[EM] Belate bit about regular election systems (but I'm not gonna get caught in a loop with Craig again...)

[David Catchpole]

Belated extras on regular election systems-

Sorry it's been so long (and it's been about 2 months since I got this
result, just after I sent a spurious one to Markus [sorry about
that...]). There's a proof to accompany this stuff, but I haven't got
around to expressing it nicely. Suffice to say it has to do with the Saari
octahedron, mathematical monotonicity and odd functions.

Here is the "basis" for the 100%, A-grade number-1 certified guaranteed
bona-fide original monorail...no, metameucil family- the family of optimal
regular single-winner election systems for which no other system exists
such that the probability of a result would be greater or equal to the old
system's result for any ensemble of votes [ from now on, "arching"]. 'Ere
dooz-

In a two-candidate case, the probability of a candidate A winning the
contest where votes are-

x*rtotal	(1-x)*rtotal
A>B		B>A

is	{ x + a ( 1 - 3x )	if x < 1/2
	{ 1/2			if x = 1/2
	{ x + a ( 2 - 3x )	if x > 1/2

where a is the determinacy coefficient, 0 <= a <= 1/3 .

a = 0 describes random dictatorship, or at least systems which have the
same results as random dictatorship in a two-candidate case. Thanks to
Pattanaik and Peleg, we know-or-at-least-conjecture-with-confidence that
random dictatorship is the "arching" member of the family for a = 0.

I have an example of a complete system for a = 1/3 , but I don't know if
(in fact, I don't think) it's "arching" per se- If there's a Condorcet
winner, the CW recieves a 2 / n+1 probability of winning (where there are
n candidates running) and the remainder recieve a 1 / n+1
probability. Otherwise, they all recieve a 1 / n probability.

Can anyone anyone help look for an algorithm that completes the metameucil
family using the cues we get from random dictatorship?

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"Never ascribe to conspiracy what can be put down to stupidity"- DJ No MC