Jobst, It was Hay Voting that I was referring to. Maybe this post contains the desired answer to your puzzle?
Kevin Venzke --- Jobst Heitzig <[EMAIL PROTECTED]> a écrit : > Dear Forest, > > you wrote: > > > I have one question, though. If best strategy is to report true > > utilities, then what do you mean by encouraging compromise? > > > > You must mean compromise in the outcome as opposed to compromise in the > > ballots. > That's true. By "compromise" I mean the transfer of two voters' share of > the winning probability from their favourite options to a common > "compromise" option in case that transfer increases both voters' > expected utility according to the ratings they provided. > > > In other words, you want the lottery to favor centrist candidates as > > much as possible without giving incentive for the voters to compromise > > through distorted ratings. > > > > Is that right? > I'm not sure. My intention was just to construct a democratic (in the > sense of equal power for all voters) method under which it was optimal > to reveal true utilities and which was hopefully more efficient than > both Hay voting and Random Ballot. Having studied D2MAC before, the idea > was natural to use two randomly drawn ballots for this. The third drawn > ballot is only used for providing the potential compromise option in a > clone-proof way, so that the whole method becomes clone-proof (another > advantage over Hay voting). > > Of course, it would be even more efficient to not draw the potential > compromise option at random (by means of a third randomly drawn ballot) > but try to find a "best" compromise given the first two randomly drawn > ballots, and also to try to find "optimal" instead of random numbers x,y > for the probability transfers. That would however destroy the incentive > to vote sincerely and introduce strategy. > > Jobst > > >> From: Jobst Heitzig > > > >> Dear friends, > >> > >> Hay voting was supposedly the first known method under which it is > >> always optimal (as judged from expected utility) to vote sincere > > ratings > >> (i.e. ratings proportional to true utility). However, it seems that it > >> is a rather inefficient method (as judged from total expected > utility), > >> even less efficient than Random Ballot. > >> > >> Here's a different, more efficient method under which it is also > always > >> optimal (as judged from expected utility) to vote sincere ratings. It > > is > >> also based on Random Ballot, but in a very different way. It is > >> essentially a Random Ballot method with an added mechanism of > automatic > >> cooperation for compromise. The basic idea is that when there is a > pair > >> of ballots showing preferences A>...>C>...>B and B>...>C>...>A, those > >> two voters can profit from cooperating and transferring part of > "their" > >> share of the winning probability from A and B to the compromise option > > C. > >> Here's the method, I call it... > >> > >> > >> RANDOM BALLOT WITH AUTOMATIC COOPERATION, Version 1 (RBAC1): > >> ------------------------------------------------------------ > >> Voters rate each option. > >> Three ballots i,j,k and two numbers x,y between 0 and 1/2 are drawn at > >> random. > >> Assume that the top-ranked options of i,j,k are A,B,C, and that i and > j > >> have assigned to A,B,C the ratings ri(A),ri(B),ri(C) and > >> rj(A),rj(B),rj(C), respectively. > >> Now check whether the inequalities > >> y * (ri(C) - ri(B)) > x * (ri(A) - ri(C)) > >> and > >> x * (rj(C) - rj(A)) > y * (rj(B) - rj(C)) > >> both hold. > >> If so, elect A, B, or C with probabilities 1/2 - x, 1/2 - y, x + y, > >> respectively. > >> Otherwise, elect A or B each with probability 1/2. > >> > >> > >> Why should it be optimal to vote sincere ratings under this method? > >> > >> Consider an arbitrary voter i with favourite option A, and some > >> arbitrary options B,C and numbers x,y between 0 and 1/2. > >> Let us designate the A,B,C-lottery with probabilities 1/2 - x, 1/2 - > >> y, x + y by L, and the A,B-lottery with probabilities 1/2 and 1/2 by > > M. > >> The only thing i can do about the election outcome is by influencing > >> whether or not "her" inequality > >> y * (ri(C) - ri(B)) > x * (ri(A) - ri(C)) > >> holds, and the only situations in which this matters at all are those > > in > >> which i is among the first two drawn ballots, the other of the two has > > B > >> top-ranked, and the third has C top-ranked. > >> As it is equally likely for i's ballot to be drawn as the first or the > >> second ballot, and as i cannot influence whether or not the other > > inequality > >> x * (rj(C) - rj(A)) > y * (rj(B) - rj(C)) > >> holds, i would therefore want "her" inequality > >> y * (ri(C) - ri(B)) > x * (ri(A) - ri(C)) > >> to hold if and only if she prefers lottery L to lottery M. > >> But the latter is the case if and only if > >> y * (ui(C) - ui(B)) > x * (ui(A) - ui(C)) > >> where ui(A),ui(B),ui(C) are i's evaluations of the true utility of the > >> options A,B,C. > >> Now x and y were arbitrary numbers, so the only way to get this > >> equivalence is to put ri(A),ri(B),ri(C) proportional to > >> ui(A),ui(B),ui(C), and perhaps adding some irrelevant constant. Q.E.D. > >> > >> > >> Note that it doesn't matter from which precise distribution x and y > are > >> drawn as long as all values from 0 to 1/2 are possible. For the sake > of > >> efficiency, one should therefore use a distribution that strongly > >> favours values near 1/2, so that cooperation will be more likely. > Also, > >> the winning probabilities can safely be changed to > >> 1/2 - x/z, 1/2 - y/z, (x+y)/z, > >> where z := 2 * max(x,y). This will increase the probability of good > >> compromises further. > >> > >> Finally, note the following important fact about the method: It is > >> perfectly democratic since it distributes power equally in the > > following > >> sense: Any faction of m voters can give "their" share m/n of the > > winning > >> probability to any option they like by simply "bullet-rating" that > >> option at one and all others at zero. > >> > >> Please send comments! > >> Jobst _____________________________________________________________________________ Ne gardez plus qu'une seule adresse mail ! Copiez vos mails vers Yahoo! Mail ---- Election-Methods mailing list - see http://electorama.com/em for list info