[EM] Maximin

[S Sosnick]

On 28-May-2011, Robert Bristow-Johnson wrote,  "is there any good reason to use minimax of 
winning votes (clipped at zero) over minimax using margins?  it seems to me that a candidate 
pairing where Candidate A just squeaks by Candidate B, but where a lotta people vote should 
have less weight than a pairing where one candidate creams the other, but fewer voters weighed 
in on it."

I think that there indeed are good reasons to focus on margins instead of on "winning votes."

One reason is simplicity.  With the maximin criterion, the winner is determined in three easy and 
easily-understood steps, as follows:

1.  From voters' rankings, determine how many votes each candidate would receive if paired 
against each of the other candidates.

2.  Determine (a) the margin of victory or defeat (that is, the number of favorable votes less the 
number of unfavorable votes) in each of those paired comparisons, (b) for each candidate, the 
minimum such margin, and (c) the maximin margin (that is, the largest of those minimum 
margins).

3.  If one candidate has the maximin margin, then that candidate wins the election.  If two or 
more candidates have the maximin margin, then choose one of them by chance.

A second reason to prefer margins is consistency.  With the maximin criterion, unlike the Schulze 
and Tideman criteria, the maximin margin in paired comparisons determines the winner whether 
or not that maximin is greater than 0.  If the maximum margin is greater than 0, then the winner 
is a Condorcet alternative.  If the maximin margin is less than 0, then the winner is the candidate 
whose worst margin is least bad.  Officials can break a tie by lot or, as in some NLRB elections, by 
nullifying and re-running the election.  

Simple and consistent--and applicable with any number of candidates.

Furthermore (and, in this respect, like--not different from--the Schulze and Tideman criteria), 
the maximin criterion lends itself to including "none of the above" on the list of candidates.  If 
that is done and a named candidate wins, then one can reasonably conclude that the winner is, 
not only preferred to the other named candidates, but also good enough to be (say) hired.  
Conversely, if "none of the above" wins the election, then one can infer that none of the named 
candidates should be hired.

--Stephen H. Sosnick (5/28/11)
 
> On May 28, 2011, at 3:41 PM, S Sosnick wrote:
> 
> >
> > On 27-May-2011, Jameson Quinn, wrote, "I agree [with Juho Laatu].   
> > If minimax is twice as likely
> > to be adopted, because it's simpler, and gives >95% of the advantage  
> > vs. plurality of the
> > theoretically-best Condorcet methods, then it *is* the best.  And  
> > besides, if we try to get
> > consensus on which is the absolutely best completion method, then  
> > almost by
> > definition, we're going to end up arguing in circles (cycles?)."
> >
> > I also agree.  More noteworthy, however, is that Nicolaus Tideman  
> > does, too.  At page 242 of
> > "Collective Decisions and Voting" (2006), he says, "If voters and  
> > vote counters have only a slight
> > tolerance for complexity, the maximin rule is the one they would  
> > reasonably choose."
> 
> will minimax of margins decide differently than ranked pairs?  if the  
> cycle has only three candidates, it seems to me that it must be  
> equivalent to ranked pairs.
> 
> is there any good reason to use minimax of winning votes (clipped at  
> zero) over minimax using margins?  it seems to me that a candidate  
> pairing where Candidate A just squeaks by Candidate B, but where a  
> lotta people vote should have less weight than a pairing where one  
> candidate creams the other, but fewer voters weighed in on it.
> 
> --
> 
> r b-j                  rbj at audioimagination.com
> 
> "Imagination is more important than knowledge."
> 
> 
> 
> 
>