[EM] How close can we get to the IIAC

[fsimmons at pcc.edu]

Arrow's Theorem is grossly misunderstood, because people have the mistaken
impression that the Independence from Irrelevant Alternatives Criterion (IIAC)
is on a par with the other criteria he mentions in his theorem.

To clear this up, let's consider the following theorem which is the essence of
Arrow's Theorem:

The Independence from Irrelevant Alternatives Criterion is incompatible with any
decisive method that satisfies the Majority Criterion in the two candidate case:

Proof:  

Suppose by way of contradiction that we have a decisive method that satisfies
both the IIAC and (in the two candidate case) the Majority Criterion.  Consider
a three candidate pairwise majority beat cycle in which  A beats B beats C beats
A, which is not a perfect three way tie, i.e. which lacks the symmetry that
would demand a three way tie even from a decisive method.   

Since the method is decisive there must be a winner.  Without loss in generality
suppose this winner to be A.  Since we assume the IIAC is satisfied, removing a
loser (B for example) cannot change the winner.  So between A and the remaining
loser C, candidate  A must still win.  But this contradicts the step "C beats A"
in the majority pairwise beat cycle.  This contradiction shows that the given
conditions are incompatible.

Note that "two candidate majority win" is extremely weak.  All reasonable
deterministic methods satisfy it, even Borda.  Any deterministic method that is
monotone in the two candidate case satisfies it.  So no reasonable, decisive,
deterministic method can satisfy the IIAC.  Therefore contrary to naive
expectations, the IIAC is not a reasonable requirement after all.

This is the real way to interpret Arrow's theorem, contrary to the popular
paraphrase "no voting method can satisfy all of the reasonable requirements of a
voting method."  The statement in quotes may be true, but it is does not
accurately paraphrase Arrow's Theorem.

Schulze's CSSD (Beatpath) method does not satisfy the IIAC, but it does satisfy
all of Arrow's other criteria, that is to say all of the reasonable ones plus
some others like Clone Independence, Independence from Pareto Dominated
Alternatives, etc.  We cannot hold the IIAC against Schulze, because no
reasonable method satisfies the IIAC.  

In short it is possible to satisfy all of Arrow's criteria simultaneously except
the IIAC.  So you cannot use Arrow's Theorem to excuse the lack of any of his
criteria, except the IIAC.  If you could trade in a couple of the other criteria
for the IIAC, it might be worth it, but you cannot make this trade, unless you
are willing to sacrifice either decisiveness or two candidate majority (not to
mention Condorcet and Monotonicity which both separately imply the two candidate
majority condition).

So IRV supporters cannot rationally blame Arrow for IRV's lack of compliances,
since its compliances are not maximal within Arrow's set of criteria.  They must
resort to Benham or Woodall to explain that IRV has maximal compliance within
some other set of criteria, not Arrow's.

It is kind of like saying that the three real algebraic equations  x+y+z=5,
x-y+2z=7, and z+12=z are incompatible.  It is true that they are incompatible,
but only because z+12 cannot equal z in the field of real numbers.  The third
equation is definitely to blame for the trouble, though it does make sense in
mod twelve clock arithmetic.

Since the IIAC is out of the question, how close can we get to the IIAC? 
Independence from Pareto Dominated Alternatives (IPDA) is one tiny step in that
direction.  Another step might be independence from alternatives that are not in
the Smith set.

A couple of years ago someone proposed that if adding a candidate changed the
winner, the new winner should  be either the new candidate or someone that beats
the new candidate pairwise.

In my next message I will consider the IIAC approximation problem from another
point of view.