[EM] Invariance to affine transformations, keeping cardinal honest

[Kristofer Munsterhjelm]

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 :-)

What would polynomial ratings look like in practice? Would they have a 
different ballot format, or ask for different data, than vNM type 
ballots? And would there exist a way for an honest voter to provide an 
unambiguous honest ballot consistent with his state/preferences?

The Ultimate Lottery sounds a bit like the Nash solution for envy-free 
division, where you maximize the product of utilities. But I don't know 
all that much about the subject.

-km