Re: [EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives
fsimm...@pcc.edu wrote: To my knowledge, so far only two monotone, clone free, uncovered methods have been discovered. Both of them are ways of processing given monotone, clone free lists, such as a complete ordinal ballot or a list of alternatives in order of approval. I think Short Ranked Pairs also passes all these. To my knowledge, Short Ranked Pairs is like Ranked Pairs, except that you can only admit XY if that will retain the property that every pair of affirmed candidates have a beatpath of at most two steps between them. See http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-November/014255.html . The definition of a short acyclic set of defeats was later changed, and the new definition is at http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-November/014258.html I also guess you could make methods with properties like the above by constraining monotone cloneproof methods to the Landau set (whether by making something like Landau,Schulze or Landau/Schulze). I'm not sure of that, however, particularly not in the X/Y case since the elimination could lead to unwanted effects. Is one of the two methods you mention UncAAO generalized? Election-Methods mailing list - see http://electorama.com/em for list info
Re: [EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives
Where's a good place to find out more about the Landau set? Is
it really
possible to have a monotone, clone free method that is
independent of non-Landau
alternatives?
It turns out that there are several versions of covering, depending on how ties
are treated. All of them including the Landau set are the same when there are
no pairwise ties (except with self).
In July of this year I gave an example that shows that no decent deterministic
monotone method can be independent from covered alternatives. The example
applies to the Landau version of uncovered. So neither Ranked Pairs nor
Beatpath nor Range restricted to Landau can monotone.
Here's the example
Suppose that we have a method that satisfies independence from non-Landau
alternatives, and that gives
greater winning probability to alternative B in this scenario
40 BCA
30 CAB
30 ABC
than in this scenario
40 DBC
30 BCD
30 CDB
as any decent method would.
Then consider the scenario
40 DBCA
30 ABCD
30 CADB
The Landau set is {A,B,C} and so by independence from non-Landau alternatives
the winner is chosen
according to the first scenario above.
Now switch A and B in the second faction to get
40 DBCA
30 BACD
30 CADB .
The Landau set becomes {D,B,C}.and so by independence from non-Landau
alternatives the winner is
chosen according to the second scenario above.
In summary, solely by raising B relative to A , the winning probability of B
decreased.
For a deterministic method the probability of B would go from one to zero.
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Re: [EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives
Hi Kristofer, you wrote: I think Short Ranked Pairs also passes all these. To my knowledge, Short Ranked Pairs is like Ranked Pairs, except that you can only admit XY if that will retain the property that every pair of affirmed candidates have a beatpath of at most two steps between them. See http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-November/014255.html . The definition of a short acyclic set of defeats was later changed, and the new definition is at http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-November/014258.html Thanks for unearthing this old idea of mine -- I had forgotten them completely already. Unfortunately, I fear Short Ranked Pairs might not be monotonic. One would habe to check. And I'm not sure your description of an algorithm for Short Ranked Pairs is valid -- after all, I only defined it abstractly by saying that one has to find the lexicographically maximal short acyclic set without giving an algorithm to find it. I propose to proceed as follows: Check how that lexicographically maximal short acyclic set can be found in the simpler case in which we define defeat strength as approval of defeating option. This will also allow us to compare the method to DMC since DMC is the result of applying ordinary Ranked Pairs with this definition of defeat strength, so applying Short Ranked Pairs to them should not be too much different. The resulting short acyclic set will contain all defeats from the approval winner to other options, but I don't see immediately whether one can somehow continue to lock in defeats similar to ordinary Ranked Pairs, skipping certain defeats that would destroy the defining properties of a short acyclic set. I somehow doubt that since that defining property is not that some configurations must not exist but than some configurations must exist. Anyway, in the case where defeat strength is approval of defeating option, all might be somewhat simpler. Jobst I also guess you could make methods with properties like the above by constraining monotone cloneproof methods to the Landau set (whether by making something like Landau,Schulze or Landau/Schulze). I'm not sure of that, however, particularly not in the X/Y case since the elimination could lead to unwanted effects. Is one of the two methods you mention UncAAO generalized? Election-Methods mailing list - see http://electorama.com/em for list info Election-Methods mailing list - see http://electorama.com/em for list info
