Dear Rob! You wrote: > Jobst, could you post mathematical definitions and intuitive > explanations for each of these sets?
The sets I mentioned are all defined in the wonderful book by Laslier on "Tournament Solutions and Majority Voting". Here are some definitions as I recall them. We assume that for each pair of options X,Y, either X>Y or Y>X. Smith Set = Top Cycle = Schwartz Set = smallest set such that everything inside beats everything outside. Uncovered Set = set of uncovered candidates = set of candidates which have beatpaths of length at most 2 to every other candidate. In this, A covers B iff A>B and, for all C, C>A implies C>B and B>C implies A>C. The covering relation is a strict (partial) ordering. Banks Set = all candidates for which there is a maximal chain of defeats having the candidate on top. A chain is a sequence A1>A2>...>Ak such that also Ai>Aj for all i<j<=k. The Banks Set is component consistent (a strong version of cloneproofness) but not monotonic as a set. Markov Set = set of candidates which have nonzero probability in the limit distribution of the following Markov process: Start with a candidate chosen uniformly at random, move to a candidate chosen uniformly at random from all candidates beating him, repeat indefinitely. (See my posting on Markov Chain approaches from this spring) Minimal Covering Set, Tournament Equilibrium Set: more complicated definition -- I will have to look this up. Inclusions: Minimal Covering Set is included in Banks Set is included in Uncovered Set is included in Smith Set. Yours, Jobst ---- Election-methods mailing list - see http://electorama.com/em for list info