[EM] PJ

Forest Simmons fsimmons at pcc.edu
Sat Jun 22 16:23:09 PDT 2019

```PJ stands for either Proportional Judgment (as opposed to Majority
Judgment) or Poetic Justice (see if you agree).

Voters submit approval ballots along with a circled favorite and a number R
between zero and one that the voter of the ballot thinks is a good approval
score given the chances for consensus.

Let P (between zero and one) be the average approval of the approval
winner.  Let Q (between zero and one) be the median of the submitted
numbers.

A ballot B is drawn.  Let R be the reasonable number marked on the ballot,
and let F be the indicated favorite.

If R>P, then elect the favorite or the approval winner with probabilities
R and (1-R) respectively.

[Note that if R=P=1, the favorite must be the same as the approval winner.]

Otherwise (If R is no greater than P) ...
... If P is in the closed interval [Q, 1], then elect the approval winner
.... else defer the decision to a second randomly drawn ballot.

I admit there is room for tweaking, but my main idea is to give incentive
for the max possible consensus.

The interesting case is when R is less than both P and Q.  In this case
both according to the voter of the random ballot and the median estimate of
reasonable possible consensus, the approval has fallen short of its
potential, so random favorite, the fall back benchmark, should be invoked.
Hence the second random ballot in this case.

Does the whole process start over again with the second randomly drawn
ballot?  Possibly, but let's keep it simple.  Just take the favorite of the
second ballot.

Why not just use F from the first random ballot in this case, with
probabilitty R, and revert to a second random ballot with probability
(1-R).  That is another possibility.  And I'm sure there are other
acceptable or even better ways to use the values of R, P, Q, and F to
decide what to do.