NML pairwise method, NML set (No Majority Losses)

[Steve Eppley]

Here are four new single-winner functions NPL (No PairLosses), 
NML (No Majority Losses), NML2, and NML3.  NPL is very similar 
to BeatsAll; the NMLs can be considered single-winner voting methods 
analogous to Smith.

I've been working on a rigorous "No Spoilers" criterion and a
redefinition of Arrow's "irrelevance" which both involve the NML set, 
but they're not complete.  I post this today because the NML2 and 
NML3 sets (below) are vaguely referenced in another message today.


NPL
---
Let O be the set of all choices.  (the set of possible outcomes)

Let P[i,j] be the number of voters who ranked choice i ahead of 
choice j, plus the number of voters who ranked i but left j unranked.

if there are choice(s) i such that, for all j<>i, P[j,i] <= P[i,j]
then 
   NPL selects these choice(s) i
else
   NPL = O


NPL is close to BeatsAll, but also includes tied choices.  NPL is
never empty, since it will select all of O rather than be empty.  
(Taken literally, the name "No PairLosses" may be objectionable,
as might the name BeatsAll.)

The NPL criterion (NPLC) is related to the Condorcet Winner criterion:
A method satisfies NPLC if and only if it selects a choice in the NPL 
set.   (Instant Runoff violates NPLC.)


NML
---
Let V be the number of people who ranked at least one choice in O.

if NPL <> O
then
   NML = NPL
else
   if there are choice(s) i such that, for all j<>i, P[j,i] <= V/2
   then 
      NML selects these choice(s) i
   else
      NML = O

Taken literally, the name "No Majority Losses" may be objectionable,
since NML will never be empty.

To satisfy a generalized definition of majority rule:  A choice which
isn't in the NML set must not win.  (The NML Criterion)

NML//Condorcet selects the same set as Condorcet.

NML//Smith and Smith//NML may select different sets, which may or
may not intersect.

Proponents of any pairwise method M or Smith//M may want to also
consider NML//M, NML//Smith//M, and Smith//NML//M.  For instance,
Smith//NML//Random is arguably better than Smith//Random, since 
Smith//Random violates the NML criterion.

* * *

NML2 and NML3 are further generalizations of majority rule.
Under certain circumstances, they treat nonvoters as if they are
voters expressing pairwise indifference over all the choices.

If anyone finds NML useful, s/he may wish to consider substituting
NML2 or NML3 for NML.


NML2
----
Let V' be the number of people who didn't rank any choices in O, but
were eligible to vote on O and did cast some ballot in some other
election held at the same time.

if NML <> O
then
   NML2 = NML
else
   if there are choice(s) i such that, for all j, P[j,i] <= (V+V')/2
   then 
      NML2 selects these choice(s) i
   else
      NML2 = O


Let E be the number of eligible voters.  E >= V+V'

NML3
----
if NML2 <> O
then
   NML3 = NML2
else
   if there are choice(s) i such that, for all j, P[j,i] <= E/2
   then 
      NML3 selects these choice(s) i
   else
      NML3 = O

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)