# [EM] "Dominant Solid Coalition" suggested criterion (and strong standard)

Mon May 8 17:35:16 PDT 2023

```Of course by analogy with "Majority", single winner versions of these
criteria could be useful,
which I suggest calling "Dominant Candidate" and "Pairwise Dominant
Candidate".

In my earlier post on this topic I omitted an "and", which I've fixed below.

Chris Benham

On 8/05/2023 2:15 pm, C.Benham wrote:

The "solid coalition score" of a subset S of candidates is the number of
ballots on which those candidates are voted together (in any order amongst themselves)
above all the candidates not in S.

The  "maximum pairwise opposition score" of subset S of candidates is
the greatest number of ballots on which a candidate outside of S is voted
above a member of S.

*If there is a subset S of one or more candidates which has a Solid
Coalition Sore that is higher than its  Maximum Pairwise Opposition Score
then the winner must come from the smallest such S.*

I think this is met by anything that meets Smith or Plurality, and
anything that meets both of Majority for Solid Coalitions and Irrelevant
Ballot Independence.

I think it should mostly supplant Majority for Solid Coalitions as an
essential standard for single-winner election methods.

The only theoretical point of keeping Majority for Solid Coalitions that
I am aware of is that it is something that a method can comply with
while meeting Later-no-Help and FBC at the cost of lots of Later-Harm and
failing Irrelevant Ballots.

I have in mind Average Ratings systems like Bucklin and MJ. No-one seems
to promote or care about Later-no-Help and that is the only criterion-compliance
advantage those methods have over IBIFA.

In addition I propose a slightly weaker version:  "Pairwise Dominant
Coalition":

*If the smallest number of ballots on which a member of candidate subset
S is voted above a candidate not in S is greater than the largest
number of ballots on which a candidate not in S is voted above a member
of S, then the winner must come from the smallest such S with
at least one member.*

These could probably have been worded more succinctly and elegantly.

Chris Benham
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