[Election-Methods] [EM] A more efficient strategy-free ratings-based method than Hay voting

[Kevin Venzke]

Jobst,

It was Hay Voting that I was referring to. Maybe this post contains the
desired answer to your puzzle?

Kevin Venzke

--- Jobst Heitzig <heitzig-j at web.de> 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


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