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

[Richard Moore]

MIKE OSSIPOFF wrote:

> So what strategy do we use for Approval in the demonstration poll?
>
> Certainly there's a good mathematical 0-info strategy, for very few
> voters, in Approval. We just don't have it yet. Bart has suggested
> that a good Approval strategy is to just guess which set of candidates
> is the best to vote for. I don't like that, it seems to me much more
> of a wild guess than estimating our sincere ratings (and relevant
> probabilities if we have probability info). But it looks as if we're
> stuck with that guessing strategy right now, for very few voters and
> 0-info. But if we say to just guess which set of voters are the best
> to vote for, that doesn't get me anywhere--I don't know about you.
> The way I'd put it is: Guess which alternative(s) you most likely need
> as a lesser-evil compromise to keep something worse from winning.

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