[EM] Safe data for unburiable relations

[Kristofer Munsterhjelm]

I thought a bit more about just what kind of data can be used to create 
a disqualification relation for a more particular resistant set, while 
retaining the unburiability by proponents of candidates outside the set.

This may be generalizable to creating unburiable methods, but I'll 
impose a broad kind of monotonicity to cover some corner cases; namely 
that lowering A or raising W never establishes A ~> W where it was not 
active, and neither raising A nor lowering W breaks A ~> W where it was 
established (more on that later). What we can use for a disqualification 
relation A ~> W is then:
	1. The first preferences of any sub-election containing A and W,
and	2. any rank preference of any sub-election not containing W.

Suppose that some data is "safe" for use to define A ~> W if A ~> W 
can't be altered by burial no matter how that data is used. Then such 
data is admissible iff it's one of the instances above.

Proof: First, that these are all safe. The first point is admissible 
because an A>W voter can't modify anybody's first preferences by burying 
W, since his ballot will always count towards A.

The second point is safe because an A>W voter can't alter anything at 
all in a sub-election that doesn't have W in it if he's restricted to 
burial.

Second, that nothing else is safe. We can't use lower ranks of 
sub-elections containing A and W, because suppose that we use kth rank 
and the burier's honest preference places W in kth place in a 
subelection. Then burying W (lowering W's rank) would decrease W's kth 
preference count and increase someone else's.

We can't use any ranks of sub-elections containing only W (not A) 
because it's possible that the burier's honest ballot ranks A first and 
W second (after the necessary eliminations to get the sub-election in 
question); then, after burying, W's first preference count is decreased 
by one and someone else's is increased by one.


The bad news is that there are some caveats that weaken the result.

First, there's the monotonicity I mentioned. If we allow W ~> A to be 
established by lowering W, then we could make an A>W's attempt at 
burying W establish W ~> A, thus directly punishing the burial. Forest's 
"bus" methods are in this vein. But I suspect this will just replace a 
burial opportunity with pushover: instead of harming W by burying W, one 
harms W by *raising* W. So I don't think we lose too much by imposing 
this monotonicity criterion - at least not if we're looking to increase 
general strategy resistance (not just burial).

Second, it's possible that one could cleverly combine unsafe data usage 
to always skirt the unsafe areas. For instance, three-candidate IFPP 
would seem to be nonmonotone because it eliminates Plurality losers, but 
the 1/3 threshold interacts with the eliminations so that the 
nonmonotone area is always out of reach. (This is why I said "no matter 
how that data is used" above.)

Third, it's strictly speaking possible to mix ballots from different 
safe subelections and still be safe. For instance, you could take the 
first voter's ballot, eliminate one set of candidates including W, then 
take the second voter's ballot and do the same, but with a different set 
of candidates including W. I've excluded such combinations because I 
can't see a way to do so without breaking neutrality or symmetry, but 
perhaps such a method exists? (Consider e.g. the "voice of reason", 
which makes its social ordering a duplicate of a particular ballot.)

Finally, chains of disqualifications make the use of category 2 *much* 
harder than it seems. Say we have a chain where A ~> B and B ~> C, so 
that A is the sole member of the resistant set for this relation.

Then if B ~> C is active because of some sub-election not containing C, 
it's possible that lowering A could break B ~> C. The result would be 
that a voter who prefers C to A could introduce C into the resistant set 
by burying the current winner, A.

This is the most serious limitation IMHO. I've tried to find a more 
general result that deals with indirect chains, but no luck so far. (If 
A ~> B were transitive, then we wouldn't have this problem.)


And a simple possible extension that deals with clone sets of size two, 
based on the ideas above:

	A indirectly disqualifies B in one step if there exists a C so that
		A disqualifies B in every subelection not containing C (category 1), and
		A beats C pairwise (category 2).

I *think* this is acyclical, but I haven't verified it.

-km