[EM] Some thoughts on Condorcet and Burial

[C.Benham]

Kristofer,

I'm sorry for being a bit tardy in replying.

> What is your definition of UMDT?

My definition of MDT is that if a set S of candidates are voted together 
above all outside-S candidates on more than one third
of the ballots, and all the members of S pairwise-beat  all the 
outside-S candidates, then the winner must be a member of S.

My definition of UMDT is that if the winner T is a member of S, then it 
must not be possible to make some outside-S candidate X
the winner just by altering some ballots that already vote X above S.

Perhaps to be a bit more strict we can replace "altering" with ' further 
down-ranking S on'.

>  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.

This example clarified for me that MDT  (and UMDT) refers to *more than* 
a third (rather than exactly a third).

It is clear to me that MDT was meant to be analogous with Mutual 
Majority (rather than "Mutual Half").

(A problem is that in English there is no word that means "more than a 
third".)

> So in neither case can this lead to nonmonotonicity if the base method
> is monotone.
>
> Sounds about right?

Yes.

Chris B.



> *Kristofer Munsterhjelm*km_elmet at t-online.de 
> <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20Some%20thoughts%20on%20Condorcet%20and%20Burial&In-Reply-To=%3C59c996f1-4936-5cd3-7026-3b04198d192e%40t-online.de%3E>
> /Wed Nov 8 05:00:27 PST 2023/
> ------------------------------------------------------------------------
> 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
>
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