[EM] Why Clone Independence?

[Colin Champion]

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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