[EM] Preliminary Chain Climbing investigation

[Kristofer Munsterhjelm]

So I implemented chain climbing as a higher order/meta method in 
quadelect to check if it is compatible with or can improve strategy 
resistance. What I've found so far (assuming my implementation is 
correct) is:

- Strategy resistant methods don't seem to preserve their resistance 
when used as the base method for chain climbing (starting from loser and 
then building a chain).

- At least for three candidates, where I have brute-force methods, there 
do exist base methods whose compositions with chain climbing are 
strategy resistant.

- However, these base methods make absolutely no sense on their own.

So the pattern seems to be that while chain climbing ensures Banks 
compliance, it nonlinearly warps the landscape of which methods are good 
and which are bad, which means that our usual heuristics won't work. It 
becomes very difficult to create a method that preserves desirable 
properties when passed through chain climbing.

As an example of strange base methods, here's a linear three-candidate 
method in the format of fpA-fpC:

f(A) = BAC + 2 BCA - ABC

The composition of this and chain climbing has about the same strategy 
resistance as Smith,IRV. And here's a nonlinear method:

f(A) = BCA * BAC

with similar performance as fpA - fpC. Neither of these strange methods 
is monotone.

On a more theoretical level, it should be possible to craft a base 
method so that the chain climbing composition always produces the same 
winner as fpA-fpC in the three candidate case, by building a set of 
implications. E.g. let f(A) = fpA - fpC be the method we want to 
replicate, and g(A) be the scoring function for the base method we want 
to pass through chain climbing, then we have
     whenever f(A) > f(B) > f(C) (i.e. A wins)
     then either
         g(C) > g(A) > g(B) or
	g(A) > g(C) > g(B)

and then similarly for every way to relabel candidates.

In the first case, B is admitted, then A beats B pairwise and is 
admitted, then C isn't because C doesn't beat B; in the second case, B 
is admitted, then C is skipped, then A is admitted.

But it's not a particularly pleasant way to design a method!

-km