[EM] Determining Approval from Interval CR ballots

[Forest Simmons]

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