[EM] Why Clone Independence?

[Forest Simmons]

You are right about the relevance of clones in spatial models.

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.

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.

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.

The Mercator distortion is the result of a projection of a curved manifold
onto a flat surface.

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.

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.

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.

It's the geometry!

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.

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.

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.

How about the candidates? Which ranking represents a given candidate?

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.

So we have a new tool uniquely adapted to the natural Universal Domain
geometry ...

...our reward for trying to understand the meaning of a clone independent
metric!

-Forest




On Fri, Feb 10, 2023, 6:48 AM Colin Champion <colin.champion at routemaster.app>
wrote:

> Kristofer - what you say is perfectly reasonable and my disagreement is
> mostly a matter of degree.
>
> I'm not persuaded that IC can be defended as a proposition of "collective
> decision making" (in Arrow's sense) in general 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.
>
> 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.
>
> 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.
>
> 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.
>
> Colin
>
>
>
> On 04/02/2023 22:26, Kristofer Munsterhjelm wrote:
>
> I seem to have forgotten to reply to this post. Well, here goes :-)
>
> On 25.01.2023 11:36, Colin Champion wrote:
>
> A couple of observations/questions.
>
> 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.
>
> 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?
>
>
> 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.
>
> Though my inuition might be wrong; I'm not entirely sure about the
> relative likelihoods here.
>
> 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.
>
>
> 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.)
>
> 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.
>
> 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).
>
> 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.
>
> All of that hinges on clone independence being "cheap", though.
>
> 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?
>
>
> 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.
>
> 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.
>
> That's also just a guess, though.
>
> -km
>
>
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