[EM] Approval Equilibrium

Forest W Simmons fsimmons at pcc.edu
Tue Jun 12 14:56:19 PDT 2007

What is an approval equilibrium?

Is it possible to deduce an approval equilibrium from sincere rankings 
or ratings?

These questions are amazingly slippery!

I won't attempt to survey the many answers that have been proposed, but 
I would like to share a line of thought that came to me after pondering 
Lomax' recent post on majority ratification.

He wrote about trying to simulate or predict (from sincere range 
ballots) what the outcome of an interactive process would be.  Small 
groups have the luxury of the interactive process, but it becomes 
expensive for large groups.

Suppose that we had a small group that repeated approval counts until 
they reached some kind of equilibrium, i.e. until they stabilized.  Is 
there some way to use sincere range ballots to predict what this 
equilibrium might be?

Of course not with 100 percent accuracy, because some voters are more 
stubborn than others, etc.

But is there a reasonable way to predict an equilibrium?

Suppose that candidate X is the equilibrium winner, and that candidate 
Y is ranked above X by 37 voters.  Then Y we would expect that when the 
approval votes stabilize with X winning, that Y would have an approval 
of about 37, since there is no point in approving anybody you like less 
than the first place candidate X.

For now let's don't worry about some of the voters rating X and Y 
exactly equal.

Continuing onward ...

If X is the approval winner, then X must have approval greater than Y, 
so X has approval greater than 37.

Similarly, if Z is ranked above X by 53 of the voters, then X must have 
more than 53 approval votes to be the approval equilibrium.

Now suppose that this number 53 is the greatest pairwise opposition of 
any candidate against X.  Then an approval of 54 for X would be 
sufficient to make X an approval equilibrium winner.

So each for each candidate X, if X can consistently get approval 
greater than his greatest pairwise opposition, then X will be an 
approval equilibrium candidate.

So the question becomes, "For which candidate X is it least difficult 
to get the approval at the required level?"

I have some possible answers, but I would like to hear yours before I 
prejudice your minds towards mine.


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