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. ---- Election-Methods mailing list - see http://electorama.com/em for list info
