I wrote:
> 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).
I should have covered also the other option of forcing clones to be separated
from each others here since that may well be required in some methods. For
example in multi-winner elections it is natural that support to one of the
clone members always means support to all members of that group. In this case a
bullet vote to A would be complemented to A>B>C (to show the separation between
{A, B} and {C}) and not to A>B=C.
This way of handling ties would change also one line in the pseudocode that I
gave. In this case it would be as follows.
- the preference relation after a clone set will be always a > relation (e.g.
A1>A2=B>A3 becomes A1>A2>A3>B, i.e. "=B" will be changed to ">B" to separate
the clone set from other candidates)
Maybe this second option should be kept as the default option since it is
safer. It limits the set of allowed votes a bit but it meets better the needs
of all methods.
Juho
On 8.8.2011, at 0.18, Juho Laatu wrote:
> 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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