The Greater Majority Method

[Steve Eppley]

Donald D wrote:
>I have come up with another single seat election method. The math
>is as follows: Each voter makes a series of preferences. The first
>two preferences of all the voters are added together. The low
>candidate is dropped. Again we add the first two preferences and
>again we drop the low candidate. We continue this routine until we
>have one candidate left - which is the winner.
>
>This math is a combination of Approval Voting and the run-off
>feature of Preference Voting. 

The math is only vaguely related to Approval voting.  In Approval 
voting, voters are limited to 3 distinct rank positions.  And the 
"None" alternative, if there is one, must be one of those which 
is ranked 2nd:
     a=b=c=...    >    None=d=e=...    >    f=g=h=...
Choices ranked 1st receive +1 point, and choices ranked 3rd receive
-1 point.

By suggesting he is combining Approval into his new proposal, Donald
appears to be making an odd assumption that when voters freely rank
the candidates, they approve their top two of the uneliminated
choices and disapprove the rest.  To me this assumption looks
irrational.

Donald's new proposal allows voters to freely rank the choices, but it 
"misinterprets" the voters' preferences by changing (each iteration) 
the '>' between the top choice and the next choice to an '='.  Except
for that wrinkle, it's the MPV method.

Note that Donald appears to have acknowledged that he was wrong about
a "need" to split votes among equally ranked candidates.  His new
method gives a full vote to each of the voter's top 2 choices.  (If
he looks at my response to the EM singlewinner poll, he'll see that I
ranked "MPV with full votes" ahead of Approval, and Approval ahead of
"MPV with split votes".  So I'm pleased Donald is shifting to full
votes, if that's what he's doing.)

-snip-
>I was trying to think of another way to determine which candidate to
>drop - this is another way.  This method considers the strength a
>candidate may have in the next set of preferences.

Donald still hasn't explained why he's limiting his search to
methods which drop one at a time.  <sigh>  Perhaps he should consider
a one-at-a-time method which drops them in order of worst "opposition
in any pairwise comparison where opposition exceeded support."

It might help Donald to save his (and our) labor if he would analyze
democratic criteria so he'll know which are important to him and why,
to help guide his search for good methods.  He should make a list.
Criterion #1 will probably be that if there's a candidate with a
majority of first choices, it should be elected.  (In Bruce's syntax:
Method M satisfies "M = Majority//M".  See below.)  But what's #2?  
What's #3?  Maybe some of Arrow's criteria?

Here are a couple of "majority rule" criteria for him to consider 
adding to his list:  
#2. If more voters ranked candidate X ahead of Y than vice versa, 
    then Y should not be elected.
#3. If a majority of the voters ranked candidate X ahead of Y, 
    then Y should not be elected.

He might also want to consider the "Truncation Resistance #2"
criterion I'm posting separately.  It provides an easy test which
will separate most of the chaff from the wheat.

>When we do consider other preferences we must face the possibility
>of a candidate having a greater majority in two preferences than the
>majority another candidate may have in the first set of preferences.
>In other words - this method will not always elect the candidate
>that may have received a majority on the tally of the first
>preferences.

Here's a slight modification which will solve that problem: 
First check to see if there's a candidate with a majority of first
choices.  If so, elect that candidate.  If not, proceed with the rest
of the algorithm (whatever it may be; this "first step" can be
prefixed to any method which is trying to adhere to majority rule). 

In Bruce's syntax, this particular combo would be called
Majority//GreaterMajority since the method "Majority" selects the
candidate with a majority of first choices if there is one, or
selects all the candidates if none has a majority of first choices.  
Since Majority//GreaterMajority = Majority//Majority//GreaterMajority, 
this combo satisfies criterion #1 above.

For most majority rule methods, prefixing Majority// doesn't change
the result.  For the exceptions, prefixing it will make a bad method 
not quite as bad.

Donald might also want to check out the iterative method which elects
the candidate with the largest so-called "majority", iterating until
at least one candidate has a "majority", where in each iteration n 
the score of a candidate is the number of voters who ranked the
candidate no worse than nth.  (Of course, like Donald's other
proposed methods, this method is inferior to Smith//Condorcet 
and Condorcet on the majority rule criteria #2 and #3 above.) 

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