[EM] re: CR/Approval and cutoffs

[Forest Simmons]

Here's the strongest argument that I can think of in defense of giving
candidates who are rated precisely at the approval cutoff level exactly
half approval:

The problem is that the CR values are discrete.  Suppose that the allowed
values are the whole numbers between zero and ten.  In this case the CR
value "six" really means six plus or minus one half, since voters cannot
"vote in the cracks"  so to speak.

If we assume that half of the ratings of "6" reflect preferences in the
upper range of this interval (between 6 and 6.5) and the other half in the
lower half of the interval, then it makes perfect sense to give half
approval for a candidate with a CR of 6 when the approval cutoff is
precisely 6.

This brings me to a proposal that I have been toying with for a long time.

When the CR resolution is small or when there are apt to be a significant
percentage of voters with approval cutoffs near the same candidates (as in
the case of voters copying "candidate cards" or party voting guides, not
to mention typical EM list examples), a small shift in the approval cutoff
can make a dramatic change in the distribution of total approval among the
candidates.

This discontinuity is the source of most (if not all) of the erratic
behavior observed in some of the methods that focus on refining the
approval cutoff.

[In the case of "Auntie" it is the only discontinuity. Other methods have
have additional discontinuities.  GIA has a discontinuity, for example,
that stems from going (in one step) from three to two candidates with
equal weight (in the approval cutoff calculation).]

We can overcome this main source of discontinuity by considering ratings
on CR ballots to be intervals of uniformly distributed values, rather than
discrete point values.

However, I propose that the two extreme values retain their point status
for two reasons:

(1) They do not contribute to the type of discontinuity we are considering
since no approval cutoff moves past either of them.

(2) In the methods under consideration it is important to respect the
wishes of voters voting only at the extremes.

As an example, suppose that we humor the sociologists by going with a
seven slot CR ballot.

The two extreme slots would count as point values of zero and 100
percent, respectively.

The other five slots would be intervals of width 20% centered at the
values 10%, 30%, 50%, 70%, and 90%.

Suppose that the approval cutoff is computed to be 70% on some ballot
where one of the candidates has been rated "slightly above average"
corresponding to "somewhere in the interval between 60% and 80%."

As discussed above, this ballot should contribute a fraction of one half
to the approval of that candidate.

Now suppose that in a similar case the approval cutoff turned out to be
72% with nothing else changed.  Wouldn't it make sense to have this ballot
contribute a fraction of 8/20 to the approval of that candidate?

In general, if the cutoff c is a fraction p/q of the way from the top of
an interval to the bottom of the interval, then any candidate rated in
that slot should get a contribution of p/q from that ballot.

In more detail ... if the interval in question is [a,b], and cutoff c is
a number in this interval, then the fraction p/q is given by

                        p/q = (b-c)/(b-a) .

It is easy to see that the value of p/q varies from zero to one as c moves
from top to bottom of the interval, with 1/2 at the center of the
interval.

In the  other example above we have a = 60%, b = 80%, and c = 72%, so

          p/q = (80-72)/(80-60) = 8/20 .

In the rare case that an approval cutoff c turns out to be precisely at
the upper or lower extreme of the CR range (one of the two point values
zero or 100 Percent), then the extreme rated candidates should still get
either full or no approval, depending on which extreme because this
convention does not violate continuity, and is probably consistent with
most voter wishes.

Forest