[EM] The Ultimate Lottery Method!
fsimmons at pcc.edu
fsimmons at pcc.edu
Mon Jan 5 11:31:03 PST 2009
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
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