[EM] SARVO-Range as lottery utility (vNM utility) method?
Forest Simmons
forest.simmons21 at gmail.com
Mon Apr 25 15:51:11 PDT 2022
El lun., 25 de abr. de 2022 4:25 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:
> 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.
>
I could be wrong but I think it should be "tied for winning."
It is interesting that this strategy can actually result in non-solid
approval coalitions on ballots ... sometimes it requires you to approve X
while leaving unapproved some candidate Y rated above X on the same ballot
... i.e. insincere strategy.
Furthermore, if estimates of both the utilities u_i and u_j, as well as of
the probabilities p_ij in question were known with a high degree of
precision, you might get away with those insincere gaps in the approval
order.
These facts reflect the fragility (anti-robustness) of the winning tie
probability based strategy.
Nevertheless, your result is highly relevant because it shows that on a
fundamental level there is a meaningful, experimental way of defining
individual utilities that are just as good as the theoretical utilities
invoked as a basis for Approval strategy.
It is equally true for the not as sensitive strategy of approving the
candidates k with above expectation utilities:
u_k >sum P_i u_i,
based on estimates of (non tie based) winning probabilities P_i, which are
still sketchy because of rampant misinformation, not to mention intentional
disinformation.
> 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
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> info
>
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