On 24.01.2023 22:04, Forest Simmons wrote:

If I were to answer the question in simple terms, I would put it like this:

• If there's a clone winner problem, then candidates may be incentivized
  to drop out of the race, which is a bad thing.
• If there's a clone loser problem, then the election can possibly go to
  whatever faction manages to muster up enough candidates to spam the
  ballot, which is worse.
• If there's a crowding problem, then the method generally behaves
  chaotically; it's a sort of IIA failure.

When we aim for clone independence, we're implicitly assuming that the
clone independence will in some way be robust; i.e. that a lack of clone
winner problems implies no incentive to exit, and a lack of clone loser
implies no incentive to entry.

James Green-Armytage showed that this implication doesn't hold for IRV.
Conversely, his research suggests that although minmax fails clone
dependence, it doesn't produce a severe nomination incentive in
practice, in either direction.

So ideally we'd be designing the methods to lack nomination incentive,
instead of designing them to pass a criterion that only implies such a
lack if it generalizes properly. But doing the former is incredibly
messy - it's both hard to verify and hard to design. Hence the stand-in
of clone independence.

And usually it works! Ranked Pairs and Schulze have very low nomination
incentive. But not always.

I would say that it's not so much that clone winner is linked to
compromising and clone loser to burial, as that they're linked to
nomination incentive. For instance, with enough candidates in impartial
culture, Ranked Pairs and Schulze are plenty susceptible to burial, even
though they're cloneproof.

(Though perhaps there is a more clear relation in say, a spatial model.
I don't know as I haven't checked.)

-km

On 1/24/23 22:32, Kristofer Munsterhjelm wrote:

A thought occurred to me: it might be that the reverse implication is
true: that we can't have vote splitting clone failure without
compromising incentive, and we can't have teaming without burial incentive.

This seems intuitively right for Plurality and Borda: suppose for
Plurality that A loses after being cloned. Then if everybody decides to
rank A1>A2>A3, then that will make A1 win again; this is a compromising
strategy for the A-voters. Conversely, in Borda, suppose that after
cloning A, A1 wins; then in at least some elections, the B>A voters
moving every A clone except A1 to equal last should make A lose again,
which is a burial strategy. These countermeasures only work if the A
voters or the not-A voters (respectively) hold a large enough share of
the votes.

But generalizing it to every method would be much harder.

And there's the obvious question: if there are implications for
vote-splitting and teaming, then what's the implication for crowding?
You'd think nonmonotonicity (due to the chaos), but nope - Kemeny has
crowding and is monotone.

-km

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?

[Yet if the rationale behind IC implicitly assumes a spatial model, the
rationale behind the consistency criterion implicitly assumes a jury
model, though both are presented as desirable properties of voting
systems in general.]

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. 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?

CJC

On 25/01/2023 00:14, Kristofer Munsterhjelm wrote:

Kristofer,

The clone dependence of Borda and Kemeny-Young are symptoms of the same
root malady ... the clone dependence distortion of the Kendall-tau metric.

You know that Kemeny-Young elects the head of the candidate ranking that
minimizes the total Kendall-tau distance from it to the ballot rankings.

The contribution of one ballot B to the Borda count of candidate X is the
distance from the ballot B ranking to the ranking with X moved to the
bottom of B ... that is the number of swaps it takes to lower X to the
bottom of B.

Clones distort the Kendall-tau metric like the rear view mirror that says
"objects may be closer than they appear."

Let's look at the example ...

60 A>B>C>D>E
40 B>C>D>E>A

The Kendall tau swap cost of moving A to the bottom of one first faction
ballot is 4 swaps per ballot ... a total Kendall-tau distance of 4*60=240
swaps.

The second faction takes zero swaps to get A to the bottom, so the Borda
total for A is 240.

Similarly, the Borda total for B is
603+404=340 total swaps.

The majority candidate A gets a lower Borda score than B!

Without the clones it would take 60 swaps to get A to the bottom of all
ballots, but only 40 swaps to move B to its faction bottom.

Note that the candidates get credit for moving through the crowd of clones.
This gives most advantage to the clone ranked bighest among its fellow
clones.

If Kendall-tau is decloned by weighting each swap with the product of the
first place scores of the candidates being swapped, then swapping A and B
yields a cost of 60*40 per swap ... while all other swaps yield zero.

So the total weighted swap cost of moving A to the bottom is 60*(6040) ...
much greater than the total cost of moving B to the bottom ... 40(60*40).

The ratio of the two costs is 60/40 ... the same as it was before the
clones were introduced.

Decloning Kendall-tau rectifies the distance distortion at the root of
Kemeny-Young crowding in a similar manner.

We have used the random ballot favorite lottery probabilities to declone
Kendall-tau. The probability distribution of any proportional lottery could
be used.

For example we could use the random implicit approval ballot lottery
probabilities.

This can be done in three ways ... counting approvals fractionally ... or
repeated drawings to narrow down to a winner ... or the Martin Harper
trick: all of B's probability (1 over the number of ballots) goes to the
candidate approved by B with the greatest approval among such candidates.

Here's another way to get approval cutoffs automatically ... on each ballot
B approve every candidate not outranked by any Smith candidate.

Then use one of the three methods in the previous paragraph to extract
proportional lottery probabilities from the resulting automatically
generated approvals.

That's enough for now!

-Forest

On Tue, Jan 24, 2023, 4:14 PM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:

I seem to have forgotten to reply to this post. Well, here goes :-)

On 25.01.2023 11:36, Colin Champion wrote:

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.

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.

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

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:

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@routemaster.app
wrote: