[EM] A monotonic DSV method for Range

[fsimmons at pcc.edu]

This method is an attempt to adapt WMA (weighted median approval), a method based on ordinal 
ballots, to Range ballots, while making judicious use of the ratings, i.e. not merely using the ordinal 
information in the cardinal ratings ballots.

Here’s an approval strategy rule that makes better use of the ratings and is more robust than the one I 
gave earlier:

On each ballot B, let r be the highest number between zero and 100 such that r percent or more of the 
random ballot probability is held by the alternatives rated at level r or above on ballot B.  Any alternative 
rated at this level or above is approved.  An alternative is approved iff it is rated at this level r or above. 

Note that as r decreases from 100 to zero the total probability of the candidates rated at level r or above 
increases from some positive value to 100 percent.  So this approval cutoff level r is somewhere between 
zero and 100 percent.

Note also that if alternative A moves up in the ratings relative to all of the other alternatives, then 
alternative A’s approval does not decrease, and none of the other alternative’s approvals increase as long 
as the random ballot probabilities do not change.  

And even if A’s advancement moves A to first place on some ballot B1 so that A’s random ballot 
probability increases at the expense of some other alternative C, and A is rated at level r(A) on Ballot 
B2, then if  A was approved before on ballot B2, then A will still be approved on ballot B2, because the 
approval cutoff r cannot more ahead of r(A) on B2, since the probability above the level of A does not 
increase.  In fact, if C is rated ahead of A on ballot B2, the probability ahead of A decreases, otherwise it 
stays the same.   



----- Original Message -----
From: 
Date: Tuesday, April 27, 2010 3:16 pm
Subject: Re: A monotonic DSV method for Range
To: ,
Cc: election-methods at lists.electorama.com,

> 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: election-methods at lists.electorama.com,
> 
> > 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.
> >
> >
> >