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

[Forest Simmons]

On Wed, Aug 16, 2023, 4:39 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> 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.
>

Can you think of a method that is so burial resistant that a sincere runoff
restricted to Smith would not appreciably improve its Sincere CW efficiency
(given |Smith|>2)?

>
> 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?
>

Basically you specify a tournament schedule in the form of a binary tree.

The voters start at the root node and by majority decision decide which
daughter branch to pursue. Recursively elect the method winner of the sub
tree the branch leads to. The boundary condition for the recursion is that
the winner of a leaf is the leaf itself.

For anything other than an election in the Society of Game Theoretic Nit
Pickers, the method should be restricted to 3 candidates, for example the
top three finishers of IRV restricted to Smith.

Suppose that when IRV is restricted to Smith, the finish order is
S1>S2>S3>...

The sincere runoff tree should be

S1 vs (S2 vs S3)

If one of these three (say C) pairwise defeats each of the other two, then
rational voters who are aware of the other voters' preferences, will elect
C.

Otherwise (still assuming rationality and preference awareness) candidate
C1 will be elected ... as can be easily (if tediously) shown in an
exhaustive (and exhausting) case-by-case analysis.

fws



> 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
>
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