[EM] Arrow's Theorem

Tue May 3 13:45:44 PDT 2011

Commenting on the criterion of Independence from Irrelevant Alternatives, Forest Simmons 
wrote, "IIAC is a totally unreasonable requirement."  Kevin Venzke added, "IIA isn't compatible 
with Condorcet.  It's not compatible with much of anything.  I take that to be the point of Arrow:  
If you want IIA you have to do some drastic things."

While I agree with those comments, I think that one can--generously but usefully--say that 
Arrow's theorem makes a different point, namely:

No election system in which voters rank candidates can have all properties that seem democratic 
or appropriate.  For example, no election system in which voters rank candidates can guarantee 
both Condorcet compliance and Later-no-harm.

The explanation is that, "democratic" implies majority rule, and--when 3 or more voters rank 3 
or more candidates--each candidate may be ranked below at least one of the other candidates 
by a majority of the voters.

--Stephen H. Sosnick (5/03/11)

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> Today's Topics:
> 
>    1. Thoughts on Arrow's Theorem and the IIAC (fsimmons at pcc.edu)
>    2. Re: Thoughts on Arrow's Theorem and the IIAC (Kevin Venzke)
> 
> 
> ----------------------------------------------------------------------
> 
> Message: 1
> Date: Mon, 02 May 2011 20:38:36 +0000 (GMT)
> From: fsimmons at pcc.edu
> To: election-methods at lists.electorama.com
> Subject: [EM] Thoughts on Arrow's Theorem and the IIAC
> Message-ID: <e3d52211166a6.4dbf164c at pcc.edu>
> Content-Type: text/plain; charset=us-ascii
> 
> In liberal arts mathematics text books Arrow's impossibility theorem is
> usually
> quoted as saying that no election method can simultaneously satisfy (1)
> neutrality, (2) anonymity, (3) decisiveness (4) monotonicty (5) the
> majority
> criterion (6) the Condorcet Criterion, and (7) the Independence from
> Irrelevant
> Alternatives Criterion (the IIAC), as though all of these requirements
> were
> equally to blame for the incompatibility, when in reality conditions one
> through
> six are perfectly compatible with each other, but condition seven is not
> even
> compatible with the existence of a Condorcet cycle.
> 
> To see why the IIAC is not compatible with the existence of a Condorcet
> cycle,
> let M be any method that satisfies the IIAC.  We will show that the only
> kind of
> winner that there can be under M is a Condorcet Winner:
> 
> Let E be an arbitrary election that is decided by M.  Let X be the winner
> of
> election E according to M.  Let Y be any of the other candidates. 
> Eliminate all
> of the other candidates one by one until only X and Y remain.  According
> to the
> IIAC, the winner is not changed at any stage of the elimination, so X is
> still
> the winner according to M when the choice is between X and Y.  Since the
> choice
> of Y was arbitrary, we see that M makes its winner X defeat each of the
> other
> candidates head to head. 
> 
> Thus we see that the IIAC is a totally unreasonable requirement.  How
> would you
> like it if somebody asked you to do something that was logically
> impossible, and
> then complained that you were imperfect for not doing it?  It's like the
> philosopher that requires god to make an immoveable object and then to
> move it,
> because (in his opinion) a perfect being would have to be capable of both
> requirements.
> 
> On the other hand there are methods that satisfy requirements one through
> six
> along with other reasonable requirements in place of the IIAC, including
> (8)
> independence from clones, (9) independence from Pareto dominated
> alternatives,
> and (10) independence from non-Smith alternatives, simultaneously.
> 
> Woodall's incompatibility theorems for various combinations of his
> criteria are
> more interesting because they spread the blame around; it's not so easy
> to
> single out a single criterion as being unreasonable.
> 
> 
> ------------------------------
> 
> Message: 2
> Date: Tue, 3 May 2011 00:45:11 +0100 (BST)
> From: Kevin Venzke <stepjak at yahoo.fr>
> To: election-methods at electorama.com
> Subject: Re: [EM] Thoughts on Arrow's Theorem and the IIAC
> Message-ID: <66458.92421.qm at web29616.mail.ird.yahoo.com>
> Content-Type: text/plain; charset=iso-8859-1
> 
> Hi Forest,
> 
> --- En date de?: Lun 2.5.11, fsimmons at pcc.edu <fsimmons at pcc.edu> a
> ?crit?:
> > In liberal arts mathematics text
> > books Arrow's impossibility theorem is usually
> > quoted as saying that no election method can simultaneously
> > satisfy (1)
> > neutrality, (2) anonymity, (3) decisiveness (4) monotonicty
> > (5) the majority
> > criterion (6) the Condorcet Criterion, and (7) the
> > Independence from Irrelevant
> > Alternatives Criterion (the IIAC), as though all of these
> > requirements were
> > equally to blame for the incompatibility, 
> 
> That is the longest list I've ever seen. I'm used to seeing 4+5+6
> replaced by Pareto perhaps. Condorcet would already imply majority,
> unless this is supposed to be mutual majority.
> 
> > when in reality
> > conditions one through
> > six are perfectly compatible with each other, but condition
> > seven is not even
> > compatible with the existence of a Condorcet cycle.
> 
> Well, we would probably expect "all properties but one" to be compatible.
> Drop out #7 and you can have a method. But drop out #6 and you still
> don't have a method.
> 
> It's clear, as you write, that IIA isn't compatible with Condorcet. It's
> not compatible with much of anything. I take that to be the point of
> Arrow: If you want IIA you have to do some drastic things. If I could
> have IIA (and for real, not on a technicality) I would certainly want 
> it...
> 
> Kevin
> 
> 
> 
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> End of Election-Methods Digest, Vol 83, Issue 2
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