[EM] SARVO-Range as lottery utility (vNM utility) method?

[Kristofer Munsterhjelm]

Some time ago I was considering cardinal methods that might work using
only lottery information (e.g. "I consider an x% chance of A, 100-x%
chance of C to be just as good as getting B for sure"). Lottery
information provides scaled utilities, i.e. variables of the form

u_A = a * absolute utility of getting A elected + b

with a > 0

for candidate A. The problem with using absolute Range is that we have
no idea what a and b are, and it's very hard to get all of society to
agree on a common a and b to calibrate the ratings so they can be
meaningfully compared.

But earlier today I was adding some information about Myerson-Weber
strategy into Electowiki. And then I noticed that it seems like
determining optimal strategy according to the M-W model makes the a and
b terms disappear!

If so, SARVO-Range based on Myerson-Weber strategy would do the job; and
we have (sort of) a principled method to do cardinal voting with lottery
utilities (von Neumann-Morgenstern ones).

Let's check.

Suppose that voter v wants to decide what rating r_i to give to
candidate i. Define the "public rated utility" u_i for candidate i as
	u_i = a * up_i + b
where up_i is the absolute utility (according to some absolute scale
that we don't know). Then we want to show that determining the optimal
Myerson-Weber strategic ratings v_i based on u_i produces the same
result as doing so for up_i. That would mean that the strategic ratings
produced by using only the lottery information are the same as the
ratings that would be produced if we were fortunate enough to know up_i
for all voters directly.

So the M-W strategy is: let
	v_i be the strategic rating we want to find
	u_i be the public utility of candidate i
	p_ij be the voter's perceived probability that i and j will be tied.

The prospective ratings are:
	R_i = SUM j != i p_ij * (u_i - u_j)

and then we choose the v vector so that
	SUM i=1..n v_i * R_i
is maximized.

So expand the prospective ratings:
	R_i = SUM j != i: p_ij * (a up_i + b - a up_j - b)
            = SUM j != i: p_ij * a * (up_i - up_j)

and for the sake of convenience, let Q_I be the alternate-universe R_i
(where we know the up values directly)
	Q_i = SUM j != i: p_ij * (up_i - up_j),

which means that R_i = a * Q_i.

If we only have the u_i values, then we want to choose the v vector to
maximize
	f(v) = SUM i = 1...n v_i * R_i
and in the alternate world,
	g(v) = SUM i = 1...n v_i * Q_i

But f(v) = a * g(v), and since a is a constant, it can't change the
location of the optimum. Hence the M-W strategy depends neither on a nor
b, as desired.

So SARVO-Range with M-W strategy accomplishes what we want.

Strictly speaking, I would have to show that it's not just a ranked
method - that it elects the good centrist in the good centrist LCR
example. But I *think* that's true.

As proposed by Warren, SARVO-Range also considers one voter at a time.
If we want homogeneity (scale invariance), it should consider
infinitesimal slices of voters at a time instead. But I don't know how
to calculate that, and the naive approach is clearly out of the
question. I need someone with calculus of variations skills!

-km