# [EM] Does the 'Independence of Irrelevant Alternatives Criterion' Imply a

Forest Simmons fsimmons at pcc.edu
Fri Apr 2 11:48:06 PST 2004

```On Fri, 2 Apr 2004, Adam Tarr wrote:

>
> >         I have to say that I don't think it makes sense for an individual to
> >prefer A to B, B to C, and C to A. It's just logically contradictory.
> >Individual preferences should be assumed to be transitive.
>
> I've argued the same thing in the past, but ultimately the same argument
> can be made without appealing to a single person's intransitivity.  For
> example:
>
> Say there is an A>B>C faction, a B>C>A faction, and a C>A>B faction.  No
> faction is a majority, or of exactly equal size to another faction.
>
> Assume that the election method in question can come up with SOME
> result.  (If the election can't come up with a result, it's not of much
> use.)  Without loss of generality, assume A wins.
>
> Now, imagine the same election without candidate B.  A majority prefer C to
> A, and they are the only two candidates, so any rational election method
> will elect C.
>
> Now add B back in.  A wins.  Therefore, IIA has been violated.
>
> This is not a rigorous proof, since I did not provide a rigorous
> justification why C should win the pairwise contest (although it is
> obvious).  But this example suffices to show, in my opinion, that no
> reasonable method will ever pass IIA.
>

This Condorcet cycle argument is the same argument I made in my original
reply to Marcos Ribeiro, and which he quoted in his 30 March posting to
the list.

Since Rob Speer was referring to that posting, I presumed that he had
access to that argument, which is why I supplied the shorter, simpler,
individual chooser argument that should be easier to understand.

The shorter argument also has the advantage of showing that in the absence
of the transitive preference axiom, even a dictatorial method cannot
satisfy the IIAC in a meaningful way.

Richard's example is the one we should be following: finding useful ways
to relax the impossibly strict IIAC.

One other small point: Rob Speer claimed that a definition cannot be
false.  Yet in mathematics when defining an operation, it is essential to
prove that the operation is "well defined."  If it turns out that the
operation is not well defined, then it must either be fixed or else

Typically (in mathematics) a false definition doesn't work out because the
definition depended too heavily on the particular representation the
proposer of the definition had in mind of the objects being operated on.

Suppose that I want to define the "size" of a rational number as the sum
of the absolute values of its numerator and denominator.

What is the size of the rational number 21/84?  Well, 21/84 = 1/4, so is
the size 21+84 ? or is the size 1+4 ?

To fix our definition (i.e. to make "size" well defined) we have to
specify which representation of the rational number we are going to use.
The most natural would be the reduced form.

So in the case of the IIAC. What representation of an electorate's
preferences did Arrow have in mind?  More importantly, what representation
of an electorate's preferences would make some relaxed version of the IIAC
useful?

Forest

```