[EM] Monotone method inspiration approach?

[Daniel Carrera]

On Tue, Jan 18, 2022 at 3:36 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> Is there a concise way of recursively describing the full
> version?
>

I think I have an idea. I'm going to let g() be the plus operator. I will
define a couple of helper functions:

h(A,B) = {1 if fpA > fpB; 0.5 if fpA = fpB; 0 if fpA < fpB}

This is just a repackaged sign function.

k(A,B,C,D,...) = H(fpA - min(fpA, fpB, fpC, ...))

This is a repackaged Heavyside function.

[A > B] = Defined the same way you did.

These provide a lot of notational convenience. Notice that the order of
arguments is significant and k() has a variable number of arguments.
Lastly, I'm going to redefine f(A) so that it too has a variable number of
arguments and the order of arguments is significant.

f(A,B,C,D,...) = score of A in the election {A,B,C,D,...}

Now I can define f() recursively:

f(A,B,C) = k(A,B,C) * ( h(B,C)*[A>C] + h(C,B)*[A>B] )

f(A,B,C,D) = k(A,B,C,D) * ( f(A,B,C) + f(A,B,D) + f(A,C,D) )

And so on.

So the score of A for the N-candidate election is k(A,B,...), which covers
the option where A is the plurality loser on this round, times the sum of
the scores of A for each of the possible (N-1)-candidate elections that
include A. This can probably be expressed with some combinatorial notation
that I haven't thought of.

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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