<div dir="auto">You are right about the relevance of clones in spatial models.<div dir="auto"><br></div><div dir="auto">Suppose that you were to take literally the distorted distances of a Mercator projection to make a decision about which location on the globe would minimize the sum of distances from the world's major airports to a proposed commerce hub.</div><div dir="auto"><br></div><div dir="auto">Kemeny-Young is a Condorcet method that finds the location (i.e. ranking) in the space of rankings, that minimizes the sum of distances from it to the voter rankings.</div><div dir="auto"><br></div><div dir="auto">The democratic relevance of the Kemeny-Young method depends on the accuracy of the Kendall-tau distance metric in the same way the likely wisdom (or lack thereof) of the location of a commerce hub would depend on the accuracy (or lack thereof) of the distorted Mercator distances.</div><div dir="auto"><br></div><div dir="auto">The Mercator distortion is the result of a projection of a curved manifold onto a flat surface. </div><div dir="auto"><br></div><div dir="auto">The distortion of the Kemeny-Young picture is due to the effect of clones. The Kendall-tau distance between two rankings is a count of the number of adjacent swaps needed to convert one ranking into another.</div><div dir="auto"><br></div><div dir="auto">When a set of clones B1 ... Bk replaces a candidate B in a voter ranking of A,B,&C the increase in swaps to get from A to C due to the mass of clones, distorts the distance ... it unduly stretches the distance from A to C relative to the diameter of the clone set.</div><div dir="auto"><br></div><div dir="auto">The clone-independent "swap cost metric" gives weights to the swaps in order to keep the "swap cost distance" the same from A to C, the same way the distance between San Francisco and Los Angeles has remained the same no matter how many new cookie-cutter towns have sprouted up between them in the post war years.</div><div dir="auto"><br></div><div dir="auto">It's the geometry!</div><div dir="auto"><br></div><div dir="auto">Donald Saari uses the same clone dependent Kendall-tau geometry in his derivation of the Borda Count ... so it suffers the same distortions. Changing to the more appropriate swap cost metric that assigns weights to the swaps instead of counting them all the same, declones Borda.</div><div dir="auto"><br></div><div dir="auto">The weights are jointly proportional to the first place counts of the two candidates being swapped. The first place count of B is partitioned among its clones ... so getting past the clone set has the same total swap cost as getting past B.</div><div dir="auto"><br></div><div dir="auto">I'm sure you can see the advantage of having a non-distorted metric on the space of ballots ... since the ballots are rankings ... and in the Universal Domain we cannot distinguish voters fron their rankings. In essence we have metrized the space of voters.</div><div dir="auto"><br></div><div dir="auto">How about the candidates? Which ranking represents a given candidate?</div><div dir="auto"><br></div><div dir="auto">If each candidate were at the top of only one ballot ranking, there would be no question. In general, a candidate position is a weighted average of voter rankings, which are easily included into the metric space: the distance from ranking R to a weighted average of ballots a1r1 ... akrk is the same weighted average a1d1 ... ask of the respective distances d1 to dk from R to r1 to r_k, respectively.</div><div dir="auto"><br></div><div dir="auto">So we have a new tool uniquely adapted to the natural Universal Domain geometry ...</div><div dir="auto"><br></div><div dir="auto">...our reward for trying to understand the meaning of a clone independent metric!</div><div dir="auto"><br></div><div dir="auto">-Forest</div><div dir="auto"><br></div><div dir="auto"><br></div><div dir="auto"><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Feb 10, 2023, 6:48 AM Colin Champion <<a href="mailto:colin.champion@routemaster.app">colin.champion@routemaster.app</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<font face="Helvetica, Arial, sans-serif">Kristofer - what you say
is perfectly reasonable and my disagreement is mostly a matter of
degree.<br>
<br>
I'm not persuaded that IC can be defended as a proposition of
"collective decision making" (in Arrow's sense)</font><font face="Helvetica, Arial, sans-serif"><font face="Helvetica, Arial,
sans-serif"> in general</font> rather than as restricted to
"certain special assumptions" (his term for a spatial model). The
property of being consecutive in all ballots is not meaningful in
itself, but only as a probabilistic indication that candidates
have some intrinsic property in common. Such a property can easily
be identified in a spatial model, but only in far-fetched cases
does a jury model have a similar property which can be inferred
from positions in ballots. Other models (I'm thinking of
Bordley's) may have candidates with no intrinsic properties at
all. Under a jury model, I think the likeliest case in which
candidates will be consecutive in all ballots is pure chance when
the number of voters is small. <br>
<br>
Under a spatial model it seems to be possible for the presence of
clones to be informative. Suppose that voters come from a
zero-mean Gaussian and that candidates come from a mixture of the
same distribution and a delta spike at the origin. Then any
candidate who has a clone can be recognised as a rightful winner.
Arrow would correctly point out that this is a piece of
information which lends itself to manipulation (a clone might be
induced to stand down), but discarding information which could
potentially be suppressed is not a sound methodology. It's like
rejecting the evidence provided by any witness who might in
principle have been persuaded not to testify.<br>
<br>
I don't claim that any of these models is remotely as useful as a
smooth spatial model, but it's worth avoiding claiming undue
generality. <br>
<br>
I'm not sure how firmly you're defending IC as a cheap
approximation to robustness to strategic nomination. You suggest
that it's Condorcet compliance rather than clone independence
which reduces nomination incentive, and I suspect you mean this in
a stronger sense than the one in which it's obvious. The median
voter theorem protects Condorcet methods against strategic
nomination in the same way as it protects them against innocent
errors. It's an imperfect protection because the theorem's
conditions won't be exactly satisfied in practice. Even so, the
differences in raw accuracy between different Condorcet methods
are so small (compared with differences in simplicity or in
resistance to tactical voting) that people don't place much weight
on them; it's likely that the same would apply to strategic
nomination. I assume that's why JGA compares non-Condorcet methods
with each other and with a representative Condorcet method.
However IC is commonly used to support a preference between
Condorcet methods, most of which seem to violate it. I suppose
different people may have different hunches as to how much good
the criterion is likely to do. <br>
<br>
Colin<br>
<br>
<br>
</font><br>
<div>On 04/02/2023 22:26, Kristofer
Munsterhjelm wrote:<br>
</div>
<blockquote type="cite">I seem
to have forgotten to reply to this post. Well, here goes :-) <br>
<br>
On 25.01.2023 11:36, Colin Champion wrote: <br>
<blockquote type="cite">A couple of observations/questions. <br>
<br>
Firstly it isn't clear to me that IC makes a lot of sense except
under a spatial model. The definition of clones is two
candidates who are consecutive in all ballots, but the concept
is only practically useful if this corresponds to some property
inherent in the candidates. Under a spatial model, two
coincident candidates will be consecutive in all ballots. (The
converse isn't clear.) The presence of clones might then arise
through cultural factors or strategic nomination. <br>
<br>
Under a jury model, if A is unmistakably better than B and C,
and B and C are unmistakably better than D, then B and C will be
consecutive in all ballots. But suppose that B and C are always
consecutive while sometimes coming above and sometimes below
both A and D. Shouldn't we assume that the consecutiveness is a
coincidence and decline to draw any conclusions from it? <br>
</blockquote>
<br>
Suppose the true order is A>B>C>D. Then if you get both
A>B>C>D and D>C>B>A, then it seems you're not in
a Kemeny type jury model, at least, because a judge has to be very
unlucky to get all of his X>Y preferences reversed. So in such
a situation, I'd say that's more evidence that you're not in a
jury model, in which case clone independence neither helps nor
hurts you. <br>
<br>
Though my inuition might be wrong; I'm not entirely sure about the
relative likelihoods here. <br>
<br>
<blockquote type="cite">Secondly, Kristofer justifies the IC
criterion as a convenient tool for designing methods which are
free from nomination incentive, saying that trying to do so
directly is "incredibly messy". However presumably one can
*measure* the susceptibility of a method to the nomination
incentive (especially if a spatial model is assumed), so this
line of thought doesn't justify accepting or rejecting a method
on account of its satisfying IC. <br>
</blockquote>
<br>
Yes, it's more about design than about testing. Testing for
nomination incentive is harder than testing for clone
independence, but perfectly doable. (That's what JGA did.) <br>
<br>
But I don't know of any theory of how to design a method to
specifically resist nomination incentive, or any model of
incentive that could easily guide method design. On the other
hand, clone independence is at least a simple criterion, so it's
easier to figure out in one's head if this or that passes or
fails. <br>
<br>
I agree that this provides no justification to optimize for clone
independence (something correlated with what we want) rather than
lack of nomination incentive (what we actually want). <br>
<br>
The most intuitive jusitification would probably be something like
"don't give the opposition anything to use against us". If clone
independence doesn't itself hinder anything desirable, then
picking it up would prevent say, FairVote from saying "but you
know, IRV is clone independent and your method isn't"; even if the
proposed method has much lower nomination incentive than IRV, it
would be preferable to not have to deal with the potential for
confusion. <br>
<br>
All of that hinges on clone independence being "cheap", though. <br>
<br>
<blockquote type="cite">Presumably there are other nomination
strategies besides nominating (or denominating) clones. JGA has
shown that minimax isn't particularly vulnerable to nomination
incentives - is it obvious that clone-independent methods are
particularly resistant? Or is it possible that clone dependence
is simply a form of error which has been identified and
taxonomised, but which is not intrinsically more important than
any other form or error? <br>
</blockquote>
<br>
From what I know, IRV has serious nomination incentive while being
clone independent, while all the cloneproof Condorcet methods also
have low nomination incentive (like most serious non-cloneproof
Condorcet methods). I would *suspect* that DAC and DSC, while
being theoretically cloneproof, also have nomination incentive,
but I don't have proof of this. <br>
<br>
So it's definitely possible that the correlation isn't
particularly strong: that it's the Condorcet rather than the clone
independence that reduces nomination incentive. In that case, I
would guess it goes something like... spatial models rarely have
huge Condorcet cycles, and when the Smith set is small, you get
free IIA against anything outside it (strategy notwithstanding);
so it doesn't particularly matter if outside-of-Smith candidates'
parties nominate a few or a lot. If that's right, then robust
clone independence (the thing that's actually correlated with
nomination incentive) would mostly matter in cases with heavily
multidimensional politics and large Smith sets. <br>
<br>
That's also just a guess, though. <br>
<br>
-km <br>
</blockquote>
<br>
</div>
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</blockquote></div>