Change the word mx to min in the second step fo the method description so that it reads ..
(2) A gets the min possible rating (say zero) if more than fifty percent of the top ratings belong to alternatives rated above A. ----- Original Message ----- From: Date: Tuesday, April 27, 2010 3:04 pm Subject: A monotonic DSV method for Range To: [email protected], > Median Probability Automated Strategy Range Voting (MPASRV) > > It is well known that optimal Range strategy is the same as > optimal Approval > strategy. But this optimal strategy is hard to automate because > (1) it depends > sensitively on hard to estimate probabilities of winning ties, > and (2) all > attempts at automating strategies based on expected ratings have > turned out to > violate monotonicity. In fact, most DSV (Designated Strategy > Voting) methods > fail Monotonicity. > > A near optimal approval strategy which depends less sensitively > (i.e. more > robustly) on probability estimates than the optimal strategy > (and based on > ordinal information only) is to approve alternative C iff the > winner is more > likely to come from among the alternatives that you like less > than C than from > among the alternatives that you prefer over C. > > Unfortunately, automating this strategy by approximating the winning > probabilities with random ballot probabilities also yields a non- > monotonicmethod. But it can be modified slightly to yield an > automated strategy Range > method that is monotonic and makes appropriate use of ratings: > > Modify each Range ballot so that for each alternative A ... > > (1) A gets the max possible rating if more than fifty percent of > the top ratings > (taken from all ballots and counted as in a random ballot probability > computation) belong to alternatives rated (on this ballot) below A. > > (2) A gets the max possible rating if more than fifty percent of > the top ratings > belong to alternatives rated above A. > > (3) Otherwise A's rating is not changed. > > Then elect the alternative with the highest average rating, > where the average is > taken over all the modified ballots. Settle any ties by use of > the random > ballot probabilities, or by random ballot itself. > > This method is monotonic. It satisfies Participation and IPDA > (Independence from > Pareto Dominated Alternatives) . It is also clone independent > in the same sense > that ordinary Approval is. > > It may seem that the method would slight candidates lacking in > first place > support. However, even when alterantive C has no first place > support, if > surrounding candidates are approved on a ballot, our process > makes sure that C > is approved also. > > To see how this works, think of a voter located in issue space. > The further the > options are from her, the lower her respective ratings for them. > Her approval > cutoff represents a "sphere" such that > > (1) half of the the voters lie inside of the sphere and half > outside, and > > (2) all of the alternatives whose Dirichlet/Voronoi regions are > containedentirely inside the sphere are approved, and those > whose regions are entirely > outside the sphere are disapproved. > > (3) those alternatives that lie right on the boundary of the > sphere get rated > according to the radius of the sphere (the smaller the radius, > the higher the > rating). > > The Voronoi/Dirichlet regions are the regions of first place > support of the > respective alternatives. In the two dimensional case they are > the colored > regions found in Condorcet and Range diagrams of Yee/Bolson > type, in contrast to > the wierd shapes found in diagrams of the same type representing > IRV elections. > For Range and Condorcet the numbers of voters in the respective > colored regions > are precisely proportional to the respective random ballot > probabilities. > > Note that our new method MPASRV automatically respects top and > bottom ratings, > so voters who think they have a better strategy can control > their own approvals > and disapprovals. > > > ---- Election-Methods mailing list - see http://electorama.com/em for list info
