Dear Steve! you wrote: > Well, first a minor point... I believe the word > "linear" is not used when there are equivalences. > A linear ordering means the same as a "strict" > ordering. When there are equivalences, the term > in the literature is usually "weak" ordering. > (I sometimes call a weak ordering a nonstrict > ordering, or just an ordering.)
You are right here of course! However, order theorists like me frequently use "order" or "ordering" as a synonym of *partial* order instead of weak ordering. Also, the term "weak ordering" is often avoided by order theorists since it misleadingly suggests that it is something weaker than a (partial) order, which it is not. Instead, we most often call that class of relations "total quasi-orders", "total" being the order-theoretic name of what others call "completeness" (while "completeness" refers to the existence of suprema and infima in ordern theory). So much about terminology -- it always gets confusing when people with different background talk, but this should be no real problem :-) > > > It may be misleading to say that because voters > tend to have transitive preferences that it's > reasonable to assume when A?C is changed to A=C > that it will also lead to A>D and C>B. So, > to avoid confusing the issues when studying > the difference between undecidedeness and > equivalence on voter behavior, I think we > should prefer an example such as this: > > > undecidedness equivalence > A C A=C > |\ /| | > | \/ | | > | /\ | | > |/ \| | > B D B=D > > > Would a voter behave differently given the > undecidedness preferences on the left instead > of the equivalence preferences on the right? > How would the undecided voter behave if allowed > to express equivalence but not undecidedness > when voting? Is the distinction significant > enough to make it worth complicating the voting > method? I'm sorry I didn't make my questions > clear earlier and I hope they are clearer now. > You made it perfectly clear, I think. As for your example, I indeed think that it will probably make no significant difference for most voters. What I suggested however in my old paper was that the expression of "undecidedness" between A and B (A?B) may be interpreted as the wish to *delegate* the decision about A,B to the other voters in order to improve the result in view of incomplete information or lack of expertise, while expression of "equivalence" of A and B (A=B) may be interpreted as the statement that they should have the same probability to be chosen at the end, so that a coin would decide between A and B if they were the only feasible alternatives. Hence, the natural decision rule for the case of just two alternatives A,B under this interpretation would... choose A if #(A>B) > #(B>A),#(A=B), choose B if #(B>A) > #(A>B),#(A=B), throw a coin if either #(A=B) >= #(A>B),#(B>A) or #(A>B)=#(B>A). When I was thinking about this differnce, however, I mostly had competitions with a small jury in mind such as dancing or design competitions. Instead, the main reason I insist on not misinterpreting undecidedness as equivalence is the fact that in contrast to me many people on this list do require preferences to be *transitive*. And with this additional provision it is well important to distinguish the two notions, as my original example shows: The "correct" preference relation A>B,C>D *is* transitive (and will thus be deemed "rational" by most people here). But the "incorrect" one with 4 equivalences instead of 4 undecidednesses is A>B,C>D,A=C=B=D=A which is *not* transitive since A >= B >= C without A >= C (it is only *quasi*-transitive)! So, the problem is, when we allow equivalence and undecidedness but require transitivity, our class of possible preference relations is that of all *partial* orders, while if we only allow equivalence it is the smaller class of all *weak* orderings. Yours, Jobst ---- Election-methods mailing list - see http://electorama.com/em for list info
