One way of looking at Woodall's DSC method is that it is designed to elect from
the clone set that
extends up to the top rank on the greatest number of ballots, i.e. kind of the
plurality winner among
clone sets.
There are two ways in which this description is not precise, but maybe we would
get a better method if
we follwed this description more closely.
(1) The solid coalitions look like clone sets on the ballots that reach up to
the top, but they don't have to
look like clone sets on the other ballots.
(2) This description doesn't tell how DSC narrows down after finding the
plurality winner solid coalition.
In fact the entire set of candidates is automatically the solid coalition that
extends to the top rank on
100% of the ballots, so for starter we need to narrow down to a proper
sub-coalition.
With regard to (1), imagine a one dimensional issue space with the candidates
distributed as follows:
A..............B1..B2..B3...................C......................D1..D2...........E
The set {B1, B2, B3} and the set {D1, D2} will be solid coalitions that
extend to the top rank on the
ballots of the voters that have a favorite among them, and they will appear as
clone sets on all of the
ballots that do not rank C first. But voters near C may well intermingle the
B's and the D's like
C >B3>D1>B2>D2>B1>E>A
This shows that a geometrical clone doesn't have to end up as a classical
ballot clone except on the
ballots of the voters that are situated in the middle of the clone set, in
which case they will appear as
solid (or assenting) coalitions that extend to the top rank.
So Woodal had the right idea for making his method clone independent.
If I uderstand correctly, Woodall invented DSC to prove a point, viz. that a
method can satisfy later no
harm, be clone free, and montone. He didn't invent the method as a serious
proposal. So I don't think
his feelings will be hurt if we suggest an improvement.
My suggestion is that once we have found the proper subset solid coalition that
extends to the top rank
on the greatest number of ballots, strike from the ballots the candidates that
are not in that coalition, and
iterate until there is only one candidate left. Elect the sole remaining
candidate.
For incomplete rankings we can modify DAC in the same way, by replacing the
term "solid" with the
term "assenting."
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