[EM] Invariance to affine transformations, keeping cardinal honest

[Forest Simmons]

Great!

One way to extend the reals to allow comparison of quantities that are
"incommensurate" with standard real ratings ... is to allow polynomials in
epsilon as ratings.

Another thought ... the Ultimate Lottery method allows ballots to be
arbitrary positively homogeneous functions of the lottery probability
variables ...

f(p1, p2,  ... p_n)

The Ultimate Lottery is the point P of real n-space that maximizes the
product of the ballots, subject to the non-negativity constraints p_k >=0,
and the normalization to unity of the Sum p_k .

The one person, one vote condition is that all of the ballots have the same
degree d of homogeneity.

f(lambda*p)=f(p)*lambda^d

-Forest

On Mon, Oct 31, 2022, 5:41 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> Good news - I'm almost done with my non-EM big project, and so I should
> be able to create some strategy simulations soon to investigate which
> Friendly variants are good :-)
>
> But here's a thought that I've talked about earlier, that came up in a
> private mail response.
>
> Suppose that von Neumann-Morgenstern utilities are the best we can do as
> far as interpersonal comparisons go.
>
> Then we can imagine that every cardinal method has an idealized
> (Platonic, so to speak) form that operates not on ballots with discrete
> ratings, but on real ratings. E.g. the idealized version of Range
> involves giving each candidate a rating on [0...1), and the candidate
> with the highest mean wins.
>
> Then vNM utilities are unique up to some affine scaling. Let u(v, X) be
> voter v's utility should X be elected, on an objective scale that we
> can't access due to interpersonal comparison problems. Then if v is
> asked to rate X, he would rate X as r(v, X) = a_v * u(v, X) + b_v, where
> a_v and b_v are the positive scaling constants for that voter.
>
> For something like Range, suppose that when v goes to vote, v chooses
> some a and b so that every rating fits on the scale, then submits that
> vote.
>
> For an idealized method, we can then propose the following invariance
> property:
>
> - Affine invariance: The outcome of the election should not depend on
> the particular a and b constants chosen by the voter.
>
> Methods that pass affine invariance plus anonymity and symmetry satisfy
> a type of one-man-one-vote because expanding your scale is equivalent to
> making the a constant larger. Hence such a method, if it accepts
> ratings, doesn't have to limit itself to a bounded scale, though the
> user interface is probably more pleasant if it does.
>
> Furthermore, these methods have another property that there exists one
> honest ballot, which solves rb-j's objection of how much a voter should
> vote his second favorite if he's honest. Every such method that I know
> of fails IIA, but that's better than the sort of "I pass IIA but the
> outcome still depends on who's in the running" thing that Range does.
>
> What methods pass this criterion?
>
> Well, ranked methods clearly do. No amount of positive affine scaling
> can turn r(v, X) > r(v, Y) into r(v, X) < r(v, Y).
>
> In addition, there's this "double normalized Range": the voter has to
> fill in a ballot where at least one candidate is rated max and at least
> one is rated min. The rules are otherwise as Range.
>
> Each forced rating gets rid of one degree of freedom, so forcing two
> candidates fixes both a and b. The intermediate candidates' ratings can
> be found as follows:
>
> Suppose Best is the best candidate, and Worst is the worst, and the
> scale is from 1 being best to 0 being worst. Then for any other
> candidate X, rate X at p if you're indifferent between getting Best with
> probability p and Worst with probability 1-p, and getting X for sure.
>
> Incentivizing honesty here would be hard, though.
>
> (Perhaps a Ranked Pairs-like algorithm is possible for triples rather
> than pairs.)
>
>
> I also had an idea that perhaps we could find the best VSE affine
> invariant method by something like my "optimum strategy" search, but
> over the space of cardinal ballots instead of ordinal ones, for a small
> number of candidates. In particular, the results for three candidates
> could be interesting.
>
> We couldn't use the idealized method because there are infinite reals on
> a bounded interval. But the observation about double normalized Range
> above suggests that we could replace the ballot format for three
> candidates with:
>         (Who's min), (Who's max), (the p value for the third candidate),
>
> or really, one of:
>         (A:  X, B:  0, C: 10)
>         (A:  X, B: 10, C:  0)
>         (A:  0, B:  X, C: 10)
>         (A: 10, B:  X, C:  0)
>         (A:  0, B: 10, C:  X)
>         (A: 10, B:  0, C:  X)
>
> for a scale of 0..10, where X is an integer between 0 and 10 inclusive.
> This gives each voter 60 different ballots to choose between (contrasted
> with 6 per voter for ordinal).
>
> So for e.g. 10 voters, 3 candidates, there are about (60+10-1) choose 10
> = 3e11 elections. There are 3003 ordinal ones. Perhaps a rating scale of
> 0..10 is too high for an integer programming search. The trick would be
> to find a quantization level that's high enough that the search is
> feasible, yet low enough that it's possible to interpolate to an
> idealized method, somewhat like I did with fpA-fpC.
>
>
> Finally, I'd say it is perhaps possible to go beyond vNM, but I'm not
> entirely sure how to do it, much less how to do it in a
> strategy-resistant manner. On the other hand, there may be limitations
> beyond vNM itself: e.g. it might be hard to tell who would be a better
> winner between Stalin and Ivan the Terrible, because they're both so
> thoroughly awful as to saturate the scale.
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20221031/2bb208c0/attachment.htm>