[EM] A variant of DSC
fsimmons at pcc.edu
fsimmons at pcc.edu
Sun Aug 7 15:25:25 PDT 2011
That "Q" in the previous subject heading was a typo.
Here's an example that illustrates the difference in Woodall's DSC and my modified version:
25 A1>A2
35 A2>A1
20 B>A1
20 C>A1
Woodall's DSC assigns 60 points to {A1, A2} and then the only other positive point coalitions that have
non-empty intersections with this set are {A2}, {A1}, {A1, B}, and {A1, C}, with respective points of 35,
25, 20 and 20. The 35 point set {A2} decides the result: A2 wins.
In my version, the 60 point coalition is the highest point proper coalition {A1, A2}, so candidates B and C
are struck from the ballots and we are left with
25 A1>A2
35 A2>A1
40 A1
This time A1 wins.
As near as I can tell, my version still has all of the advantages of DSC, including later-no-harm, clone
independence, monotonicity, etc.
Note that Woodall and I get the same result for
25 A1>A2
35 A2>A1
40 D>A1
namely, that A1 wins. But if you split the D faction in half, you get the original scenario above. It seems
to me that A1 should continue to win, but classical DSC switches to A2 without any good reason. In
other words, it lacks a certain kind of consistency that our modified version has.
Jameson,
the only advantage of DSC over DAC is that DAC does not satisfy later-no-harm. In the context of
chicken this would keep the bluffer from truncating, but to no avail; the plurality winner (with 48 points)
would win, since (singleton) it would form the highest point solid coalition all by itself.
Under DAC the bluffer would truncate but would still form an assenting coalition with the guy who did not
truncate her, but not a solid coalition. An even bigger assenting coalition would be the plurality winner
together with the bluffer. Of these two, only the bluffer would be in the second largest coalition, so the
bluffer would win under DAC.
----- Original Message -----
From:
Date: Saturday, August 6, 2011 3:13 pm
Subject: AQ variant of DSC
To: election-methods at lists.electorama.com,
> 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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