On Sun, 7 Nov 2004, Steve Eppley wrote: > If James used the term Borda to refer to any scoring rule, > which is a possible interpretation of his premise about > "no matter how" the points are allocated to successive > preferences, then his supposition is wrong. Plurality > rule is a scoring rule where S1 = 1 and S2=S3=...=Sn=0, > and it always elects a candidate ranked first by an > absolute majority, if such a candidate exists. Any scoring > rule such that S1 >= 2 * (S2 + S3 + ... + Sn) has that > property. (So do the scoring rules where n = 2 and > S1 > S2.)
Maybe I shouldn't beat a dead horse (voting method?), but that condition on scoring rules electing a 1st-by-majority is wrong. As a counter-example, consider 3 candidates with S1=2,S2=1,S3=0. This satisfies the condition on S1 >= 2*(S2+...Sn), but with the following ballots it doesn't elect the 1st-choice majority: 51: A>B>C 49: B>C>A A: 102 B: 149 C: 49 Another way to see that its wrong: scoring rules produce the same result if one replaces S_i by M*S_i+L, M>0, but S1>=2*(S2+...SN) changes when such a transformation is made to the S_i. I think that plurality is the only scoring rule that also elects a candidate first-choice by a majority. Consider S1=1, S2=s, S3=0, and fraction 0.5+f voting A>B>C, 0.5-f voting B>C>A. A: 0.5+f B: 0.5-f+s*(f+0.5) A-B = 2f-s*(f+0.5) If f < 1/(4/s-2), then A loses despite getting more than half the votes. This shows that there is no scoring rule that ensures this form of majority rule for the N=3 case other than plurality. (Replacing S_i with L*S_i+M for L,M independant of i, L>0, does not affect the outcome, so setting S1=1,Sn=0 does not lose any generality other than the silly rule S1=S2=S3) For N>3, the same result holds: use the same example, with C replaced by N-2 clones of C. -wjs /-----------------------------------------\ | Warren Schudy | | WPI Class of 2005 | | Physics and computer science major | | AIM: WJSchudy email: [EMAIL PROTECTED] | | http://users.wpi.edu/~wschudy/ | \-----------------------------------------/ ---- Election-methods mailing list - see http://electorama.com/em for list info