[EM] A link between max A>B and "maximally beneficial elimination"

[Kristofer Munsterhjelm]

In my last post, I mentioned the idea of making A's score so that when 
candidates who are not A or X or Y are eliminated, the result of the A 
vs X vs Y first preference Copeland contest gives A the highest score; 
with the idea that this would preserve monotonicity because if A is 
raised, then A's first preferences is never lowered in any reduced 
election, so the max operator ensures monotonicity.

In that post I said that the elimination shouldn't eliminate the 
Plurality winner.

But here's a fun part: if we disregard this constraint, and try to 
"clean up" IRV in the same manner, we get the max A>B method that I 
stumbled across a while ago, where A's score is the maximum landslide 
virtual runoff he's involved in, and the candidate with the highest such 
score wins.

Why does that happen? Let E be some elimination order that doesn't 
include either A or some other candidate X. A's IRV score under the 
elimination order E is the Plurality score he obtains when the 
elimination order is followed and only A and X remain. And A's 
unqualified score is A's score with the elimination ordering that 
maximizes A's score.

It's pretty easy to see that the order that one eliminates everybody but 
A and X doesn't matter for determining A's score; in the end only A and 
X matter. And since only A and X are left, A's score under an 
elimination order that spares X is simply A>X (in the absence of any 
equal-rank).

So the task of finding an elimination order that maximizes A's score is 
reduced to finding some X so that A>X is maximized. Which is what the 
max A>B method does!

The interesting part about this is that max A>B's strategy vulnerability 
is all compromise and no burial. Thus it seems that (at least in this 
case), trying to rescue monotonicity by creating an optimized 
elimination order -- rather than just eliminating directly -- doesn't 
produce burial incentive. So creating a custom elimination order for 
some method in such a way that the elimination order never leads to 
monotonicity, might be a way to get a monotone burial-resistant method.

Unfortunately, naively porting this to Friendly Voting or first 
preference Copeland doesn't produce anything useful. Take Friendly 
Voting (really Friendly Cover):
f(A) = (sum over all candidates X weakly covers: fpA) - (sum over all 
candidates Y beating A pairwise: fpY)

Then the elimination schedule method with no constraints reduces to: A's 
score is f(A) with everybody but two candidates eliminated, and these 
two candidates chosen so that A's score is maximized.

If A is in the Smith set and there are at least two candidates outside 
it, then A's score will be equal to the number of voters, which is 
completely uninformative. And this also shows that the "making the case 
for A" approach (produce an elimination schedule that maximizes A's 
score) isn't cloneproof.

But it's an interesting primitive. Perhaps something good can be done 
with it by moving Condorcet detection outside, or restricting what 
candidates the max can be taken over, or doing leximax as a tiebreaker...

I suspect that the following methods are all monotone:
	- max A>B
	- f(A) = choose a set S of candidates to maximize the score when first 
all candidates in S are eliminated and then Friendly Cover is done on 
the reduced election -- regardless of the size of S

and that the "choose a set S" hybrid methods with the base method being 
IRV (or first preference Copeland) is also burial resistant, no matter 
the size of S.

Perhaps even a Carey-esque primitive would work, something like "choose 
a subset S with more than 1/3 of the first preferences, then eliminate 
them, then choose a subset Q with more than half, then eliminate all but 
some X, then A>X is A's score". But Carey fails mutual majority, so it 
has its limits.