[EM] disappointment vs. regret
Forest Simmons
fsimmons at pcc.edu
Tue Feb 18 13:51:04 PST 2020
Warren Smith reminds us from time to time that range voting minimizes in
some sense something called "Bayesian Regret" which is the difference
between the social utility of the "best candidate" and the one elected by
sincere ballots.
A related but different concept is what I call "ballot disappointment,"
which tries to quantify the disappointment for one voter in one step of a
sequential elimination method.
Suppose that an elimination step replaces candidate X with candidate Y, as
the new "champion." How much disappointment does that incur for the
sincere voter of a ranked preference ballot B?
For example if the method must eliminate candidate X in favor of some
candidate Y that covers X, it would be nice if Y were the candidate (among
those covering X) that incurred the least total disappointment for this
transition.
Here's my proposal. For each candidate Z let f(Z) be the first place (that
is random favorite) probability for candidate Z over the entire set of
ballots.
Then on ballot B the disappointment in going from candidate X to candidate
Y is ...
....zero if Y is ranked ahead of or equal to X
else the sum (over all candidates Z ranked ahead of Y) of f(Z).
So if Y is ranked higher than X on ballot B, then the voter of ballot B has
little cause for complaint, otherwise the disappointment is the probability
that a better Y would have been chosen by random ballot.
This is the foundation of my new Landau method based on ranked preference
ballots with out the need for approval cutoffs.
Ordinarily my proposal for the initial candidate in the sequence would be a
candidate chosen by random ballot or else the approval winner, but I'm
studiously avoiding requiring the voters to make approval judgments, and I
want to have a deterministic version of the method, as well. So in the next
message I have a deterministicsolution that does not require voter to make
approval judgments..
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