Dear participants, suppose "d[X,Y]" with X<>Y is the number of voters who strictly prefer candidate X to candidate Y. Then the "Smith set" is the smallest non-empty set of candidates with d[A,B] > d[B,A] for each candidate A of this set and each candidate B outside this set.
A "chain from candidate A to candidate B" is an ordered set of candidates C(1),...,C(n) with the following three properties: 1. C(1) is identical to A. 2. C(n) is identical to B. 3. d[C(i),C(i+1)] - d[C(i+1),C(i)] > 0 for each i = 1,...,(n-1). A "Schwartz winner" is a candidate A who has chains at least to every other candidate B who has a chain to candidate A. The "Schwartz set" is the set of all Schwartz winners. The term "innermost unbeaten set" is another term for "Schwartz set." Example: Suppose that there are 3 candidates; candidate A pairwise beats candidate B; candidate B pairwise beats candidate C; and there is a pairwise tie between candidate A and candidate C. Then the Smith set is ABC and the Schwartz set is A. ****** A "Landau winner" (aka an "uncovered candidate") is a candidate A such that for each other candidate B at least one of the following two statements is valid: 1. d[A,B] >= d[B,A]. 2. There is a candidate C with d[A,C] >= d[C,A] and d[C,B] >= d[B,C]. ****** The "Landau set" (aka "uncovered set", aka "Fishburn set") is the set of all Landau winners. Markus Schulze ---- Election-methods mailing list - see http://electorama.com/em for list info
