[EM] Some thoughts on Condorcet and Burial

[Kristofer Munsterhjelm]

On 11/8/23 05:07, C.Benham wrote:
> In my last EM post I included Smith//DAC in a list of Condorcet methods 
> "that meets mono-raise".
> 
> A knowledgeable correspondent has cast doubt on this claim, and I admit 
> that I can't prove that it does.
> 
> But I am fairly sure that any failure example needs there to me more 
> than three candidates in the top cycle, and if I'm right about that
> then I'm not concerned enough to scratch it (at least) as a quite
> burial-resistant curiosity that is far less absurd than "elect the
> member of the Smith set that is voted the least desirable".

I suspect that you're right, and this holds for monotone methods in 
general. Here's the reasoning:

Suppose we have an ABCA cycle in Smith//X, where X is some monotone 
method. Our strategy to show nonmonotonicity is to shrink the Smith set 
to make the winner change, since raising A can never grow the Smith set. 
(Note that there may be more candidates *outside* the Smith set, but 
they're all irrelevant for our purposes.)

Suppose that we try to kick B off the Smith set. But this is impossible 
since raising A can never alter B>C or C>B, and it can only increase 
A>B. Since we already have A>B, raising A can't kick B off the set.

Okay, so suppose that we try to kick C off the Smith set since C>A. But 
if we reverse this, then we have both A>B and A>C, which would make A 
the Condorcet winner. Hence kicking C off the Smith set won't work.

So in neither case can this lead to nonmonotonicity if the base method 
is monotone.

Sounds about right?

> So this method meets both Double Defeat and Unburiable Mutual Dominant 
> Third.  I doubt that an acceptable Condorcet method
> can get more Burial resistant than that.

What is your definition of UMDT? I had some trouble trying to generalize 
DMTCBR to an actual DMT set criterion, so it would be interesting to know.

The problem I encountered was that, if the method also passes DMT 
(without which DMT burial resistance would be kind of strange), then the 
pre-burial winner W is part of the innermost DMT set. And then when 
voters who prefer X to W downrank W, they often change the relative 
order of candidates within the DMT set on their ballots.

E.g. if A, B, C are also part of the DMT set, then X>W>A>B>C>D voters 
changing their votes to X>A>B>C>D>W change their pairwise preference 
between, for instance, A and W. Even though they prefer a non-DMT 
candidate to the DMT set, their burial of the winner changes the 
relative order within the set.

If we require that such alterations - burials *within* the DMT set - 
should not matter, then the Smith-IRV hybrids fail since they're not 
absolutely unburiable, e.g.:

1: A>C>B
1: B>C>A
1: C>B>A

C is the CW, then

1: A>C>B
1: B>A>C <- burying C under A
1: C>B>A

is a perfect tie and every candidate has equal chance of winning.

Perhaps something like "the new winner should not be preferred by the 
buriers to the old winner". Or "should not be someone who was outside 
the DMT set before the burial started"? But I'm not sure what properly 
captures the strong type of resistance that the Smith-IRV hybrids pass 
and that leads to generally low strategic vulnerability.

-km