In the proof paragraph after step 6 below, the constancy of the sign of the difference of the probability above and below X before and after the averaging step is only true for X=A_k , precisely where it is needed. I had written ...
By the way, your method of adjusting only the relative likelihood of the current winner to the other candidates is growing on me. Here's a specific version adapted to ordinal ballots: 1. Calculate the random ballot lottery L_0 . 2. Use Joe Weinstein's weighted median (approve X iff L_0 gives more probability to lower ranked than higher ranked candidates) strategy on each ballot to determine the approval winner A_0 . 3. Let L'_0 be the lottery that gives A_0 one hundred percent of the probability, and let L_1 be the average of L_0 and L'_0. 4. Use weighted median based on L_1 to determine A_1. 5. etc. 6. In finitely many steps the sequence A_0, A_1, ... will converge to (i.e. get stuck on) the winning candidate A. Proof of the assertion in step 6 is the same as Jobst's proof in the case of Cardinal ballots, because in Joe Weinstein's strategy the difference in total probability above X and below X has the same sign before and after the averaging step, so A_k's approval doesn't change from stage k to stage (k+1). [Therefore the current approval champ beats the previous one only by setting a new approval record. This cannot happen more than a finite number of times when there are a finite number of voters.] Forest ________________________________ From: Jobst Heitzig [mailto:[EMAIL PROTECTED] Sent: Wed 8/3/2005 11:03 PM To: Simmons, Forest Cc: election-methods-electorama.com@electorama.com Subject: Re: [EM] 0-info approval voting, repeated polling, and adjusting priors Dear Forest! You wrote: > At each successive stage we would base the new lottery calculation on a > weighted average of all the old cutoffs. In other words, the cutoff on > each ballot is adjusted slightly towards the most recent lottery > expected value before calculating the new lottery probabilities. One must know the individual cardinal utility functions for this, it seems. An alternative would be to switch from expected utility 0-info strategy to median utility 0-info strategy (aka Weinstein's strategy) so that only rankings would have to be known. Jobst -------------- next part -------------- An HTML attachment was scrubbed... URL: /pipermail/election-methods-electorama.com/attachments/20050804/6c948f56/attachment.html ------------------------------ _______________________________________________ Election-methods mailing list [EMAIL PROTECTED] http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com End of Election-methods Digest, Vol 14, Issue 5 ***********************************************
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