[EM] How close can we get to the IIAC

[Kristofer Munsterhjelm]

fsimmons at pcc.edu wrote:

> 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.  

A nitpick: Schulze doesn't satisfy Independence from Pareto-dominated 
Alternatives. Steve Eppley gives an example on his site:

1: A>D>E>C>B
5: A>D>E>B>C
3: A>B>D>E>C
2: B>A>D>E>C
2: B>D>E>C>A
6: C>B>A>D>E
4: C>A>B>D>E
5: D>E>C>A>B
2: D>B>E>C>A

D Pareto-dominates E. If E is removed, Schulze elects A, but if not, 
Schulze elects B.

If Schulze satisfied IPDA, the method would be even better than it is, 
but the difference may amount to nothing in the context of large public 
elections.

Schulze does satisfy local IIA, i.e. independence from alternatives not 
in the Smith set, by virtue of always electing from the Smith set.

> 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.

That's an interesting way to counter the IRV supporters; I hadn't 
thought of it, probably because I've considered the later proof of 
Arrow's (the one with unanimity) rather than the earlier one with 
monotonicity.

> 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.

If our only objective is to grow "independence from *" to as large a 
space as possible (Independence from Smith-dominated alternatives, IPDA, 
...), then we may have to let go of other objectives on the way. The 
very statement that "IIAC is out of the question" is really just a 
variant of that, where we say that IIAC demands too much. Therefore, 
there has to be some limit to when we'll no longer permit the tradeoff.

Where is that limit? The question depends on the relative value of 
criteria. For instance, say independence of non-Dutta candidates implies 
nonmonotonicity - I don't know if it does, but for the sake of the 
argument... then a Condorcet method that satisfies INDC would be neat, 
but I wouldn't be willing to give up monotonicity for it.

Pressed to reason why, I would likely say something that in public 
voting settings, the additional protection conferred by INDC over say 
independence of non-Smith or Pareto-dominated alternatives is slight, 
but the loss of monotonicity is great, for the latter means that the way 
the method determines whether a candidate is "good" is itself suspect.

> 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.

I think Woodall has stated a weak version of IIA, as well. Ah yes, here 
it is:

Weak-IIA. If x is elected, and one adds a new candidate y ahead of x on 
some of the ballots on which x was first preference (and nowhere else), 
then either x or y should be elected.

( http://www.mcdougall.org.uk/VM/ISSUE3/P5.HTM )