[EM] That was MMT1. MMT2 is like MTAOC. Replacing the word "any" in the definitions.

[MIKE OSSIPOFF]

MMT1:

I'm not changing the definition of MMT that I posted last night. 
I'll call it "MMT1". 

It's even more demanding in its reciprocity
requirements than was the original (one-paragraph-definition) MTAOC.
It was that over-demandingness that led me to propose my
pseudocode-defined MTAOC.

MMT2:

MMT2 is like the pseudocode-defined MTAOC, with all middle
ratings conditional.

Its definition differs from that of MMT1 only in a few words
in its "mutual-majority candidate set".

MMT2 definition:

A "mutual majority" candidate set is a set of candidates who are each
rated above bottom by each member of the same majority of the voters--
where that set includes at least one top-rated candidate on the ballot
of every voter in that majority.

If there are one or more mutual-majority candidate sets, then the winner
is the most top-rated candidate who is in a mutual-majority candidate set.

If there are no mutual-majority candidate sets, then the winner is the most
top-rated candidate.

[end of MMT2 definition]

MMT3, likewise, differs only by a few words in its definition of a
mutual-majority candidate set.

MMT3 definition:

A "mutual majority"candidate set is a set of candidates who are each 
rated above bottom by each member of the same majority of the voters--
where that set includes all of the top rated candidates of each ballot
of the voters in that majority.

If there are one or more mutual-majority candidate sets, then the winner
is the most top-rated candidate who is in a mutual-majority candidate set.

If there are no mutual-majority candidate sets, then the winner is
the most top-rated candidate.

[end of MMT3 definition]

As I said, MMT2 is like the pseudocode-defined MTAOC, where all
middle ratings are conditional.

MMT3 is like my initial one-paragraph-definition MMT. It's too demanding
in its reciprocity-requirement.

MM1 is more demanding still.

Henceforth, when I say "MMT", without a distinguishing number, 
I will be referring to MMT2.

Likewise, when I say "MMPO", without a distinguishing number,
I'll be referring to MMPO2.

I'll refer to my initial, one-paragraph-definition, MTAOC as MTAOC1

I'll refer to my pseudocode-defined MTAOC as MTAOC2.

When I say "MTAOC" without a distinguishing number, 
I'll be referring to MTAOC2.

The advantage of MMT is its brevity of definition. 

The advantage of MTAOC is its flexibility, in allowing the option 
of giving, to any particular candidate, an unconditional middle rating.

The word "any" is ambiguous:

"I can beat _anyone_ at chess. Can you beat anyone at chess?"

Therefore I replace "any" in my definition of MMT1 (which isn't
my favorite MMT version, and isn't what I mean by "MMT").

For MMT1, here's how I define a mutual-majority candidate set:

A "mutual-majority candidate set is a set of candidates who are
each rated above bottom by each member of the same majority of the
voters--where that set includes every candidate rated above bottom
by one or more members of that majority of voters.

I repeat, though, that I prefer MMT2, and MMT2 is what I mean when
I say "MMT" without a distinguishing number.


Mike Ossipoff