[EM] A thought about manipulability

[Kristofer Munsterhjelm]

As the Condorcification proof shows, if you elect someone else than the 
(absolute) CW, there will exist at least one majority who all prefer 
someone else to the winner, and (if majorities can force outcomes) they 
can compromise to get that candidate elected.

But one subtlety that I missed: this means that if the honest election 
is a cycle, there's always a strategy. There may be a "cycle to cycle" 
strategy, but there is definitely a "cycle to CW" (or rather, cycle to 
majority winner) strategy.

I think it's been proven (though I don't remember who did it) that with 
impartial culture, as the number of voters goes to infinity, the 
fraction of elections that are a Condorcet cycle approaches one. If so, 
then a consequence of the above is that *every* method with the majority 
forcing property is manipulable in almost every case, under impartial 
culture with enough voters.

That's surprising. Though again, maybe not?

It suggests that impartial culture is too strong, at least with lots of 
voters -- it's too close to a tie to be realistic.

(But then again, we usually don't consider this sort of "forced 
compromise" in a cycle to be a type of strategy that anyone can reliably 
pull off. Intuitively, a method electing a non-CW seems to be open to 
strategy to get the CW elected, but with a honest cycle it feels more 
like the electorate is too divided to pull a cycle to non-cycle strategy 
off.)

-km