[EM] uses of truncation
Forest W Simmons
fsimmons at pcc.edu
Thu Mar 15 17:29:02 PDT 2007
Various methods that make use of approval have alternative versions
that use truncation as the approval cutoff.
This suggests the concept of a virtual candidate "trunc" that is ranked
below the lowest ranked real candidate on each ballot, but above any
(and all) truncated candidates.
How could trunc be used?
As mentioned before trunc could be used as an approval cutoff.
What else?
Suppose trunc is included with the other candidates in some method like
Beatpath, and trunc turns out to be the method winner. Then ... (fill
in the blank).
Speaking of Beatpath, for each real candidate C, let C(1) be the
strength of the strongest beatpath from C to trunc. Let C(2) be the
strength of the strongest beatpath from trunc to C. The winner is the
real candidate C for which the difference C(2)-C(1) is the largest,
i.e. for which C(1)-C(2) is the smallest.
UncTrunc:
If trunc is uncovered, then the real candidate that has the greatest
pairwise opposition to trunc is the winner, i.e. the candidate that is
ranked on the greatest number of ballots wins in this case.
Else initialize a list with trunc, and as long as the current top
member T of the list is uncovered, add to the top of the list the
candidate (from among those that cover T) that scores the most pairwise
votes against T. The candidate that ends up at the top of the list is
the winner.
Note that X covers trunc iff X beats every candidate that is ranked on
fewer than half of the ballots.
If trunc is uncovered, then every real candidate X is beaten by some
real candidate Y that is ranked on fewer than half of the ballots. But
this can happen only if X is also ranked on fewer than half of the
ballots. Which means that X is also beaten by trunc. In other words,
if trunc is uncovered, then trunc is the beats all candidate.
Forest
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