[EM] Mutual majority burial resistance

[Kristofer Munsterhjelm]

Dominant mutual third burial resistance is a nice thing to have, but I
find it quite hard to reason about. So how about considering an easier
criterion?

Mutual majority burial resistance: Suppose that the method passes mutual
majority and X wins, being in the innermost mutual majority set. Then
voters who prefer some Y not in this set to X can not make Y win by
lowering X on their ballots. (Either nothing happens or Z wins instead.)

If the mutual majority set is size one (i.e. a majority winner), then
every majoritarian method is burial resistant. This because the voters
who prefer X to W have no way to influence W's first preference count by
lowering W, since they must by definition be ranking X above W.

That's also why Plurality is burial immune.

Is there any way to generalize this to larger mutual majority sets? I
was originally thinking of something like DAC, with a recursive proof
that enlarging a solid coalition with majority support couldn't get the
candidate the buriers wanted elected, elected... but I couldn't make the
proof work.

How does IRV do it? I imagine it's the usual pattern where using
elimination makes something that applied to a single candidate (in the
case above, the majority criterion candidate can't be buried) apply to a
whole set. As Plurality is unaffected by burial, the elimination order
is also unaffected by burial. And thus at some point, all but one member
of the mutual majority set has been eliminated, and then the
single-candidate burial immunity comes into play. (I'm not *entirely*
convinced that the elimination order is unaffected by burial, but it
probably is; I'd just have to make a better proof.)

But we can't use elimination because that destroys monotonicity (unless
we could somehow engineer a proper look-ahead). That's why I was looking
at solid coalitions as an alternative, since DAC/DSC is a monotone
alternative to IRV.

Any ideas in general?

-km