[EM] Approval Equilibrium

[Abd ul-Rahman Lomax]

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?

In another post, I respond specifically to Mr. Simmon's post.

Here I want to explore the question itself, in this manner. Given a 
Range ballot, I will assume 0-100 Range, what I'm now calling Range 
100, under what conditions can we deduce that a majority has approved 
the winner?

Right off, I want to assert that there are no conditions where we can 
prove that the majority approves the winner; this is because it is 
possible that the voters approve of *no* winner, and if they have no 
opportunity to reject the election, the most we could say is that we 
have elected, perhaps, the least-disapproved winner. There is no 
substitute for an explict acceptance.

The ballot could, however, have an explicit rating that is set by the 
voter, meaning "I hereby ratify the election of any candidate whom I 
have rated above this level." (The same could be done with a ranked 
method, which turns the ranked method into, possibly, a more 
sophisticated form of Approval.)

We could also assume 50% as an approval level. Theoretically, this is 
a neutral rating: the election is about what the voter expected. 
However, it is better to allow the voter to set the approval level, 
because strategic concerns will cause the voter to normalize and 
truncate, thus shifting what would theoretically be, in a 
zero-knowledge vote, a "just what I expected" outcome, in one 
direction or the other, away from midrange.

If voters set an approval cutoff, which is probably best as a "this 
rating or higher" cutoff, then we know, at least, the voter's opinion 
about it when the ballot was cast. If we get an explicit approval by 
a majority of voters, from this method, of the Range winner, we have, 
I'd submit, utterly no problem, and ratifying that winner 
automatically without further process is without serious issue.

But what if no candidate meets this standard? By analogy with many 
current top-two elections, it would be best to hold a runoff. And if 
we are dead-set against that?

Obviously, we will need to start lowering the Approval cutoff. How to 
do this? Obviously, we should do so minimally. We would lower the 
cutoff until a majority-ratified winner appears. AT this point, 
though, we would need to decide *why* we are searching for some other 
winner than the Range winner? We aren't going to satisfy the Majority 
Criterion. If our goal is to select the best winner, we are better 
sticking with the Range winner.

But if our goal is to approach majority approval of the Range winner, 
if possible, and if not, we will then have a runoff, it's another 
story. And what seems to me as an extremely simple solution is to 
determine if there is a candidate who pairwise beats the Range 
winner. If there is, there is clearly a need for some kind of further 
process, for the majority has indicated a preference for someone 
else. Selecting that someone else will lower the expected utility 
inferred from the Range ballots, which is undesirable. And choosing 
the Range winner is choosing someone who would lose in a 
zero-knowledge two-candidate election, and even in an election with 
some knowledge.

It's clear to me that the simplest trigger for a runoff would be the 
existence of a pairwise winner, who beats the Range winner. This is a 
situation where we need a judge to weigh the merits of each candidate 
in relationship to the electorate, there is no general answer, in 
fact. And the judge is, properly, the electorate. We have described 
methods which might avoid such a runoff under some conditions, but 
not under all, by allowing the electorate to explicitly ratify a 
range of candidates in advance. The consequences of this on voting 
strategy should be examined, my unverified intuition is that it is 
harmless; that is, it might wrongfully withhold permission to ratify, 
but would not wrongfully grant it, thus erring on the side of caution.

If a voter says, on the ballot, (a Range ballot, that is important) 
"I would accept this election outcome," rarely, I think, would the 
voter turn around and say, "I now realize from all the other votes 
that I could have refused to accept this winner and I would then get 
my favorite, someone I have a slight preference for, so I'm going to 
reject this winner and subject the electorate to an overall loss." 
Some would, to be sure, but it would generally be enough that *some* 
of those who support the majority favorite would reverse their votes, 
thus allowing ratification of the Range winner.

This is why pure ratification is possibly superior to a top-two 
runoff. But the down side, perhaps, is that it could leave the 
election unresolved after two polls. If we make that easy with 
something like Delegable Proxy or Asset Voting, it's not a problem, 
and it makes the original ratification simple. Elsewhere I note that 
if the delegation of voting proxy authority is in a separate 
election, with the voter having free choice in choosing an elector to 
represent him or her, we have the best of both worlds, the accuracy 
of direct democracy [and it's possible with DP for this to be full 
direct democracy, but not with Asset] with the efficiency of 
representative democracy.