[EM] Joe Weinstein's Approval Strategy Idea

[Forest Simmons]

Remember that Joe W. once suggested approving as far down your preference
order as you can without exceeding 50 percent probability of the winner
coming from your approved set.

Whether or not you should approve the next candidate (the one that would
tip the scales to more than fifty percent) depends on how far it would tip
the scales, how much you like that candidate, etc.

I would like to hear some ideas for rules of thumb to help decide whether
or not to approve this borderline candidate.

Joe's idea is sound from an information theoretic point of view.  The
expected information from a Bernoulli (i.e. binary) random variable is
greatest when the two values are equally likely.

For this reason, to get the most information about a person from a
True/False test or questionnaire, you should ask questions that (a priori)
have a fifty/fifty chance of being answered True.  If you ask questions
for which you already know the likely answers, then in all likelihood you
won't learn much about your respondent.

As a math instructor I like to gear my tests so that the average student
has a fifty/fifty chance of correctly solving a typical problem.

This is the best kind of test for separating the sheep from the goats, but
students don't like them, so I usually put in several easy problems near
the front for psychological purposes.

Back to Joe's idea: It seems to me that the idea of his strategy is to
maximize the probability that your ballot will be positively pivotal in
determining the winner.  In other words, this strategy actualizes your
potential voting power.

In the case of many viable candidates, no single candidate has a large
probability of winning, so each voter can approach the fifty/fifty
optimum, and each voter that makes use of this strategy is equally likely
to be positively pivotal in the outcome.

I say "positively" pivotal, because if one voter is pivotal, then all are
pivotal, but not necessarily in the direction that they would desire.

Take plurality, for example.  Suppose that A beats B by one vote, but you
voted for C.  Your vote (or lack of vote) was pivotal in preventing a tie
between A and B. But it could not be considered positively pivotal if you
actually preferred B to A.

In Approval suppose that A beats B by one vote and that you approved
neither or both, then you missed your chance of positively affecting the
outcome.  Joe's strategy minimizes the probability that you will end up
with this kind of regret.

With the above idea of minimizing this kind of regret in mind, I would
like to make the following rule of thumb suggestion:

Include or exclude the borderline candidate according to whether or not it
makes the approved/unapproved winning odds closer to or further from
fifty/fifty.

Suppose for example, that p1, p2, and p2 are the winning probabilities for
candidates C1, C2, and C3, and that your preference order is C1>C2>C3.

When should you approve C2 ?

If you want to maximize the probability that your vote is positively
pivotal, then approve C2 only if (p1+p2) is closer to 50% than p1 is.

If both are equally close to 50% then approve C2 only if C2 has above
average utility for you.

For example, suppose that p1=.3 and p2=.4, then (p1+p2)=.7 which is the
same distance (.2) from .5 as p1 is.  If candidate C2's utility is more
than fifty percent of the max, then approve C2.

In practice it is difficult to know the probabilities p1, p2, p3, etc.
reliably.  But what I have in mind is more along the line of Lorrie
Cranor's Declared Strategy Voting (DSV).

Remember the DSV CRAB race?  CRAB stands for Cumulative Repeated Approval
Balloting.  You submit your CR ballot, check the box for your favorite
CRAB strategy, and the DSV machine implements your CRAB strategy for you.

Joe's strategy could be the default strategy, for example.

Where would the p1, p2, p3, etc. values come from?

Well during the typical CRAB race, various candidates cycle through first
place.  The percentage of time spent in first place (so far) is an
estimate of that candidate's probability of winning.  Or (alternatively)
the p values (having started equal) adjust (according to Bayesian
principles, for example) as the race advances.

That's the general idea.  There's lots of room for imagination in the
details.

Forest


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