# [EM] suggested improvement on Mutual Majority criterion/set (add-plump proofed MM)

C.Benham cbenhamau at yahoo.com.au
Tue Jan 24 08:47:57 PST 2012

```I've decided to bin  (i.e.I now withdraw) my suggested "Add-Top Proofed
Mutual Majority" as something highly desirable or a real improvement on
plain Mutual Majority.

I defined it thus:

> *If the number of ballots on which some set S of candidates is  voted
> strictly above all the candidates outside S is greater than the number
> of ballots on which all the members of  S are voted below equal-top
> (i.e. strictly below some/any outside-S candidate), then the winner must
> come from S.*

I've discovered that it's incompatible with the Condorcet criterion.

2: A>B=C=D=E
1: A>X
1: B>X
1: C>X
1: D>X
1: E>X

The criterion says that the winner must come from {A B C D E}, but X is
the CW (pairwise beating all the other candidates 4-3).

Following a suggestion from Kevin Venzke, I now instead propose

*If  the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which any outside-S candidate is voted strictly above any
member of S, then the winner must come from S.*

Kevin gives this demonstration of how it differs from regular Mutual
Majority:

28: A>B
27: B>A
45: C>D

Both MM and APPMM say that the winner must come from {A B}, but if we
add 12A ballots APPMM says the same thing while MM now says nothing.

I haven't found or thought of any method that meets both of
Mono-add-Plump and regular Mutual Majority (aka Majority for Solid
Coalitions) but fails APPMM.

Chris Benham

I wrote (13 Jan 2012):

On 21 Dec 2011 I proposed this criterion:

> *The winner must come from the smallest set S of candidates about which
> the following is true: the number of ballots on which all the members
> (or sole member) is voted strictly above all the non-member candidates
> is greater than the number of ballots on which a (any)   non-member
> candidate is voted strictly above all the members of S.*

That is fairly clear, but the wording could perhaps be improved, say:

*If the number of ballots on which some set S of candidates is  voted
strictly above all the candidates outside S is greater than the number
of ballots on which all the members of  S are voted below equal-top
(i.e. strictly below some/any outside-S candidate), then the winner must
come from S.*

I tentatively suggested the name "Add-Top Proofed Solid Coalition
Majority".   A bit less clumsy would be "Add-Top Proofed Mutual
Majority". Maybe there is a better name that either does without the
word "Majority" or includes another word that qualifies it.  For the
time being I'll stick with Add-Top Proofed Mutual Majority (ATPMM)

I gave this example:

45: A>B
20: A=B
32: B
03: D

My criterion says that the winner must be A, but Mike Ossipoff's MTA
method elects B.

I did endorse MTA as an improvement on MCA, but since it (and not MCA)
fails this (what I consider to be very important) criterion (and is also
a bit more complicated than MCA) I now withdraw
that endorsement.  I still acknowledge that MTA may be a bit more
"strategically comfortable" for voters, but I can't give that factor
enough weight to make MTA acceptable or win its comparison with MCA.

Chris Benham

```