[EM] What are some simple methods that accomplish the following conditions?

C.Benham cbenham at adam.com.au
Sat Jun 1 08:13:27 PDT 2019

> In the example profiles below 100 = P+Q+R, and  50>P>Q>R>0.  One 
> consequence of these constraints is that in all three profiles below 
> the cycle A>B>C>A will obtain.
> (3) elect candidate B given
> P: A
> Q: B>>C  (or B>C)
> R: C>>B. (or C>B)

In your profile (3),  isn't B simply the Condorcet winner?   (And so 
there is no "cycle A>B>C>A")

You presumably have in mind that the ballot will allow voters indicate 
an approval threshold in their rankings. In that case one method fills 
the bill is good old
Approval Sorted Margins:


I think that method is somewhat better at resisting Burial than 
Smith//Approval(explicit), which in this April 2002 EM post by Adam Tarr 
is called
"Approval-Completed Condorcet":
> The following are the sincere preferences of my example electorate: 
> 49: Bush>Gore>Nader 12: Gore>Bush>Nader 12: Gore>Nader>Bush 27: 
> Nader>Gore>Bush**
> Say that some of the Gore>Bush>Nader voters were extremely 
> non-strategic and decided to approve both Bush and Gore. So the votes 
> now look like: 49: Bush>>Nader>Gore 6: Gore>Bush>>Nader 6: 
> Gore>>Bush>Nader 6: Gore>>Nader>Bush 6: Gore>Nader>>Bush 27: 
> Nader>Gore>>Bush Now, Bush wins the approval runoff 55-51-33. This is 
> where ACC's favorite betrayal scenario comes in. Since Bush wins the 
> approval vote, the only way the majority can guarantee a Gore win is 
> to make Gore the initial Condorcet winner, which requires that the 
> Nader camp vote Gore in first place.

Where  Smith//explicit Approval  fails,  Approval Sorted Margins easily 
elects the sincere Condorcet winner.

Gore's approval score is 51 and Nader's is 33. Both adjacent pairs (B-N 
and N-B) are pairwise out of order.  The gap between 55 and 51
is (much) smaller than that between 51 and 33, so we flip the order of 
that pair to give the final order N>B>G which has no adjacent
pair out of order pairwise.

Also giving the same result would be to use Approval(explicit) Margins 
as the measure of defeat strength in a traditional Condorcet method
like Schulze or Ranked Pairs or Smith//MinMax.

Chris Benham

On 31/05/2019 8:03 am, Forest Simmons wrote:
> In the example profiles below 100 = P+Q+R, and 50>P>Q>R>0.  One 
> consequence of these constraints is that in all three profiles below 
> the cycle A>B>C>A will obtain.
> I am interested in simple methods that always ...
> (1) elect candidate A given the following profile:
> P: A
> Q: B>>C
> R: C,
> and
> (2) elect candidate C given
> P: A
> Q: B>C>>
> R: C,
> and
> (3) elect candidate B given
> P: A
> Q: B>>C  (or B>C)
> R: C>>B. (or C>B)
> I have two such methods in mind, and I'll tell you one of them below, 
> but I don't want to prejudice your creative efforts with too many ideas.
> Here's the rationale for the requirements:
> Condition (1) is needed so that when the sincere preferences are
> P: A
> Q: B>C
> R: C>B,
> the B faction (by merely disapproving C without truncation) can defend 
> itself against a "chicken" attack (truncation of B) from the C faction.
> Condition (3) is needed so that when the C faction realizes that the 
> game of Chicken is not going to work for them, the sincere CW is elected.
> Condition (2) is needed so that when  sincere preferences are
> P: A>C
> Q: B>C
> R: C>A,
> then the C faction (by proactively truncating A) can defend the CW 
> against the A faction's potential truncation attack.
> Like I said, I have a couple of fairly simple methods in mind. The 
> most obvious one is Smith\\Approval where the voters have control over 
> their own approval cutoffs (as opposed to implicit approval) with 
> default approval as top rank only.The other method I have in mind is 
> not quite as simple, but it has the added advantage of satisfying the 
> FBC, while almost always electing from Smith.
> ----
> Election-Methods mailing list - see https://electorama.com/em for list info

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