[EM] Preliminary simulation of fpA-{sum, max} fpC and compromise vs burial incentive

[Kristofer Munsterhjelm]

A quick note on the contingent vote:

The Heaviside formulation seems to break ties in a way that combines 
better with Smith, and Smith// than the "obvious" way I programmed it in 
the simulator. See the pdf for results with 29 voters, 4 candidates; the 
Heaviside method is called "TEST".

The "obvious" way is: if A is in the top two by Plurality count and 
beats someone who is in the other tier (i.e. ranks second by Plurality 
and beats someone who ranks first, or vice versa), then A is a winner.

The Heaviside works like this:

A's score =
	sum over B being some other candidate
		(product over C being some other candidate, neither A nor B
			H(fpA - fpC) * H(fpB - fpC)
		) * A>B

elect the candidate with the maximum score;

where H is the standard Heaviside function (with H(0) = 0.5), and fpX is 
the number of first preferences for X.

The idea being that we're checking all pairs {A, B} for whether they're 
contenders to the top two. If they are, then A must have a higher first 
preference than everybody but B -- that's what the H(fpA-fpC) term 
checks; and B must also have a higher first preference than everybody 
but A, and that's what the H(fpB - fpC) term checks.

So if A and B are truly the top two, then the H(...) product evaluates 
to one, and then A gets score equal to A>B and B gets score equal to 
B>A, and whoever beats the other pairwise wins.

H(0)=1 gives slightly worse results, but H(0)=0 is much worse.

(Incidentally, the nonmonotonicity is obvious when the method is 
described as above: the H(fpB-fpC) term is not monotone in increasing 
fpA. It could be made monotone by replacing it with something like H(fpA 
+ fpB - 2 fpC), which would give something that has about the same 
strategy resistance as ordinary fpA-max fpC, but alas, it doesn't 
generalize any better than fpA-max fpC does).

-km
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