LOE-3 (was Re: Lesser of two evils)

[Steve Eppley]

Rob L wrote:
-snip-
>  If Candidate X would win an election against Y using a given
>  election method, those that supported X over Y should not be  
>  afraid of Y winning merely because they (X supporters) vote their 
>  preference of a separate Candidate Z over X, still ranking Y 
>  below X.
>
>This I think states Lesser of Two Evils concisely, without overreaching. 

I'm not sure what you mean by not overreaching.  I think you've
captured the lesser of evils principle (at least for the case
where there's a majority x>y; voters who believe z>x>y ought not face
a dilemma even if there's only a plurality x>y), expressed from the
point of view of the individual voter so it's easy to understand.  
But there are scenarios where even Condorcet and Smith//Condorcet
violate that strong form you've written: if X's largest pairwise
defeat happens to be Z, then one's vote for Z>X may cause X's 
largest pairwise defeat to be larger than Y's largest pairwise 
defeat.  Here's an example:

  33 voters:  X > Y > Z
  35 voters:  Y > Z > X
  32 voters:  Z ? X > Y   <--- Note the '?'.  a dilemma?

  X loses to Z  (33 < 35+?)
  Z loses to Y  (32 < 68)
  Y loses to X  (35 < 65)     <--- a majority who rank X over Y

So to make a criterion which doesn't overreach, it would need to
include some condition limiting the scenarios.  The condition
would need to be reasonable and not arbitrarily favor some methods, 
as Don pointed out.

The weakest condition (which implies strongest criterion) is
probably directly derivable from what I wrote above: "... assuming 
X's largest pairwise defeat isn't to Z or that Y's largest pairwise 
defeat is larger than X's largest pairwise defeat..."  But this kind 
of wording appears to favor Condorcet and begs the question about
whether a different condition could favor another method like 
Copeland or IR.  So I think Don would object to this condition prima
facie, even though it excludes few scenarios.  Maybe it would pass 
his acceptance testing if the part about the *largest* defeats is 
removed: "... assuming there isn't some majority of all the voters 
who prefer Z to X..."  But then the criterion doesn't cover as many 
scenarios.

Here's an attempt at a useful criterion, which includes some
wording--excess baggage, technically--which focusses attention on
the dilemma from the point of view of the voter.  The first 3 lines
are an loe principle, and the last 3 lines are a condition which
excludes an unusual scenario.  This is really just Mike's GMC
criterion with the extra phrase "no matter how many voters rank z
better than x" thrown in.

    LOE-3
    -----
    If a majority of the ballots rank candidate x better than 
    candidate y, then y cannot win no matter how many ballots rank 
    candidate z better than x, 
    unless all of the candidates in the Smith set have at least one 
    pairwise defeat in which some (not necessarily the same) majority 
    of the ballots rank the pair-winner better.

* *

Would it be desirable to write a LOE criterion which doesn't
stipulate that a majority ranks X>Y?  Merely a plurality?  The
scenario-limiting condition would then need to be more complex
to be useful, I think.

Here's another LOE formulation I thought about for awhile, which is 
even more general than Rob's:

   If X wins with a given set of ballots, then either X or Z will win
   if some or all of the X=Z and X>Z rankings are changed to Z>X 
   (all other rankings being unchanged).

It's easy for Copeland and IR to fail this, but there are some
scenarios where Condorcet and Smith//Condorcet can fail it as well
(so it needs to be made more limited in a reasonable way).  Copeland
can fail it by flipping the X vs Z pairing from a win for X to a loss
for X.  IR can fail it by eliminating X before Z.  Not coincidentally, 
Copeland and IR fail it in an example we've kicked around a lot:
  45:  Y
  20:  X
  35:  Z ? X > Y

* *

By the way, I think the '2' should be removed from 'LO2E' when we
post about it to the ER list.  I've heard the dilemma often
described as "lesser of evils" so I think people will recognize the
term without the 2.  I'm using the heuristic "shorter is better."

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