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

[Joshua Boehme]

On 4/21/25 3:43 PM, Kristofer Munsterhjelm wrote:

> Very nice. I think you're right. (Strictly speaking, we can probably remove 
> the first condition, because if fpA=fpB, then as long as B>A > fpA in 
> expectation, we can still change a tie between fpA and fpB into a decisive 
> victory for B, which is an improvement for B.)
> 
> However, other obvious voting model - impartial anonymous culture or IAC - 
> fails the criteria above. As a result, most of the methods I've been looking 
> at converge to a fractional manipulability value for three candidates for IAC.

This could still lead somewhere interesting...

Let A be the honest frontrunner of a given plurality election (i.e., fpA > 
fpB for all B != A).

The election is manipulable iff there exists a B != A such that B>A >= fpA.

That is, it is NOT manipulable iff for all B != A, fpA > B>A.

This is an interesting situation, as it's akin to being a Condorcet winner 
but even more so. A>B > fpA, so this condition implies A is a strong 
Condorcet winner, but the converse doesn't hold (e.g., a centrist candidate 
could be everyone's #2 pick but no one's #1).

In other words, in a plurality election, there's either a "super" Condorcet 
winner decisively above all the other candidates, or it's manipulable by 
strategic voters.

(Maybe this was already a known result somewhere, though.)