[EM] An interesting little compromise criterion

[Kevin Venzke]

Hello,

When trying to define strategy criteria about compromise there are some difficulties. The
obvious counterpart to Later-no-harm is the strong form of FBC, which isn't compatible with
majority favorite. As a result, I usually measure "how often" methods appear to violate
strong FBC, rather than being able to apply a pass/fail test.

If we assume a method satisfies majority favorite, and say that if compromising voters could
create a majority favorite, this ensures or prevents some result, then we create
contradictions. I'm not sure if anyone ever proposed "majority defeat disqualification"
as a *criterion*; it doesn't work well, but could lead one to the Schwartzified version,
CDTT.

I came up with CCE (for "Condorcet Compromise Extension") based on the assumption that a
method would elect a CW when there is one. So, if X can be made the CW then any candidate
Y who is beaten by X, shouldn't be elected, so that the X>Y voters don't have the incentive
to compromise. This too leads to cycles of claims, necessitating a Schwartzified version I
call the CCE "top tier" or CCETT.

Here is a different idea. First consider this "bullet vote criterion" (BVC): If all voters
bullet vote, then the winner is the candidate who gets the most votes.

Few methods fail this. A method would have to be random or perhaps not admit bullet voting
as valid.

Consider then this "CompBV" criterion (meaning "compromise criterion, assuming BVC"):

If all voters preferring X to Y were to bullet vote for X, and this would result in an
election with only bullet votes, *and* X would then have the most bullet votes, then Y must
not be elected.

Example:

29: A>D
28: B
23: C>A
20: D

B can't win because A beats B and A>B voters can create a BV-only election that A wins. And
D can't win similarly, due to the A votes from the A>D and C>A voters. A is the CW, and so
is fine to elect.

C is not barred from winning, because their C>A votes prevent the creation of a BV-only
election by the voters preferring A, B, or D to C.

Some comments:
1. In many scenarios this will impose no restrictions, because if any pairwise Y>X or Y=X
voters aren't bullet-voting then CompBV says nothing. So this isn't suitable as part of a
method definition (sort of like clone independence isn't).

Since the criterion is often indifferent, then to have value, one would have to hope that
satisfying CompBV in the few, "simplest" scenarios would somehow mean that some
compromise-related merit propagates also to all other scenarios. I'm not sure how believable
that is. I'll come back to it.

2. The criterion isn't actually purely about compromise. Consider:

8: A>B
5: B
7: C

CompBV requires that A win, even though the modification the A>B voters can make is to
truncate, not compromise.

3. This criterion does not appear to give rise to cyclic claims! So you don't need a
Schwartzified version.

That is to say, several methods appear to satisfy CompBV. If there were cyclic claims then
every method should fail it.

The compliant methods are:
1. Raynaud(WV), i.e. successively eliminate the losers of the strongest defeats
2. the LNHarm-compatible Borda interpretation I mentioned
3. my "suggested" CCETT methods, like CCE-Schulze etc. (See votingmethods.net/cce)

I was a bit disappointed this list was so short. But surprised by what was on it.

All three of these violate Plurality, although the CCE methods satisfy what I call "weak"
Plurality, under which a voter's own first preferences can't serve to disqualify the voter's
own non-last preferences.

(The question occurred to me of whether CCETT inherently implies CompBV, but that is not so:

470: A>C>B>D
341: C>B>D>A
122: B>D>A>C
 67: D

CCETT allows any winner, but CompBV disallows D because of the B>D voters.)

Methods that did very well but not perfectly (in simulations measuring CompBV) include
MinMax(WV)-like methods, a couple of methods where candidates need only contend with their
max opposition candidate, and FP chain climbing where you seed with the first preference
loser.

Non-Condorcet implicit-approval-heavy methods did quite badly. With 3 candidates Bucklin
was the single worst method that satisfies majority favorite, far worse than FPP.

But the other question is whether CompBV compliance corresponds at all to the rate of strong
FBC failures (a more general compromise measurement). It's very mixed. The CCETT methods are
about the best compromise methods I have. So are the WV-likes. But all the other methods I
mentioned are mediocre at compromise.

So I don't really know the significance. It was just interesting to see a non-cyclic
compromise criterion.

Kevin