[EM] disappointment vs. regret

[Forest Simmons]

Now to review the "Minimum Disappointment Covering Enhancement" method:

Ranked ballots with the possibility of equal ranking and truncation are
required, along with access to or the ability to compute (from the ballots)
a random favorite distribution.

An initial candidate x0 is obtained by some election method based on ranked
preference ballots of the type mentioned above. It could be the method
described in my previous message, or it could be River, Random Ballot,
Greatest Implicit Approval, or any other monotonic, clone-free method.

1. If x0 is uncovered, then elect x0.

2. Otherwise, among all of the candidates that cover x0 replace x0 with the
one that minimizes the sum of squares (over all ballots) of the ballot
disappointments for the transition from the previous value to the updated
value of x0.

 3. Return to step 1.

Remember that the ballot disappointment in the transition from candidate x
to candidate y is zero if y is ranked ahead of x, else it is the
probability that random favorite would pick a candidate ranked ahead of y.

On Tue, Feb 18, 2020 at 1:51 PM Forest Simmons <fsimmons at pcc.edu> wrote:

> 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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