Hello Forest, > Most of the credit should be yours; in fact, the proof and all of the > ingredients are yours. I hurried to post the message this morning, because I > was sure that you were going to beat me to it! I would certainly believe you > if > you said that you had already thought of the same thing but didn't have time > to > post the message before I did.
Well, I don't. I was never before thinking of asking separately some additional information from the voters which is already contained in the ratings part of their ballot (namely what the favourite is), in order to transfer the strategic incentives away from the ratings to this separate information. And that is exactly the genial part! Now that the general technique is clear, we can easily derive a lot of similar methods, also some that only incorporate a very small amount of chance, for those who don't like chance processes in voting methods. The general technique is this: Ask for ratings and some additional information. Use the additional information and to determine -- independently from all ratings -- two possible winners or winning lotteries, at least one of which must be a lottery of at least two options in which the probabilities can vary. Then use the ratings to decide between these two possibilities in some monotonic way (e.g., using unanimity as in your proposal, or some qualified majority, or even Random Ballot, or whatever). Then strategy-freeness in the ratings part follows from the fact that they are only used in a monotonic binary choice between lotteries which are not known before. For example: Method "Range top-3 runoff" (RT3R) =================================== 1. Each voter separately supplies a "nomination" range ballot and a "runoff" range ballot. 2. From all "nomination" ballots, determine the options A,B,C with the top-3 total scores a>b>c. 3. Let L be the lottery in which B wins with probability p = max(0,(2b-a-c)/(b-c)) and C wins with probability 1-p. 4. Let q be the proportion of "nomination" ballots on which the lottery L has an expected rating below the rating of A on that ballot. 5. Option A wins if, on at least the same proportion q of all "runoff" (!) ballots, the lottery L has an expected rating below the rating of A on that ballot. Otherwise B wins with probability p and C wins with probability 1-p. Notes: p is so designed that it can take all values between 0 and 1 but will be the larger the lower c is, in order to get a large expected rating of the final winner. Of course, the formula for p could be modified in all kinds of ways. The expected total "nomination" rating of the final winner is at least a - 2(a-b), so if the race between A and B is close (i.e., a-b is small), we have quite an efficient outcome. On the other hand, if A clearly beats B, it will win with a high probability since the true proportion of voters who prefer A to B and C is probably larger than can be seen from the strategically used nomination ballots. What do you think? Yours, Jobst ---- Election-Methods mailing list - see http://electorama.com/em for list info