[EM] Invariance to affine transformations, keeping cardinal honest

[Forest Simmons]

On Tue, Nov 1, 2022, 2:36 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> On 11/1/22 04:58, Forest Simmons wrote:
> > 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
>
> I think you'll have to explain that in more steps :-)
>

In due time ...

>
> What would polynomial ratings look like in practice? Would they have a
> different ballot format, or ask for different data, than vNM type
> ballots?


A polynomial can be represented as a sequence of coefficients, each of them
a rating on a scale of zero to 99, with an understanding that the k-th
coefficient is multiplied by epsilon^k.

Addition is by addition of the sequences, which, in this case are functions
from non negative integers into the whole numbers between zero and 99. So
you add them the way you add functions.

Comparisons of magnitudes are by ignoring the higher degree terms except
when the lower degree terms are tied.

And would there exist a way for an honest voter to provide an
> unambiguous honest ballot consistent with his state/preferences?
>

Since affine combinations of polynomials are by addition, multiplication,
and division by positive reals, I don't see a problem ... we need to
explore it to see if there are unanticipated problems. But I'm sure this
has already been done.

>
> The Ultimate Lottery sounds a bit like the Nash solution for envy-free
> division, where you maximize the product of utilities.


In this case the product of the expectations of the respective ballot
ratings.  I don't know if Nash thought of this election methods
application, but we called it the Nash Lottery in our second voting lottery
article.
The Ultimate Lottery is a generalization.
In the Nash lottery a ballot is a function of the form

p-->f(p)=r•p

So obviously f(lambda p)=lambda f(p), which shows f to be homogeneous of
degree one.

p-->(Sum (a_k*p_k)^n)^n

Is also homogeneous of degree one.

As n approaches infinity, this L_n norm approaches the sup norm
Max(a_k*p_k), which is also homogeneous of degree one.

When proving the Nash lottery to be a proportional lottery, you define the
Lagrangian to be
Sum log(f(p))-lambda(Sum(p_k)-1), and set its gradient to zero for
stationarity. The lambda is the lagrange multiplier for constraining total
probability to unity.

Follow your nose and the Nash allocation of probabilities falls right into
your lap.

After going through that for the Nash Lottery, do the same thing for the
Ultimate Lottery, making use of Euler's Partial Differential Equation for
homogeneous functions when it is needed; You can derive it for yourself by
taking the gradient of both sides of the homogeneity condition ...

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

All of this is somewhere in the EM List archives about ten years back. But
it'll be easier to recreate it by following your nose, than to find it in
the archives. We never did publish it; our paper was already too long and
detailed for the reviewers.


But I don't know
> all that much about the subject.
>

The wikipedia article on homogeneous functions is good ... including their
treatment of Euler's Theorem.

The main thing for our immediate purpose is that you can multiply a ratings
ballot by any positive number without changing its contribution to the Nash
or Ultimate lottery probability.

>
> -km
>
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