[EM] Approval Cutoffs

[Forest Simmons]

Suppose that you have complete access to a set of ballots that rank or
rate the candidates in a single winner race, and it is your duty to
recommend an approval cutoff to the voter of each ballot, so that they are
not wasting their ballots by voting too (opti- or pessi-) mystically :-)

How could you do this without displaying favoritism towards any of the
candidates?

The trouble is that most approval strategies depend on probabilities, but
the minute each voter follows your recommendation, the winner is a matter
of certainty, not probability.

Suppose, for example, that your recommended cutoffs would result in
candidate A winning the election.  In order to not display favoritism
towards A you adjust your recommendation.

With the new recommendation candidate B takes the place of A as winner, so
you make a further adjustment.

With this new recommendation C becomes the winner, so another adjustment
is made, which results in A being the winner again.

These considerations illustrate the difficulties inherent in methods
that convert ranked or rated ballots into approval ballots.

Ideally we would like a method such that if A wins based on the assigned
cutoffs, the voters would still consider that the assigned cutoff was
their best strategy, even knowing that A was certain to win.

For example, if A wins and all of the cutoffs were at A, inclusive or not
depending on whether or not the second placer was considered inferior to
A, then the all of the rational voters could be satisfied that their
approval cutoffs were at the right place.

But as we learned from Cumulative Repeated Approval Balloting (CRAB)
simulations, this kind of approval equilibrium is not always possible.

So what do we do?

One possibility is to try to minimize the number of voters that feel
(after the fact) that their approval cutoff was at the wrong place,
perhaps taking into account how far they would have moved the cutoff if
they had known who the winner was going to be.

Another possibility is to do some kind of simulation like the CRAB race
mentioned above, and then base the approval cutoffs on the empirical
probabilities that come out of the simulation.  It seems to me that
these empirical probabilities would be more accurate and reliable than
polling information.


Any thoughts about this?

Forest