[EM] suggested improvement on Mutual Majority criterion/set

[C.Benham]

I now see that I erred in how I defined my new suggested variant of the 
Mutual Majority (aka Majority for Solid Coalitions) criterion, so 
disregard what I previously wrote (pasted below this message)  starting 
with "Preliminary definitions").

Instead I now propose this:

*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.*

It has occurred to me that this concept isn't just invulnerable to 
Mono-add-Plump. It is also invulnerable to Mono-add-Top. So my tentative 
suggested name is "Add-Top Proofed Solid Coalition Majority".

It  is a bit stronger than normal  Mutual Majority criterion.

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

Mutual Majority only says that the winner must be A or B, but "ATPSCM" 
says that the winner must be A. Both Kevin Venzke's ICA and Mike 
Ossipoff's MTA methods elect B

40: A>B
35: B
25: C

Mutual Majority here says nothing, but ATPSCM  says (here like Minimal 
Defense) that the winner must be A or B. Fortunately here there is no 
discernible practical problem because no method wants to elect C.

Chris Benham


Chris Benham wrote (13 Dec 2011):

Back in December 2008  I criticised  Marcus Schulze's  "beatpath 
Generalized Majority Criterion"  (which says in effect that if any
candidate X has a majority-strength beatpath to candidate Y then Y can't
win unless Y has a majority-strength beatpath back to X)  in part
because the concept is vulnerable to Mon-add-Plump. That is, extra
ballots that plump for candidate A can cause A to fall out of the set of
candidates that  the criterion specifies are qualified to win.

Then it was pointed out to me that to some extent the Mutual Majority
(aka Majority for Solid Coalitions) criterion has the same problem. A
candidate X can be in the set of candidates that are qualified to win
and then some extra ballots that plump for X are added and then the set
of candidates the criterion specifies are qualified to win expands to
include one or more new candidates. X doesn't actually fall out of the
set (as with beatpath GMC), but nonetheless according to the criterion
X's case has been weakened by the new ballots that plump for X.

I propose a replacement for Mutual Majority which addresses this problem
and also unites it with Majority Favourite.

Preliminary definitions:

A "solid coalition" of candidates of size N is a set S of (one or more)
candidates that on N number of ballots have all been voted strictly
above all outside-S candidates.

Any given solid coalition A's  "rival solid coalitions" are only those
that contain a candidate not in A.

Statement of criterion:

*If one exists, the winner must come from the smallest  solid coalition
of candidates that is bigger than the sum of all its rivals.*

[end criterion definition]

This wording could perhaps be polished, and I haven't yet thought of a
name for this criterion and resulting set. (Any suggestions?)

It might be possible to use the set as part of  an ok voting method.


Chris Benham