[EM] A more general chicken dilemma

[Kristofer Munsterhjelm]

On 2025-09-28 18:22, Kevin Venzke wrote:
> Hi Kristofer,
> 
> Kristofer Munsterhjelm via Election-Methods <election-methods at lists.electorama.com> a écrit :
>> I've been investigating different criteria lately, and the chicken
>> dilemma criterion seems to apply to very few elections in general (as a
>> proportion of the number of elections possible with a given number of
>> voters).
>>   
>> So I was wondering if the following would be a good generalization:
>>   
>> The following is required for an election to be applicable:
>>       - There are three candidates and three factions, call them A, B, and C.
>> Each faction prefers its corresponding candidate to everybody else (but
>> may rank others coequal top based on the method, e.g. if it's Approval).
>>       - The number of C-voters who prefer A to B is equal to the number who
>> prefer B to A (i.e. they have no meaningful preference between the two).
>>       - If the A-voters and B-voters express a full ballot, then A wins.
>>   
>> Then for every applicable election:
>>       - If the B-voters truncate above A, then B must not win, otherwise the
>> criterion is failed.
>>   
>> What do you think?
> 
> I think it's probably fine, but have you really broadened things much?

I might have thought it broadened things more than it did.

My intent was to remove two conditions that didn't seem relevant. First, 
that A and B obtain a majority of the first preferences, and second, 
that the C voters all truncate above both A and B.

The first isn't actually weakened much because suppose that fpA + fpB < 
v/2, i.e. they don't have a majority. Then C does, and must win by the 
majority criterion, so the generalized CD criterion would only apply to 
methods that fail the majority criterion in this case.

But the second may be useful. Instead of requiring that the C-voters 
plump for C, now they can be any combination of C>A>B, C>B>A, C>A=B, and 
C=A=B as long as they don't systematically favor one of A and B.

> For purposes of testing compliance, you may find it difficult to cover all cases
> of who is *or might be* an A-voter or B-voter, if they are ranking candidates
> equal-top.

Yes. A votes-only interpretation would be something like:

Let x be the number of voters who rank A=B>C.

Then for all allocations of voters to the A bloc and B bloc so that:
	number in A bloc = y + (A>B>C) + (A>B=C)
	number in B bloc = z + (B>A>C) + (B>C>A)
	x = y+z
	x >= 0, y >= 0, z >= 0
the above must hold.

This is rather more cumbersome to deal with. The real question is what 
it would mean for an A=B>C voter to "truncate above A". We *could* say 
that this means "plumping for B" in that case, but it's not quite as 
elegant.

> 
> It is interesting, seeing it described like this, to wonder why the C voters
> matter at all.
> 
> Your phrasing also makes one of my criticisms less intuitive. I tend to think it's
> blatant cheating to require
> 
> - If the B-voters truncate above A, then B must not win
> 
> and not also require
> 
> - If the A-voters truncate above B, then A must not win

I think that follows from symmetry, because the criterion doesn't say 
anything about the relative sizes of A and B; so you could just swap the 
labels of A and B and the criterion would then apply to the A-voters.

But you're right, we could also require this directly, and it would make 
it more clear what it's about.

> because I conceive of the situation as an A/B mutual majority which has to hang
> together or else C wins as punishment.
> 
> But with your wording it's not clear that the B voters are even playing a role in
> A's victory. If A can win single-handedly, why would those voters owe anything to
> candidate B?
> 
> To me, a criterion with this name and concept should just say something like:
> 
> "If mutual majority doesn't apply, then the first-preference winner wins."
> (otherwise follow mutual majority)
> 
> This waives the punishment if the first preference winner is part of the supposed
> mutual majority, but otherwise it probably gets the punishment right.

Something like this almost works:

"If there exists a mutual majority, and someone in that mutual majority 
wins, then voters in that mutual majority who prefer some other 
candidate to the winner can't change the winner to the one they prefer 
by truncating above the actual winner"

But the problem is that the set of all winners is also a mutual majority 
(a mutual unanimity, even), so this is very close to plain immunity to 
truncation and thus is very strong.

-km