A thought about manipulability
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
It seems to me that impartial culture is a very tenuous assumption for
public elections ... even less realistic than Yee diagram culture which is
generically cycle free ... not just free of a top cycle.
The Yee diagram is based on multivariate normal distributions, perhaps not
too realistic ... but the Central Limit Theorem would seem to make it more
(not less) realistic as the size of the election increases.
Remember the governor recall election in California that Arnold won ...
there were several dozens of candidates. Since the ballots were "vote one"
style there could be no ballot cycle.
But who could plausibly argue that a top cycle would have been likely ...
if only complete preferences had been elicited, and there had been even
more candidates?
Yes, inconsequential ties among lower popularity candidates might become
more common with more candidates, just as tied lower candidates in a Yee
diagram might become more common.
In that context all candidates at the same distance from the center of the
distribution are tied with each other ... but not likely for first place.
There is only one point at the precise center... but infinitely many points
at any positive radius r from the center ... the bigger the radius, the
more room for tied candidates ... for what that's worth (if anything).
On Sun, Jun 25, 2023, 4:07 PM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:
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
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On 6/26/23 02:14, Forest Simmons wrote:
It seems to me that impartial culture is a very tenuous assumption for
public elections ... even less realistic than Yee diagram culture which
is generically cycle free ... not just free of a top cycle.
Yes; spatial models are probably the closest model we have, although
they could be improved - e.g. voters may have a distribution of belief
of the position of some given candidate, not just a point estimate, so
that complete unknowns' positions are, well, unknown. Then they may be
risk averse, which leads to a penalty to candidates who for some reason
or other haven't got their position out there.
The question is more, I think, whether impartial culture is a good
challenge for methods... a sort of "very hard difficulty" mode because
it's more liable to produce near-ties than real elections, with the idea
that if a method can handle anything IC has to throw at it, then it can
definitely handle the real world...
... or if IC contains distortions that make it the wrong model to use,
so that optimizing for strategy resistance with IC may lead to more
real-world strategy vulnerability.
Complicating this is of course that don't even know how hardened a
method has to be. If it's as weak as Plurality, we know it leads to
Duverger. (I also think that IRV's center squeeze failure shows that it
isn't good enough either.) But would Minmax or Ranked Pairs lead to
stability and/or multi-party rule, or do single-member districts
inherently lead to concentration? It's hard to tell, though top two
runoffs in France (and Greek historical Approval use) seems to suggest
that not all single-winner methods produce two-party rule. And Debian's
use of Schulze seems to (weakly) suggest it's less likely to blowing up
in one's face than IRV. At least such explosions aren't severe enough to
lead to calls for a repeal.
If everything single-member does produce k-party rule then we don't
really need to care about strategic situations involving more than k
viable candidates. If honest Condorcet cycles are not just rare today
(obviously they're rare with two-party rule), but will always be rare,
then there's not much need to care about VSE for multi-Smith set
situations. Etc...
But we don't know.
-km