In my previous message I wrote ... > Ballots are ratings with a minimum possible rating of zero.
> Ballots with all zero ratings are thrown out as not valid. > The lottery probabilities are chosen so as to maximize the product of the expected ratings over the ballots. > This method is (1) monotone, (2) clone free, and (3) gives proportional probability to stubborn voters. (4) Each ballot has equal weight in determining the winning probabilities. (5) Good opportunities for cooperation are not wasted by this method. (6) There is little if any incentive for insincere ratings. Let's use the Lagrange multiplier method to find the lottery that maximizes the product of the expected ballot ratings: Let ProdE represent the product of the expected ratings, and let SumP represent the sum of the lottery probabilities. Then a necessary condition for maximality of ProdE is the stationarity of the expression L = log(ProdE) - Lambda*SumP as the lottery probabilities are varied subject to the constraint SumP = 1. Setting to zero the partial derivative of L with respect to the lottery probability p(k) of the k_th alternative (i.e. candidate number k) verifies claim (4) in that each ballot b contributes to p(k) precisely the quantity (1/N)*p(k)*b(k)/E(b) where N = Lambda is the number of ballots, b(k) is ballot b's rating of alternative k, and E(b) is ballot b's expected rating E(b) = p(1)*b(1) + p(2)*b(2) + ... So we see that the total probability contributed by ballot b to the lottery is exactly 1/N . Property (3) follows as a corollary, since a stubborn faction of M voters that rate only alternative k above zero will contribute a fraction M/N to the winning probability of alternative k. I'll leave properties (1) and (2) as an exercise, and defer properties (5) and (6) to another message. Jobst, has this been done before? If not, let's write it up and submit it for publication. Forest ---- Election-Methods mailing list - see http://electorama.com/em for list info