MIKE OSSIPOFF a écrit : > Steph-- > > [a shift key isn't working, and so some letters may be incorrectly > uncapitalized. i'll use brackets for parentheses.] > > here's your example: > > A family of examples with (11+2x) (x>=0) voters and 3 candidates is: > 2+x: A > 2: A > B > C > 2: B > A > C > 1+x: B > C > A > 4: C > Ranked pairs with winning votes produces: > A (6+x) > C (5+x) , B (5+x) > C (4) and A (4+x) > B (3+x). > A is the Condorcet winner and wins. > Margins and relative margins produce of course the same result. > If I am one of the two B > A > C voter, my 2nd (A) > choice harms my favorite 1st choice (B). > The proof is, if I and my co-thinker vote B only: > 2+x: A > 2: A > B > C > 2: B (truncated !) > 1+x: B > C > A > 4: C > Ranked pairs with winning votes produces: > B (5+x) > C (4), C (5+x) > A (4+x) and A (4+x) > B (3+x) can't lock. > B wins now. > > i reply: > > i assume that you're saying that in your example x can have any > nonzero value without changing the result. For simplicity i let X = 1. > > maybe i made an error, but by my count, B is the BeatsAll winner, > regardless of whether the BAC voters truncate.
You are wrong there. A always beats B by one vote (4+x)-(3+x) =1. Yes, regardless of the BAC voters truncating, it does not change. With X=1, A (5) > B (4). > Would you re-post the example that you intended, preferably with > exactly-determined numbers of voters? With X=1: 3: A 2: A > B > C 2: B > A > C 2: B > C > A 4: C Ranked pairs with winning votes produces: A (7) > C (6) , B (6) > C (4) and A (5) > B (4). A is the Condorcet winner and wins. Margins and relative margins produce of course the same result. If I am one of the two B > A > C voter, my 2nd (A) choice harms my favorite 1st choice (B). The proof is, if I and my co-thinker vote B only: 3: A 2: A > B > C 2: B (truncated !) 2: B > C > A 4: C Ranked pairs with winning votes produces: B (6) > C (4), C (6) > A (5) and A (5) > B (4) can't lock. B wins now. The other example I am referring to at: http://groups.yahoo.com/group/election-methods-list/message/10107 uses X=0. > if you say that truncation can take victory from a CW in wv, you're > probably intending the old example in which a poorly-supported CW loses > to truncation. i've many times said that truncation can work against > a cw when there's lots of indifference toward that cw. When there's > no majority voting the cw over the candidate who steals the election. > A candidate can be CW even if he beats the other candidates with > sub-majority defeats. And if he beats y with a submajority defeat, > i make no claim that y can't steal the election by truncation. Please define a poorly-supported CW. If the definition is criteria (wv or rm) independent it should be really helpful. If it is not, we need to compare the sizes of the poorly-supported CW (wv) class to the size of the same class using (rm). This is why the probability of a CW being a poorly-supported CW depending on the criteria matters. > if that's your point, then you're right. it's something that i've > often said. > > please note that SFC is about a CW who is preferred to > candidate y by a majority who vote sincerely. Re-read SFC & GSFC. > Those criteria, though they don't mention truncation, tell what i > claim for wv when no one order-reverses. > > mike ossipoff All votes I gave previously were sincere preferences. Only the truncation that leads to a better result for the truncaters is unsincere as you were doing. The CW I used (namely A) was a "CW who is preferred to candidate y (namely B or C) by a majority who vote sincerely". Thus if this is the definition of a non "poorly-supported CW", my counter-example fits... If SFC & GSFC apply to non-poorly supported CW, please explain why the sincere CW (namely A) can get stolen with an unsincere truncation in my examples. I will consider "non poorly-supported CW" as a sincere CW. I hope it is what you mean. The only conclusion I was able to obtain by myself, was that your SFC & GSFC analysis was good when there was no truncation (nor sincere, neither unsincere) present. In this particular case, margins, relative margins and winning votes all produce the same results. Still I think you are right in assuming the probability of gain from an unsincere truncation is lower with winning votes (maybe null as you say, I am not sure). I was able to prove it is null for 3 candidates, not further yet. But your analysis seems to be built on the assumption only one truncation (unsincere) occurs. I think many can happen, sincere ones as much as multiple unsincere (many voters could expect an improvement) or both types. I have just proved to you that this probability is not null using (wv) with multiple truncations. Even, relative margins would protect the CW in these examples (except for X=0 where a tie occurs). Thus the probability to steal a CW when all sincere votes (including sincere truncations) distributions are considered may be lower or higher with relative margin than with winning votes. I do not know. But I think sincere truncations increase with an increasing number of candidates. Finally, once we know that because of sincere truncations, unsincere truncations can lead to (rare I hope) special cases where a CW can get stolen, it increases the probability of unsincere truncations occuring. This is why I try to evaluate the probability of truncation occuring and then the probability of being able to steal a CW in overall cases. Steph. ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em