[EM] Intuitive argument that FPTP manipulability approaches certainty in impartial culture

[Kristofer Munsterhjelm]

Suppose, for any given election, we relabel candidates so that they are 
in sorted order by first preferences, i.e. A has the most first 
preferences, then B, then C, and so on.

A wins in the honest election.

Then the election is manipulable if, when every X>A voter ranks X first, 
the winner changes from A to X.

For a three-candidate election, specifically, we can manipulate if
	fpA > fpB and (B>A) > fpA,
because then all B>A voters rank B first and fpB becomes equal to B>A 
which is greater than fpA.

Now consider the impartial culture with c candidates and n voters. Every 
first preference count is a binomial with expectation n/c, while every 
pairwise preference has expectation n/2.

Since n/2 > n/c when we have more than two candidates, in the limit of n 
approaching infinity, we should have (B>A) > fpA with probability one. 
Thus Plurality is almost always manipulable under impartial culture with 
enough voters.

(What's needed to make this formal rather than just intuitive is an 
argument that uses the variance of the binomials to show that p((B>A) > 
fpA) = 1 in the limit of n approaching infinity.)

I suspect that certain manipulability under IC is true of at least every 
three-candidate weighted positional system closer to Plurality than 
Borda, but proving that would be pretty tough. I did some calculations 
that, if I didn't mess them up, shows that it's *not* true for 
Antiplurality.

-km