# [EM] Benham-DAC/DSC

Kristofer Munsterhjelm km_elmet at t-online.de
Tue Jun 14 09:54:23 PDT 2022

```On 14.06.2022 01:25, Kevin Venzke wrote:
> Hi Kristofer,
>
> Le dimanche 12 juin 2022, 06:35:17 UTC−5, Kristofer Munsterhjelm <km_elmet at t-online.de> a écrit :
>> Here's a modification to DAC and DSC inspired by Benham's method:
>>
>> Intersect coalition sets as in DAC and DSC. But before intersecting with
>> a set, check if the current set contains a candidate who beats everybody
>> else in it pairwise. If so, elect this candidate.
>>
>> It's clearly Condorcet (the initial set is the set of all candidates).
>> It's probably not DMTBR (because DAC and DSC aren't.) Is it Smith? For a
>> Smith failure to happen, the set at some stage needs to have more than
>> one Smith candidate, and then they all get eliminated by the next
>> intersection. I don't know if that's impossible - or possible.
>
> Neither variant seems to satisfy DMTBR. I think DAC itself can only fail
> DMTCBR if you'd say that Bucklin can.

It seems like it fails Condorcet itself, oops. (or my simulation code is

100: A>B>C>D
101: A>D>B>C
100: C>B>D>A
100: D>B>C>A
100: D>C>A>B

The coalitions should be:

501: ABCD
201: A
200: BCD
200: D
...

so A gets elected, but D is the DMTC (and Condorcet winner). It's
reminiscent of how Plurality fails DMTCBR and requires "if both first
and second place are >1/3 fpp, then elect whoever pairwise beats the
other" to pass.

I didn't check if I could make A into the Condorcet loser (here there's
a loser's cycle between A, B, and C).

As for Bucklin, it doesn't pass DMTC itself, e.g.:

2: A>B>C>D
1: B>C>A>D
1: B>D>A>C
1: C>D>A>B

A is the CW and DMTC, but Bucklin elects B.

But even when it initially elects the DMTC, it seems vulnerable to
burial (this example also works for Smith,Bucklin which always elects
the DMTC):

2: A>B>C>D
2: B>A>C>D
1: C>A>B>D

A is the CW and DMTC and wins. Then,

2: A>B>C>D
2: B>C>D>A   <-- burial
1: C>A>B>D

Every candidate is member of the Smith set and the Bucklin winner is B.

I imagine that methods that pass only one of the LNHs could be
vulnerable to burial because:
- suppose the method fails later-no-harm but does pass later-no-help.
Then it's possible that your honest completion of the ballot would harm
your first preference, but the burial ballot (or plain truncation) wouldn't;
- suppose the method fails later-no-help but passes later-no-harm. Then
burial ballot does.

So I would say Bucklin fails because burial works. It's not the only
thing that works (truncation also works), but I don't think it needs to be.

> The DAC version seems to be a lot like DMC or approval-elimination Condorcet.

I guess I can sort of see that? You said DAC behaves like Bucklin, which