Ok, I agree that you need a concrete enough description to check the properties
of the method.
If the tree is (((A,B),C),D), then all of them are explicit clones at top level
(trivial), A, B and C are explicit clones, and also A and B are explicit clones
within those larger clone groups.
If you vote D>B>C>A, that is clone compliant with clone groups {A, B, C, D} and
{A, B, C} but not with {A, B}. In the examples that I gave I assumed that the
first occurrence of one of the members is the place where all the clones should
be. That means that after B you must have A. The corrected vote would be
D>B>A>C. As with the traditional clone definition, now all the clones stand
next to each others.
You could read the vote e.g. so that after you have ranked D (one of the
clones) first, you must rank A, B and C next. Although I didn't describe the
process explicitly (only as an example), if you don't say anything about the
others, the completion procedure should add them as equal after D (=> D>A=B=C).
One thing that I did not cover explicitly is how to handle equality. I guess it
is ok not to require clones to be separated from others but just require them
to be next to each others. What I mean is that if A and B are the only clones
and there are three candidates {A, B, C}, then e.g. a bullet vote A would be ok
since we consider A>B=C to be clone compliant enough although A is here better
than B, and C is equal to B (i.e. closer to B in some sense).
You wrote about inferring structure from the votes. I however assumed that the
trees would be agreed and announced already before the actual election day.
Voters would be expected to respect that structure and not try to separate
clones from each others. You could also derive the trees from the given votes,
but of course that would be a more complex thing to do, and you would have to
violate/modify some voter opinions that were cast without knowing that they
violate the order in the post-derived tree. It is also hard to say which
candidates should be declared as clones and which ones not. There is no
requirement to have a full binary tree here.
(One could also avoid violating any voter's opinion if one would declare only
those candidates as clones that meet the traditional very strict clone
definition (= those who are next to each others in every vote). But that would
be quite unnecessary since all the clones would already be their correct
places.)
I don't think there is risk of losing monotonicity in the predecared tree +
preprocessing + Condorcet approach if we assume that the Condorcet method is
monotonic and the preprocessing rule just limits the set of allowed candidate
orderings in the input votes. If we correct erroneous votes to clone compliant
votes and that causes the result to change in a nonmonotonic way, that should
maybe not be considered to be a violation of nonmonotonicity. If that is a
problem, then we could just reject all badly formulated votes and not count
them in any statistics. In that sense the method is just plain Condorcet with
some strict rules on which votes are legal.
Yes, the SODA approach to the chicken problem is tree-like. The predeclared
tree and limited set of acceptable votes approach could be seen as one straight
forward and simple approach that can be used also as a measure stick to see how
much other methods can improve from that.
I'll write also some pseudocode to make the vote correcting / complementing
process more explicit.
- derive clone sets from the candidate tree (every branch of the tree is a
clone set whose members are all the candidates in that branch)
- read every vote starting from the highest ranked candidate
- if some candidate is not followed (without interruptions) by all the other
candidates of a clone set whose member this candidate is, the vote must be
corrected (or complemented if the omission of other clones was intentional)
- start corrections from the smallest clone set, and then continue with the
bigger ones (note that every clone set is a subset of all the other larger
clone sets that include this candidate)
- lift the other members of the clone set next to this candidate, maintain
their relative order and preference relation between them (>, =)
- the preference relation after the clone set will be the one that preceded the
first non-clone-set-member below this candidate (e.g. A1>A2=B>A3 becomes
A1>A2>A3=B, i.e. "=B" stays although the candidate before B changed)
- note that after solving all smaller clone sets a larger clone set will not
change the order of the already moved candidates since they are automatically
part of the clone set, and already "well ordered"
- note that when the vote reading process moves (after rearranging some clone
sets) forward, some later candidates may cause changes in the order of the
(later) already moved candidates (e.g. the already collected A* clone set in
vote A1>A2>A3>A4>B could still become A1>A2>A4>A3>B if A2 and A4 form a clone
set {A2, A4})
(I hope that's detailed enough. Please point out errors. I'm getting too tired
to check :-).)
Juho
On 7.8.2011, at 22.22, Jameson Quinn wrote:
>
> I think the "explicit clone preprocessing of the votes + Condorcet"
> description that I gave below is a quite accurate definition of a method that
> both eliminates the clone problems and has rich ballots (rich enough to take
> position also on the order within the competing branch).
>
> I still think you have to spell things out more for us. If the tree is
> (((A,B),C),D) and I vote DBCA, what does my vote get corrected to? And I can
> easily think of several variations of how to preprocess votes into clone
> trees. In general, I think methods which try to infer structure from votes
> are tricky. Either you're risking nonmonotonicity by reading in more than is
> really there, or you could end up just reinventing a complicated way to
> restate DSC/DAC.
>
> Note that part of the SODA solution for the chicken dilemma -- that is, the
> enforcably-mutual preferences between candidates -- is tree-like. So I can
> see the potential advantages of trees, I just don't think it's fair to claim
> benefits for a method that's not well-described enough for us to construct
> pathologies.
>
> JQ
>
>
> ps. By the way, can anyone explain to me a scenario where DSC would be better
> than DAC? I understand that with full rankings they're equivalent, but I
> don't see when DSC is better.
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