[EM] Comments--few voters, 0-info

[Forest Simmons]

Here's my two cents worth, which might be subtitled

"A Poor Man's Guide to Zero Information Approval Voting"

Consider that your utilities for the candidates have a mean, median, and
midrange. (If the utilities are distributed evenly as Borda Count seems
to assume, then all three of these are the same number. Usually they are
different.)

Don't approve anybody below your mean utility, since that would lower your
expected utility.

Don't approve anybody below the midrange, since under zero information
there is no reason to vote for mediocre candidates (let alone scum).

Don't approve anybody below the median, since with zero information you
have no reason to dilute the power of your vote.

Finally, if there is one gap in the utilities that is significantly larger
than all of the other gaps, don't approve anybody below that gap.

Approve everybody not ruled out above, unless a gut feeling tells you to
knock one or two more off the bottom of the resulting approved list.

That's it. I know it's not "optimal" but I believe it is adequate.

Of course, we should continue to search for a simple rule of thumb that is
close to optimal, since such a search can turn up unexpected gems of
knowledge both closely and distantly related to our immediate objective.


On Sat, 17 Mar 2001, Richard Moore wrote:

> I thought some more about what I previously wrote (copied below) and
> think a few amendments are in order.
> 
> First, the approximation formula I gave assumes the low point of the utility
> scale is at zero, which I should have mentioned.
> 
> Second, I think that, for the value of N in the formula, a better choice than
> the number of voters would be the number of approval votes (all voters
> and all candidates) that are expected to be cast by the other voters. So if
> we expect every other voter to bullet vote, N is the number of voters, but
> if we expect them to approve multiple candidates, then it will be greater.
> Also, if we expect any abstentions, N would be reduced accordingly.
> 
> Third, the formula should be modified if there are more than three candidates.
> Specifically, the formula as given is an estimate of the effect of our first
> choice on the threshold level for our second choice. But if we then choose
> to approve our second choice, we need to consider the effects of both
> first and second choice on our third choice, and so on. I'm not sure of the
> best way to do it. My current guess is, for the nth choice, we could use
> 
>            (n-1)*(S-M)
> M +    -------------
>              (N+n-1)
> 
> This is based on the observation that if we have already approved n-1
> choices and the other voters have contributed N choices, our portion
> of the total is (n-1)/(N+n-1). Again, it's only an approximation even
> with these refinements.
> 
>  -- Richard
> 
> 
> Richard Moore wrote:
> 
> > The effect of having only a few voters is that the threshold shifts upward
> > from the mean. I would be happy with a good first-order approximation
> > to the threshold. If N is the number of voters, then the error resulting
> > from the above-the-mean strategy is reduced as N increases. This
> > suggests adding a term that would be inversely proportional to N.
> > Maybe if we study the problem some more, the approximation can
> > be refined. For instance, maybe it should be inversely proportional
> > to 1/(N+1). I note that if N=1 (you are the only voter), this expression
> > gives 50%; multiplying this by the full utility scale (100) and adding that
> >
> > value to the mean (50) of the top and bottom candidates gives 100, so
> > this matches the obvious expectation that you should never vote your
> > second choice if you are the only voter.
> >
> > If the mean is something other than 50, then you would have to
> > stretch or squeeze the adjustment value so that you don't end
> > up with values outside the utility scale.
> >
> > Until we get some more solid math, I am going to suggest the
> > following first-order approximation for zero-info elections with
> > N voters (including oneself). If S is the full-scale utility value, and
> > M is the mean utility of the other candidates, then the threshold
> > utility for a candidate would be approximately:
> >
> >            (S-M)
> > M +    -------
> >            (N+1)
> >
> >  -- Richard
> 
>