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## Re: An election identical to its reversal, and consequences.

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• Dear Warren D. Smith, at your website, you claim that every ranking method violates either resolvability or reversal symmetry. However, there are many ranking
Message 1 of 15 , Mar 22, 2013
Dear Warren D. Smith,

at your website, you claim that every ranking method violates
either resolvability or reversal symmetry. However, there are
many ranking methods that satisfy both criteria (e.g. the
Schulze method, the Kemeny-Young method, Tideman's ranked
pairs method, and the Borda method).

It is true that it is possible to create examples (like in
the paper by Ivars Peterson) where a concrete ranking method
is indecisive and range voting is decisive. But it is also
possible to create examples where range voting is indecisive
and a ranking method is decisive.

Markus Schulze
• ... --I do not believe I made that claim. Where? ... --I think you are misunderstanding. (One of) your notions of resolvability is, in the LIMIT of
Message 2 of 15 , Mar 23, 2013
--- In RangeVoting@yahoogroups.com, Markus Schulze <markus.schulze@...> wrote:
>
> Dear Warren D. Smith,
>
> at your website, you claim that every ranking method violates
> either resolvability or reversal symmetry.

--I do not believe I made that claim. Where?

> However, there are
> many ranking methods that satisfy both criteria (e.g. the
> Schulze method, the Kemeny-Young method, Tideman's ranked
> pairs method, and the Borda method).

--I think you are misunderstanding. (One of) your notions of
"resolvability" is, in the LIMIT of infinite set of votes each with
some "noise", the probability of a tie goes to zero.

Continuum range voting obeys the far stronger claim that, with ANY
nonempty FINITE set of votes each
with some "noise," the probability of a tie is zero.

This is any advantage of range voting versus, not only Schulze method, but actually versus every ranking-based voting method at all that does not suffer a reversal paradox.

> It is true that it is possible to create examples (like in
> the paper by Ivars Peterson) where a concrete ranking method
> is indecisive and range voting is decisive. But it is also
> possible to create examples where range voting is indecisive
> and a ranking method is decisive.
> Markus Schulze

--that is misleading. There simply is no probability>0 continuum range voting election
(with noisy votes) which is "indecisive."
It is possible to create probability=0 examples -- but who cares?
• ... --another formulation (closer to Schulze s and more precise) is this. Schule considers the set of all possible N-voter C-candidate elections. In the limit
Message 3 of 15 , Mar 23, 2013
> --I think you are misunderstanding. (One of) your notions of
> "resolvability" is, in the LIMIT of infinite set of votes each with
> some "noise", the probability of a tie goes to zero.
>
> Continuum range voting obeys the far stronger claim that, with ANY
> nonempty FINITE set of votes each
> with some "noise," the probability of a tie is zero.
>
> This is any advantage of range voting versus, not only Schulze method, but actually versus every ranking-based voting method at all that does not suffer a reversal paradox.

--another formulation (closer to Schulze's and more precise) is this.
Schule considers the set of all possible N-voter C-candidate elections. In the limit
N-->infinity with C>0 held fixed.

He shows the Schulze method obeys the property that the proportion of those elections which are tied (indecisive), goes to zero. This is also true of many other voting methods,
for example plain plurality. But there are some methods, such as Copeland, for which this is FALSE. It also is false for "highest median ranking wins."

For continuum range voting, not only does it obey that property, but it obeys the far stronger property, that with N>0 and C>0 BOTH held fixed, the proportion
of possible elections which are indecisive, is zero.

This stronger property is false for Schulze voting. It is also false for every ranking-based voting system which avoids "reversal paradox." It also is false for every ranking-based voting system obeying "neutrality."

One of the problems with the ranking-based voting system crowd, is that they are
so trapped in the "rankings are the only possible kind of vote" false-mindset, that they
don't even notice when somebody points out that entire trap need sot be avoided.
Ranking-based voting voting systems are a design mistake, in my view, and this is one reason why.
• ... This is correct, but a tie is not a bad outcome. With Range, and with some other methods, it indicates that the electorate is not ready to make a
Message 4 of 15 , Mar 23, 2013
At 10:16 AM 3/23/2013, WarrenS wrote:

>Continuum range voting obeys the far stronger claim that, with ANY
>nonempty FINITE set of votes each
>with some "noise," the probability of a tie is zero.

This is correct, but a "tie" is not a bad outcome. With Range, and
with some other methods, it indicates that the electorate is not
ready to make a decision. What to do then is up to special
tie-breaking rules, which can include a coin toss, or, better from
the point of view of majority rule, a runoff.

Warren is here making a strong claim for "continuum range voting,"
which is probably impractical. For those who are not familiar, this
is range with votes being any rational number between 0 and 1 full vote.

>This is any advantage of range voting versus, not only Schulze
>method, but actually versus every ranking-based voting method at all
>that does not suffer a reversal paradox.

I'm unconvinced that this is an advantage. It's a formal rule
satisfaction, for a rule that was just made up. Like a lot of voting
systems criteria, it *sounds* reasonable.

The example given on Warren's page, a generalized HAK election,
generates a tie between the candidates who are mid-preference,
because in order to demonstrate reversal symmetry, it must.

Warren claims that renge voting "solves the problem," but if you
apply the equivalent three-rating range voting, you get an A/K tie.
Condorcet methods produce a three-way tie.

The issue has exposed for me, more clearly, the "assumptions" about
voter preferences that underlie how voting systems work. The behavior
of plurality and IRV assume that voters have a strong preference for
their favorite, and a weak preference -- or no preference -- between
all other candidates. IRV then, when it eliminates a candidate, is
only eliminating a weakly preferred candidate, the harm to election
utility would be small. We, then, analyze IRV with models that do not
satisfy the assumption, and, naturally, IRV fails, badly, and in
real-world situations where this "strong preference for favorite"
don't hold, it has actually held. Plurality essentially has the same
assumption.

IRV *works* in a limited situation which violates the assumption,
that of a minor party spoiler. When this candidate is eliminated,
what is happening is that, if we think of the IRV election as a range
election, the votes for the eliminated candidate are renormalized,
pulling the low assumed rating of the second-ranked candidate for
minor party voters up to one full vote.

Analysis of voting systems has generally ignored runoff voting,
because it's not deterministic from the original ballot set, and it's
messy to analyze, because voters can change their minds with more
information, or simply time to think, and the set of voters in the
runoff may be different than in the primary. Thus an *extremely
common* voting method, that of continuing to vote until a majority
favors an option, is ignored.

Notice that vote-splitting does not affect majority, even when it is
vote-for-one. We have suggested using Approval for runoff voting,
and, strictly speaking, to demonstrate an explicit majority choice,
multiple majorities should *not* be resolved by awarding the win to
the candidate with the most votes. However, that might be done in the
name of efficiency; nevertheless, it introduces possible pathology.
My sense is that a sane method might look at the margin, and go to a
runoff when it is close.

Imagine the HAK election in a vote-for-one top two runoff system. H
does not gain a majority, so there is a runoff. The problem is how to
choose which of A and K goes into the runoff. Whichever one goes into
the runoff -- it might be chosen by lot -- that one, if preferences
are maintained, will win, 2:1, over H. A better primary system would
see that H cannot win a runoff, from the expressed preferences, and
would eliminate H. However, that assumes that the preference
strengths are equal.

I have been suggesting hybrid systems, particularly for use with
majority-required runoff systems, i.e., systems that seek majority
approval and that only terminate in a second election if no majority
can be found, two elections being considered enough.

It's totally clear and should be uncontroversial that a Range or
Score ballot collects the maximum useful information from the voter.
It's been shown -- by Warren -- that with many candidates, Borda is
equivalent to Range, and it's easy to see that a Range ballot is a
ranked ballot with equal ranking allowed. If the resolution is high
enough, a Range ballot can express full ranking. And if a ranked
ballot only allows a limited number of ranks, it likewise involves
equal ranking, at least at the bottom.

Forcing voters to rank candidates when they have no discriminable
preference is introducing noise into the system, and very possibly
bias, especially with relatively unknown candidates. ("I don't like
that name.")

Almost all voting systems set up equal-ranking bottom.
Full-rank-required IRV does not, that's an exception, which would
clearly be rejected in the U.S. (It was rejected when Oklahoma tacked
a second preference requirement into its Bucklin implementation.) So
there is nothing strange about equal ranking, it's just that it's
assumed that this only happens at the bottom. But real voters will
often have *some preference* of a set of voters over full bottom, yet
find it difficult to determine the preference, which indicates that
the set of candidates at that preference level is actually equal --
or close to equal -- in real preference.

A hybrid system can, in two rounds, satisfy the Majority Criterion,
the Condorcet Criterion, and approximate -- closely -- what I'll call
the Range Criterion, i.e., the maximization of expressed social utility.

> > It is true that it is possible to create examples (like in
> > the paper by Ivars Peterson) where a concrete ranking method
> > is indecisive and range voting is decisive. But it is also
> > possible to create examples where range voting is indecisive
> > and a ranking method is decisive.
> > Markus Schulze
>
>--that is misleading. There simply is no probability>0 continuum
>range voting election
>(with noisy votes) which is "indecisive."

Sure. Now, see if you can get real voters to vote in a continuum.
Here is how you could do it: you randomize the votes, i.e, adding a
spread to them over the intermediate range, so if the voter has
voted, say 0.4, the vote is spread over 0.35 - 0.45. But ... this is
equivalent to tossing a coin to resolve a tie in other supposedly
indecisive systems. Add that rule, abe to handle multiple ties, and
they all become decisive. With an open process.

In any case, collecting Range data in elections seems to be something
that could be highly desirable, even if the data is not fully used in
determining the winner.

I've pointed out that the classic Buckln ballot was essentially a
Range 4 ballot over the approved range, with equal rationg bottom for
all canidates, and midrange being considered minimal approval. That
system prohibited multiple voting in ratings 4 and 3, but allowed it
in rating 2. It allowed skipping ratings. It *was* Range, analyzed in
a particular way that looks for a majority, among higher ratings
first, and that completed based on Approval criteria when there was
no majority found.

That system could be improved by

1. Adding the rating value of 1, a disapproved rank/rating, but
preferred over bottom.
2. Allowing equal rating of candidates, which is more realistic and
which can make the system handle more candidates.
3. Holding a runoff when no majority approval can be found.
4. Holding the runoff between, in this order of preference, until two
have been chosen:
A. Range winner.
B. Pairwise winner over the Range winner. If more than one, the
one with the highest total score advances.

(many variations are possible. We have not had much indpendent
analysis of the possibilities.)
5. Allow write-in votes in the runoff, and use an advanced system for
the runoff, like Bucklin, which allows write-in votes in addition to
lower-ranked votes for a named candidate, and can also allow those
who prefer a named candidate to also approve a specific write-in.

From a democratic perspective, majority with repeated ballot, until
majority is found, without eliminations (new nominations, strictly,
with every ballot) is very solid and effective. It's Condorcet
compliant in the final election.

However, it can be made more *efficient*, and that's what Approval
Voting can do, and likewise, even better, Range. A range ballot
provides maximum information to the voters in the next ballot.

We have tended to confuse the generation of information, on which
decisions can be based, with the actual method of making the choice.
Range methods are used all the time, *informally*. Yet for each
person, a decision is made -- or the decision is postponed. People
have a personal threshold for determining Approval cutoff, and they
vary it depending on conditions. Many people will postpone a decision
until circumstances indicate a loss from further delay, in which case
they will lower their approval cutoff until the sense is developed
that one choice is superior, or at least more likely to be superior.

Bucklin, then, with its declining approval cutoff, simulates a series
of repeated elections with declining approval cutoff, quite what
student of Approval have described would happen, as voters become
more intent on generating a result.

So the initial round has minimum approval rating of 4, then it's 3,
then it's 2. This can be pushed toward more detailed rating, and it
can be extended into the unapproved zone, until a Bucklin ballot
becomes a full ranked/rating ballot. (i.e., "full" in the sense that
the entire span of individual perceived utilities is covered.).

We need to distinguish between the ballot system, how voters express
preferences, and the analysis. Obviously analysis can affect voter
strategy, and some analytical methods even encourage reversed
ratings, but when equal rating/ranking is allowed, generally, true
reversal is not needed. There are exceptions with systems that
eliminate candidates, where it may be necessary to Favorite Betray to
allow a better choice in the next round.

But decent systems don't eliminate candidates until a winner is declared.
• ... If equal ranking and skipped ranks are disallowed, I agree. Rather than a mistake, I d call it a limiting assumption that disallows a clear expression of
Message 5 of 15 , Mar 23, 2013
At 10:39 AM 3/23/2013, WarrenS wrote:

>Ranking-based voting voting systems are a design mistake, in my
>view, and this is one reason why.

If equal ranking and skipped ranks are disallowed, I agree.

Rather than a "mistake," I'd call it a limiting assumption that
disallows a clear expression of preference strength; therefore the
information adequate to begin to estimate the social utility winner is missing.

Ranked methods that disallow equal ranking seem to do so based on the
idea that it's easy for voters to make pairwise choices. That is
something that can be a useful idea, but that fails, under common
circumstances, if equal ranking is prohibited.

Procedurally, voters can clearly rank some candidates over others,
but not necessarily all.

What is weird is that almost all systems allow equal-ranking bottom.
So why not equal ranking at other ranks?

A range ballot does express rankings, if the voter so chooses.
Identical problem, in fact. Do you rate two candidates the same or do
you make a choice, preferring one over the other. With some systems,
strategic considerations will loom large.
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