far
> intelligent elections. Approval stuffs complex relationships into a
> binary value, thus distorting the outcome. Range has the potential
> for overcoming this. Approval could pave the way.
• ... --point taken. Another thing I should remark about this (as long as I am constructing cheap oversimplified mathematical examples to demonstrate the Nursery
Message 16 of 24 , Dec 13, 2005
> Warren, I think we should talk about Range "scores" or "results", not
> "votes". I think applying the word "vote" to a range score is
> confusing, because we usually think of votes as binary - either you
> vote for someone or you don't.

--point taken.

oversimplified mathematical examples to demonstrate the Nursery Effect):
it is entirely possible for Approval and Range to deliver different winners
like this:

1. say the "true value" of A is 49% and of B is 51%.
2. suppose all voters know that on average, BUT due to noise, lies,
confusion, etc, they each have some distribution of their honest range votes
on both A and B, where that distribution has mean 49 or 51.
3. Suppose the median on A's distribution is above 50 and on B's is
below 50.
4. Then if all range voters vote "honestly" and all approval voters round off
their range honest scores to 0 to 100 (closest) then B will win with range
and A with approval.

Now it could be argued that in this case, the range result is "correct"
and that the approval difference from range was purely due to bias introduced
by the forced rounding. This sort of thing (well, there are numerous variants
of it) is presumably exactly the reason why honest range voting
experimentally (in computer sims)
has a lot better Bayesian Regret than honest approval voting,
and why, more generally, range voting with finer "grain size" has
better Bayesian Regret than RV with coarser grain size, enough so
that 0-99 granularity seems experimentally worth it.

wds.
• ... Guess what. He was an obscure, fringe candidate. ... Geez, you just acknowledged that Range scores were a different vegetable, then you compare them by
Message 17 of 24 , Dec 14, 2005
At 02:09 AM 12/13/2005, Jan Kok wrote:
>
>Badnarik got 1/102 as many votes as Kerry. He still looks like an
>obscure, fringe candidate.

Guess what. He was an obscure, fringe candidate.

>Badnarik 9%, Kerry 55% average Range scores.
>
>but it's an order of magnitude better than how he fared under Approval
>or Plurality.

Geez, you just acknowledged that Range scores were a different
vegetable, then you compare them by assuming that the relative scores
can be directly compared, in other words, that there is a linear
scale. But there is a little detail: Approval cutoff. Further, low
Range scores, where high scores are present (above 50 is high), are
much worse then the number might make it seem, for someone who thinks
Badnarik is a terrible candidate, but better than the absolute worst,
might give Badnarik a nonzero rating.

In fact, a low Range score indicates no particular level of support,
beyond it being possible to say that a nonzero score indicates that
the electorate does not unanimously think that the candidate is the
worst possible....

Median Range might be more accurate. But probably the best data would
come from knowing what each voter considered his or her approval
cutoff. Candidates rated above that cutoff would be considered
supported, even if the data was not actually used to determine the winner.

The practical usage of this data would be fundraising. I don't think
a 9% Range rating would be particularly impressive. A large donor
would probably want to see the actual ballot data and look at the distribution.

Consider this: if for every 10 voters, one voted 100% for Badnarik
and the rest voted zero, you'd have a 9% rating. That's actually a
decent rating.

But many voters who really don't like Badnarik but who prefer him to,
though they would never vote for him or contribute to his cause, even
though they don't support him at all except as a lesser of evils, and
way down the scale at that. Suppose that *all* voters gave Badnarik a
rating of 9%.

Indeed, if this was Range11, with votes possible from 0-10, then all
these voters would be giving Badnarik one notch above the absolute minimum.

Not a good showing at all. No nursery effect for him. His supporters
may continue to support him, that's all.

>People make those candidate-to-candidate comparisons all the time.
>Otherwise, what would be the point of taking and publishing polls?
>So, I claim that taking the ratio of the voting results for different
>candidates under a given voting method, as I have done above, is a
>reasonable thing to do. We could call that ratio (a given candidate's
>poll or vote results / winner's poll or vote results) the "prominence"
>of that candidate. Candidates with higher prominence in a campaign
>have gotten and will continue to get more attention from the public
>than candidates with low prominence.
>
>WE are students of voting methods and voting reformers. What we do
>all the time on these discussion groups is compare different voting
>methods in just about every way imaginable. I claim that it is quite
>reasonable to compare the _prominence_ of a given candidate or
>candidates among different voting methods. For example:
>
>Badnarik's prominence: Plurality 1/158, Approval 1/102, Range 1/6.
>Nader's prominence: Plurality 1/133, Approval 1/6, Range 1/2.2 .
>Cobb's prominence: Plurality 1/507, Approval 1/30, Range 1/11.
>
>Warren's point, which I agree with, is that _all_ the alternative
>candidates stand to gain more prominence under Range Voting than under
>Approval.
>
> > Nothing like this is true of Range 9%. If *all* voters voted 9% for
> > Badnarik, and I suspect that if you look at the ballots, most voters
> > voted close to 100% for candidates they preferred, but certainly much
> > higher than 50%, a 9% range rating is *lousy*. Unless you analyze the
> > ballot numbers themselves, you cannot derive *any* support for
> > Badnarik from 9% Range. It could mean that voters entirely range
> > between utter hatred and serious disapproval of Badnarik. How is this
> > suppose to create a "nursery effect"?
> >
> > Individual ballot analysis, however, could come up with better
> > information. If 9% of the voters actually *preferred* Badnarik, that
> > would be a very strong showing!
>
>I don't think the average voter is going to do such a careful analysis
>I don't thing some voter, Joe, would look at Badnarik's 0.6% Approval
>(say it's a poll result) and decide to find out more about Badnarik,
>but in a parallel universe using Range Voting, the same Joe would look
>at Badnarik's 9% Range score and decide NOT to find out more, because
>9% is supposedly a lousy score.
>
> > Contrary to what Warren asserts, it can't be said that Badnarik, and
> > the other candidates, did significantly "better" with Range than with
> > Approval. The Approval result is far more solid.
>
>They do better in terms of prominence, which is what matters to the
>alternative candidates and their supporters.
>
> > >This effect has the very important consequence that third
> parties (who are the
> > >ones who care about, i.e. are the most motivated about,
> > >voting reform) will want RV far more than they want AV, because
> > >they do not like getting far fewer votes.
>
>Warren, I think we should talk about Range "scores" or "results", not
>"votes". I think applying the word "vote" to a range score is
>confusing, because we usually think of votes as binary - either you
>vote for someone or you don't.
>
>I think the way to describe the difference between voting methods
>without defining "prominence" is to say something like "Range Voting
>allows alternative candidates to get much better vote results
>(compared to the winner's results) than other voting methods." Or,
>"My exit poll study showed that Badnarik would have gotten a Range
>score of 9, compared to a score of 40 for Bush. Under Plurality,
>Badnarik got 0.38% of the vote, and Bush gor 50.3%. Thus, under Range
>Voting, it seems that Badnarik would have gotten MUCH better, more
>impressive results relative to Bush, than he did in the actual 2004
>Plurality election."
>
>Cheers,
>- Jan
>
> > > And that means you can hope
> > >to get a unified dedicated group of 100,000 troops trying to get RV
> > >in the USA.
> > >But you cannot hope that for AV. And that means only voting systems
> > >satisfying the Nursery effect can hope for adoption in the USA.
> >
> > Warren claims this. Given that it is extremely easy to see through
> > this claim -- I'm just talking about the alleged Nursery Effect as
> > having been proven by Warren's study -- he's off on a windmill-tilt.
> >
> > But my purpose in writing here is to point out something that may
> > have been missed. In every state there are not only the third
> > parties, but also often one of the *major* parties is in the
> > minority. And, *in that state*, that party is being hurt by the
> > present system, as it bleeds votes to third parties.
> >
> > Approval, indeed, does resolve this problem with minimum fuss.
> >
> > Warren's argument holds no weight at all unless he can convince the
> > third parties that it is in their interest to promote Range, and even
> > that is small stuff compared to convincing, in a state, the minority
> > party among the top two. Approval has a basic equity argument in its
> > favor that might allow multiple paths to implementation. Essentially,
> > there is no coherent reason for *not* allowing overvoting. And if you
> > allow overvoting, you have Approval. All it takes is removing a
> > sentence or two from the Election Code. Count the votes as they are.
> > No changes in ballots or election machines (which are all capable of
> > counting multiple votes, they have to be, because there are elections
> > with more than one winner, it is all a matter of how one sets the
> > options for a particular election).
> >
> > Range is essentially an improved form of Approval. Any voter can vote
> > Range as if it were Approval; and many will, under present
> > conditions. But Range allows an additional option: voting an
> > intermediate vote. Further, it thus incorporates in the ballot far
> > intelligent elections. Approval stuffs complex relationships into a
> > binary value, thus distorting the outcome. Range has the potential
> > for overcoming this. Approval could pave the way.
>
>
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>
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> * To unsubscribe from this group, send an email to:
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>----------
• ... Warren, that seems quite unlikely to me; for the median to differ significantly from the mean, there would have to be some bias in the system, essentially
Message 18 of 24 , Dec 14, 2005
At 01:05 PM 12/13/2005, warren_d_smith31 wrote:
>1. say the "true value" of A is 49% and of B is 51%.
>2. suppose all voters know that on average, BUT due to noise, lies,
>confusion, etc, they each have some distribution of their honest range votes
>on both A and B, where that distribution has mean 49 or 51.
>3. Suppose the median on A's distribution is above 50 and on B's is
>below 50.

Warren, that seems quite unlikely to me; for the median to differ
significantly from the mean, there would have to be some bias in the
system, essentially some bias that does not affect higher-granularity
Range but does affect yes-no decisions.

Further, to carp a little, the difference between 49 and 51 in a
Range vote is probably pretty meaningless. In fact, I'd expect both
of these voters to rate the candidates as 50. It really is a toss-up;
that voters would have such a fine line of discrimination between
candidates is pretty unlikely.

>4. Then if all range voters vote "honestly" and all approval voters round off
>their range honest scores to 0 to 100 (closest) then B will win with range
>and A with approval.
>
>Now it could be argued that in this case, the range result is "correct"
>and that the approval difference from range was purely due to bias introduced
>by the forced rounding. This sort of thing (well, there are
>numerous variants
>of it) is presumably exactly the reason why honest range voting
>experimentally (in computer sims)
>has a lot better Bayesian Regret than honest approval voting,

There is really a fly in the ointment of this discussion. You stated
that all voters know, on the average, that A is better than B. That's
not how you stated it, but it is implied. So why would these voters
not vote, on the average, for A? You simply assert that enough of
them would vote for B, due to "noise," to shift the election. You do
not show any mechanism for that unlikely outcome.

I don't think you can have it both ways. The noise can't be random,
and roundoff error does not explain what you suggested could occur.

>and why, more generally, range voting with finer "grain size" has
>better Bayesian Regret than RV with coarser grain size, enough so
>that 0-99 granularity seems experimentally worth it.

Non sequitur. Given that it will become more precise due to noise,
0-9 is probably enough.

(By "noise," I don't mean negative campaiging, I mean simple
variation in where similar voters will vote the same candidate. So if
half of a voter group rate a candidate as 6 and half use 7, that
group will average 6.5
• 1. I do not understand Lomax s improved and different redefinition of median . He says he will not discuss the case with median=max or min, but in fact that
Message 19 of 24 , Dec 14, 2005
1. I do not understand Lomax's improved and different
redefinition of "median". He says he will not discuss the
case with median=max or min, but
in fact that is exactly the case that matters the most
and hence is the most essential to discuss.

If Lomax has a new statistical quantity in mind which is
better than both mean and median for our purposes,
then by all means explain/define it simply &
clearly and explain why it is better. I admit to a certain skepticism
there really is something better than these two
old favorites which has been floating
arund unnoticed all this time, but could be.

2. Lomax considers it "unlikely" that median and mean
could differ significantly. In fact that is
commonplace and occurs in "skewed" distributions,
i.e. having a nonzero mean-centered third moment.
Also, it occurs (massively) with approval-vote-style data,
as we've just been discussing ad nauseum!
• ... Ah, this is irritating. I didn t say that I will not discuss [the extreme cases] as a refusal, but merely due to lack of time in the writing of that post.
Message 20 of 24 , Dec 15, 2005
At 10:19 PM 12/14/2005, warren_d_smith31 wrote:
>1. I do not understand Lomax's improved and different
>redefinition of "median". He says he will not discuss the
>case with median=max or min, but
>in fact that is exactly the case that matters the most
>and hence is the most essential to discuss.

Ah, this is irritating. I didn't say that "I will not discuss [the
extreme cases] as a refusal, but merely due to lack of time in the
writing of that post. I think I gave sufficient information that
Warren, certainly, or others, could apply the concept to the extreme cases.

>If Lomax has a new statistical quantity in mind which is
>better than both mean and median for our purposes,
>then by all means explain/define it simply &
>clearly and explain why it is better. I admit to a certain skepticism
>there really is something better than these two
>old favorites which has been floating
>arund unnoticed all this time, but could be.

The two old favorites have their application. Yet the objection to
Median analysis for Range voting is that Median doesn't work. It
doesn't work because of a certain limitation when fed discrete data.
It would work fine if the data were continuous, if, as in the real
world, somehow, voters expressed an exact number that was subject to
the chaotic influence of existence, so that no two numbers matched exactly.

Median is totally well defined, and very, very useful, in the
standard definition. It just needs a little extension to be useful in
the analysis of Range data where the granularity is finite and ties
can easily be produced.

>2. Lomax considers it "unlikely" that median and mean
>could differ significantly. In fact that is
>commonplace and occurs in "skewed" distributions,

Which is what I said. I don't expect distributions to be skewed in
the manner assumed by Warren. Yes, I could be wrong. It was just a comment.

>i.e. having a nonzero mean-centered third moment.
>Also, it occurs (massively) with approval-vote-style data,
>as we've just been discussing ad nauseum!

That has not been shown.

Now, as to the analysis of the extremes. Consider this a first
attempt. All that is necessary is that the data be spread. It could
actually be spread by a very small margin. What is important is that
it be spread so that every vote is a discrete and unique number. The
most obvious and simple way to do this is to spread it over the
half-interval evenly. However, unlike the other votes, this does
*not* produce the same mean and median for the data at that interval,
but I do think it behaves correctly. It rounds off correctly.

The only issue, really, is how it behaves in interaction with votes
for other candidates who are not rated by the same number of voters.

(and there is still the issue of what to do with blank votes.)

But I'd be happy to see that it behaves well if applied to matched
numbers of votes, i.e., no incomplete ballots.

My intuition is that the simple, even spread that I've suggested will
behave properly.

For one thing, it should produce the same results as Approval, if all
voters vote Approval style.

It is ironic that Mr. Brown seems to have missed that this is
precisely what he said he wanted: a system that takes Range data and
automatically casts votes to reflect that data. What this does is to
determine an optimal approval cutoff from the Range data, and then it
essentially uses that cutoff to convert the Range data into approval
data. Where the critical votes (at the median Range) are tied, it
spreads the data in such a way that the recalculated median generates
the same Approval votes as would exist if voters voted above and
below that median in Approval style.

I'm saying that it appears that this approach, as far as what is on
the wiki, has been completely overlooked. Rather, median was rejected
simply because of ties due to the integral nature of the data.

(By the way, using higher granularity range does not solve the
problem, because many voters will round off, or will vote approval
style, an extreme form of rounding off. Thus simple median ties would
remain quite common, as Warren noted.)

(I'm asking the resident math professor to approach this question
creatively, instead of dismissively. Is there a solution to the
problem? Have I suggested it or an approach to it? If not, *is there
another solution*?)
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