Benham-DAC/DSC
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.
-km
On 6/12/22 1:34 PM, Kristofer Munsterhjelm wrote:
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.
I guess I should clarify: Begin with S being the set of all candidates.
At each step, if there's a Condorcet winner among the candidates in S,
elect him. Let S_C be the solid/acquiescing coalition with the most
support that's not already been examined. If S intersect S_C is
non-empty, let S equal S_C. In any case mark S_C as examined, then repeat.
In other words, the "current set" that's to be checked for a CW limited
to its members is the set that's the result of all the prior
intersections, not the currently greatest-support unexamined coalition set.
-km
On 6/12/22 1:38 PM, Kristofer Munsterhjelm wrote:
On 6/12/22 1:34 PM, Kristofer Munsterhjelm wrote:
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.
I guess I should clarify: Begin with S being the set of all candidates.
At each step, if there's a Condorcet winner among the candidates in S,
elect him. Let S_C be the solid/acquiescing coalition with the most
support that's not already been examined. If S intersect S_C is
non-empty, let S equal S_C. In any case mark S_C as examined, then repeat.
Wow! It really isn't my day today. "If S intersect S_C is non-empty, let
S equal S intersect S_C". Let's hope that's all!
-km
Hi Kristofer,
Le dimanche 12 juin 2022, 06:35:17 UTC−5, Kristofer Munsterhjelm km_elmet@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.
The DSC variant violates Smith because the strongest coalition excluding at
least one candidate is often a singleton containing the plurality winner. This
could be the Condorcet loser.
For the DAC variant I haven't been able to find a Condorcet loser failure,
although I'm not sure logically why there couldn't be one. It's probably less
likely that a set with few candidates could have a lot of strength relative to
sets with more, making it less common to see multiple disqualifications in the
same step.
The DAC version seems to be a lot like DMC or approval-elimination Condorcet.
Kevin
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@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
bad)
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
it's possible that a honest ballot doesn't help your favorite but a
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
behaves like gradual approval.
-km
On 14.06.2022 18:54, Kristofer Munsterhjelm wrote:
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@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
bad)
Er, nevermind. It would just elect D directly from the first coalition
of every candidate - my simulator was bad and of course its trace was
also bad for the same reason.
-km