Peter de Blanc schrieb: > On Wed, 2007-02-07 at 00:47 +0100, Jobst Heitzig wrote: >> Then you should be able to provide a thorough proof that it is optimal >> to express rankings proportional to your true utilities, by showing the >> respective derivatives to be zero. >> >> Please do so, since I still question that they are! > > Jobst, I can do this for you but it would take me a while to do. I'm not > very good at typesetting HTML and the formulae are very ugly and > complicated. > > Let me ask you: in the original writeup for "the n-Substance problem," > do you believe that: > > 0. the pricing rule given satisfies the criterion given (ie that it is > optimal to purchase quantities proportional to utility densities)? > > All I did to get the formulae from Hay Voting from there was: > > 1. I let the substances be transfers of voting mass between candidates > (there are n choose 2 such possible transfers) > 2. I calculated exactly how large each transfer would be by assuming > that the size of the transfer would be proportional to the difference in > utility between candidates. > 3. I then calculated how much voting mass each candidate would be left > with after all the transfers > > I thought that each of the steps was adequately justified. If you have a > problem with one particular step, then it would be easier for me to try > to clarify that. > > - Peter de Blanc OK, let me do it for you then. Given expressed ratings r1...rn, your formula was
pi = (sqrt(n-1) - fi) / n sqrt(n-1) where fi = c sum_j (rj-ri) = c (t - n ri) where c = 1 / sum_ij (ri-rj)² and t = sum_j rj Given true utilities u1...un, this results in expected utility E = sum_i (pi ui). If this would me at a local maximum for ri proportional to ui, each derivative gk := (d/drk)E(r1=a*u1,...rn=a*un) should be zero. But gk is proportional to (d/drk) sum_i (fi ui) = c² [ (t - n uk) sum_ij (ui-uj)² + sum_i (t - n ui) 4 (t - n uk) ] = c² [ (t - n uk) sum_ij (ui-uj)² ] which is zero for each k if and only if uk = t/n, for each k, that is, if all uk are equal. This is certainly not the case in general, hence putting ri = a ui for whatever a is *not* optimal. Yours, Jobst ---- election-methods mailing list - see http://electorama.com/em for list info
