[EM] Monotone method inspiration approach?

[Kristofer Munsterhjelm]

On 18.01.2022 10:29, Kristofer Munsterhjelm wrote:
> On 18.01.2022 01:57, Daniel Carrera wrote:
>> On Mon, Jan 17, 2022 at 4:40 PM Kristofer Munsterhjelm
>> <km_elmet at t-online.de <mailto:km_elmet at t-online.de>> wrote:
>>
>>>     These if-then clauses may be difficult to put into mathematical
>>>     expression form, but I found this (which may seem a bit ugly at first):
>>>             f(A) = g(H(fpA - fpC) * H(fpB-fpC) * A>B,
>>>                      H(fpA - fpB) * H(fpC-fpB) * A>C)
>>>
>>>     where H is the step function H(x) = 1 if x > 0, 0 otherwise,
>>
>>
>> Btw, if you want to make a plot of that, I recommend you replace the
>> Heavyside function with one that is equal to 0.5 when x is exactly 0.
>> Otherwise, when fpB == fpC you'll see f(A) suddenly crash to 0 for no
>> apparent reason and you'll be left scratching your head.
> 
> That's deliberately left unspecified because when fpB == fpC, that means
> that two candidates tie for last, and then IRV's in big trouble. I
> *think* using second preferences to break the tie would then work, but
> as this situation is measure zero I didn't want to further complicate my
> expression by including it.
> 
> In practice, this tie dependence can further amplify IRV's chaotic
> behavior. Warren calls it "near-tie nightmares".

Come to think of it, perhaps that would make for an interesting way to
handle ties in IRV. A first approximation would be this: when there are
no ties, X's score is X's pairwise strength against whatever Y is left
in the final round if X isn't eliminated before the final round,
otherwise 0. And if B and C is tied, then X's score is one half his
score if B got eliminated + one half his score if C got eliminated.

It's not quite the same thing as H being the symmetric (odd) Heaviside
function because the above explanation doesn't handle three-way ties
(fpA-fpC = fpB-fpC = 0). But perhaps it could still be useful for
something. Is there a concise way of recursively describing the full
version?

This is all assuming g() is the plus operator. Different results happen
if g is say, max.

-km