# [EM] Single-candidate DMTBR idea

Forest Simmons forest.simmons21 at gmail.com
Sat Mar 12 22:02:38 PST 2022

```El sáb., 12 de mar. de 2022 4:45 p. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 3/12/22 6:58 PM, Forest Simmons wrote:
> > Kristofer,
> >
> > 45 ABC
> > 35 BCA
> > 25 CAB
> >
> > Each of the A and B factions has more than a third of the votes.
> > Candidate A defeats B pairwise.
> >
> > Almost every respectable method except TACC (as well as most
> > non-respectable methods) agree that candidate A should have the greatest
> > winning probability.
> >
> > But some nagging doubt persists ... whence the Condorcet Cycle?
> >
> > A general scalene triangle has a longest side, and the endpoints of that
> > side are further from each other than they are from the vertex V
> > opposite that side, which means that the V faction favorite cannot be
> > the rational, sincere last choice of any of the three factions.
> >
> > And yet the voted ballots in our above three faction example give each
> > candidate a turn at last place.
> >
> > Somebody's lowest preference is either mis-triangulated or
> mis-represented.
>
> I don't get what you mean here. Certainly it's possible for honest
> Condorcet cycles to exist. Warren gave an example ...

Warren and I gave an example an of a set of four factions in a single plane
that induces a cycle geometrically. This cannot happen with only three
factions ... if there is a cycle, then the preferences are inconsistent
with the lengths of the sides of the triangle.

A three dimensional issue space can give rise to a cycle but those
preferences are not based on metrics/distances between factions.

of candidates
> evaluated on three issues, say corruption, domestic policy, and foreign
> policy. Each faction cares primarily about one dimension, and the
> candidates have positions on each issue that leads to a cyclical
> majority (e.g. candidate A is incorruptible, has awful domestic policy,
> and okayish foreign policy).
>
> Honest cycles also exist in 2D spatial models, e.g. Poundstone's example
> https://www.rangevoting.org/PoundstoneCondCyc.png.
>

But not when limited to 3 ballot factions.

>
>
> If the cycle is false, though, then any faction could have done the
> burial. You say the A faction is the only group that has anything to win
> by conducting the burial, so they must have done it. However, there's a
> bit of battle-of-wits logic here. Suppose that the method did elect C by
> this logic. Then it's possible that the C voters, knowing this,
> engineered the cycle (honest is C>B>A) in order to push their winner
> from B (their second choice) to C (their first).

The burial of C by A isn't the only possibility ... just the most likely to
succeed ... so the one most in need of checking.

If the check confirms B>C as sincere, then B wins, so the burial of B did
not pay off for C.

Okay, so C can't win
> because of second-order reasoning.

C can and will win if it was buried by A, because it is a finalist in the B
vs C honesty check.

Don't make this too hypothetical. The proposed implementation is tweaked
DMC:

The Implicit Approval order is A>B>C. The nominal DMC winner W is A,
because A is not pairwise defeated by any candidate with greater implicit
approval. The other finalist is C, the candidate that would be the DMC
winner if W did not defeat it pairwise.

The tweaked method elects the sincere winner between W=A and C, which is C
if C was the sincere CW, else A, the DMC winner with ballots taken at face
value.

The tweaked method changes the DMC winner only if it detects and elects a
better DMC winner (a sincere CW in this three faction example) while
exposing an insincere order reversal.

Of course, this tweaked version of DMC is not the only possibility for
exploiting the potential for a sincere, binding, binary choice between two
finalists.

I already suggested a tweaked version of TACC that makes use of the same
device.

Do any other applications come to mind?

-Forest

And A can't win because of
> first-order reasoning. So B must win, right? But then it's possible that
> the B faction knew this (honest: B>A>C) and buried A to make B win.
>
> So my point would be that since there's a Condorcet cycle, any Condorcet
> method (no matter who wins) will be open to burial. One could argue that
> the sensible methods do the right thing and elect the candidate whose
> defector coalition has to be the largest for this to be a successful
> burial: a method that elects A is fooled by a faction of 45 voters
> executing burial, but if the method were to elect C, it could be fooled
> by a faction of 25 voters, which is worse.
>
> In a way, that's what DMTBR says: there's no way for a Condorcet method
> to be absolutely immune to burial, so the best thing we can do is to
> make some set of candidates immune to being buried by candidates outside
> of that set, and then try to make that set as small as possible. And I
> suspect that 1/3 is the best possible...
>
> At least without doing something clever with UD or repeated balloting.
> I'm not sure how a second ballot question would help, because there's no
> reason for an A>B>C burier to not also "bury" by indicating B>C where
> honest is C>B... so I may be missing something. Duple rules (like Random
> Pair) are IIRC only strategy-proof if the pair is decided independently
> of the voters' input.
>
> In the vein of DSV, imagine that I take some Condorcet method plus top
> two and make the DSV procedure fill in the second ballot information so
> that it's consistent with (or strategically advantageous given) the
> first ballot ranking. Then either the combined method is not Condorcet
> (and it's not surprising that it would resist burial better), or it's
> subject to the same limitations as above, I would think...
>
> I would guess the answer is that the combined method isn't Condorcet,
> because there would be a tension between burying the honest CW so that
> the second round consists of your favored candidate and someone who's
> going to lose - and burying too far which means that someone intolerable
> wins the second round. Perhaps most UD solutions are like Approval:
> there may be a Nash (or core) equilibrium around the honest CW, but the
> setting benefits whoever has got the most complete information, and the
> potential backfire can get very unpleasant indeed.
>
> -km
>
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