Nondictatorial & Nonmanipulable axioms (was Re: York's
Bruce Anderson
landerso at ida.org
Sat May 11 04:16:35 PDT 1996
On May 10, 5:37am, Steve Eppley wrote:
> Subject: Re: Nondictatorial & Nonmanipulable axioms (was Re: York's
> Bruce Anderson wrote:
> >Steve Eppley wrote:
> >> My understanding is that axiom A4 (below) limits the voting method
> >> to using only ranking info from the voters the way Condorcet does,
> >> ignoring any rating (a.k.a. weighting, a.k.a. intensity) info.
> >
> >No, it's A2 that does this.
>
> Here's A2 again (slightly restated to eliminate the reference to A1):
>
> A2. (Unrestricted domain) Every individual preference relation
> that satisfies the following two assumptions is admissible.
> (Completeness) For every pair of outcomes o1 and o2,
> either o1 is liked at least as much as o2, or o2
> is liked at least as much as o1.
>
> (Transitivity) For any three outcomes o1, o2, and o3,
> if o1 is liked at least as much as o2,
> and o2 is liked at least as much as o3,
> then o1 is liked at least as much as o3.
>
> You wrote that A2 is what limits the tally algorithm to using
> ranking info only (prohibits more detailed voter ratings of the
> choices). I don't see this.
It's the requirement that the domains of social choice functions and of social
preference functions consist of the preference relations of individuals, and A2
is the closest I could get to that from the portion of Ordeshook that you
quoted.
> If I vote {A=100, B=60, C=0}, for example, then A >= B, B >= C, and
> A >= C. So the first assumption appears to be satisfied.
>
> And the weighted ballot appears to satisfy the transitivity
> assumption as well: A >= B, B >= C, and A >= C.
>
> How could a weighted ballot with a typical tally algorithm (like
> simply summing preferences) fail either of these two assumptions?
Weights do indeed induce allowable preference relations, which is (in my
opinion) why postulating weighted voting methods doesn't really get around the
conundrums raised by Arrow and Gibbard-Satterthwaite. However, as (I think)
these theorems are usually stated, it's the voter's preference relations that
the voting methods are assumed to act upon, not any weights that give rise to
these preference relations.
> >> And that the purpose of this limitation is to prevent voters from
> >> manipulating the outcome by strategically misrepresenting their
> >> ratings.
> >
> >That's Gibbard-Satterthwaite, not Arrow.
>
> But here's the paragraph in Ordeshook which follows A4:
>
> "Put simply, A4 requires that if the set of feasible outcomes is
> restricted to the pair x and y, if x and y are the only two outcomes
> that the group can consider, then the social preference between x
> and y depends only on individual preferences over {x,y} and not on
> individual preferences over any larger set, including sets of
> outcomes that have not even been offered to the public for
> evaluation. In effect, this condition prohibits expressions of
> intensity of preferences over the set {x,y} by referring to some
> alternative not under consideration. Although this might not seem
> distasteful, outcomes can be manipulated, as we argue later, by
> persons strategically misrepresenting their true intensities."
>
> I'm not fluent in this terminology. Doesn't the prohibition on
> expressing intensities imply ranked ballots only? And doesn't
> Ordeshook clearly say that removal of this Arrow axiom would allow
> manipulation by strategic misrepresentation of intensities?
>
>-- End of excerpt from Steve Eppley
The key is: "A4 requires that if x and y are the only two outcomes
> that the group can consider ... this condition prohibits expressions of
> intensity of preferences over the set {x,y} by referring to some
> alternative not under consideration.
In other words -- in an election involving x, y, u, v, and w -- x can't finish
ahead of y just because (say) x beat each of u, v, and w, pairwise, by 90% to
10%, while y lost to each of u, v, and w, pairwise, by 10% to 90%; it can only
depend on how x and y fare against each other pairwise, because u, v, and w
could be declared ineligible.
It's like saying, say, in PAC-10 football: suppose each team were to play each
other team once, and suppose UCLA finishes 8-1, beating every other team but
USC; and USC finishes 7-2, losing only to Arizona and ASU; and every other team
finishes 5-4 or worse. Then Arrow would say that the PAC-10 champion cannot
depend on whether or not Arizona and ASU are found to have used ineligible
players and, as a result (say), their games are not counted towards determining
the championship. It's absurd. Either those games count, and UCLA is the
champion; or the games don't count (i.e., they are forfeited), and USC is the
champion. But no one familiar with sports would ever argue that there should be
the same winner (UCLA or USC) whether or not the Arizona and ASU games count in
the standings. Yet that's what A4 requires.
Basically, the theorem depends on the axioms, not on Ordeshook's, Arrow's, or
anyone else's interpretation of the axioms. And different interpretations can
cast quite different lights on the reasonability of A4. Yes, A4 precludes
dependence on truely irrelevant alternatives; but it also precludes dependence
on Smith winners as well -- and the latter, not the former, is why it can't be
satisfied by reasonable voting methods.
Bruce
More information about the Election-Methods
mailing list