Matthias Urlichs wrote:
Hi, So who did come up with the mistake "Schultz", and did they eat too many peanuts? ;-) Anthony Towns:The correct restatement is something more like: { x | forall y: y >> x --> x >>> y }Or, in understandable language: The Schwartz set is the innermost unbeaten set, or the smallest set of candidates such that none are beaten by any candidate outside the set.
I think we need to come up with better, understandable, language. Let me make sure I understand the Schwartz set.Is it accurate to say that if x is in the Set, and y>>x, then y is in the set?
Is it accurate to say that if there is no y such that x>>y (i.e., x defeats nothing), then x is NOT in the set?
If we have options A, B, C, D, an A>>B,C,D, B>>C>>D>>B, then {B,C,D} is not the Schwartz Set because A>>B? A is in the Schwartz Set because there is no x such that x>>A?
I'm trying to figure out where x>>>y comes into play with the Schwartz set. What is the Schwartz set for: A==B; A>>C,D,E; B>>C,D,E; C>>D>>E>>C?I coul make an argument that {A} is the Schwartz set. It's an unbeaten set, and the smallest of all possible unbeaten sets (tied with {B}, etc. The same argument could be made for {B}. I think the answer is {A, B}, but I don't have any real justification for it.
As for resolution, I'd adopt the SSD method, which states 1- Determine the Schwartz set. 2- Drop the smallest defeat(s) within the set. 3- Repeat 1-3 until there's a winner. See http://electionmethods.org/CondorcetEx.htm. It explains all of this in reasonably easy-to-understand language.