[EM] Ordinal ballots to approval ballots via Pereira and Weinstein

[Forest Simmons]

In a recent posting to the EM I gave a graphical description of how the
Pereira transformation decomposes a range ballot into a weighted sum of
approval ballots.

Now I want to show how to start with ranked preference style ballots and
end up with a weighted sum of approval ballots.

First we need a probability distribution for the candidates.  This
distribution can be based on pre-election polls, or it can be based on some
statistics from a random sample of the ballots, or from the entire set of
voted ballots.  To the extent that we us the voted ballots to come up with
the distribution, we can consider this method a "DSV" method.

To illustrate the variety of possibilities, I suggest averaging to
distributions together: the first distribution is the ordinary random
ballot distribution.  The second one that we will call the "implicit
approval random ballot distribution."

We're assuming that voters can both truncate and equal rank, whether top,
bottom, or any other rank.

Under this assumption, when a voter ranks a candidate above bottom, it
shows a certain minimum level of approval that we call "implicit approval."

For the "implicit approval random ballot" probabilities, we contemplate an
expriment that starts with drawing a ballot at random.  The set of
candidates that are not truncated (i.e. implicitly approved) is the initial
value of a set S.  If this set contains more than one member, more ballots
are drawn until we get a ballot that implicitly approves one or more of
these tmembers of S.  We update S by eliminating the candidates that are
not implicitly approved by this new ballot.  We keep drawing more ballots
to narrow down S as far as possible (hopefully to a singleton).  Then with
a uniform distribution on the remaining members of S, we draw at random the
name of one candidate from S.

The probability that candidate C is the one drawn by this experiment is the
probability associated with C in the "implicit approval random ballot"
probability distribution.

I suggested above that we average this distribution with the distribution
given by the more common random ballot probability distribution.

In my next post I will show how to use such a distribution to change the
ballots into range ballots (based on an idea of Joe Weinstein).

To Be Continued...
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