[EM] Automated Approval methods (was Single Contest)

[fsimmons at pcc.edu]

This kind of approach has been experimented with for a long time by Rob LeGrand, and there doesn't 
seem to be any good way to make it monotone.

Here's a very conservative and simple approach that may have some value in some context, if not this 
one:

For each rating ballot b approve the top N candidates where N is the (rounded) sum of the ballot b 
ratings of all of the candidates divided by the maxRange value  

Let S be the sum over candidates X of the ballot ratings b(X) .

Then N is S divided by maxRange, rounded to the nearest whole number (or rounded to even when 
exactly halfway between floor and ceiling of S/maxRange).

The N highest rated candidates on ballot b are approved.

If these approvals are used to elect an approval winner, the method is montone and as clone free as 
possible for automated approval.  (It can split clone sets at the approval boundary on a ballot).

Here is a possible heuristic for the method:

If the ballot b ratings are normalized (by dividing by maxRange) and taken to represent probabilities, so 
that b(X) is the probability that candiadte X would correctly represent the ballot b voter on a random 
question, then the sum S is the expected number of candidates that would agree with this voter on a 
random question.

So why not approve the top S voters, since they are the most likely to be the ones that would agree with 
the voter?

Note that this is a zero information strategy, and for all I know, it could well be zero-info-optimal by some 
criterion or other.  The usual zero info strategy is to assume that all of the candidates are equally likely 
to win, and to approve above expectation on that basis, but the insertion of lots of clones can radically 
change those probabilities.

This kind of reminds me of the rule that Kristofer suggested for how many winners there should be in a 
PR election when that number hasn't been decided ahead of time.

> Date: Sun, 24 Jul 2011 20:01:48 +0100 (BST)
> From: Kevin Venzke 
> Hi Kristofer,
> 
> --- En date de?: Dim 24.7.11, Kristofer Munsterhjelm 
> a ?crit?:
> > > I also tried implementing the most obvious (I suppose)
> > method: Take the
> > > ratings and conduct simulated approval polling, either
> > for some
> > > determined or semi-random number of iterations, or
> > until someone wins
> > > twice in a row. This doesn't test as well as I thought
> > it would though.
> > 
> > What Approval strategy do you use?
> 
> I always use "better than expectation" when it is allowed to 
> assume the
> voters know the method is approval. (Which is just to say that 
> the main
> sim, when during pure Approval, can't use "better than expectation.")
> 
> I put a tiny amount of "average utility of all candidates" into the
> expectation just to try to avoid the situation where your 
> favorite won
> all the polls so therefore you don't approve him.
> 
> Kevin Venzke