Sen (1999a:250-253) contains a short summary discussion on his view on the Theorem. First I quote him and then give my comment. Sen states:
“The Arrow Theorem does not in fact show what the popular interpretation frequently takes it to show. It establishes, in effect, not the impossibility of rational choice, but the impossibility that arises when we try to base social choice on a limited class of information.”
This is not correct. Using the information provided by pairwise voting results, we can decide to a tie (deadlock, indifference) when such might arise. It is the adoption of the APDM axiom that, wickedly, turns this indifference into an inconsistency. The APDM does not mean lack of information, it only corrupts the information that exists.
“At the risk of oversimplification, let me briefly consider one way of seeing the Arrow theorem. Take the old example of the “voting paradox,” with which eighteenth-century French mathematicians such as Condorcet and Jean-Charles de Borda were much concerned. If person 1 prefers option x to option y and y to z, while person 2 prefers y to z and z to x, and person 3 prefers z to x and x to y, then we do know that the majority rule would lead to inconsistencies. In particular, x has a majority over y, which has a majority over z, which in turn enjoys a majority over x. Arrow’s theorem shows, among other insights it offers, that not just the majority rule, but all mechanisms of decision making that rely on the same informational base (to wit, only individual orderings of the relevant alternatives) would lead to some inconsistency or infelicity, unless we simply go for the dictatorial solution of making one person’s preference ranking rule the roost.”
Locating the problem in the informational base is erroneous. Clearly, majority decision does not lead to inconsistencies, for it is the use of the APDM axiom that does so - and we don’t need it for majority decisions. The Arrow Theorem does not show that there are inconsistencies for all mechanisms - we namely can use mechanisms without APDM.
“This is an extraordinarily impressive and elegant theorem — one of the most beautiful analytical results in the field of social science. But it does not at all rule out decision mechanisms that use more — or different — informational bases than voting rules do. In taking a social decision on economic matters, it would be natural for us to consider other types of information.”
I don’t know about “extraordinarily impressive and elegant”. Condorcet came up with his paradox, as earlier people came up with paradoxes when dividing by zero, as Bertrand Russell had his set-paradox, and as the Cretian Epimenides said “All Cretians are liars.” Arrow’s Theorem solves the Condorcet paradox by showing that we must not use APDM - though Arrow apparently did not realise that. The theorem is basic, and we must be glad that we have it, as APDM apparently can cause a lot of confusion, as the last 50 years have shown.
“Indeed, a majority rule — whether or not consistent — would be a nonstarter as a mechanism for resolving economic disputes. Consider the case of dividing a cake among three persons, called (not very imaginatively) 1, 2, and 3, with the assumption that each person votes to maximize only her own share of the cake. (This assumption simplifies the example, but nothing fundamental depends on it, and it can be replaced by other types of preferences.) Take any division of the cake among the three. We can always bring about a “majority improvement” by taking a part of any one person’s share (let us say, person 1’s share), and then dividing it between the other two (viz., 2 and 3). This way of “improving” the social outcome would work — given that the social judgment is by majority rule — even if the person thus victimized (viz., 1) happens to be the poorest of the three. Indeed, we can continue taking away more and more of the share of the poorest person and dividing the loot between the richer two—all the time making a majority improvement. This process of “improvement” can go on until the poorest has no cake left to be taken away. What a wonderful chain, in the majoritarian perspective, of social betterment!”
Remember that Sen writes this book for a general audience of economists who will not have gone deeper in social choice theory. Though Sen now relates basic truisms, his reasoning nevertheless is a bit off. Indeed, Western democracies tend to have property rights and a “status quo” rule, and a Madisonian philosophy that democracy actually exists to protect the minorities. We use all kinds of additional information, in order to settle problems of fairness and equity. Thus the majority rule is not suggested for the raw form that Sen uses as an example. Then, crucially, when Sen suggests that this example clarifies that we must use more information to solve the Arrow paradox, then this is a non-sequitur. His argument becomes seductive, since the reader is seduced into thinking that, indeed, we use more information. But the truth is that we use this additional information to solve equity matters, and not to solve the Arrow inconsistency.
“Rules of this kind build on an informational base consisting only of the preference rankings of the persons, without any notice being taken of who is poorer than whom, or who gains (and who loses) how much from shifts in income, or any other information (such as how the respective persons happened to earn the particular shares they have). The informational base for this class of rules, of which the majority decision procedure is a prominent example, is thus extremely limited, and it is clearly quite inadequate for making informed judgments about welfare economic problems. This is not primarily because it leads to inconsistency (as generalized in the Arrow theorem), but because we cannot really make social judgments with so little information.