[EM] Lp-ball range

[Warren Smith]

>FW Simmons:
Range voting is cardinal ratings with certain constraints on the
possible ratings, namely that they have to fall within a certain
interval or "range" of values, and usually limited to whole number
values.

Ignoring the whole number requirement, we could specify a constraint
for an equivalent method by simply limiting the maximum of the
absolute values of the ballot scores.  Call this "method infinity."

We could get another (non-equivalent) system by limiting the sum of
the absolute values of the scores.  Call this "method one."

Yet another system is obtained by limiting the sum of the squared
values of the scores.  Call this method two.

Other methods are obtained by limiting the sum of the p powers of the
absolute values of the scores. In this scheme method two corresponds
to p=2, and methods infinity and one, respectively, are the limits of
method p as p approaches infinity or one.

Suppose that there are three candidates.  Then graphically the
constraints for the three respective methods corresponding to p equal
to infinity, one, and two, turn out to be a cube, an octahedron, and a
ball with a perfectly spherical boundary, respectively.

The optimal strategies for methods infinity and one generically
involve ballots represented by corners of the cube and octahedron,
respectively.

In the case of method infinity, this means that all scores on a
strategically voted ballot will be at the extremes of the allowed
range, i.e. method infinity is strategically equivalent to Approval.

In the case of method one, the corners represent the ballots that
concentrate the entire max sum value in one candidate, and since
negative scores are allowed, this method is strategically equivalent
to the method that allows you to vote for one candidate or against one
candidate but not both.  I don't think anybody has studied this method

--it was proposed about 30-40 years ago and served as an inspiration
for approval voting (mentioned in Brams book I think, I think the name
"negative voting" perhaps was used?)

> but in the case of only three candidates it is the same as Approval.

--it is equivalent to it...  votes:  A gets +1, B and C get 0; or A
gets -1, B and C get 0

> The unit ball for method two has no corners or bulges (which all other values of p involve), so the strategy is not so obvious. But if Samuel Merrill is right, then in the zero information case, the optimum strategy for method two is to vote appropriately normalized sincere utilities.

--wrong.
Your best strategy for any of these methods is, you identify the two
"frontrunners", you vote max for one and min for the other, and then
if you have any freedom left, you start considering the other N-2
candidates.  With L-infinity voting you get approval.
With L1 or L2 voting you get either plurality or antiplurality as the
two kinds of strategic vote.

ONLY with p=infinity (among all p>=1) is there any freedom of choice
left after you max/min the two frontrunners.  This is the ONLY
method in this class whee strategic voters can express order N bits of
information.

This begins to explain why range voting has a unique status among all
"COAF" voting methods
(based on convex "balls" defining the allowed votes)... I pointed all
this out some 11 years back...



-- 
Warren D. Smith
http://RangeVoting.org