# [EM] Arrow's Theorem.

Alex Small asmall at physics.ucsb.edu
Tue Jul 15 09:25:18 PDT 2003

```Eric Gorr said:
> If Approval fails Arrow's Theorem in any example of an Approval
> election, it simply fails Arrow's Theorem. Nothing more needs to be
> said.

A theorem says "Given these assumptions, this statement is true."  Arrow
assumes that the outcome is uniquely determined by a set of preference
ballots (usually without equal rankings allowed, ar at least with
non-binary preferences).  Given that assumption, as well as 3 or more
candidates, pareto, and non-dictatorship, Arrow proves that IIA is
impossible to satisfy.

Approval doesn't satisfy the assumption about an outcome uniquely
determined from preference ballots.  Therefore, we cannot assert from
Arrow's Theorem that Approval satisfies IIA.

As to whether Approval does or does not satisfy IIA, the answer all
depends on the assumptions you make concerning voter behavior and
strategy.  Some people say this is a bad thing:  Approval outcomes are
unpredictable.  Others says it's a good thing:  The dynamics of an
Approval election are determined entirely by voter behavior, not by
intrinsic mathematical constraints

So, here's how it works:

Approval is NOT covered by Arrow's Theorem.
Approval MIGHT satisfy IIA.

Alex

```