[EM] new working paper: (edit/second thought)
Kristofer Munsterhjelm
km-elmet at broadpark.no
Sat Feb 19 12:31:57 PST 2011
Kevin Venzke wrote:
> Hi Kristofer,
>
> --- En date de : Sam 19.2.11, Kristofer Munsterhjelm <km-elmet at broadpark.no> a écrit :
>> It also seems possible to bury using Bucklin. Say that your
>> sincere preference is A > B > C > D, and that B
>> wins in the second round, but if you could somehow keep B
>> from winning, then A would win in the third. Then
>> dishonestly burying B, say by voting A > C > D > B,
>> would help.
>>
>> A method that passes LNHarm doesn't have this problem,
>> AFAIK, because later preferences cannot harm your earlier
>> preferences. Your chance of having A win is the same whether
>> you vote A > B > C > D or A > D > C > B.
>
> Actually I don't think this is true. This example shows that an A loss
> voting A>B>C>D means that voting A alone is also an A loss. I can't see
> any guarantee from LNHarm that A>D or A>D>C or A>D>C>B can't make A win.
> And LNHarm/LNHelp don't apply to shuffled lower preferences.
The A>B>C>D vote makes A lose because it gives B the vote it needed to
win. So a bare-A vote would also deprive B of this ballot, assuming that
it's not a kind of ER-Bucklin where the bare A ballot means every other
candidate gets a point at second place.
In this context, voting > B > C > D makes A lose. If you're going to be
immune from that, later preferences shouldn't harm earlier ones, i.e.
the method should pass LNHarm.
Let's try that again. If you prefer A to B, and B wins, then the point
of burial is to make A win instead of B by moving B lower down in ranks.
However, if the method passes LNHarm, then, to quote Woodall's
definition, "adding a later preference should not harm any candidate
already listed". In other words, because later ranks can't harm A, both
A>B>C>D (the honest ballot) and A>C>D>B (the maximally burying ballot)
has the same effect on whether or not A wins, which is the effect that a
bare-A vote gives. Since you can't rig the field in favor of A by
rearranging later ranks, and burial tries to get A to win by doing just
that, LNHarm secures a method against this kind of burial.
The LNHelp-failure type burial seems to be an "untruncating" sort of
burial. You have a candidate A and you want A to be helped as much as
possible by later candidates, so you add a bunch of them after the
A-bullet vote. If the method meets LNHarm but not LNHelp, you don't fill
later ranks to harm B in particular, but to help A. Adding B last is
just a precaution so that B won't be helped in turn by candidates ranked
lower than him.
I think JGA only used complete ballots, in which case the only type of
burial that can happen is the one that LNHarm secures against. I'm not
sure about this, however.
If a method meets both LNHelp and LNHarm, then there's no point at doing
any of what we've called burial. Pushing B to the bottom won't make A
win, and filling stuff after a truncated ballot won't make A win either.
Yet that pair comes at a great price: one can't have all of LNHelp,
LNHarm, mutual majority, and monotonicity (see
http://www.mcdougall.org.uk/VM/ISSUE6/P4.HTM). Since Smith implies
mutual majority, any Smith-constrained version of a method that meets
both LNHs will fail to be monotone.
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