# [EM] DSC = DOC, meets ENHarm

Kevin Venzke stepjak at yahoo.fr
Sun Jan 18 14:27:01 PST 2004

```On Dec 29, I posted a method called "Descending Opposing Coalitions":
http://groups.yahoo.com/group/election-methods-list/message/12966

I have since realized that DOC is equivalent to Descending Solid Coalitions.
Suppose there are four candidates, ABCD.  A ballot opposing the set CD is
necessarily solidly commited to AB, the inverse of the opposed set.  This
means that the sets in DOC will be in precisely the same order (but inverted)
and have precisely the same effect as the sets in DSC.

This means that DSC already passes Earlier-No-Harm.  (DAC, analogously,
should pass Earlier-No-Help.)

An example might help show this.  Suppose my ballot is A=B>C=D.  DSC counts
my ballot towards sets AB and ABCD.  If I vote A>B>C=D, DSC counts my ballot
towards A alone, also.  This allows my ballot to "hook up" with other ballots
which place A first, but perhaps some other candidate second.  So the only
effect the inserted ">" can have is that {A} rises higher in the list, which
can only be a helpful thing.

I don't mean to suggest that LNH properties imply their ENH counterparts.
That seems to be clearly false, looking at any sort of equal-rankings IRV.

What is most interesting to me about DSC, is that it is like FPP corrected
for Clone-Winner problems.  It's an elegant (but very minimal) improvement:

8 A>B
5 B>A
7 C

DSC identifies 13 voters for {A,B}, and essentially finds that that set is
the plurality winner (at least, at this point): C can't win now.  It next
encounters the 8 voters for {A}, causing A to win.

But note that the C voters can't make B win by voting C>B.  And also:

10 A
9 C>B
9 D>B
...
9 Y>B
9 Z>B

Elects A.  So it's a minimal improvement.

Kevin Venzke
stepjak at yahoo.fr

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