[EM] Approval Equilibrium
Abd ul-Rahman Lomax
abd at lomaxdesign.com
Wed Jun 13 18:52:26 PDT 2007
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....
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