In liberal arts mathematics text books Arrow's impossibility theorem is usually quoted as saying that no election method can simultaneously satisfy (1) neutrality, (2) anonymity, (3) decisiveness (4) monotonicty (5) the majority criterion (6) the Condorcet Criterion, and (7) the Independence from Irrelevant Alternatives Criterion (the IIAC), as though all of these requirements were equally to blame for the incompatibility, when in reality conditions one through six are perfectly compatible with each other, but condition seven is not even compatible with the existence of a Condorcet cycle.
To see why the IIAC is not compatible with the existence of a Condorcet cycle, let M be any method that satisfies the IIAC. We will show that the only kind of winner that there can be under M is a Condorcet Winner: Let E be an arbitrary election that is decided by M. Let X be the winner of election E according to M. Let Y be any of the other candidates. Eliminate all of the other candidates one by one until only X and Y remain. According to the IIAC, the winner is not changed at any stage of the elimination, so X is still the winner according to M when the choice is between X and Y. Since the choice of Y was arbitrary, we see that M makes its winner X defeat each of the other candidates head to head. Thus we see that the IIAC is a totally unreasonable requirement. How would you like it if somebody asked you to do something that was logically impossible, and then complained that you were imperfect for not doing it? It's like the philosopher that requires god to make an immoveable object and then to move it, because (in his opinion) a perfect being would have to be capable of both requirements. On the other hand there are methods that satisfy requirements one through six along with other reasonable requirements in place of the IIAC, including (8) independence from clones, (9) independence from Pareto dominated alternatives, and (10) independence from non-Smith alternatives, simultaneously. Woodall's incompatibility theorems for various combinations of his criteria are more interesting because they spread the blame around; it's not so easy to single out a single criterion as being unreasonable. ---- Election-Methods mailing list - see http://electorama.com/em for list info
