Suppose a method passes LNHarm but not LNHelp, and in the election, W
wins. Then what this says is that, for X-first voters, a full ballot
that happens to rank X first is never worse than one that ranks X first
and then truncates. However, it doesn't say whether a ballot that ranks
every candidate with W in second place is any worse (or better) than one
that ranks every candidate with W last. So such a method may still
reward burial.
Now suppose that a method passes LNHelp but not LNHarm. Then in the
scenario above, an X>W voter who wants to flip the election from X to W
can get the full effect of burial by simply voting X alone, since
further ballots will not help X get elected. However, assuming that the
initial election is honest with the X>W voters voting full ranks, burial
might still work. That is, if burying W makes X win, then truncating the
ballot below X also gets X to win; but it may still be the case that an
untruncated ballot with W last gets X the victory whereas an untruncated
ballot with W second doesn't.
If the method passes both, then it's burial immune because whether X
wins is completely unaffected by later preferences. So someone who votes
X ahead of W is already bringing as much strength to bear against W
(short of strategizing above the current rank of X, i.e. compromising).
Unfortunately, as Woodall shows, we can't have all of LNHarm, LNHelp,
mutual majority, and monotonicity. But the LNHs aren't necessary for
burial immunity: e.g. voting for more candidates could still make X
lose, but the particular pattern of burying W might still not have an
effect.
(And in any case, we can't have total burial immunity with Condorcet...
but perhaps there's some kind of set analog of LNH that would help get
some idea of what needs to be done to make a method MMBR or DMTBR. "If X
is not in the mutual majority set and W is for some ballot configuration
of X>W voters, then the ordering of candidates below X by those voters
can't affect whether X wins", perhaps. But that's still a very hard
criterion to pass!)
Suppose a method passes LNHarm but not LNHelp, and in the election, W
wins. Then what this says is that, for X-first voters, a full ballot
that happens to rank X first is never worse than one that ranks X first
and then truncates. However, it doesn't say whether a ballot that ranks
every candidate with W in second place is any worse (or better) than one
that ranks every candidate with W last. So such a method may still
reward burial.
Now suppose that a method passes LNHelp but not LNHarm. Then in the
scenario above, an X>W voter who wants to flip the election from X to W
can get the full effect of burial by simply voting X alone, since
further ballots will not help X get elected. However, assuming that the
initial election is honest with the X>W voters voting full ranks, burial
might still work. That is, if burying W makes X win, then truncating the
ballot below X also gets X to win; but it may still be the case that an
untruncated ballot with W last gets X the victory whereas an untruncated
ballot with W second doesn't.
If the method passes *both*, then it's burial immune because whether X
wins is completely unaffected by later preferences. So someone who votes
X ahead of W is already bringing as much strength to bear against W
(short of strategizing *above* the current rank of X, i.e. compromising).
Unfortunately, as Woodall shows, we can't have all of LNHarm, LNHelp,
mutual majority, and monotonicity. But the LNHs aren't *necessary* for
burial immunity: e.g. voting for more candidates could still make X
lose, but the particular pattern of burying W might still not have an
effect.
(And in any case, we can't have total burial immunity with Condorcet...
but perhaps there's some kind of set analog of LNH that would help get
some idea of what needs to be done to make a method MMBR or DMTBR. "If X
is not in the mutual majority set and W is for some ballot configuration
of X>W voters, then the ordering of candidates below X by those voters
can't affect whether X wins", perhaps. But that's still a very hard
criterion to pass!)
