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. >> ---- >> Election-Methods mailing list - see http://electorama.com/em for list info > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info
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