[EM] A monotonic DSV method for Range

[fsimmons at pcc.edu]

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-monotonic
method.  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 contained
entirely 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.