Richard Moore rmoore4 at home.com
Sat Mar 17 23:05:00 PST 2001

```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

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