[EM] Condorcet-Kemeny clarifications

[Kristofer Munsterhjelm]

On 11/5/21 1:34 AM, Richard, the VoteFair guy wrote:
> On ..., Forest Simmons wrote:

>  > You invoked Arrow, but Arrow never said that Clone Winner and IIAC 
> were incompatible.
> 
> I referred to Arrow's theorem as an example of a broader concept. That 
> concept is that if a method has a zero failure rate for specific 
> fairness criteria, then there are specific other criteria that cannot 
> have a zero failure rate.
> 
> As you point out, there are proofs that support this concept for 
> specific combinations of characteristics.
> 
> In my opinion we don't need to wait for more theorems to extend this 
> concept to the broader concept: There are going to be other combinations 
> of fairness criteria for which getting a zero failure rate for one of 
> them means that we cannot get a zero failure rate for another one.

There are also going to be combinations of fairness criteria where it's 
perfectly possible to get both.

And my objection is that it's not clear which one it is.

Even in the strict "pass or fail" regime, we know there are some 
combinations that are incompatible (e.g. Condorcet and Participation). 
There are others that are compatible (e.g. Condorcet and monotonicity). 
We know this from impossibility proofs or the existence of methods that 
do both.

But it seems pretty shaky to conclude from evidence of IRV (which is not 
a particularly good method) that the properties you're looking at are of 
the former category, not the latter. This would be like concluding, 
because IRV passes Condorcet loser and fails monotonicity, and Plurality 
passes monotonicity but fails Condorcet loser, that Condorcet loser and 
monotonicity are necessarily incompatible.[1]

So if you trust the numbers of your simulation, at least do some tests 
with worthy challengers to Kemeny -- by which I mean something closer to 
River or Ranked Pairs, not IRV.

(Although your numbers may themselves be inaccurate if they show Borda 
to have 100% clone independence for two candidates -- depending on just 
what that means, as I brought up in my other post.)

> I'm attempting to use measurements to quantify the answer to the 
> question: "How close can we get to identifying a method that has a nice 
> balance of low failure rates across the most important fairness criteria?"
> 
>  > Clone winner failure is not about electing the wrong clone ... it's
>  > about none of the clones of the erstwhile winner being elected.
>  > ...
>  > That is the spoiler problem.
> 
> As I understand it, the word "spoiler" overlaps with the clone 
> independence (CI) criterion and the IIA (independence of irrelevant 
> alternatives) criterion. Specifically:
> 
> * CI refers to the effect of adding candidates.
> 
> * IIA refers to the effect of removing candidates.
> 
> I think the word spoiler can refer to either an added candidate or a 
> removed candidate changing the results, right?

My point was that if you give me a clone independence failure, I can 
translate it directly into IIA failure. So any method that's clone 
independent lacks these particular IIA failures; and any method that 
passes IIA would automatically also pass clone independence.

The translation is pretty easy: you just flip the two elections in the 
pair of elections that demonstrate the clone failure, so that instead of 
adding a candidate and the winner then changing, you remove a candidate 
and the winner changes.

The given directions used to determine failure aren't set in stone 
either. For instance, JGA talks about "candidate exit incentive", where 
allies of a candidate have an incentive to leave to help that candidate 
win. Apart from the notion being more generalized than clones (allowing 
for near-clones), this is vote-splitting, but -- significant in this 
context -- it goes *from* the election with more candidates, *to* the 
election with fewer. Just like IIA.

So if you want an IIA-like phrasing of clone independence, it would be: 
"A clone candidate who didn't win shouldn't change the outcome by 
dropping out". It shouldn't matter if vote-splitting happens because 
someone unwisely decided to join the race (adding) or was averted only 
because someone found out he had to leave (removing).

I agree that whether you consider clone failure to be adding or removing 
candidates can influence how you count the proportion of elections that 
are vulnerable -- I'll say a bit more about that in another post about 
the proportion of two-candidate Borda elections susceptible to clone 
failure, later. But however the counting is done, fixing clone 
dependence also, as a consequence, fixes a subset of IIA failures; 
because there's a direct way to go from one to the other.

-km

[1]  Even worse is giving up the search for a monotone Condorcet 
loser-passing method because clearly they must be incompatible; or even 
deliberately making a method fail Condorcet loser in the hope of making 
it pass monotonicity.