[EM] Approval Equilibrium

[Abd ul-Rahman Lomax]

This is only an approach and a description of my thoughts as I 
explore the question raised by Mr. Simmons.

At 05:56 PM 6/12/2007, Forest W Simmons wrote:
>What is an approval equilibrium?
>
>Is it possible to deduce an approval equilibrium from sincere rankings
>or ratings?

It is possible to predict to some degree, but not to deduce, because 
of the interactive effects I have mentioned. If we assume fixed 
preferences on the part of the electorate, that they are going to 
stick with their rankings and ratings no matter what, it gets easier. 
We would assume, then, that the ratings or rankings are sincere and 
accurate, and then use them.

Rankings, though, don't contain enough information to clearly 
anticipate Approval, unless an approval cutoff has been specified, in 
which case we should then be able to find a maximum approval 
candidate, rather easily. An obvious example of the inadequacy of 
ratings alone would be an election where the voters actually approve 
of no candidates, and, given the opportunity, would reject the entire 
election, preferring None of the Above over electing any one of them.

It is possible to assume an Approval cutoff in a rating system, at 
midrange. I have elsewhere argued that a rational Range vote would be 
truncated such that irrelevant candidates are excluded, at first, and 
the limits of the Range set at max and min for the favorite and the 
lest preferred relevant candidate(s); the rest are then tacked in 
where they fall. So we know that a 100% rating implies Approval, for 
sure, but if the voter has rationally rated the remainder of the 
candidates, preference strength and strategic considerations may make 
an approval vote, in some sense, be any vote above midrange, and we 
might consider exact midrange to be a maximally grudging approval....

>These questions are amazingly slippery!

Indeed. There is really no perfect substitution for full deliberative 
process. But Range could get pretty close.

>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.

Unless you have Asset Voting or Delegable Proxy. In which case full 
deliberative process in a representative body that is chosen with 
full representation (almost) becomes possible. A real runoff, 
however, is a small aspect of deliberative process that is commonly 
seen as justified with a large electorate. A rule that would allow 
the favorite candidate on a ballot to recast the vote on behalf of 
the voter, i.e., it's asset voting, would provide a very good 
compromise. Better, perhaps: a separate "office" which is elected in 
the same election, that of electoral stand-in (i.e., proxy) for the 
voter to allow a runoff to be conducted at minimum expense, with 
candidates for the office disqualified and write-ins allowed.

Thus we could have our deliberative cake without the cost and 
distorting inconvenience of a general election. I argued that this 
inconvenience would shift results in a runoff toward the Range 
winner, but that is just an effect that is possibly socially useful, 
not necessarily a desirable characteristic of the method. I'd rather 
not rely on inconveniencing voters!

>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?

I think so, with reasonable accuracy. The problem is, however, that 
the ballots will be strategically distorted, in general, even within 
what I would call "sincere" votes. Take a three-candidate election 
and let the voters rate the candidates; then remove one candidate, 
the voters would generally shift their ratings unless they are voting 
Approval style. We expect a two-candidate election under Range to be, 
generally, the same as Plurality.

>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?

The approach that occurs to me uses preference strength. At first 
glance, it seems Mr. Simmons is going to take a different tack. But I 
suspect that he will bring it back in when he gives us his own solution.

I'd assume that the votes represent real preference strength. The 
votes for irrelevant candidates don't matter, only real frontrunners 
need be considered. And I'd suggest renormalizing the votes for at 
least part of the process. Renormalizing them allows the votes that 
remain among the real contenders to be more representative of what we 
would see if the other candidates hadn't been there in the first place.

However, if a voter ranks A at 100, B at 99, C at 98, and D at 0, we 
clearly have a voter who approves of A, B, and C. We know that this 
voter, almost certainly, would approve any of the three top. So, yes, 
it is quite slippery.

But the information is there to do a reasonable job in predicting. 
Just not a perfect one.

>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.

If X and Y are the remaining contenders, equal rating is an 
abstention. Whether it is a piece of majority approval or not 
depends, though, on the rating. If it is high, it's an approval. If 
it is low, it's disapproval.

In any case, if X is approved, so is Y, if the ratings are equal.


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

Yes.

>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.

Yes, sufficient.

>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.

I'm finding it difficult to keep what Mr. Simmons is saying in my 
head. In fact, when I tried to parse it carefully, I failed.

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

Since I didn't understand what was said before, I may totally miss 
the question. It seems there are undefined terms or descriptions.

So, pending possible clarification from him, let me return to the 
initial question. Is it possible to deduce an Approval equilibrium 
from a set of Range votes? As I've noted, I'm skeptical, but what 
*could* we do?

Suppose there is a pairwise winner, A. If we define a Prospective 
Approval Margin for A and B as being the vote margin in the pairwise 
election between A and B, then suppose we find the maximized 
PAM(A,B), i.e., we find the candidate B who loses to A by the 
greatest margin. What does this mean?

Well, this is a measure of the election quality of A over B. However, 
it does not include preference strength. We could maximize PAM(A), 
and yet it could be for two candidates who were almost clones. If one 
of these was the pairwise winner, it would be quite likely that one 
of them would be the equilibrium Approval winner. In the other 
direction, if the preferences were maximally strong, then all we know 
is that B isn't going to be the equilibrium winner.

And what I get from this, which is merely a plan of approach, is that 
the probability that a pairwise winner is the equilibrium approval 
winner is proportional to the rating distance between the two 
candidates, summed over all voters. And to go further than this, to 
refine it and correct it, is more than I can do tonight....