[EM] Arrow's theorem and cardinal voting systems

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[EM] Arrow's theorem and cardinal voting systems

Rob Lanphier
Hi folks,

As some of you might have seen, Electowiki is a lot more active than
it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
that has reservations is regarding how some advocates talk about
Arrow's theorem.  I'm hoping you all can do one of the following:
a)  change my view about Arrow's theorem, -or-
b)  offer me some help in better articulating my view about Arrow's theorem.

Many Score voting[1] activists claim that cardinal methods somehow
dodge Arrow's theorem.  It seems to me that *all* voting systems (not
a mere subset) are subject to some form of impossibility problem.
Arrow's impossibility theorem deserved great acclaim for subjecting
all mainstream voting systems of the 1950s to mathematical rigor, and
it's clear that his 1950 paper and 1951 book profoundly influenced
economics and game theory for the better.  His 1972 Nobel prize was
well deserved.  It seems that it has become fashionable to find
loopholes in Arrow's original formulation and declare the loopholes
important.  Even if the loopholes exist, talking up those loopholes
doesn't seem compelling, given the subsequent work by other theorists
broaden the scope beyond Arrow's version.

But, what the heck, let's actually talk about Arrow's original
formulation.  I believe Score voting fails unrestricted domain:
<https://en.wikipedia.org/wiki/Unrestricted_domain>

In particular, let's say that 90% of voters prefer candidate A over candidate B:
90:A>B
10:B>A

Arrow posits that there should only be one way to express that, and
Score fails it.  In Score, it's possible to sometimes pick A, and
sometimes pick B, depending on the score values on the ballots.  If
Score *always* chose either A or B, then it would pass Universality.

Score advocates claim that this isn't a bug, it's a *feature*.  If
(for example), voters for A only mildly prefer A over B, but voters
for B strongly detest A, then the correct social choice is B.
However, it doesn't seem practical to inflict this level of nuance on
voters.  I suspect that the first election where the Condorcet winner
is beaten by a minority-preferred candidate (e.g. like what happened
in Burlington 2009 [2]) will result in a repeal (like what happened in
Burlington).  Back to the A/B example above, It's hard to imagine
voters would consider the selection of "B" to be fair in a large
election.

It's fine to hold the opinion that Universality is an uninteresting
criterion, and that therefore, Arrow's set of criteria isn't very
interesting.  For example, a few years ago, we went through a phase
where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
interesting criterion, and advocating for Condorcet variants that meet
that criterion.  Regardless, just because we find one criterion less
compelling than another, we should talk accurately about the failed
criterion.

My way of thinking about Arrow's theorem (and being thankful for it)
is to think of it like the physics of voting systems.  For example, in
real-world physics, a "perfect" vehicle is impossible, because it's
impossible to meet these criteria:
* Goes faster than the speed of light
* Has infinite capacity
* Has a luxurious and comfortable passenger cabin
* Fits in a small coat pocket
* Is easy to produce
* Is cheap (or even free)

Just because a perfect vehicle is not possible, I'm glad
transportation innovation didn't stop with Ford's Model T.  Of course,
automobile sellers compete on the tradeoffs between the criteria
above, and much public policy debate is about mode-of-transport
tradeoffs between planes, trains and automobiles (and bicycles, and
scooters, and and and...).  We need public policy debates around
election method tradeoffs, too.

I'm hoping we can try to stop trying to declare clever loopholes in
Arrow's theorem, and just acknowledge the reality that *all* voting
systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
central insight (there's no "perfect" system given perfectly
reasonable criteria) is valid, and that it's only on the specifics of
the exact criteria chosen for the 1951 proof that might be flawed.  I
believe that election method activists should speak (and write) with
clarity about the tradeoffs involved.  Whenever I see someone
gleefully declare that Arrow's theorem doesn't apply to their voting
method (and imply perfection), the credibility of the writer drops
*precipitously* in my mind.

Am I wrong?

Rob

p.s. I've been meaning to write this email for a while.  What inspired
me to finally write it has been reading the current state of
Electowiki and Wikipedia articles on the topic, like the "Arrow's
impossiblity theorem" article on Electowiki[4]

[1]: https://electowiki.org/wiki/Score_voting
[2]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election
[3]: https://electowiki.org/wiki/IIAC
[4]: https://electowiki.org/wiki/Arrow%27s_impossibility_theorem
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Re: [EM] Arrow's theorem and cardinal voting systems

James J Faran
About Score voting failing Unrestricted Domain:

Part of the confusion of those advocating score and you is not a confusion on anyone's part, but rather a difference in what each considers a preference.  (It's not possible to have a good reasoned argument until both sides agree on what the words mean.)  Score voters would say

A:100; B:95; C:0

and

A:100; B:5; C:0

are different preferences, but you seem to say that these are both A>B>C and so are the same.  I would say the Electowiki page on Unrestricted Domain needs to be edited to include both possibilities, but I can't be bothered.

You also seem to think that most voters would not be able to understand that sort of nuance.  You may be right there, especially in today's political climate (especially in the United States?), where there are two sides and the other side is always demonized.

Note that any new voting system will almost always try to be replaced by the loser under the new system.  ("The current government is illegitimate!  If it wasn't for the biased voting system we would have won!" -- cf. the recently revived call for the elimination of the U. S. Electoral College after Mr. Trump won with a minority of the popular vote.)  If the winner can't keep support, the losing side will be able to push through a change.

However, a question:  If we had the following score ballots:

9000:  A:100; B:95; C:0
1000:  B:100; C:85; A:0

giving A a score of 900,000 and B a score of 955,000, hence a victory for B, would there really be enough antipathy to B to cause outrage?  All the A voters seemed to think B was pretty good.  Of course (see above), the losing side could always complain.  Anyone wedded to Condorcet winners would be outraged.  And, of course, no real world election would end up like this.  Score may be a little too ripe for manipulation.  Gibbard-Satterthwaite, anyone?

Jim Faran


________________________________________
From: Election-Methods <[hidden email]> on behalf of Rob Lanphier <[hidden email]>
Sent: Thursday, January 9, 2020 6:17 PM
To: Election Methods
Subject: [EM] Arrow's theorem and cardinal voting systems

Hi folks,

As some of you might have seen, Electowiki is a lot more active than
it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
that has reservations is regarding how some advocates talk about
Arrow's theorem.  I'm hoping you all can do one of the following:
a)  change my view about Arrow's theorem, -or-
b)  offer me some help in better articulating my view about Arrow's theorem.

Many Score voting[1] activists claim that cardinal methods somehow
dodge Arrow's theorem.  It seems to me that *all* voting systems (not
a mere subset) are subject to some form of impossibility problem.
Arrow's impossibility theorem deserved great acclaim for subjecting
all mainstream voting systems of the 1950s to mathematical rigor, and
it's clear that his 1950 paper and 1951 book profoundly influenced
economics and game theory for the better.  His 1972 Nobel prize was
well deserved.  It seems that it has become fashionable to find
loopholes in Arrow's original formulation and declare the loopholes
important.  Even if the loopholes exist, talking up those loopholes
doesn't seem compelling, given the subsequent work by other theorists
broaden the scope beyond Arrow's version.

But, what the heck, let's actually talk about Arrow's original
formulation.  I believe Score voting fails unrestricted domain:
<https://en.wikipedia.org/wiki/Unrestricted_domain>

In particular, let's say that 90% of voters prefer candidate A over candidate B:
90:A>B
10:B>A

Arrow posits that there should only be one way to express that, and
Score fails it.  In Score, it's possible to sometimes pick A, and
sometimes pick B, depending on the score values on the ballots.  If
Score *always* chose either A or B, then it would pass Universality.

Score advocates claim that this isn't a bug, it's a *feature*.  If
(for example), voters for A only mildly prefer A over B, but voters
for B strongly detest A, then the correct social choice is B.
However, it doesn't seem practical to inflict this level of nuance on
voters.  I suspect that the first election where the Condorcet winner
is beaten by a minority-preferred candidate (e.g. like what happened
in Burlington 2009 [2]) will result in a repeal (like what happened in
Burlington).  Back to the A/B example above, It's hard to imagine
voters would consider the selection of "B" to be fair in a large
election.

It's fine to hold the opinion that Universality is an uninteresting
criterion, and that therefore, Arrow's set of criteria isn't very
interesting.  For example, a few years ago, we went through a phase
where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
interesting criterion, and advocating for Condorcet variants that meet
that criterion.  Regardless, just because we find one criterion less
compelling than another, we should talk accurately about the failed
criterion.

My way of thinking about Arrow's theorem (and being thankful for it)
is to think of it like the physics of voting systems.  For example, in
real-world physics, a "perfect" vehicle is impossible, because it's
impossible to meet these criteria:
* Goes faster than the speed of light
* Has infinite capacity
* Has a luxurious and comfortable passenger cabin
* Fits in a small coat pocket
* Is easy to produce
* Is cheap (or even free)

Just because a perfect vehicle is not possible, I'm glad
transportation innovation didn't stop with Ford's Model T.  Of course,
automobile sellers compete on the tradeoffs between the criteria
above, and much public policy debate is about mode-of-transport
tradeoffs between planes, trains and automobiles (and bicycles, and
scooters, and and and...).  We need public policy debates around
election method tradeoffs, too.

I'm hoping we can try to stop trying to declare clever loopholes in
Arrow's theorem, and just acknowledge the reality that *all* voting
systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
central insight (there's no "perfect" system given perfectly
reasonable criteria) is valid, and that it's only on the specifics of
the exact criteria chosen for the 1951 proof that might be flawed.  I
believe that election method activists should speak (and write) with
clarity about the tradeoffs involved.  Whenever I see someone
gleefully declare that Arrow's theorem doesn't apply to their voting
method (and imply perfection), the credibility of the writer drops
*precipitously* in my mind.

Am I wrong?

Rob

p.s. I've been meaning to write this email for a while.  What inspired
me to finally write it has been reading the current state of
Electowiki and Wikipedia articles on the topic, like the "Arrow's
impossiblity theorem" article on Electowiki[4]

[1]: https://electowiki.org/wiki/Score_voting
[2]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election
[3]: https://electowiki.org/wiki/IIAC
[4]: https://electowiki.org/wiki/Arrow%27s_impossibility_theorem
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Re: [EM] Arrow's theorem and cardinal voting systems

fdpk69p6uq
In reply to this post by Rob Lanphier

On Thu, Jan 9, 2020, 6:17 PM Rob Lanphier wrote:
Hi folks,

As some of you might have seen, Electowiki is a lot more active than
it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
that has reservations is regarding how some advocates talk about
Arrow's theorem.  I'm hoping you all can do one of the following:
a)  change my view about Arrow's theorem, -or-
b)  offer me some help in better articulating my view about Arrow's theorem.

Many Score voting[1] activists claim that cardinal methods somehow
dodge Arrow's theorem.  It seems to me that *all* voting systems (not
a mere subset) are subject to some form of impossibility problem.
Arrow's impossibility theorem deserved great acclaim for subjecting
all mainstream voting systems of the 1950s to mathematical rigor, and
it's clear that his 1950 paper and 1951 book profoundly influenced
economics and game theory for the better.  His 1972 Nobel prize was
well deserved.  It seems that it has become fashionable to find
loopholes in Arrow's original formulation and declare the loopholes
important.  Even if the loopholes exist, talking up those loopholes
doesn't seem compelling, given the subsequent work by other theorists
broaden the scope beyond Arrow's version.

But, what the heck, let's actually talk about Arrow's original
formulation.  I believe Score voting fails unrestricted domain:
<https://en.wikipedia.org/wiki/Unrestricted_domain>

In particular, let's say that 90% of voters prefer candidate A over candidate B:
90:A>B
10:B>A

Arrow posits that there should only be one way to express that, and
Score fails it.  In Score, it's possible to sometimes pick A, and
sometimes pick B, depending on the score values on the ballots.  If
Score *always* chose either A or B, then it would pass Universality.

Score advocates claim that this isn't a bug, it's a *feature*.  If
(for example), voters for A only mildly prefer A over B, but voters
for B strongly detest A, then the correct social choice is B.
However, it doesn't seem practical to inflict this level of nuance on
voters.  I suspect that the first election where the Condorcet winner
is beaten by a minority-preferred candidate (e.g. like what happened
in Burlington 2009 [2]) will result in a repeal (like what happened in
Burlington).  Back to the A/B example above, It's hard to imagine
voters would consider the selection of "B" to be fair in a large
election.

It's fine to hold the opinion that Universality is an uninteresting
criterion, and that therefore, Arrow's set of criteria isn't very
interesting.  For example, a few years ago, we went through a phase
where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
interesting criterion, and advocating for Condorcet variants that meet
that criterion.  Regardless, just because we find one criterion less
compelling than another, we should talk accurately about the failed
criterion.

My way of thinking about Arrow's theorem (and being thankful for it)
is to think of it like the physics of voting systems.  For example, in
real-world physics, a "perfect" vehicle is impossible, because it's
impossible to meet these criteria:
* Goes faster than the speed of light
* Has infinite capacity
* Has a luxurious and comfortable passenger cabin
* Fits in a small coat pocket
* Is easy to produce
* Is cheap (or even free)

Just because a perfect vehicle is not possible, I'm glad
transportation innovation didn't stop with Ford's Model T.  Of course,
automobile sellers compete on the tradeoffs between the criteria
above, and much public policy debate is about mode-of-transport
tradeoffs between planes, trains and automobiles (and bicycles, and
scooters, and and and...).  We need public policy debates around
election method tradeoffs, too.

I'm hoping we can try to stop trying to declare clever loopholes in
Arrow's theorem, and just acknowledge the reality that *all* voting
systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
central insight (there's no "perfect" system given perfectly
reasonable criteria) is valid, and that it's only on the specifics of
the exact criteria chosen for the 1951 proof that might be flawed.  I
believe that election method activists should speak (and write) with
clarity about the tradeoffs involved.  Whenever I see someone
gleefully declare that Arrow's theorem doesn't apply to their voting
method (and imply perfection), the credibility of the writer drops
*precipitously* in my mind.

Am I wrong?

Rob

p.s. I've been meaning to write this email for a while.  What inspired
me to finally write it has been reading the current state of
Electowiki and Wikipedia articles on the topic, like the "Arrow's
impossiblity theorem" article on Electowiki[4]

[1]: https://electowiki.org/wiki/Score_voting
[2]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election
[3]: https://electowiki.org/wiki/IIAC
[4]: https://electowiki.org/wiki/Arrow%27s_impossibility_theorem
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Re: [EM] Arrow's theorem and cardinal voting systems

robert bristow-johnson
In reply to this post by James J Faran


> On January 9, 2020 11:12 PM Faran, James <[hidden email]> wrote:
>
>  
> About Score voting failing Unrestricted Domain:
>
> Part of the confusion of those advocating score and you is not a confusion on anyone's part, but rather a difference in what each considers a preference.  (It's not possible to have a good reasoned argument until both sides agree on what the words mean.)  Score voters would say
>
> A:100; B:95; C:0
>
> and
>
> A:100; B:5; C:0
>
> are different preferences, but you seem to say that these are both A>B>C and so are the same.  I would say the Electowiki page on Unrestricted Domain needs to be edited to include both possibilities, but I can't be bothered.
>
> You also seem to think that most voters would not be able to understand that sort of nuance.  You may be right there, especially in today's political climate (especially in the United States?), where there are two sides and the other side is always demonized.
>
> Note that any new voting system will almost always try to be replaced by the loser under the new system.  ("The current government is illegitimate!  If it wasn't for the biased voting system we would have won!" -- cf. the recently revived call for the elimination of the U. S. Electoral College after Mr. Trump won with a minority of the popular vote.)  If the winner can't keep support, the losing side will be able to push through a change.
>
> However, a question:  If we had the following score ballots:
>
> 9000:  A:100; B:95; C:0
> 1000:  B:100; C:85; A:0
>
> giving A a score of 900,000 and B a score of 955,000, hence a victory for B, would there really be enough antipathy to B to cause outrage?  All the A voters seemed to think B was pretty good.  Of course (see above), the losing side could always complain.  Anyone wedded to Condorcet winners would be outraged.  And, of course, no real world election would end up like this.  Score may be a little too ripe for manipulation.  Gibbard-Satterthwaite, anyone?
>

words have meaning.  "preference" without a quantitative adjective is Ranked ballot.

"strong preference" vs. "weak preference" implies a Score ballot.

my question that i have asked the Score Voting or Approval Voting advocates years ago remains: "How much should I score my second choice?"  or "Should I approve my second choice or not?"  that tactical question faces the voter in a Score or Approval election the second he/she steps into the voting booth.  but not so for the ordinal Ranked ballot.

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Re: [EM] Arrow's theorem and cardinal voting systems

robert bristow-johnson
In reply to this post by James J Faran
i forgot something...

> On January 9, 2020 11:12 PM Faran, James <[hidden email]> wrote:
>
>
> Note that any new voting system will almost always try to be replaced by the loser under the new system.  ("The current government is illegitimate!  If it wasn't for the biased voting system we would have won!" -- cf. the recently revived call for the elimination of the U. S. Electoral College after Mr. Trump won with a minority of the popular vote.)  If the winner can't keep support, the losing side will be able to push through a change.
>

well, assuming that you can't change the rules of an election once it's decided, the "losing side" can only advocate changing the rules for future elections.  but the circumstances will not be the same and sometimes the losing side will hurt themselves in advocating changing the rules.

an example in Vermont is the 2014 gubernatorial election in which the GOP candidate would almost certainly have won if the state practiced Ranked-Choice Voting of some form.  it's the GOP who opposed RCV the most.  i opined about that in this commentary: https://vtdigger.org/2014/11/11/robert-bristow-johnson-ways/ 

the loser under the new system now might be the winner under the same system later.

--
 
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Re: [EM] Arrow's theorem and cardinal voting systems

Rob Lanphier
In reply to this post by fdpk69p6uq
Hi fdpk69p6uq,

I have to confess that your reply in this thread bothers me a little
bit.  More inline below....

> On Thu, Jan 9, 2020, 6:17 PM Rob Lanphier wrote:
>> Many Score voting[1] activists claim that cardinal methods somehow
>> dodge Arrow's theorem.  It seems to me that *all* voting systems (not
>> a mere subset) are subject to some form of impossibility problem.

On Thu, Jan 9, 2020 at 8:43 PM <[hidden email]> wrote:
> Didn't Arrow agree that rated systems aren't included?
>
> https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/

Are you saying that a single interview of an 90-year-old Kenneth Arrow
by a much younger interviewer with a possible bias on the matter
should be the final word on this subject?  How much of his
professional credibility did Arrow stake on that interview?  How much
preparation for that line of questioning do you think Arrow gave for
that interview?  Who's going to interview Gibbard?  Who's going to
interview Satterthwaite?

It was really clear to me a the point that Aaron Hamlin tried to get
Dr. Arrow's take on the "favorite betrayal criterion" that Dr. Arrow
wasn't actively reading this mailing list, and instead was enjoying
his waning years.  ;-)  It's interesting that Arrow had been here in
the Bay Area for a very long time, and witnessed San Francisco (and
Oakland, and others) move to Instant Runoff, and that he clearly had
some reservations about IRV.  He also seemed more concerned with
Electoral College and California's jungle primary than he did with any
of the municipal election reforms.

I really appreciate that Aaron did that interview before Dr. Arrow
died, and I told Aaron I was a little jealous, and was kicking myself
for not seeking Arrow out myself after I moved to San Francisco in
2011.  Given that, my assessment of the interview above probably seems
a bit harsh.  All I'm saying is that the interview wasn't set up for
the level of scrutiny necessary to establish rote truth.  I'm not
willing to accept that interview as definitive proof that Arrow's key
insight on voting systems is strictly limited to ordinal voting
systems, and that cardinal voting systems are provably free of any
sort of impossibility paradox.

Rob
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Re: [EM] Arrow's theorem and cardinal voting systems

Kristofer Munsterhjelm-3
In reply to this post by Rob Lanphier
On 10/01/2020 00.17, Rob Lanphier wrote:

> Hi folks,
>
> As some of you might have seen, Electowiki is a lot more active than
> it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
> that has reservations is regarding how some advocates talk about
> Arrow's theorem.  I'm hoping you all can do one of the following:
> a)  change my view about Arrow's theorem, -or-
> b)  offer me some help in better articulating my view about Arrow's theorem.
>
> Many Score voting[1] activists claim that cardinal methods somehow
> dodge Arrow's theorem.  It seems to me that *all* voting systems (not
> a mere subset) are subject to some form of impossibility problem.
> Arrow's impossibility theorem deserved great acclaim for subjecting
> all mainstream voting systems of the 1950s to mathematical rigor, and
> it's clear that his 1950 paper and 1951 book profoundly influenced
> economics and game theory for the better.  His 1972 Nobel prize was
> well deserved.

First a little pet peeve of sorts: It's not a Nobel prize. The Nobel
foundation says as much at the very bottom of
https://www.nobelprize.org/nomination/economic-sciences/ :-)

> It seems that it has become fashionable to find
> loopholes in Arrow's original formulation and declare the loopholes
> important.  Even if the loopholes exist, talking up those loopholes
> doesn't seem compelling, given the subsequent work by other theorists
> broaden the scope beyond Arrow's version.
>
> But, what the heck, let's actually talk about Arrow's original
> formulation.  I believe Score voting fails unrestricted domain:
> <https://en.wikipedia.org/wiki/Unrestricted_domain>
>
> In particular, let's say that 90% of voters prefer candidate A over candidate B:
> 90:A>B
> 10:B>A
>
> Arrow posits that there should only be one way to express that, and
> Score fails it.  In Score, it's possible to sometimes pick A, and
> sometimes pick B, depending on the score values on the ballots.  If
> Score *always* chose either A or B, then it would pass Universality.

I agree. Arrow's theorem requires unrestricted domain to work, and what
unrestricted domain says (as far as I know it) is that the inputs are
lists of orderings, that every such list must be admissible, and that's
all that's required. I.e. "The rankings, all rankings, and nothing but
the rankings".

> Score advocates claim that this isn't a bug, it's a *feature*.  If
> (for example), voters for A only mildly prefer A over B, but voters
> for B strongly detest A, then the correct social choice is B.
> However, it doesn't seem practical to inflict this level of nuance on
> voters.  I suspect that the first election where the Condorcet winner
> is beaten by a minority-preferred candidate (e.g. like what happened
> in Burlington 2009 [2]) will result in a repeal (like what happened in
> Burlington).  Back to the A/B example above, It's hard to imagine
> voters would consider the selection of "B" to be fair in a large
> election.

Sure, you might say that unrestricted domain is uninteresting. If you
do, and you design a voting system that doesn't pass unrestricted
domain, then Arrow won't apply. You could well get IIA out of it (with
certain assumptions). That doesn't mean that Arrow's is wrong, it just
means that it no longer applies.

I think that it's a good idea to think: "is my objection about strategy
or not?" If it's about strategy-proofness, the right theorem isn't
Arrow's, it's Gibbard's. And Gibbard's theorem holds for Range as well,
so Range doesn't get around it. So just because Range is outside of the
scope of Arrow, that doesn't mean that it's invulnerable to strategy.

And, to go on a bit of a tangent, I think that ordinary (normalized)
Range has both a cardinal and ordinal component to it. The ordinal
component might violate IIA.

Consider, for instance, something like: election one has a pro-war
candidate and an anti-war candidate, and their positions are otherwise
pretty much the same. But election two has ten pro-war candidates
ranging from economic left to right and ten anti-war candidates ranging
from economic left to right. It may be the case that in election one,
people judge the candidates by whether their war stance agree, and in
election two, people judge the candidates more heavily by left vs right
than by war stance. The distribution of the candidates affects what the
voters consider important features, and thus irrelevant candidates could
alter who the winner is.

On the other hand, "The Possibility of Social Choice" (Sen) suggests
that only very weak utility-like comparisons are required to salvage
IIA. So who knows? Perhaps with a method that's not Range, you can have
it even with the scenario above.

> It's fine to hold the opinion that Universality is an uninteresting
> criterion, and that therefore, Arrow's set of criteria isn't very
> interesting.  For example, a few years ago, we went through a phase
> where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
> interesting criterion, and advocating for Condorcet variants that meet
> that criterion.  Regardless, just because we find one criterion less
> compelling than another, we should talk accurately about the failed
> criterion.
>
> My way of thinking about Arrow's theorem (and being thankful for it)
> is to think of it like the physics of voting systems.  For example, in
> real-world physics, a "perfect" vehicle is impossible, because it's
> impossible to meet these criteria:
> * Goes faster than the speed of light
> * Has infinite capacity
> * Has a luxurious and comfortable passenger cabin
> * Fits in a small coat pocket
> * Is easy to produce
> * Is cheap (or even free)
>
> Just because a perfect vehicle is not possible, I'm glad
> transportation innovation didn't stop with Ford's Model T.  Of course,
> automobile sellers compete on the tradeoffs between the criteria
> above, and much public policy debate is about mode-of-transport
> tradeoffs between planes, trains and automobiles (and bicycles, and
> scooters, and and and...).  We need public policy debates around
> election method tradeoffs, too.

The feeling I get from it is roughly:

- Wouldn't it be nice if we had perfection even under honesty?
- Well, here's one reason we can't have perfection.
- So have fun with the complexity.

That is, Arrow's theorem is useful to say what you can't have. In your
vehicle analog, we would want to find out how to create that perfect
FTL-equipped free car if we could. Arrow's theorem just tells us that we
can stop looking. (And Gibbard's is like: even if your "vehicle" moves
the world instead of moving itself, thus making Arrow no longer apply,
you still can't get perfection under strategy.)

> I'm hoping we can try to stop trying to declare clever loopholes in
> Arrow's theorem, and just acknowledge the reality that *all* voting
> systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
> central insight (there's no "perfect" system given perfectly
> reasonable criteria) is valid, and that it's only on the specifics of
> the exact criteria chosen for the 1951 proof that might be flawed.  I
> believe that election method activists should speak (and write) with
> clarity about the tradeoffs involved.  Whenever I see someone
> gleefully declare that Arrow's theorem doesn't apply to their voting
> method (and imply perfection), the credibility of the writer drops
> *precipitously* in my mind.

It does seem that there are no perfect methods, even if we can
circumvent Arrow, and even if we don't care about strategy. For
instance, Approval requires calculation even by honest voters, and for
Range it's not even obvious what the one honest vote *is*.

It would be useful to get a proof of this, but I wouldn't know where to
start. So the observation that it's impossible to attain perfection must
be inductive rather than deductive, i.e. we have a hunch because nothing
has attained perfection until now.

I think the problem ultimately is: someone thinks "Arrow implies
imperfection, so if I get rid of Arrow, I have perfection". They confuse
implication for equivalence.
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Re: [EM] Arrow's theorem and cardinal voting systems

Toby Pereira
In reply to this post by Rob Lanphier
Arrow published a mathematical theorem so presumably everything was rigorously defined and not open to interpretation, so that would include unrestricted domain. On the Wikipedia it says "In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed." And while it might be defined differently and more precisely in Arrow's paper, I wouldn't say score fails by that definition. But it doesn't really matter anyway. You can define it in a way that score fails or define it in a way that it doesn't apply to score. Similarly you could define a condition where degrees of liking must be allowed, and ranked methods would fail that. And as has been said, it's not as if Arrow's Theorem is the be all and end all. All methods have their own problems, and whether they happen to be covered by one particular theorem is neither here nor there.

But what I would say is that I consider Arrow's Theorem to be possibly the most overrated and overstated theorem of all time. If you look at the criteria that ranked methods must fail one of according to Arrow's Theorem, most of them are just criteria that any remotely reasonable method would pass. The theorem, stated more informally, is basically that with a few reasonable background assumptions, all ranked-ballot methods fail independence of irrelevant alternatives. Which is interesting enough itself, except that this was known for centuries anyway from the Condorcet Paradox. If head to head A beats B, B beats C, and C beats A, then any winner in the three-way election has to overturn one of the head to head results as a result of an irrelevant alternative being added.

Toby

On Thursday, 9 January 2020, 23:17:56 GMT, Rob Lanphier <[hidden email]> wrote:


Hi folks,

As some of you might have seen, Electowiki is a lot more active than
it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
that has reservations is regarding how some advocates talk about
Arrow's theorem.  I'm hoping you all can do one of the following:
a)  change my view about Arrow's theorem, -or-
b)  offer me some help in better articulating my view about Arrow's theorem.

Many Score voting[1] activists claim that cardinal methods somehow
dodge Arrow's theorem.  It seems to me that *all* voting systems (not
a mere subset) are subject to some form of impossibility problem.
Arrow's impossibility theorem deserved great acclaim for subjecting
all mainstream voting systems of the 1950s to mathematical rigor, and
it's clear that his 1950 paper and 1951 book profoundly influenced
economics and game theory for the better.  His 1972 Nobel prize was
well deserved.  It seems that it has become fashionable to find
loopholes in Arrow's original formulation and declare the loopholes
important.  Even if the loopholes exist, talking up those loopholes
doesn't seem compelling, given the subsequent work by other theorists
broaden the scope beyond Arrow's version.

But, what the heck, let's actually talk about Arrow's original
formulation.  I believe Score voting fails unrestricted domain:

In particular, let's say that 90% of voters prefer candidate A over candidate B:
90:A>B
10:B>A

Arrow posits that there should only be one way to express that, and
Score fails it.  In Score, it's possible to sometimes pick A, and
sometimes pick B, depending on the score values on the ballots.  If
Score *always* chose either A or B, then it would pass Universality.

Score advocates claim that this isn't a bug, it's a *feature*.  If
(for example), voters for A only mildly prefer A over B, but voters
for B strongly detest A, then the correct social choice is B.
However, it doesn't seem practical to inflict this level of nuance on
voters.  I suspect that the first election where the Condorcet winner
is beaten by a minority-preferred candidate (e.g. like what happened
in Burlington 2009 [2]) will result in a repeal (like what happened in
Burlington).  Back to the A/B example above, It's hard to imagine
voters would consider the selection of "B" to be fair in a large
election.

It's fine to hold the opinion that Universality is an uninteresting
criterion, and that therefore, Arrow's set of criteria isn't very
interesting.  For example, a few years ago, we went through a phase
where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
interesting criterion, and advocating for Condorcet variants that meet
that criterion.  Regardless, just because we find one criterion less
compelling than another, we should talk accurately about the failed
criterion.

My way of thinking about Arrow's theorem (and being thankful for it)
is to think of it like the physics of voting systems.  For example, in
real-world physics, a "perfect" vehicle is impossible, because it's
impossible to meet these criteria:
* Goes faster than the speed of light
* Has infinite capacity
* Has a luxurious and comfortable passenger cabin
* Fits in a small coat pocket
* Is easy to produce
* Is cheap (or even free)

Just because a perfect vehicle is not possible, I'm glad
transportation innovation didn't stop with Ford's Model T.  Of course,
automobile sellers compete on the tradeoffs between the criteria
above, and much public policy debate is about mode-of-transport
tradeoffs between planes, trains and automobiles (and bicycles, and
scooters, and and and...).  We need public policy debates around
election method tradeoffs, too.

I'm hoping we can try to stop trying to declare clever loopholes in
Arrow's theorem, and just acknowledge the reality that *all* voting
systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
central insight (there's no "perfect" system given perfectly
reasonable criteria) is valid, and that it's only on the specifics of
the exact criteria chosen for the 1951 proof that might be flawed.  I
believe that election method activists should speak (and write) with
clarity about the tradeoffs involved.  Whenever I see someone
gleefully declare that Arrow's theorem doesn't apply to their voting
method (and imply perfection), the credibility of the writer drops
*precipitously* in my mind.

Am I wrong?

Rob

p.s. I've been meaning to write this email for a while.  What inspired
me to finally write it has been reading the current state of
Electowiki and Wikipedia articles on the topic, like the "Arrow's
impossiblity theorem" article on Electowiki[4]

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Re: [EM] Arrow's theorem and cardinal voting systems

Steve Eppley
In reply to this post by Rob Lanphier
I think the spirit of Rob's question is (or should be):  Can any plausibly democratic voting method satisfy this Independence criterion: "Assuming voters' preferences don't change, the winner must not change if another candidate chooses not to compete." (Let's call methods plausibly democratic if they don't privilege any candidates or voters.  In other words, the Neutrality and Anonymity criteria in the literature of social choice theory.)

None can satisfy that Independence.

The last time I checked, advocates of Range Voting acknowledge that when there are only two candidates, optimal voting strategy is to give the highest possible score to the voter's most preferred candidate and the lowest possible score to the voter's least preferred candidate. (Some of those advocates may even think that's sincere voting, bless their little hearts.)  Since that's such an obvious strategy, it's reasonable to assume at least some of the voters will eventually learn to use it (just as many voters have learned to vote for a compromise to help defeat a "greater evil" given Plurality Rule).  Now consider an example: Suppose that given a majoritarian voting method such as Plurality Rule, Rock would beat Scissors, Scissors would beat Paper, and Paper would beat Rock, by a narrow majority in each pairing.  Suppose also that Rock would be the winner if Rock, Paper and Scissors compete given Range Voting (or Approval).  Which one wins if only Rock and Paper compete
given Range Voting (or Approval)?  Obviously, Paper can win, since the majority who prefer Paper can elect Paper using their optimal voting strategy.  Thus the winner can change from Rock to Paper when Scissors doesn't compete, which violates Independence.

Given the fact that strategic voting is possible, many (most?) criteria definitions are "naive."  For instance, a "Condorcet winner given sincere voting" can lose given a voting method that satisfies the Condorcet criterion.  I hope we can all agree that the spirit of the Condorcet criterion is that the sincere Condorcet winner should win (when it exists).  Call that the Sincere Condorcet criterion.  No plausibly democratic voting method can satisfy Sincere Condorcet, but some voting methods can perform better on it than others, by making defensive strategies easier and/or more palatable, or offensive strategies riskier.

I presume the same is true regarding Independence: some plausibly democratic voting methods perform better than others on Independence, even though none satisfy Independence.

For the criterion that matters most to me, I don't have a rigorous definition.  Here's a non-rigorous definition:  The voting method should give candidates who want to win a strong incentive to take positions that the voters themselves would collectively choose given a well-functioning direct democracy... even on issues that most voters don't care strongly about.  Here's how I relate that to voting methods like Maximize Affirmed Majorities (MAM), which facilitate competition, count all pairwise majorities, and pay attention to the sizes of the majorities:  Suppose candidate Alice wants to win, and is considering taking position p on some issue.  Although she knows a majority of the voters prefer alternative q over p, her wealthy campaign donors favor p and most voters care more about other issues.  Given a voting method like MAM, the risk to Alice is that by advocating p, she would create an opportunity for another candidate Bob to enter the race, take position q and copy
Alice's positions on all other issues.  The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice.  Defeating Alice.  A deterrent against taking unpopular positions to benefit donors.

--Steve

On 1/9/2020 6:17 PM, Rob Lanphier wrote:

> Hi folks,
>
> As some of you might have seen, Electowiki is a lot more active than
> it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
> that has reservations is regarding how some advocates talk about
> Arrow's theorem.  I'm hoping you all can do one of the following:
> a)  change my view about Arrow's theorem, -or-
> b)  offer me some help in better articulating my view about Arrow's theorem.
>
> Many Score voting[1] activists claim that cardinal methods somehow
> dodge Arrow's theorem.  It seems to me that *all* voting systems (not
> a mere subset) are subject to some form of impossibility problem.
> Arrow's impossibility theorem deserved great acclaim for subjecting
> all mainstream voting systems of the 1950s to mathematical rigor, and
> it's clear that his 1950 paper and 1951 book profoundly influenced
> economics and game theory for the better.  His 1972 Nobel prize was
> well deserved.  It seems that it has become fashionable to find
> loopholes in Arrow's original formulation and declare the loopholes
> important.  Even if the loopholes exist, talking up those loopholes
> doesn't seem compelling, given the subsequent work by other theorists
> broaden the scope beyond Arrow's version.
>
> But, what the heck, let's actually talk about Arrow's original
> formulation.  I believe Score voting fails unrestricted domain:
> <https://en.wikipedia.org/wiki/Unrestricted_domain>
>
> In particular, let's say that 90% of voters prefer candidate A over candidate B:
> 90:A>B
> 10:B>A
>
> Arrow posits that there should only be one way to express that, and
> Score fails it.  In Score, it's possible to sometimes pick A, and
> sometimes pick B, depending on the score values on the ballots.  If
> Score *always* chose either A or B, then it would pass Universality.
>
> Score advocates claim that this isn't a bug, it's a *feature*.  If
> (for example), voters for A only mildly prefer A over B, but voters
> for B strongly detest A, then the correct social choice is B.
> However, it doesn't seem practical to inflict this level of nuance on
> voters.  I suspect that the first election where the Condorcet winner
> is beaten by a minority-preferred candidate (e.g. like what happened
> in Burlington 2009 [2]) will result in a repeal (like what happened in
> Burlington).  Back to the A/B example above, It's hard to imagine
> voters would consider the selection of "B" to be fair in a large
> election.
>
> It's fine to hold the opinion that Universality is an uninteresting
> criterion, and that therefore, Arrow's set of criteria isn't very
> interesting.  For example, a few years ago, we went through a phase
> where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
> interesting criterion, and advocating for Condorcet variants that meet
> that criterion.  Regardless, just because we find one criterion less
> compelling than another, we should talk accurately about the failed
> criterion.
>
> My way of thinking about Arrow's theorem (and being thankful for it)
> is to think of it like the physics of voting systems.  For example, in
> real-world physics, a "perfect" vehicle is impossible, because it's
> impossible to meet these criteria:
> * Goes faster than the speed of light
> * Has infinite capacity
> * Has a luxurious and comfortable passenger cabin
> * Fits in a small coat pocket
> * Is easy to produce
> * Is cheap (or even free)
>
> Just because a perfect vehicle is not possible, I'm glad
> transportation innovation didn't stop with Ford's Model T.  Of course,
> automobile sellers compete on the tradeoffs between the criteria
> above, and much public policy debate is about mode-of-transport
> tradeoffs between planes, trains and automobiles (and bicycles, and
> scooters, and and and...).  We need public policy debates around
> election method tradeoffs, too.
>
> I'm hoping we can try to stop trying to declare clever loopholes in
> Arrow's theorem, and just acknowledge the reality that *all* voting
> systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
> central insight (there's no "perfect" system given perfectly
> reasonable criteria) is valid, and that it's only on the specifics of
> the exact criteria chosen for the 1951 proof that might be flawed.  I
> believe that election method activists should speak (and write) with
> clarity about the tradeoffs involved.  Whenever I see someone
> gleefully declare that Arrow's theorem doesn't apply to their voting
> method (and imply perfection), the credibility of the writer drops
> *precipitously* in my mind.
>
> Am I wrong?
>
> Rob
>
> p.s. I've been meaning to write this email for a while.  What inspired
> me to finally write it has been reading the current state of
> Electowiki and Wikipedia articles on the topic, like the "Arrow's
> impossiblity theorem" article on Electowiki[4]
>
> [1]: https://electowiki.org/wiki/Score_voting
> [2]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election
> [3]: https://electowiki.org/wiki/IIAC
> [4]: https://electowiki.org/wiki/Arrow%27s_impossibility_theorem
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> Election-Methods mailing list - see https://electorama.com/em for list info


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Re: [EM] Arrow's theorem and cardinal voting systems

Kristofer Munsterhjelm-3
In reply to this post by robert bristow-johnson
On 10/01/2020 05.46, robert bristow-johnson wrote:

> my question that i have asked the Score Voting or Approval Voting
> advocates years ago remains: "How much should I score my second choice?"
> or "Should I approve my second choice or not?"  that tactical question
> faces the voter in a Score or Approval election the second he/she steps
> into the voting booth.  but not so for the ordinal Ranked ballot.

I tended to phrase this as: Approval requires that the voters engage in
manual DSV (or earlier, that they "dither" - in the sense of reducing a
full-color image to black-and-white). Approval satisfies so many
properties on paper by placing the burden on the voter instead.

But I've recently found a paper that puts this very clearly:
https://www.jstor.org/stable/1955800 (use Sci-Hub if you don't have
access). The paper shows that if the voter preferences aren't naturally
dichotomous (a bunch of equal ranked candidates at top above a bunch of
equal ranked candidates at bottom), then the Condorcet winner may end up
being dead last by approval score; and that for a particular type of
strategy, any candidate may be an equilibrium Approval winner.

To quote from the conclusion: "Strategic calculations are endemic to AV
even though all of the votes considered are called sincere. Given the
literature's emphasis on approval and disapproval sets - even the name,
approval - and the fact that voting for all approved candidates and no
others is optimal for dichotomous preferences, one gets the false
impression that AV will eliminate strategic thinking and voting. The
results here show that this is far from true, however."
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Re: [EM] Arrow's theorem and cardinal voting systems

fdpk69p6uq
In reply to this post by Rob Lanphier
On Thu, Jan 9, 2020 at 11:46 PM robert bristow-johnson wrote:
"strong preference" vs. "weak preference" implies a Score ballot.


my question that i have asked the Score Voting or Approval Voting advocates years ago remains: "How much should I score my second choice?"

If asked to rank three ice cream flavors, my preference would be Strawberry > Chocolate > Garlic. 

If then asked to choose between:

1. Chocolate
2. A mystery box with a 75% chance of containing Strawberry and a 25% chance of containing Garlic

I would choose #1, which shows that:

A. My preference for Chocolate > Garlic is significantly stronger than my preference for Strawberry > Chocolate.
B. If voting honestly, I should give Chocolate at least a 4 out of 5 on a Score ballot.

The odds can then be varied, to narrow in on a more precise rating, which is essentially what we all do internally when we rate a movie or restaurant or product or student or respond to a Likert scale survey, etc.

Of course, this is imprecise, but so is forcing voters to rank many candidates when they are indifferent between some of them.


If Vanilla and French Vanilla were both on the same ballot, I would be indifferent between them.  Forcing me to choose between them and then arbitrarily assigning the same weight to this very weak preference that was applied to my Chocolate > Garlic preference would be rather undemocratic, no?  http://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html

Organisms don't have ordered lists of equal-strength preferences in their brains.  They have fuzzy estimates of utility that they then convert to rankings when necessary.

"The majority judgement experiment proves that the model on which the theory of social choice and voting is based is simply not true: voters do not have preference lists of candidates in their minds. Moreover, forcing voters to establish preference lists only leads to inconsistencies, impossibilities and incompatibilities." https://hal.archives-ouvertes.fr/hal-00243076/document#page=40
 
that tactical question faces the voter in a Score or Approval election the second he/she steps into the voting booth.  but not so for the ordinal Ranked ballot.

From what I've been told (though I haven't read and understood it myself), Gibbard's theorem proves that ALL voting systems require voters to make tactical decisions, no matter whether they are ranked or rated or otherwise.



On Fri, Jan 10, 2020 at 2:02 AM Rob Lanphier wrote:
that cardinal voting systems are provably free of any
sort of impossibility paradox.

I've never heard anyone claim that they are.  The claim is simply that Arrow's theorem, in particular, doesn't apply to cardinal systems, which Arrow seems to agree with in that interview (and I don't know why that would be a big deal or why he shouldn't be trusted to interpret his own theorem).  Satterthwaite's also doesn't. It's Gibbard's theorem, specifically, that applies to the general case of all conceivable voting systems: https://politics.stackexchange.com/a/14245/10373

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Re: [EM] Arrow's theorem and cardinal voting systems

Kristofer Munsterhjelm-3
In reply to this post by Steve Eppley
On 10/01/2020 12.41, Steve Eppley wrote:

> For the criterion that matters most to me, I don't have a rigorous definition.  Here's a non-rigorous definition:  The voting method should give candidates who want to win a strong incentive to take positions that the voters themselves would collectively choose given a well-functioning direct democracy... even on issues that most voters don't care strongly about.  Here's how I relate that to voting methods like Maximize Affirmed Majorities (MAM), which facilitate competition, count all pairwise majorities, and pay attention to the sizes of the majorities:  Suppose candidate Alice wants to win, and is considering taking position p on some issue.  Although she knows a majority of the voters prefer alternative q over p, her wealthy campaign donors favor p and most voters care more about other issues.  Given a voting method like MAM, the risk to Alice is that by advocating p, she would create an opportunity for another candidate Bob to enter the race, take position q and copy
> Alice's positions on all other issues.  The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice.  Defeating Alice.  A deterrent against taking unpopular positions to benefit donors.

How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
ranked above A2 on more ballots than A2 is ranked above A1, then if the
original winner was A, the new winner should be A1.

That most voters care about other issues than p vs q means that Alice
and Bob should be near-clones, since "Alice but with q" is a slight
improvement to "Alice with p", but not enough of an improvement that
some other candidate is ranked between A1 and A2.

If voters care more about q vs p, then A1 and A2 will no longer be
near-clones, but hopefully the method should generalize robustly from
the clone case so that it follows the spirit of the criterion.
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[EM] An incentive to take positions a direct democracy would choose (was Re: Arrow's theorem and cardinal voting systems)

Steve Eppley
On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:

> On 10/01/2020 12.41, Steve Eppley wrote:
>> For the criterion that matters most to me, I don't have a rigorous definition.  Here's a non-rigorous definition:  The voting method should give candidates who want to win a strong incentive to take positions that the voters themselves would collectively choose given a well-functioning direct democracy... even on issues that most voters don't care strongly about.  Here's how I relate that to voting methods like Maximize Affirmed Majorities (MAM), which facilitate competition, count all pairwise majorities, and pay attention to the sizes of the majorities:  Suppose candidate Alice wants to win, and is considering taking position p on some issue.  Although she knows a majority of the voters prefer alternative q over p, her wealthy campaign donors favor p and most voters care more about other issues.  Given a voting method like MAM, the risk to Alice is that by advocating p, she would create an opportunity for another candidate Bob to enter the race, take position q and copy
>> Alice's positions on all other issues.  The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice.  Defeating Alice.  A deterrent against taking unpopular positions to benefit donors.
> How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
> ranked above A2 on more ballots than A2 is ranked above A1, then if the
> original winner was A, the new winner should be A1.
>
> That most voters care about other issues than p vs q means that Alice
> and Bob should be near-clones, since "Alice but with q" is a slight
> improvement to "Alice with p", but not enough of an improvement that
> some other candidate is ranked between A1 and A2.
>
> If voters care more about q vs p, then A1 and A2 will no longer be
> near-clones, but hopefully the method should generalize robustly from
> the clone case so that it follows the spirit of the criterion.

If by "How about this?" you're suggesting satisfaction of that clone criterion ("... the new winner should be A1") implies satisfaction of my non-rigorous criterion ("create a strong incentive to take positions the voters would choose"), I don't see why that would be so. 

Instant Runoff would elect A1 in that clone criterion's scenario, yes?  But Instant Runoff doesn't create the incentive.  To the contrary, Instant Runoff defeats candidates who advocate compromises that voters would collectively choose, and makes those candidates & positions appear unpopular.  Instant Runoff rewards extremists the same way Plurality Rule does, because it counts at most one of the majorities, which can be a coalition of minorities.  For example, a minority who want abortion banned, plus a minority who want immigrants deported, plus a minority who want guns unregulated, plus a minority who want capital gains taxes slashed, etc, can together add up to a majority.

--Steve

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Re: [EM] An incentive to take positions a direct democracy would choose (was Re: Arrow's theorem and cardinal voting systems)

Kristofer Munsterhjelm-3
On 11/01/2020 17.46, Steve Eppley wrote:

> On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:
>> On 10/01/2020 12.41, Steve Eppley wrote:
>>> For the criterion that matters most to me, I don't have a
>>> rigorous definition.  Here's a non-rigorous definition:  The
>>> voting method should give candidates who want to win a strong
>>> incentive to take positions that the voters themselves would
>>> collectively choose given a well-functioning direct democracy...
>>> even on issues that most voters don't care strongly about.
>>> Here's how I relate that to voting methods like Maximize Affirmed
>>> Majorities (MAM), which facilitate competition, count all
>>> pairwise majorities, and pay attention to the sizes of the
>>> majorities:  Suppose candidate Alice wants to win, and is considering
>>> taking position p on some issue.  Although she knows a majority of the
>>> voters prefer alternative q over p, her wealthy campaign donors favor p
>>> and most voters care more about other issues.  Given a voting method
>>> like MAM, the risk to Alice is that by advocating p, she would create an
>>> opportunity for another candidate Bob to enter the race, take position q
>>> and copy Alice's positions on all other issues.  The larger the
>>> majority who prefer q over p, the larger the majority who would
>>> tend to rank Bob over Alice.  Defeating Alice.  A deterrent
>>> against taking unpopular positions to benefit donors.

>> How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
>> ranked above A2 on more ballots than A2 is ranked above A1, then if the
>> original winner was A, the new winner should be A1.
>>
>> That most voters care about other issues than p vs q means that Alice
>> and Bob should be near-clones, since "Alice but with q" is a slight
>> improvement to "Alice with p", but not enough of an improvement that
>> some other candidate is ranked between A1 and A2.
>>
>> If voters care more about q vs p, then A1 and A2 will no longer be
>> near-clones, but hopefully the method should generalize robustly from
>> the clone case so that it follows the spirit of the criterion.

> If by "How about this?" you're suggesting satisfaction of that clone
> criterion ("... the new winner should be A1") implies satisfaction of my
> non-rigorous criterion ("create a strong incentive to take positions the
> voters would choose"), I don't see why that would be so.
>
> Instant Runoff would elect A1 in that clone criterion's scenario, yes?

I don't think it would in every such scenario. Consider this election pair:

Before cloning:

110: A
100: X>A
100: Y>A

X and Y are eliminated and then A wins.

After cloning:

110: A2>A1
100: X>A1>A2
100: Y>A1>A2

First A1 is eliminated, and then X and Y are eliminated, and then A2
wins. But A1 is the CW and beats A2 pairwise 200-110.

If the q-preferring majority ranks A1 and A2 low enough, then IRV may
exclude A1 before it gets to determine who should win of A1 and A2. It's
the usual center squeeze.

Does that make the clone criterion more suited to your purposes, or
would it have to be stronger? I suppose the clone criterion is a sort of
local optimum criterion (if Alice exists, then Bob can copy all of
Alice's positions except the one a majority dislikes, and overtake
Alice), while your non-rigorous criterion is a global optimum criterion.

(In passing, I think I see that LIAA + clone independence implies this
clone criterion, as well.)
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Re: [EM] An incentive to take positions a direct democracy would choose (was Re: Arrow's theorem and cardinal voting systems)

Steve Eppley
On 1/11/2020 2:42 PM, Kristofer Munsterhjelm wrote:

> On 11/01/2020 17.46, Steve Eppley wrote:
>> On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:
>>> On 10/01/2020 12.41, Steve Eppley wrote:
>>>> For the criterion that matters most to me, I don't have a
>>>> rigorous definition.  Here's a non-rigorous definition:  The
>>>> voting method should give candidates who want to win a strong
>>>> incentive to take positions that the voters themselves would
>>>> collectively choose given a well-functioning direct democracy...
>>>> even on issues that most voters don't care strongly about.
>>>> Here's how I relate that to voting methods like Maximize Affirmed
>>>> Majorities (MAM), which facilitate competition, count all
>>>> pairwise majorities, and pay attention to the sizes of the
>>>> majorities:  Suppose candidate Alice wants to win, and is considering
>>>> taking position p on some issue.  Although she knows a majority of the
>>>> voters prefer alternative q over p, her wealthy campaign donors favor p
>>>> and most voters care more about other issues.  Given a voting method
>>>> like MAM, the risk to Alice is that by advocating p, she would create an
>>>> opportunity for another candidate Bob to enter the race, take position q
>>>> and copy Alice's positions on all other issues.  The larger the
>>>> majority who prefer q over p, the larger the majority who would
>>>> tend to rank Bob over Alice.  Defeating Alice.  A deterrent
>>>> against taking unpopular positions to benefit donors.
>>> How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
>>> ranked above A2 on more ballots than A2 is ranked above A1, then if the
>>> original winner was A, the new winner should be A1.
>>>
>>> That most voters care about other issues than p vs q means that Alice
>>> and Bob should be near-clones, since "Alice but with q" is a slight
>>> improvement to "Alice with p", but not enough of an improvement that
>>> some other candidate is ranked between A1 and A2.
>>>
>>> If voters care more about q vs p, then A1 and A2 will no longer be
>>> near-clones, but hopefully the method should generalize robustly from
>>> the clone case so that it follows the spirit of the criterion.
>> If by "How about this?" you're suggesting satisfaction of that clone
>> criterion ("... the new winner should be A1") implies satisfaction of my
>> non-rigorous criterion ("create a strong incentive to take positions the
>> voters would choose"), I don't see why that would be so.
>>
>> Instant Runoff would elect A1 in that clone criterion's scenario, yes?
> I don't think it would in every such scenario. Consider this election pair:
>
> Before cloning:
>
> 110: A
> 100: X>A
> 100: Y>A
>
> X and Y are eliminated and then A wins.
>
> After cloning:
>
> 110: A2>A1
> 100: X>A1>A2
> 100: Y>A1>A2
>
> First A1 is eliminated, and then X and Y are eliminated, and then A2
> wins. But A1 is the CW and beats A2 pairwise 200-110.
>
> If the q-preferring majority ranks A1 and A2 low enough, then IRV may
> exclude A1 before it gets to determine who should win of A1 and A2. It's
> the usual center squeeze.
>
> Does that make the clone criterion more suited to your purposes, or
> would it have to be stronger? I suppose the clone criterion is a sort of
> local optimum criterion (if Alice exists, then Bob can copy all of
> Alice's positions except the one a majority dislikes, and overtake
> Alice), while your non-rigorous criterion is a global optimum criterion.
>
> (In passing, I think I see that LIAA + clone independence implies this
> clone criterion, as well.)

You're right that Instant Runoff fails "clone A1 should win."

I don't know whether its satisfaction implies satisfaction of "the incentive to take positions the voters would choose."  My election method analysis skills are very rusty.

I don't recall LIAA.  I assume you mean LIIA (Local Independence of Irrelevant Alternatives, promoted by Peyton Young).

There appears to be a flaw in that clone criterion.  Suppose 3 clones majority cycle: Bob > Alice > Charlie > Bob.  The premise of the "clone A1 should win" criterion could hold: In the "original" scenario where Bob doesn't run, Alice wins.  We don't have enough information to show that Bob will win if Bob runs too.  Alice could still win if the Bob>Alice majority is the smallest of the three cyclic majorities. (When I described my thinking about the incentive in MAM, I wrote: "The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice.")  But that clone failure isn't necessarily a failure of the voting method to create the strong incentive.  My hunch is that typically, candidates like Alice won't be able to rely on a Bob>Alice majority being the smallest in a cycle, when taking positions on issues.  The chance that Bob>Alice won't be smallest in a cycle is a risk to be avoided, all else being equal.

Thanks for spending time on this.  I hope you can continue.

--Steve
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Re: [EM] Arrow's theorem and cardinal voting systems

Rob Lanphier
In reply to this post by James J Faran
Hi Jim,

Thanks for the thoughtful response!  I *want* to respond to everyone,
but I suppose I'll just circle back on respond to yours now.  More
inline...

On Thu, Jan 9, 2020 at 8:12 PM Faran, James <[hidden email]> wrote:

> About Score voting failing Unrestricted Domain:
>
> Part of the confusion of those advocating score and you is not a confusion on anyone's part, but rather a difference in what each considers a preference.  (It's not possible to have a good reasoned argument until both sides agree on what the words mean.)  Score voters would say
>
> A:100; B:95; C:0
>
> and
>
> A:100; B:5; C:0
>
> are different preferences, but you seem to say that these are both A>B>C and so are the same.

Hmm, that seems to be an interesting way of putting it.  I agree that
this difference is at the heart of the matter.  It seems that Arrow's
proof also relies on these being the same, and I believe the criterion
stipulates that they *must* be the same.  That doesn't mean Arrow's
theorem doesn't apply to systems that allow for this extra
information; it just means that those systems don't meet that
criterion.

> I would say the Electowiki page on Unrestricted Domain needs to be edited to include both possibilities, but I can't be bothered.

Well, I'm not going to edit it for you ;-P

I suppose it would be good to cover many of the points of this
discussion on that page, but I'd prefer to come to a shared
understanding before I make any edits.

> You also seem to think that most voters would not be able to understand that sort of nuance.  You may be right there, especially in today's political climate (especially in the United States?), where there are two sides and the other side is always demonized.

Well, it's not the first time the United States had problems with
partisanship causing things to get personal:
* https://en.wikipedia.org/wiki/George_Washington%27s_Farewell_Address
* https://en.wikipedia.org/wiki/Burr%E2%80%93Hamilton_duel
* https://en.wikipedia.org/wiki/Petticoat_affair
* https://en.wikipedia.org/wiki/Caning_of_Charles_Sumner
* https://en.wikipedia.org/wiki/American_Civil_War

I don't think the problem is today's political climate (here in the
USA or elsewhere).  The problem is with getting enough people to agree
that the system is fair.

> Note that any new voting system will almost always try to be replaced by the loser under the new system.  ("The current government is illegitimate!  If it wasn't for the biased voting system we would have won!" -- cf. the recently revived call for the elimination of the U. S. Electoral College after Mr. Trump won with a minority of the popular vote.)  If the winner can't keep support, the losing side will be able to push through a change.
>
> However, a question:  If we had the following score ballots:
>
> 9000:  A:100; B:95; C:0
> 1000:  B:100; C:85; A:0
>
> giving A a score of 900,000 and B a score of 955,000, hence a victory for B, would there really be enough antipathy to B to cause outrage?  All the A voters seemed to think B was pretty good.

I think the answer would be "yes", once the people who were passionate
about A discovered that A was preferred to B 9:1 by 9,000 out of the
10,000 voters, and that it was only those people who rated crackpot C
as "85" who swung the election.  As others have pointed out on the
list, it's probably not a good idea for a voting system to push these
mathematical nuance problems onto voters.

> Of course (see above), the losing side could always complain.

Well, sure, but when the losing side has a point, that's a problem for
the advocates for the electoral system in question.

> Anyone wedded to Condorcet winners would be outraged.  And, of course, no real world election would end up like this.  Score may be a little too ripe for manipulation.  Gibbard-Satterthwaite, anyone?

Like I said in my original email, I think the great thing about these
impossibility theorems (like Arrow's and Gibbard-Satterthwaite) is
that they demonstrate cases where tradeoffs will be necessary.  Your
exaggerated example is helpful in coming to a shared understanding of
a edge case in the system that could also be manifested in a more
realistic example.

Rob
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Re: [EM] Arrow's theorem and cardinal voting systems

Forest Simmons
In reply to this post by Rob Lanphier
Rob,

Thanks for starting this great thread!

The "no perfect car" analogy is good.  More definite is the "no 100 percent efficient internal combustion engine" analogy that follows from the second law of thermodynamics.  It applies to all kinds of engines, but that doesn't mean that internal combustion is as good as it gets.

If Gibbard-Satterthwaite tells us that we cannot have all of the nice properties we want in one election method, that doesn't mean that one method is as good as the next.

It follows from Arrow that we cannot have the Majority Criterion and the IIAC at the same time, but there are many decent methods (like River) that do satisfy the MC, and a bunch of other nice properties, like Monotonicity, Clone Independence, the Condorcet Criterion, and Independence from Pareto Dominated Alternatives, as well as the basic Neutrality and Anonymity fairness criteria.

The way to think of Arrow's "Dictator" theorem is that it is extremely hard to get a rankings based method with even minimal decency conditions (like non-dictatorship) without scuttling the IIAC.

In other words, no decent ordinal based method can satisfy the IIAC, which is the same point of view that Toby and Eppley expressed.  It comes down to the mere existence of a Condorcet Cycle.  Here's the subtle part that most people don't understand.  Condorcet Cycles can exist in the preference schedules of an election even if the election method makes no mention of Condorcet, for example even in IRV/Hare/STV/RCV elections:

45 A>B>C
20 B>C>A
35 C>A>B

There exists a majority preference cycle A>B>C>A even though it causes no problem for IRV, since B is eliminated and then C is the majority winner between the two remaining candidates.

Now let's check the IIAC.  Suppose that A, one of the losers withdraws from the race.  Then the winner changes from C to B, since B beats C by a majority.  This shows that IRV does not satisfy the IIAC, because removing a loser from the ballot changes the winner.

But this is not just a problem for IRV, it's a problem for any method that respects the Majority Criterion; if the method makes A the winner, then removing B changes the winner.  If it makes B the winner, then removing C changes the winner.  If it makes C the winner, then (as we saw in the case of IRV above) removing A changes the winner. to B.

So Arrow's "paradox" can be considered as forcing us to realize that the IIAC is not a realistic possibility in the presence of ordinal ballots because such ballots allow us to detect oairwise (head-to-head) preferences, and when it comes down to a single pair of candidates the Majority Criterion says the pairwise winner must be chosen,

However, as someone mentioned, Approval Voting avoids this "paradox" once the ballots have been submitted, since the Approval winner A is always the "ballot CW," and in two different ways:(1) For any other candidate X, candidate A will be rated above X on more ballots than not, and (2) A's approval score will be higher than the sore of any other candidate.  From either point of view, if we remove a loser Y from the ballots, then A will still be the winner according to the same ballots with Y crossed out.

That's at the ballot level.  But if Y withdrew before the ballots were filled out, it could change the winner, because if Y were the only approved candidate for a certain voter before the withdrawal, that voter might decide to lower her personal approval cutoff before submitting her ballot.  Or she could raise the cutoff if Y had been the only disapproved candidate.

As others have mentioned in this discussion, Approval Voting externalizes the problem of the IIAC from being a decision problem for the method itself to a strategical decision problem for the voter. A voter might think of that as an unfair burden.

One answer to this problem could be DSV (Designated Strategy Voting): You submit your sincere ratings, and the DSV machine applies a strategy of your choice or a default strategy to transform the ballots into approval style ballots.  Rob LeGrand explored some of the possibilities and limitations of this approach in his master's thesis.  He doesn't claim to have exhausted the possibilities.  (I also have some ideas in this vein that still need exploring.)

What constitutes a "sincere rating."  One approach to that has already been mentioned in the ice-cream flavor context in this thread. Another is to use as a rating for candidate X your subjective probability that on a typical issue of any significance candidate X would support the same side you support.

It's not just Approval that requires some hard thinking in conjunction with filling out the ballots. Ranking many candidates (think about the number of candidates in the election that propelled Schwarznegger into office) may be just as burdensome as trying to decide exactly which candidates to mark as approved. In Australia you can get around this difficulty by copying "candidate cards" or by voting the party line. Presumably these experts are reflecting state of the art strategy in their rankings ... the strategy that is indispensable for optimum results according to Gibbard-Satterthwaite.  This is not just a problem of Approval, though it may seem worse in Approval.  In actuality, aoproval and score/range are the only commonly used methods where optimal strategy never requires you to "betray " your favorite.

To cut the Gordian knot of this complexity Charles Dodgson (aka Lewis Carroll) suggested what we now call Asset Voting. Each Voter delegates her vote to the candidate she trusts the most to rep[resent her in the decision process. Since write-ins are allowed, she can write in herself if she doesn't trust anybody else to be her proxy.  These proxies get together with their "assets"  (delegated votes) and choose a winner by use of some version of Robert's Rules of Order.

Which criteria are satisfied by this method? Does Gibbard Satthethwaite have anything to say about it? How about Arrow?  For that matter does first past the post plurality satisfy the IIAC? (No more or less than Approval in reality.)

Let's talk about Gibbard-Satterthwaite.  Is there any incentive for a person to delegate as proxy someone other than her favorite? 

If we are talking representative democracy, then why would you want to delegate your vote to candidate B when candidate A was the one you trusted most to represent you in making important decisions once in office?

All of the "problems" with the method are essentially externalized to the deliberations governed by"Robert's Rules of Order" in the smoke filled room.

Gibbard-Satterthwaite is taken to say that it is impossible to obtain sincere preferences or sincere utilities from voters in the context of full information (or disinformation) elections.  Yet it turns out to be relatively easy; you just need to separate the ballot into two parts.  The first part requires strategic voting to pick the two alternatives as finalists.  The second part is used solely to choose between these two options.  (In the case of cardinal ballots the finalists are lotteries.)  A version of the uncertainty principle obtains here: if you use the sincere ballots for any other instrumental purpose than to choose between the two finalists, then you almost certainly destroy their sincerity.

However there would be no problem comparing tthe sincere part with the strategical part to get statistics about voters' willingness to vote insincerely in choosing the finalists.

Another avenue that has been barely explored is the use of chance to incentivize consensus when there is a potential for it.

For example, suppose that preferences are

60 A>C>>>>B
40 B>C>>>>A

Under approval voting the A faction has a strong incentive to downgrade C and vote 60 A>>>>>C>B making A the (insincere) approval winner as Gibbard-satterthwaite would predict.

However, if the rules said that in the absence of a full consensus approval winner, the winner would be chosen by random ballot, then (assuming rational voters voting in their own interest) C would be the sure outcome; no rational voter in either faction would prefer random ballot expectations over a sure deal on C.

Jobst Heitzig is the pioneer in this area.

In sum, Arrow, et.al. should not constitute a nail in the coffin of creative progress in Election Methods.  IMHO that is an important message we need to send if we want to attract new talent.

Forest





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Re: [EM] Arrow's theorem and cardinal voting systems

Kristofer Munsterhjelm-3
In reply to this post by fdpk69p6uq
On 11/01/2020 06.24, [hidden email] wrote:
> On Thu, Jan 9, 2020 at 11:46 PM robert bristow-johnson wrote:

>>    my question that i have asked the Score Voting or Approval Voting
>>    advocates years ago remains: "How much should I score my second choice?"
>
>
> If asked to rank three ice cream flavors, my preference would be
> Strawberry > Chocolate > Garlic. 
>
> If then asked to choose between:
>
> 1. Chocolate
> 2. A mystery box with a 75% chance of containing Strawberry and a 25%
> chance of containing Garlic
>
> I would choose #1, which shows that:
>
> A. My preference for Chocolate > Garlic is significantly stronger than
> my preference for Strawberry > Chocolate.
> B. If voting honestly, I should give Chocolate at least a 4 out of 5 on
> a Score ballot.
>
> The odds can then be varied, to narrow in on a more precise rating,
> which is essentially what we all do internally when we rate a movie or
> restaurant or product or student or respond to a Likert scale survey, etc.
>
> Of course, this is imprecise, but so is forcing voters to rank many
> candidates when they are indifferent between some of them.

That sounds like it's still a normalized ballot, so that "honest Range"
with this calibration would fail Steve Eppley's independence. It doesn't
solve the problem of Range having many sincere ways to express the same
ballot, either, though it does narrow down which ballots are honest/sincere.

Suppose we have an ice cream election and (for the sake of the argument)
you're exactly indifferent between a 75:25 lottery between Strawberry
and Garlic, and a certain choice of Chocolate. (Suppose also that you're
risk neutral, because risk aversion is not the point.)

Then what we know is that
        0.75 * utility(Strawberry) + 0.25 * utility(Garlic) = utility(Chocolate)

This is a linear equation with three unknowns. We need two more to
unambiguously determine the values. A standard zero and a standard unit
will do.

A standard unit is reasonable to have, because multiplying every unknown
by some constant preserves all the lottery-based equations, so someone
who likes exaggerating his scale (e.g. by saying "OMG, this is the best
thing ever!" every time he sees something good) will have more influence
than someone who likes to be economical with his values.

But it's hardly clear how to *find* that standard zero and unit. For
e.g. a pizza election, you could say the standard zero is no pizza at
all and the standard unit is a Margherita (say). But for a political
election? And strictly speaking, the scale would have to be unbounded so
that it can both accommodate people who don't particularly like pizza
and people who have lived their whole lives for the purpose of getting a
pizza.

So the point is that the lack of a single reference honest ballot for
Range is a due to cardinal utilities being very hard to calibrate
between people. And if you can't say what the one honest ballot is, then
there will still be ambiguity in any cardinal system as to what
constitutes sincerity, and how you should rate your choices.

You may try to define a single honest ballot for a semi-cardinal method
that automatically normalizes the endpoints to max and min value, so
that the two unknowns are given and the voter can just use the lottery
method to fill in the remaining data. But if you do so, then a
two-candidate election becomes a majority election and you're back to
Eppley's independence failure example. So in some sense, the
impossibility is "tight" - if you want IIA, the ballots must be
independently calibrated. If you let them be relatively calibrated even
a little, IIA goes away.

That doesn't mean that rankings are better than ratings, period. But a
ranked ballot makes it possible to have a single honest ballot without
needing to standardize it -- at the expense of ranked methods failing
IIA. And the ambiguity of rated ballots makes honesty and strategy blur
together. Since no election method can know if you've chosen the right
scale, it seems like honesty and strategy will always be blurred
somewhat together, no matter the cardinal method.

> If Vanilla and French Vanilla were both on the same ballot, I would be
> indifferent between them.  Forcing me to choose between them and then
> arbitrarily assigning the same weight to this very weak preference that
> was applied to my Chocolate > Garlic preference would be rather
> undemocratic, no? 
> http://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html

To some degree, what I said above also holds for truncation and
equal-rank, but there it doesn't seem to be as serious a problem. It
would be interesting to find out why.

Perhaps truncation and equal rank are convenience features, so the voter
says "determining who wins of these is worth less to me than the effort
it is to rank those candidates, so I'll let someone else decide". That
might be a decision that a voter can do without needing to do any
absolute calibration. But if so, the problem with cardinal ballots is
then not that there are many honest ballots *as such*, but rather that
the voter is required to make a strategic effort.

> Organisms don't have ordered lists of equal-strength preferences in
> their brains.  They have fuzzy estimates of utility that they then
> convert to rankings when necessary.

(I have a suspicion that what we really have are utility vectors, and
what we call "utility" is more like a norm of these. But I have no
proof, and it's sort of besides the point.)

>     that tactical question faces the voter in a Score or Approval
>     election the second he/she steps into the voting booth.  but not so
>     for the ordinal Ranked ballot.
>
>
> From what I've been told (though I haven't read and understood it
> myself), Gibbard's theorem proves that ALL voting systems require voters
> to make tactical decisions, no matter whether they are ranked or rated
> or otherwise.

All deterministic ones, to be precise :-) And Gibbard doesn't say voters
*need* to do it - it only says that a voter who wants to maximize the
impact of his vote needs to do so.
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Re: [EM] Arrow's theorem and cardinal voting systems

Forest Simmons
In reply to this post by Forest Simmons
Just a couple of additional thoughts:

Besides Arrow and Gibbard-Sattherwaite we have lots of criterion incompatibility results from Woodall, and defensive strategy criteria from Mike Ossipoff, Steve Eppley, et. al..  In particular we never worried about Later No Help and later No Harm until Woodall came along. Venzke and Benham picked up the torch and brought Woodall into the EM Listserv discussion.

Many EM contributors have clarified which combinations of various criteria of more practical than academic stripe are compatible or not: Participation, FBC, Precinct Summability, Chicken, etc. 

In particular, we now know through the work of Ossipoff, Venzke, Benham, and others that the Chicken Defense and Burial Defense (against CW burial) are incompatible in the presence of Plurality and the the FBC, unless we allow an explicit approval cutoff or some other strategic switch on the ballots.  Standard ordinal ballots are not adequate for this even when truncation and equal rankings (including equal top) are allowed.   A non-standard ballot that allows us to get compatibility to all of these except the CC is MDDA(sc) which is Majority Defeat Disqualification Approval with symmetric completion below the approval cutoff.  This method also satisfies other basic criteria such as Participation, Clone Independence, Mono-Raise, Mono-Add, and IDPA, for example.

In the context of the current discussion, the approval cutoff or some equivalent strategic switch is essential for the compatibility of chicken resistance and burial resistance..  No strategy, no compatibility. So basically there is no decent method that is resistant to both Burial of the CW and Chicken offensives.  (IRV is chicken resistant and has a form of burial resistance, but routinely buries the CW unless voters strategically betray their favorite to save the CW. Furthermore, it fails mono-raise.)

Before collaborative efforts of EM List members there was no known clone independent, monotonic method for electing from the uncovered set.  The closest thing was Copeland, which is clone dependent.

Again the main point is that Arrow, and Gibbard-Satterthwaite are not the "end of history" for election methods, just like the collapse of the USSR was not the end of history as Fukuyama once proclaimed or Thatcher's famous TINA "there is no alternative" (to capitalism).  Arrow and G-S give very valuable insights and help us avoid cul-de-sacs, but they are not the last word in election methods progress.  The "end of history" and TINA slogans are an excuse for giving up prematurely for lack of imagination. We cannot allow Arrow and G-S to become excuses for lack of imagination in Election Methods.  What if Yee had given up before inventing the beautiful Yee diagrams that constitute an Electo-Kaleidoscope for the study of election methods analogous to the telesope and the electron microscope in astronomy as instruments in other branches of knowledge?

On Mon, Jan 13, 2020 at 3:32 PM Forest Simmons <[hidden email]> wrote:
Rob,

Thanks for starting this great thread!

The "no perfect car" analogy is good.  More definite is the "no 100 percent efficient internal combustion engine" analogy that follows from the second law of thermodynamics.  It applies to all kinds of engines, but that doesn't mean that internal combustion is as good as it gets.

If Gibbard-Satterthwaite tells us that we cannot have all of the nice properties we want in one election method, that doesn't mean that one method is as good as the next.

It follows from Arrow that we cannot have the Majority Criterion and the IIAC at the same time, but there are many decent methods (like River) that do satisfy the MC, and a bunch of other nice properties, like Monotonicity, Clone Independence, the Condorcet Criterion, and Independence from Pareto Dominated Alternatives, as well as the basic Neutrality and Anonymity fairness criteria.

The way to think of Arrow's "Dictator" theorem is that it is extremely hard to get a rankings based method with even minimal decency conditions (like non-dictatorship) without scuttling the IIAC.

In other words, no decent ordinal based method can satisfy the IIAC, which is the same point of view that Toby and Eppley expressed.  It comes down to the mere existence of a Condorcet Cycle.  Here's the subtle part that most people don't understand.  Condorcet Cycles can exist in the preference schedules of an election even if the election method makes no mention of Condorcet, for example even in IRV/Hare/STV/RCV elections:

45 A>B>C
20 B>C>A
35 C>A>B

There exists a majority preference cycle A>B>C>A even though it causes no problem for IRV, since B is eliminated and then C is the majority winner between the two remaining candidates.

Now let's check the IIAC.  Suppose that A, one of the losers withdraws from the race.  Then the winner changes from C to B, since B beats C by a majority.  This shows that IRV does not satisfy the IIAC, because removing a loser from the ballot changes the winner.

But this is not just a problem for IRV, it's a problem for any method that respects the Majority Criterion; if the method makes A the winner, then removing B changes the winner.  If it makes B the winner, then removing C changes the winner.  If it makes C the winner, then (as we saw in the case of IRV above) removing A changes the winner. to B.

So Arrow's "paradox" can be considered as forcing us to realize that the IIAC is not a realistic possibility in the presence of ordinal ballots because such ballots allow us to detect oairwise (head-to-head) preferences, and when it comes down to a single pair of candidates the Majority Criterion says the pairwise winner must be chosen,

However, as someone mentioned, Approval Voting avoids this "paradox" once the ballots have been submitted, since the Approval winner A is always the "ballot CW," and in two different ways:(1) For any other candidate X, candidate A will be rated above X on more ballots than not, and (2) A's approval score will be higher than the sore of any other candidate.  From either point of view, if we remove a loser Y from the ballots, then A will still be the winner according to the same ballots with Y crossed out.

That's at the ballot level.  But if Y withdrew before the ballots were filled out, it could change the winner, because if Y were the only approved candidate for a certain voter before the withdrawal, that voter might decide to lower her personal approval cutoff before submitting her ballot.  Or she could raise the cutoff if Y had been the only disapproved candidate.

As others have mentioned in this discussion, Approval Voting externalizes the problem of the IIAC from being a decision problem for the method itself to a strategical decision problem for the voter. A voter might think of that as an unfair burden.

One answer to this problem could be DSV (Designated Strategy Voting): You submit your sincere ratings, and the DSV machine applies a strategy of your choice or a default strategy to transform the ballots into approval style ballots.  Rob LeGrand explored some of the possibilities and limitations of this approach in his master's thesis.  He doesn't claim to have exhausted the possibilities.  (I also have some ideas in this vein that still need exploring.)

What constitutes a "sincere rating."  One approach to that has already been mentioned in the ice-cream flavor context in this thread. Another is to use as a rating for candidate X your subjective probability that on a typical issue of any significance candidate X would support the same side you support.

It's not just Approval that requires some hard thinking in conjunction with filling out the ballots. Ranking many candidates (think about the number of candidates in the election that propelled Schwarznegger into office) may be just as burdensome as trying to decide exactly which candidates to mark as approved. In Australia you can get around this difficulty by copying "candidate cards" or by voting the party line. Presumably these experts are reflecting state of the art strategy in their rankings ... the strategy that is indispensable for optimum results according to Gibbard-Satterthwaite.  This is not just a problem of Approval, though it may seem worse in Approval.  In actuality, aoproval and score/range are the only commonly used methods where optimal strategy never requires you to "betray " your favorite.

To cut the Gordian knot of this complexity Charles Dodgson (aka Lewis Carroll) suggested what we now call Asset Voting. Each Voter delegates her vote to the candidate she trusts the most to rep[resent her in the decision process. Since write-ins are allowed, she can write in herself if she doesn't trust anybody else to be her proxy.  These proxies get together with their "assets"  (delegated votes) and choose a winner by use of some version of Robert's Rules of Order.

Which criteria are satisfied by this method? Does Gibbard Satthethwaite have anything to say about it? How about Arrow?  For that matter does first past the post plurality satisfy the IIAC? (No more or less than Approval in reality.)

Let's talk about Gibbard-Satterthwaite.  Is there any incentive for a person to delegate as proxy someone other than her favorite? 

If we are talking representative democracy, then why would you want to delegate your vote to candidate B when candidate A was the one you trusted most to represent you in making important decisions once in office?

All of the "problems" with the method are essentially externalized to the deliberations governed by"Robert's Rules of Order" in the smoke filled room.

Gibbard-Satterthwaite is taken to say that it is impossible to obtain sincere preferences or sincere utilities from voters in the context of full information (or disinformation) elections.  Yet it turns out to be relatively easy; you just need to separate the ballot into two parts.  The first part requires strategic voting to pick the two alternatives as finalists.  The second part is used solely to choose between these two options.  (In the case of cardinal ballots the finalists are lotteries.)  A version of the uncertainty principle obtains here: if you use the sincere ballots for any other instrumental purpose than to choose between the two finalists, then you almost certainly destroy their sincerity.

However there would be no problem comparing tthe sincere part with the strategical part to get statistics about voters' willingness to vote insincerely in choosing the finalists.

Another avenue that has been barely explored is the use of chance to incentivize consensus when there is a potential for it.

For example, suppose that preferences are

60 A>C>>>>B
40 B>C>>>>A

Under approval voting the A faction has a strong incentive to downgrade C and vote 60 A>>>>>C>B making A the (insincere) approval winner as Gibbard-satterthwaite would predict.

However, if the rules said that in the absence of a full consensus approval winner, the winner would be chosen by random ballot, then (assuming rational voters voting in their own interest) C would be the sure outcome; no rational voter in either faction would prefer random ballot expectations over a sure deal on C.

Jobst Heitzig is the pioneer in this area.

In sum, Arrow, et.al. should not constitute a nail in the coffin of creative progress in Election Methods.  IMHO that is an important message we need to send if we want to attract new talent.

Forest





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Re: [EM] Arrow's theorem and cardinal voting systems

Richard Lung
This is obviously a very learned summary that would require considerable study to do justice to, much more than my old head is capable of, quite apart from severe personal distresses, that have been over-whelming our lives.
Yet, just the other day, after a half century of amateur study, to my surprise, my naive physicists mathematical text crawled across the finishing line of publication. Only the last chapter is of direct interest to electoral mathematicians. It contains an explanation of how to conduct a two-dimensional election: FAB STV 2D.
The two dimensions are Representation and Arbitration. The latter graphs at 90 degrees (neutrally) to the former, and the count is conducted without disturbing the normal one-dimensional count that is FAB STV. But taken together as a complex variable the count is according to the rules of complex variables.

from
Richard Lung.


On 15/01/2020 20:57, Forest Simmons wrote:
Just a couple of additional thoughts:

Besides Arrow and Gibbard-Sattherwaite we have lots of criterion incompatibility results from Woodall, and defensive strategy criteria from Mike Ossipoff, Steve Eppley, et. al..  In particular we never worried about Later No Help and later No Harm until Woodall came along. Venzke and Benham picked up the torch and brought Woodall into the EM Listserv discussion.

Many EM contributors have clarified which combinations of various criteria of more practical than academic stripe are compatible or not: Participation, FBC, Precinct Summability, Chicken, etc. 

In particular, we now know through the work of Ossipoff, Venzke, Benham, and others that the Chicken Defense and Burial Defense (against CW burial) are incompatible in the presence of Plurality and the the FBC, unless we allow an explicit approval cutoff or some other strategic switch on the ballots.  Standard ordinal ballots are not adequate for this even when truncation and equal rankings (including equal top) are allowed.   A non-standard ballot that allows us to get compatibility to all of these except the CC is MDDA(sc) which is Majority Defeat Disqualification Approval with symmetric completion below the approval cutoff.  This method also satisfies other basic criteria such as Participation, Clone Independence, Mono-Raise, Mono-Add, and IDPA, for example.

In the context of the current discussion, the approval cutoff or some equivalent strategic switch is essential for the compatibility of chicken resistance and burial resistance..  No strategy, no compatibility. So basically there is no decent method that is resistant to both Burial of the CW and Chicken offensives.  (IRV is chicken resistant and has a form of burial resistance, but routinely buries the CW unless voters strategically betray their favorite to save the CW. Furthermore, it fails mono-raise.)

Before collaborative efforts of EM List members there was no known clone independent, monotonic method for electing from the uncovered set.  The closest thing was Copeland, which is clone dependent.

Again the main point is that Arrow, and Gibbard-Satterthwaite are not the "end of history" for election methods, just like the collapse of the USSR was not the end of history as Fukuyama once proclaimed or Thatcher's famous TINA "there is no alternative" (to capitalism).  Arrow and G-S give very valuable insights and help us avoid cul-de-sacs, but they are not the last word in election methods progress.  The "end of history" and TINA slogans are an excuse for giving up prematurely for lack of imagination. We cannot allow Arrow and G-S to become excuses for lack of imagination in Election Methods.  What if Yee had given up before inventing the beautiful Yee diagrams that constitute an Electo-Kaleidoscope for the study of election methods analogous to the telesope and the electron microscope in astronomy as instruments in other branches of knowledge?

On Mon, Jan 13, 2020 at 3:32 PM Forest Simmons <[hidden email]> wrote:
Rob,

Thanks for starting this great thread!

The "no perfect car" analogy is good.  More definite is the "no 100 percent efficient internal combustion engine" analogy that follows from the second law of thermodynamics.  It applies to all kinds of engines, but that doesn't mean that internal combustion is as good as it gets.

If Gibbard-Satterthwaite tells us that we cannot have all of the nice properties we want in one election method, that doesn't mean that one method is as good as the next.

It follows from Arrow that we cannot have the Majority Criterion and the IIAC at the same time, but there are many decent methods (like River) that do satisfy the MC, and a bunch of other nice properties, like Monotonicity, Clone Independence, the Condorcet Criterion, and Independence from Pareto Dominated Alternatives, as well as the basic Neutrality and Anonymity fairness criteria.

The way to think of Arrow's "Dictator" theorem is that it is extremely hard to get a rankings based method with even minimal decency conditions (like non-dictatorship) without scuttling the IIAC.

In other words, no decent ordinal based method can satisfy the IIAC, which is the same point of view that Toby and Eppley expressed.  It comes down to the mere existence of a Condorcet Cycle.  Here's the subtle part that most people don't understand.  Condorcet Cycles can exist in the preference schedules of an election even if the election method makes no mention of Condorcet, for example even in IRV/Hare/STV/RCV elections:

45 A>B>C
20 B>C>A
35 C>A>B

There exists a majority preference cycle A>B>C>A even though it causes no problem for IRV, since B is eliminated and then C is the majority winner between the two remaining candidates.

Now let's check the IIAC.  Suppose that A, one of the losers withdraws from the race.  Then the winner changes from C to B, since B beats C by a majority.  This shows that IRV does not satisfy the IIAC, because removing a loser from the ballot changes the winner.

But this is not just a problem for IRV, it's a problem for any method that respects the Majority Criterion; if the method makes A the winner, then removing B changes the winner.  If it makes B the winner, then removing C changes the winner.  If it makes C the winner, then (as we saw in the case of IRV above) removing A changes the winner. to B.

So Arrow's "paradox" can be considered as forcing us to realize that the IIAC is not a realistic possibility in the presence of ordinal ballots because such ballots allow us to detect oairwise (head-to-head) preferences, and when it comes down to a single pair of candidates the Majority Criterion says the pairwise winner must be chosen,

However, as someone mentioned, Approval Voting avoids this "paradox" once the ballots have been submitted, since the Approval winner A is always the "ballot CW," and in two different ways:(1) For any other candidate X, candidate A will be rated above X on more ballots than not, and (2) A's approval score will be higher than the sore of any other candidate.  From either point of view, if we remove a loser Y from the ballots, then A will still be the winner according to the same ballots with Y crossed out.

That's at the ballot level.  But if Y withdrew before the ballots were filled out, it could change the winner, because if Y were the only approved candidate for a certain voter before the withdrawal, that voter might decide to lower her personal approval cutoff before submitting her ballot.  Or she could raise the cutoff if Y had been the only disapproved candidate.

As others have mentioned in this discussion, Approval Voting externalizes the problem of the IIAC from being a decision problem for the method itself to a strategical decision problem for the voter. A voter might think of that as an unfair burden.

One answer to this problem could be DSV (Designated Strategy Voting): You submit your sincere ratings, and the DSV machine applies a strategy of your choice or a default strategy to transform the ballots into approval style ballots.  Rob LeGrand explored some of the possibilities and limitations of this approach in his master's thesis.  He doesn't claim to have exhausted the possibilities.  (I also have some ideas in this vein that still need exploring.)

What constitutes a "sincere rating."  One approach to that has already been mentioned in the ice-cream flavor context in this thread. Another is to use as a rating for candidate X your subjective probability that on a typical issue of any significance candidate X would support the same side you support.

It's not just Approval that requires some hard thinking in conjunction with filling out the ballots. Ranking many candidates (think about the number of candidates in the election that propelled Schwarznegger into office) may be just as burdensome as trying to decide exactly which candidates to mark as approved. In Australia you can get around this difficulty by copying "candidate cards" or by voting the party line. Presumably these experts are reflecting state of the art strategy in their rankings ... the strategy that is indispensable for optimum results according to Gibbard-Satterthwaite.  This is not just a problem of Approval, though it may seem worse in Approval.  In actuality, aoproval and score/range are the only commonly used methods where optimal strategy never requires you to "betray " your favorite.

To cut the Gordian knot of this complexity Charles Dodgson (aka Lewis Carroll) suggested what we now call Asset Voting. Each Voter delegates her vote to the candidate she trusts the most to rep[resent her in the decision process. Since write-ins are allowed, she can write in herself if she doesn't trust anybody else to be her proxy.  These proxies get together with their "assets"  (delegated votes) and choose a winner by use of some version of Robert's Rules of Order.

Which criteria are satisfied by this method? Does Gibbard Satthethwaite have anything to say about it? How about Arrow?  For that matter does first past the post plurality satisfy the IIAC? (No more or less than Approval in reality.)

Let's talk about Gibbard-Satterthwaite.  Is there any incentive for a person to delegate as proxy someone other than her favorite? 

If we are talking representative democracy, then why would you want to delegate your vote to candidate B when candidate A was the one you trusted most to represent you in making important decisions once in office?

All of the "problems" with the method are essentially externalized to the deliberations governed by"Robert's Rules of Order" in the smoke filled room.

Gibbard-Satterthwaite is taken to say that it is impossible to obtain sincere preferences or sincere utilities from voters in the context of full information (or disinformation) elections.  Yet it turns out to be relatively easy; you just need to separate the ballot into two parts.  The first part requires strategic voting to pick the two alternatives as finalists.  The second part is used solely to choose between these two options.  (In the case of cardinal ballots the finalists are lotteries.)  A version of the uncertainty principle obtains here: if you use the sincere ballots for any other instrumental purpose than to choose between the two finalists, then you almost certainly destroy their sincerity.

However there would be no problem comparing tthe sincere part with the strategical part to get statistics about voters' willingness to vote insincerely in choosing the finalists.

Another avenue that has been barely explored is the use of chance to incentivize consensus when there is a potential for it.

For example, suppose that preferences are

60 A>C>>>>B
40 B>C>>>>A

Under approval voting the A faction has a strong incentive to downgrade C and vote 60 A>>>>>C>B making A the (insincere) approval winner as Gibbard-satterthwaite would predict.

However, if the rules said that in the absence of a full consensus approval winner, the winner would be chosen by random ballot, then (assuming rational voters voting in their own interest) C would be the sure outcome; no rational voter in either faction would prefer random ballot expectations over a sure deal on C.

Jobst Heitzig is the pioneer in this area.

In sum, Arrow, et.al. should not constitute a nail in the coffin of creative progress in Election Methods.  IMHO that is an important message we need to send if we want to attract new talent.

Forest





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