[EM] Determining Approval from Interval CR ballots

[Forest Simmons]

I want to point out that this method of deciding approval contributions
depends on having at least two candidates with non-zero probability
(except for the extreme CR zero width intervals).

Consequently you can only get 100% probability for a single candidate as a
limiting case or in a special case where there is enough information in
the extremes of the CR interval to decide the winner.

Because of this I must retract my claim that a CW would necessarily be an
equilibrium winner.

Forest

On Tue, 14 Oct 2003, Forest Simmons wrote:

> Most voters cannot calculate precise utilities, so it makes sense to
> consider ratings as having a plus or minus error associated with them.
>
> This point of view pays mathematical dividends by restoring continuity
> to the relationship between winning probabilities and approval ratings of
> the candidates, thereby increasing the stability of methods that attempt
> to determine equilibrium approval ratings from CR ballots.
>
> This posting is (mainly) concerned with how to determine an interval
> ballot's contribution to the approval rating of a candidate, given the
> winning probabilities of the candidates.
>
> An interval CR ballot is any ballot that allows voters to assign
> candidates to intervals of ratings.
>
> An interesting question which we will consider in another posting is
> what default convention should be used for converting ordinary CR ballots
> into interval ballots.  Should the interval endpoints be halfway between
> the CR values, or should they overlap, and if so, how much, etc.?
>
> In this posting let's assume that all intervals (except possibly those
> representing the extreme ratings) have non zero length.
>
> Suppose that p1, p2, ... pN, are the respective probabilities of
> candidates 1 thru N winning the election.
>
> How much approval should ballot B contribute to the total approval of
> candidate number one? [Once we know the general answer to this question,
> we can apply the same process to the case of the other N-1 candidates.]
>
> Let c1, c2, ... cN be the respective centers of the candidates' intervals
> on ballot B.
>
> Calculate the (conditional) expected winning CR value on the condition
> that candidate one does not win:
>
>                       E = (p2*c2 + ... + pN*cN)/(p1 + ... + pN)
>
> The fraction of the first candidate's interval that lies above this value
> is the fraction that ballot B should contribute to the approval of
> candidate one.
>
> Here's the reasoning: we ask ourselves if we want to support candidate one
> given the value of E.  If E is entirely below candidate one's true CR
> value, then we would like to fully support candidate one, since if
> candidate one were to win, it would give us more CR than the expected CR
> value if candidate one did not win.
>
> [The last sentence of the above paragraph is the crux of the matter, hence
> the sentence that needs to be mulled over and pondered until digested
> fully.]
>
> If E is above the true CR value of candidate one, then we would like to
> with hold our support.
>
> If E falls somewhere in candidate one's interval, then we don't know if E
> is above or below the true CR value for this candidate, which could be
> anywhere in the interval with equal likelihood [the method could be
> adapted to some other distribution of probability, but let's not get weird
> here.], so we support candidate one according to the probability that
> his/her true CR value lies above E.
>
> As long as this rule is used for all (except possibly the extreme CR
> intervals), then ballot B's contribution to candidate one's approval will
> vary continuously with the values of p1, ... pN.
>
> Note that we have not ruled out the use of overlapping intervals.  Such
> intervals might be useful in modeling the fact that not all voters can
> decide for sure whether they prefer candidate X to candidate Y, but they
> are leaning in that direction.
>
> The respective intervals could be 75 plus or minus 10, and 60 plus or
> minus 10 with an overlap of 5 CR points, for example.
>
> Two intervals could have the same center but different widths depending on
> the confidence the voter has in his/her CR assessment.
>
> The ballot could allow the voter to choose between several widths or just
> two choices such as "wide" verses "narrow," with one being twice the width
> of the other, say.
>
> This also addresses the "dark horse" problem.  The candidate that you
> despise gets an interval of zero width centered on zero.  You might assign
> a dark horse candidate to an interval centered on zero but with positive
> width reflecting the uncertainty associated with the (to you) relatively
> unknown candidate.
>
> That's enough for now. I know this is very brief, so questions are welcome
> on the parts that need clarification.
>
> Forest
>
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