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<p>I think it would have been more helpful to appreciate that there
is no way round (multiple) order in the vote, whether it be by
x-voting, party voting or cardinal voting, which has the
disadvantage of having little or no usage. (A similar handicap
applies to trying to exclude proportion from the count, without
disproportionate results. It is "an exercise in futility" as Enid
Lakeman said of "affirmative gerrymandering.")<br>
</p>
<p>As regards x-voting, in the UK, big parties and small parties
alike have been straining, as far back as can be remembered, at
least since WW2, to persuade electors not to elect, but to exclude
either small or big parties. (HG Wells said in 1912, in The Labour
Unrest, We no longer have elections, only Rejections.)</p>
<p>Moving to party voting does not solve the problem. Anika Freden
distinguished four main kinds of strategic voting with party list
systems. What you find in the Cold War divide of elections into
Simple Plurality and Closed Lists, or their amalgamation into a
Double X-vote, is that the voters are denied all but a
single-preference X-vote, and pressured not to waste it on an
election!</p>
<p>Elections could not banish ordinal voting but they have turned
voting into a class system of double standards, where contriving
politicians defectively order choices for their leaders, but do
not permit a ranked choice for the general voters.<br>
</p>
<p>Cardinal voting confounds an individual vote with a collective
count. It is not for equal voters to say how many votes they think
a candidate should have. That is a collective result of the count.
And as Enid Lakeman said, cumulative votes count against each
other.</p>
<p>Richard Lung.<br>
</p>
<p><br>
</p>
<div class="moz-cite-prefix">On 17/02/2024 03:47, Closed Limelike
Curves wrote:<br>
</div>
<blockquote type="cite"
cite="mid:CA+euzPhXH1ydjH=zqiAo3FBZ6GsAnD6imL0WBjD=sunZ-py1dw@mail.gmail.com">
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<div dir="ltr">
<div dir="ltr">First, I'd like to
thank Kristofer for his
wonderful response+addition to
this discussion. :-)</div>
<div dir="ltr"><br>
</div>
<div dir="ltr">I made most of the
edits on Wikipedia, and I'm
happy to talk about how we could
try and make them more neutral.
My goal wasn't to be a cardinal
partisan, although I'll admit
I'm generally a supporter. I'm a
big fan of some of the newer
Condorcet methods (like Ranked
Pairs) as well, and I think the
difference between these and
cardinal methods is likely
pretty small in practice.</div>
<div dir="ltr"><br>
</div>
<div dir="ltr">Rather than
advocating any particular voting
system, my goal was to nip some
common misunderstandings about
these theorems in the bud.
Mostly these relate to the
applicability of some of these
theorems (especially Arrow's) to
cardinal systems. It sounds like
in doing so, I might have
introduced a framing that gives
the opposite misimpression (that
cardinal systems are somehow
immune to <i>any</i> kind of
unpleasant behavior, when
they're clearly not).</div>
<div dir="ltr"><br>
</div>
<div dir="ltr">
<div>Here's what I think is
important for people to
understand on each of these
topics:</div>
<div>
<ul>
<li><b>Arrow's theorem:</b> Within
the Arrovian paradigm (a
function aggregates
individual preferences to
give us
social preferences), any
rule that satisfies IIA
(and therefore coherence)
is cardinal.</li>
<li><b>Gibbard-Satterthwaite:</b> It's
impossible to guarantee
honesty (no preference
reversals) for any ordinal
voting system with >2
candidates (original) or
any cardinal system with
>3 candidates (WDS
extension).</li>
<ul>
<li><i>Comment on
semi-honest rankings</i>:
I think honesty in
rankings and honesty in
ratings are both
valuable (but distinct)
notions of honesty, and
it's reasonable to
separate them.
Satterthwaite's original
theorem focused on
ordinal systems, however
(assuming rankings
throughout). Because of
that, I interpret the
theorem as being about
ordinal honesty, which
score voting happens to
satisfy for the
3-candidate case.</li>
<li>
<div><i>Comment on
revelation principle</i>:
You're completely
correct. I
misinterpreted the
textbook I've been
working from as
claiming something
stronger than it
actually was, and I'll
fix this ASAP.</div>
</li>
</ul>
<li><b>Gibbard's theorem: </b>Within
the game-theoretic
paradigm
(reported individual
preferences are the
results of a game, not the
thing we actually care
about), perfect guaranteed
honesty is impossible for
any voting system.</li>
<ul>
<li><i>Honest mechanisms: </i>I
do think we want to be
clear on the distinction
between social
choice mechanisms and
voting systems. Some
mechanisms (like VCG)
can be efficient and
still guarantee honesty
if monetary incentives
are available.</li>
</ul>
</ul>
</div>
<div>
<div>By the way, I'd be very
interested in a source on
strategy implying IIA
violations, so I can add it
to the article! </div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Thu, Feb 15, 2024 at
10:00 AM <<a
href="mailto:election-methods-request@lists.electorama.com"
target="_blank" moz-do-not-send="true"
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Today's Topics:<br>
<br>
1. Impossibility on Wikipedia: Arrow, Gibbard, and
Satterthwaite<br>
(Rob Lanphier)<br>
2. Re: Impossibility on Wikipedia: Arrow, Gibbard, and<br>
Satterthwaite (Richard Lung)<br>
3. Re: Impossibility on Wikipedia: Arrow, Gibbard, and<br>
Satterthwaite (Kristofer Munsterhjelm)<br>
<br>
<br>
----------------------------------------------------------------------<br>
<br>
Message: 1<br>
Date: Wed, 14 Feb 2024 23:13:28 -0800<br>
From: Rob Lanphier <<a href="mailto:roblan@gmail.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">roblan@gmail.com</a>><br>
To: <a href="mailto:election-methods@lists.electorama.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">election-methods@lists.electorama.com</a><br>
Subject: [EM] Impossibility on Wikipedia: Arrow, Gibbard, and<br>
Satterthwaite<br>
Message-ID:<br>
<CAK9hOY=nJQ1QstfHi-6mh42H=_<a
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<br>
Hi folks,<br>
<br>
I'm going to send a similar email here to the EM list that I
recently<br>
sent to several folks who hang out in academic circles. The
answer I<br>
received from the academic circles was valuable, but I also
think that<br>
folks on this mailing list can provide a different (and
useful)<br>
perspective.<br>
<br>
I've long taken it for granted that impossibility theorems
like<br>
Arrow's theorem and Gibbard's theorem mathematically prove
that there<br>
are always going to be important electoral criteria that will
be<br>
mutually exclusive in ANY credible electoral system. I've
been at<br>
peace with that for a long time, much in the same way that I'm
at<br>
peace with mutually exclusive criteria for my transportation
needs<br>
(e.g. I should take something with more carrying capacity than
a<br>
bicycle to go shopping for large furniture, no matter how good
the<br>
bike is). The physics of electoral systems and the physics of
the<br>
real world have certain mathematical rules that are tough to
get<br>
around.<br>
<br>
Since the Center for Election Science (<<a
href="https://electionscience.org" rel="noreferrer"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://electionscience.org</a>>)<br>
started getting momentum and having some electoral success in
the late<br>
2010s, there's been a push to distinguish between "cardinal
voting"<br>
and "ordinal voting" as the top of the hierarchy
distinguishing all<br>
voting systems. Since the ballot is what people see, that's<br>
understandable, I suppose. However, in my mind, the ballots
don't<br>
matter as much as the tallying method, and moreover, it's
possible to<br>
use cardinal voting ballots and then tally them using systems
that<br>
some folks classify as "ordinal" systems.<br>
<br>
In discussions with electoral reform folks over the past few
years,<br>
I've been learning about Arrow, Gibbard, and Satterthwaite,
and trying<br>
to document what I've learned on Wikipedia and electowiki.<br>
<br>
In editing Wikipedia articles related to election methods in
the past<br>
few years, it seems there are three theorems that have made
the rounds<br>
with regards to impossibility theorems:<br>
<br>
1. Arrow's impossibility theorem (published in 1951):
basically the<br>
granddaddy of impossibility theorems, which seemingly only
applies to<br>
ordinal voting methods.<br>
<<a
href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem</a>><br>
2. Gibbard's theorem (published in 1973): generalizes Arrow's
theorem<br>
to apply to pretty much every social choice function<br>
<<a
href="https://en.wikipedia.org/wiki/Gibbard%27s_theorem"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Gibbard%27s_theorem</a>><br>
3. The Gibbard?Satterthwaite theorem (published in 1978): a
more<br>
specific version of Gibbard's theorem which apparently only
applies to<br>
ordinal systems, and focuses on strategic voting<br>
<<a
href="https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem</a>><br>
<br>
What bothers me of late is a recent change that's been made to
the<br>
Gibbard-Satterthwaite article. I'll quote the most bothersome<br>
addition/replacement that's in the "Gibbard?Satterthwaite
theorem"<br>
article as of this writing:<br>
> The theorem does not apply to cardinal voting systems
such as score<br>
> voting or STAR voting, which can often guarantee honest
(or semi-honest)<br>
> rankings in cases covered by the Gibbard-Satterthwaite
theorem,[4] nor<br>
> does it apply to decision mechanisms other than
ranked-choice voting.<br>
> Gibbard's theorem provides a weaker result that applies
to such<br>
> mechanisms.<br>
><br>
> The Gibbard-Satterthwaite theorem is often misunderstood
as claiming<br>
> that "every voting system encourages dishonesty" or the
related adage<br>
> that "there is no best voting system." However, such
interpretations are<br>
> not correct; by the revelation principle, there exist
many (deterministic,<br>
> non-trivial) voting systems that allow for honest
disclosure (outside the<br>
> class of ranked-choice voting systems).<br>
<br>
It seems disingenuous to say that all of these voting systems
don't<br>
apply to cardinal systems if there is some way to vote
"honestly"<br>
(whatever that means). Strategy and honesty are not mutually<br>
exclusive, and cardinal systems like "score voting" require
voters to<br>
be very strategic as part of their voting calculus. As noted
above,<br>
Condorcet tallying methods can be used to tally "cardinal
ballots" and<br>
"ordinal ballots", since both express the preferences.<br>
<br>
I'll quote what one of the folks in the academic circles
stated:<br>
> Since it seems implausible to suppose that one person?s
cardinal<br>
> evaluations have meaning in comparison to another
person?s evaluations,<br>
> it is implausible to suppose that there is such a thing
as an honest cardinal<br>
> evaluation of candidates. If there is such a thing as an
honest cardinal<br>
> evaluation of candidates, then opportunities to benefit
from dishonest<br>
> evaluations of candidates are rife in systems based on
cardinal<br>
> evaluations, while they are likely to be quite rare under
Condorcet-<br>
> consistent ranking-based voting systems.<br>
<br>
This assertion more-or-less comports with my opinion. While I
don't<br>
think that systems that insist on ranking-based ballots
(ordinal<br>
ballots) are ALWAYS superior to systems that rely on simple
addition<br>
of rating-based ballots (cardinal ballots), I think the
implicit<br>
rankings are at least as important as the explicit ratings. I<br>
generally think of STAR voting as "Condorcet lite", because,
for two<br>
finalists "candA" and "candB", the final runoff doesn't pay
attention<br>
to whether:<br>
scenario 1 ) "candA" has 5 stars and "candB" has 4 stars on
ballot #1234<br>
or<br>
scenario 2) "candA" has 1 star, and "candB" has 0 stars on
ballot #1234<br>
<br>
In the end, in both scenarios, ballot #1234 counts in full for<br>
"candA", which seems fair to me. Regardless, I've frequently
found<br>
myself distrusting hardcore cardinal advocates when I see
changes like<br>
the one made to English Wikipedia's "Gibbard?Satterthwaite
theorem"<br>
article.<br>
<br>
Are cardinal voting advocates correct to continually claim
that<br>
Arrow's, Gibbard's, and Satterthwaite's theorems don't apply
to their<br>
favorite voting methods? Is there a useful distinction to be
drawn<br>
between Gibbard's 1973 theorem and the "Gibbard-Satterthwaite
theorem"<br>
published in 1978? Is the distinction I draw above correct?<br>
<br>
Rob<br>
<br>
<br>
------------------------------<br>
<br>
Message: 2<br>
Date: Thu, 15 Feb 2024 16:02:57 +0000<br>
From: Richard Lung <<a
href="mailto:voting@ukscientists.com" target="_blank"
moz-do-not-send="true" class="moz-txt-link-freetext">voting@ukscientists.com</a>><br>
To: Rob Lanphier <<a href="mailto:roblan@gmail.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">roblan@gmail.com</a>>,<br>
<a
href="mailto:election-methods@lists.electorama.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">election-methods@lists.electorama.com</a><br>
Subject: Re: [EM] Impossibility on Wikipedia: Arrow, Gibbard,
and<br>
Satterthwaite<br>
Message-ID: <<a
href="mailto:c84dcd1e-0f80-4f4c-920b-c0cf25b4ebb3@ukscientists.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">c84dcd1e-0f80-4f4c-920b-c0cf25b4ebb3@ukscientists.com</a>><br>
Content-Type: text/plain; charset=UTF-8; format=flowed<br>
<br>
<br>
Whether you are right or no, there is no conensus on the
matter. Theorem <br>
Arrow is like a Cold War between electoral systems. It
acknowledges <br>
ordinal votes as a basis for elections, in a denigratory sort
of way, <br>
but over-looks the count beyond crude plurality. Ever since, <br>
mathematicians have demonstrated it takes more than
mathematics to have <br>
a good understanding of elections. Formerly, it was not so,
when they <br>
acted freely as enthusiasts, but the institutionalisation of
election <br>
studies appears to have robbed them of any independent
critical sense.<br>
<br>
That understanding, I gather from his parliamentary speeches
on "Mr <br>
Hare's system," is what John Stuart Mill had, the greatest
philosopher <br>
of science in the 19th century. The Hare-Mill tradition, that
has <br>
continued to the present day, has been by-passed. In so doing,
social <br>
choice theory reveals its provincialism. A Nobel prize or so,
to give <br>
away, is not a proof. A theorem is only as good as the
assumptions on <br>
which it is based. And theorem Arrow compares to a critique of
a bicycle <br>
on the basis of the short-comings of a unicycle. It does not
deal with <br>
the democratic necessity of a proportional count as well as an
ordinal <br>
vote. Simple plurality is "maiorocracy" or the tyranny of the
majority, <br>
as Mill and Lani Guinier said.<br>
<br>
It is not apparent what decisive argument the social choice
school have <br>
that they can take to the voters, for whom elections are
supposed to be <br>
meant, and has not been so for 70 years. It is not even
apparent that, <br>
after 70 years, they have any idea of, or even belief in, a
standard <br>
model of democratic election.<br>
<br>
Richard Lung.<br>
<br>
<br>
On 15/02/2024 07:13, Rob Lanphier wrote:<br>
> Hi folks,<br>
><br>
> I'm going to send a similar email here to the EM list
that I recently<br>
> sent to several folks who hang out in academic circles.
The answer I<br>
> received from the academic circles was valuable, but I
also think that<br>
> folks on this mailing list can provide a different (and
useful)<br>
> perspective.<br>
><br>
> I've long taken it for granted that impossibility
theorems like<br>
> Arrow's theorem and Gibbard's theorem mathematically
prove that there<br>
> are always going to be important electoral criteria that
will be<br>
> mutually exclusive in ANY credible electoral system.
I've been at<br>
> peace with that for a long time, much in the same way
that I'm at<br>
> peace with mutually exclusive criteria for my
transportation needs<br>
> (e.g. I should take something with more carrying capacity
than a<br>
> bicycle to go shopping for large furniture, no matter how
good the<br>
> bike is). The physics of electoral systems and the
physics of the<br>
> real world have certain mathematical rules that are tough
to get<br>
> around.<br>
><br>
> Since the Center for Election Science (<<a
href="https://electionscience.org" rel="noreferrer"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://electionscience.org</a>>)<br>
> started getting momentum and having some electoral
success in the late<br>
> 2010s, there's been a push to distinguish between
"cardinal voting"<br>
> and "ordinal voting" as the top of the hierarchy
distinguishing all<br>
> voting systems. Since the ballot is what people see,
that's<br>
> understandable, I suppose. However, in my mind, the
ballots don't<br>
> matter as much as the tallying method, and moreover, it's
possible to<br>
> use cardinal voting ballots and then tally them using
systems that<br>
> some folks classify as "ordinal" systems.<br>
><br>
> In discussions with electoral reform folks over the past
few years,<br>
> I've been learning about Arrow, Gibbard, and
Satterthwaite, and trying<br>
> to document what I've learned on Wikipedia and
electowiki.<br>
><br>
> In editing Wikipedia articles related to election methods
in the past<br>
> few years, it seems there are three theorems that have
made the rounds<br>
> with regards to impossibility theorems:<br>
><br>
> 1. Arrow's impossibility theorem (published in 1951):
basically the<br>
> granddaddy of impossibility theorems, which seemingly
only applies to<br>
> ordinal voting methods.<br>
> <<a
href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem</a>><br>
> 2. Gibbard's theorem (published in 1973): generalizes
Arrow's theorem<br>
> to apply to pretty much every social choice function<br>
> <<a
href="https://en.wikipedia.org/wiki/Gibbard%27s_theorem"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Gibbard%27s_theorem</a>><br>
> 3. The Gibbard?Satterthwaite theorem (published in 1978):
a more<br>
> specific version of Gibbard's theorem which apparently
only applies to<br>
> ordinal systems, and focuses on strategic voting<br>
> <<a
href="https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem</a>><br>
><br>
> What bothers me of late is a recent change that's been
made to the<br>
> Gibbard-Satterthwaite article. I'll quote the most
bothersome<br>
> addition/replacement that's in the "Gibbard?Satterthwaite
theorem"<br>
> article as of this writing:<br>
>> The theorem does not apply to cardinal voting systems
such as score<br>
>> voting or STAR voting, which can often guarantee
honest (or semi-honest)<br>
>> rankings in cases covered by the
Gibbard-Satterthwaite theorem,[4] nor<br>
>> does it apply to decision mechanisms other than
ranked-choice voting.<br>
>> Gibbard's theorem provides a weaker result that
applies to such<br>
>> mechanisms.<br>
>><br>
>> The Gibbard-Satterthwaite theorem is often
misunderstood as claiming<br>
>> that "every voting system encourages dishonesty" or
the related adage<br>
>> that "there is no best voting system." However, such
interpretations are<br>
>> not correct; by the revelation principle, there exist
many (deterministic,<br>
>> non-trivial) voting systems that allow for honest
disclosure (outside the<br>
>> class of ranked-choice voting systems).<br>
> It seems disingenuous to say that all of these voting
systems don't<br>
> apply to cardinal systems if there is some way to vote
"honestly"<br>
> (whatever that means). Strategy and honesty are not
mutually<br>
> exclusive, and cardinal systems like "score voting"
require voters to<br>
> be very strategic as part of their voting calculus. As
noted above,<br>
> Condorcet tallying methods can be used to tally "cardinal
ballots" and<br>
> "ordinal ballots", since both express the preferences.<br>
><br>
> I'll quote what one of the folks in the academic circles
stated:<br>
>> Since it seems implausible to suppose that one
person?s cardinal<br>
>> evaluations have meaning in comparison to another
person?s evaluations,<br>
>> it is implausible to suppose that there is such a
thing as an honest cardinal<br>
>> evaluation of candidates. If there is such a thing as
an honest cardinal<br>
>> evaluation of candidates, then opportunities to
benefit from dishonest<br>
>> evaluations of candidates are rife in systems based
on cardinal<br>
>> evaluations, while they are likely to be quite rare
under Condorcet-<br>
>> consistent ranking-based voting systems.<br>
> This assertion more-or-less comports with my opinion.
While I don't<br>
> think that systems that insist on ranking-based ballots
(ordinal<br>
> ballots) are ALWAYS superior to systems that rely on
simple addition<br>
> of rating-based ballots (cardinal ballots), I think the
implicit<br>
> rankings are at least as important as the explicit
ratings. I<br>
> generally think of STAR voting as "Condorcet lite",
because, for two<br>
> finalists "candA" and "candB", the final runoff doesn't
pay attention<br>
> to whether:<br>
> scenario 1 ) "candA" has 5 stars and "candB" has 4 stars
on ballot #1234<br>
> or<br>
> scenario 2) "candA" has 1 star, and "candB" has 0 stars
on ballot #1234<br>
><br>
> In the end, in both scenarios, ballot #1234 counts in
full for<br>
> "candA", which seems fair to me. Regardless, I've
frequently found<br>
> myself distrusting hardcore cardinal advocates when I see
changes like<br>
> the one made to English Wikipedia's
"Gibbard?Satterthwaite theorem"<br>
> article.<br>
><br>
> Are cardinal voting advocates correct to continually
claim that<br>
> Arrow's, Gibbard's, and Satterthwaite's theorems don't
apply to their<br>
> favorite voting methods? Is there a useful distinction
to be drawn<br>
> between Gibbard's 1973 theorem and the
"Gibbard-Satterthwaite theorem"<br>
> published in 1978? Is the distinction I draw above
correct?<br>
><br>
> Rob<br>
> ----<br>
> Election-Methods mailing list - see <a
href="https://electorama.com/em" rel="noreferrer"
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<br>
Message: 3<br>
Date: Thu, 15 Feb 2024 18:48:22 +0100<br>
From: Kristofer Munsterhjelm <<a
href="mailto:km_elmet@t-online.de" target="_blank"
moz-do-not-send="true" class="moz-txt-link-freetext">km_elmet@t-online.de</a>><br>
To: Rob Lanphier <<a href="mailto:roblan@gmail.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">roblan@gmail.com</a>>,<br>
<a
href="mailto:election-methods@lists.electorama.com"
target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">election-methods@lists.electorama.com</a><br>
Subject: Re: [EM] Impossibility on Wikipedia: Arrow, Gibbard,
and<br>
Satterthwaite<br>
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<br>
On 2024-02-15 08:13, Rob Lanphier wrote:<br>
> Hi folks,<br>
> <br>
<br>
> Are cardinal voting advocates correct to continually
claim that<br>
> Arrow's, Gibbard's, and Satterthwaite's theorems don't
apply to their<br>
> favorite voting methods? Is there a useful distinction
to be drawn<br>
> between Gibbard's 1973 theorem and the
"Gibbard-Satterthwaite theorem"<br>
> published in 1978? Is the distinction I draw above
correct?<br>
<br>
My understanding and opinion is this:<br>
<br>
You have three different "main" impossibility theorems:<br>
<br>
- Arrow's says that no deterministic reasonable ordinal voting
method <br>
can pass IIA. This means that regardless of whether voters are
honest or <br>
strategic, it's possible that A's win over B depends on not
just how <br>
many voters prefer A to B, but also how many prefer X to Y.
This is only <br>
for cardinal methods.<br>
<br>
- Gibbard-Satterthwaite says that no deterministic reasonable
ordinal <br>
method is strategy-proof: there always exists at least one
election <br>
where at least one voter has an incentive to adjust his
preferences <br>
based on how others are voting.<br>
<br>
- Gibbard's theorem extends this to a much broader class of
election <br>
methods that includes cardinal methods, thus also implying
that no <br>
deterministic cardinal voting method is strategy-proof.<br>
<br>
(In particular, both of Gibbard's theorems are about
strategy.)<br>
<br>
The part about the revelation principle is incorrect and seems
to be <br>
using a very specific definition of honesty, inspired by
Warren Smith. <br>
To my knowledge, what the revelation principle says is this:<br>
<br>
- Say that a participant in a mechanism employs strategy if
the <br>
information he submits to the mechanism depends on the actions
of the <br>
other participants or his belief about them.[1]<br>
<br>
- Say that a mechanism only uses honesty if no participant has
an <br>
incentive to employ strategy.<br>
<br>
- Then, if there exists a mechanism where people employ
strategy to <br>
drive it to an optimal or equilibrium outcome, then there also
exists a <br>
mechanism that only uses honesty that reaches the same outcome
or <br>
equilibrium.<br>
<br>
Consider it like this: suppose you're involved in a court
case, and you <br>
hire a lawyer. You tell the lawyer the truth and the lawyer
comes up <br>
with whatever strategy that will advance your interests, given
the <br>
evidence and information about the other party. That's a
system where <br>
you or your lawyer use strategy to drive the system to a
particular state.<br>
<br>
But consider a hypothetical legal system using an AI judge.
This AI has <br>
a lawyer interface to each party; upon hearing the truth from
each <br>
party, it then simulates a court case the way it would proceed
with <br>
virtual lawyers who employs strategy based on the honest
information. <br>
The AI comes with a proof that your virtual lawyer won't
incriminate <br>
you. You would then interact honestly with the system,
represented by <br>
the judge, and it would strategize internally.<br>
<br>
Thus for any system that requires strategy to get an
equilibrium, there <br>
exists another system where you can be honest. It just absorbs
the <br>
"lawyer component" into itself.<br>
<br>
The problem is that there is no such equilibrium for
deterministic <br>
voting methods. That's implied by Gibbard, because otherwise,
you could <br>
create a strategy-proof method by first creating a method that
invites a <br>
particular type of strategy, and then embodying a "lawyer
component" <br>
into it to perform that strategy.<br>
<br>
If you try to do this, as far as I understand it, you get a <br>
nondeterministic method since the equilibrium is a mixed
strategy. And, <br>
as we know, there exist strategy-proof nondeterministic
methods, so that <br>
fits.<br>
<br>
This would be like: sometimes, your lawyer says "if we do
this, then no <br>
matter what the other party does, we win". But other times he
says "if <br>
we focus on these elements and the other party focuses on
those, then we <br>
win". Like rock-paper-scissors, which strategy you should play
depends <br>
on the strategy the other guy is going to use. So you play
them at <br>
random in such a way that the other guy can't guess what
you're doing. <br>
That gives a nondeterministic method.<br>
<br>
Okay.<br>
<br>
So now about IIA.<br>
<br>
The seeming advantage that cardinal methods have over ordinal
ones is <br>
that they pass IIA. In Range, if A has score 100 and B has
score 50, <br>
then A will continue to beat B even if we remove every other
candidate.<br>
<br>
But I've always been of the opinion that this IIA compliance
is either <br>
illusory or a distraction, because it doesn't answer what we
really care <br>
about. And that is whether the presence of a candidate who
doesn't win <br>
changes the outcome.<br>
<br>
Some cardinal proponents say that methods like Range have
multiple <br>
honest ballots: there are many ways to vote that are all
consistent with <br>
your preference ordering. But a voter has to choose which
honest ballot <br>
to cast, and there's no externally fixed scale (what exactly
does ten <br>
points mean? What does zero mean?).[2] Thus the selection of
candidates <br>
who run will affect the scale, which means that some voters
would change <br>
their ratings based on who's running - even if those
additional <br>
candidates don't win. So these methods fail what we could call
"de facto <br>
IIA", for lack of a better term.<br>
<br>
For instance, suppose the election starts off with two
pro-democracy <br>
candidates. Then a number of authoritarians enter the race.
It's likely <br>
that in Approval, some voters who would've approved of one of
the <br>
pro-democracy candidates but not the other, would now approve
both to <br>
mark their distaste for authoritarianism and keep the
authoritarians <br>
from winning.<br>
<br>
So in my opinion, ordinal methods are merely honest about
their <br>
limitations. Their logic says: "okay, I can't know if his
10/10 is the <br>
same as her 10/10 or her 5/10. I'll accept that this means I
must fail <br>
IIA, instead of seeming to pass it by passing the buck to the
voters <br>
that they must use a fixed scale that's not affected by who's
in the race."<br>
<br>
So ordinal methods clearly fail IIA, and aren't strategy-proof
(by <br>
Gibbard-Satterthwaite). Cardinal methods pass IIA (but it
doesn't mean <br>
what one may think it means) and aren't strategy-proof (by
Gibbard's <br>
theorem).<br>
<br>
Both ordinal and cardinal methods may pass de facto IIA for
subsets of <br>
elections. E.g. Condorcet methods pass IIA as long as there is
a CW, <br>
because adding a candidate either makes that candidate the new
CW (hence <br>
he's not irrelevant) or the current winner stays a CW.
Similarly, <br>
Approval passes de facto IIA with dichotomous preferences
where every <br>
voter has a class of OK candidates and a class of not-OK
candidates, and <br>
the boundary between OK and not OK doesn't depend on who's in
the race. <br>
(Such voters may sometimes approve everybody or nobody.)<br>
<br>
But in general: both cardinal and ordinal methods are
susceptible to <br>
strategy. And both cardinal and ordinal methods may have the
winner <br>
change from A to B as a consequence of C entering the race.<br>
<br>
-km<br>
<br>
[1] Strictly speaking we would also want "honesty" to have
some <br>
connotation of "being the actual information being asked for".
Say <br>
you're dealing with a system that asks you what you like the
least and <br>
then gives it to you. You would answer its question with a
thing that <br>
you want to be given, so that answer stays the same no matter
what other <br>
people interacting with it would say. So by my definition that
wouldn't <br>
be strategy, but common-sense would say that it is not honest
either.<br>
<br>
Warren's definition of honesty is somewhat based on this idea,
but it <br>
goes too far in the other direction. It doesn't consider
ballots where <br>
your ranking stays the same as under honesty but your scores
don't, as <br>
being strategic. I think in part that's due to the difficulty
in <br>
comparing utilities, but we *can* hold rated voting to a
higher <br>
standard. Ask and I'll elaborate - this post is long enough
:-)<br>
<br>
[2] As a side note: we probably can make *some* observations
of other <br>
people's utilities even if we don't have a fixed scale. For
instance, I <br>
can probably reason that a candidate who would put you in a
prison camp <br>
would be a much worse choice from your perspective than one
who would <br>
arrange a party; and that the difference in utilities would be
much <br>
greater than say, between a candidate who holds a week-long
party and <br>
one who holds a two-week long party.<br>
<br>
Some cardinal proponents also refer to von Neumann-Morgenstern
utilities <br>
as a way of making comparisons between people's strength of
preference: <br>
basically using lotteries to determine how much more a voter
prefers one <br>
choice to another. But such scales still need to be normalized
because <br>
they always have two unknown variables per voter. Methods that
do the <br>
normalization so as to give each voter the same strength fail
IIA. Not <br>
doing such renormalization can make the method pass IIA but
they still <br>
don't pass "de facto IIA".<br>
<br>
See e.g. <br>
<a
href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#Incomparability_between_agents"
rel="noreferrer" target="_blank" moz-do-not-send="true"
class="moz-txt-link-freetext">https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#Incomparability_between_agents</a>
<br>
for the need for normalization.<br>
<br>
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End of Election-Methods Digest, Vol 235, Issue 30<br>
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