[EM] Unifying DMTC and DMTCBR, and method X criterion

[Kristofer Munsterhjelm]

On 8/15/23 02:24, Forest Simmons wrote:
> Great criterion as far as it goes ... great at not rewarding buriers, 
> but bad at electing buried candidates ... if I understand it.
> 
> Hence the need for a sincere runoff:
> 
> 40 A>B(Sincere A>C)
> 35 B>C
> 25 C>A
> 
> The sincere CW is C, which cannot be elected because 25 is less than 
> 100/3, if I understand the proposed critersion.

Let's see: A has more than 1/3, and A beats B pairwise. So B is 
disqualified. B has more than 1/3 and beats C pairwise, so C is 
disqualified. Hence A must be elected.

That's right. Here the burial actually succeeds because the A voters 
prefer A to C, the sincere CW. This proves that the burial resistance 
isn't absolute. But note that this also happens to Smith,IRV and Smith,IFPP.

IRV and IFPP themselves elect A even in the sincere scenario.

> But a top three sincere runoff of the form
> 
> A vs (B vsC)
> 
> will elect C assuming rational voters informed of the true preferences.

I'm not entirely sure about the notation. Could you reacquaint me with 
the concept of a sincere top-k, k>2 runoff?

But on a more general note, I would say that it's not always necessary 
to require that a criterion that removes the incentive for some kind of 
strategy, to be able to gracefully recover if the strategy is used anyway.

Monotone methods all make pushover strategy irrelevant. However, if a 
particularly monotone method were to elect B, then A>B>C voters decide 
to downrank A to last place (pushover) and then it switches to electing 
C instead, I wouldn't consider that a particularly severe weakness with 
the method. Being able to recover the sincere winner anyway would be a 
nice deluxe option, but it's not make-or-break, I wouldn't say.

Or consider Warren's attempt to generalize DH3 to every Condorcet method 
by saying that if all the factions go on a burial spree, then every 
Condorcet method will at the end elect the dark horse. He says something 
like:

Consider

37:C>A>B>D
37:C>B>A>D
32:A>B>C>D
32:A>C>B>D
31:B>A>C>D
31:B>C>A>D

Every method elects C. But the A and B voters don't like that, so they 
uprank D to second place:

37:C>A>B>D
37:C>B>A>D
32:A>D>B>C
32:A>D>C>B
31:B>D>A>C
31:B>D>C>A

Now A wins in Minmax, Schulze, etc. Simultaneously, the C voters are 
saying they need protection against either the A or B voters doing so, 
so they too bury A and B under D. However, they didn't expect that both 
would be doing it at the same time, so what happens is:

37:C>D>A>B
37:C>D>B>A
32:A>D>B>C
32:A>D>C>B
31:B>D>A>C
31:B>D>C>A

And now D is the CW: cue the explosion stock effect.

Under a method that passes my aforementioned criterion, this can't work, 
because post-burial, the criterion bars A and B from being elected, and 
C has the highest Minmax score of the remaining two, hence the burial 
does nothing.

Warren could then argue that "but if everybody just does what's 
intuitive, then they all rank D second, and then there's still a big 
boom!". But I would think that knowing that burial doesn't work and 
might easily backfire would tend to temper such ideas.

-km