The most complex property seems to be good old rationality. It appears that we better introduce the information set or knowledge base I(.) and state the condition that it must contain the Arrow Theorem. Then:
ARa agents are rational (they accept logic, [106] have a preference ordering, are morally consistent (DA), and are educated on Arrow’s Theorem (I(~a)))
The I(~a) condition is a novel aspect, that, however, should not come as a surprise, given what we said in the introduction. There is a difference between a learning process and a result. In a common classroom or used-car-salesman strategy, people are goaded into buying some axioms as reasonable and attractive, and then burn themselves, which teaches them. This may be called rational from the viewpoint of learning. This paper however concentrates on the after-learning-rationality, the kind of rationality that makes learning so worthwhile.
How does Arrow’s original approach relate to the inclusion of I(~a) ? Arrow (1950, 1951, 1963) has no incorporation of learning - though he later has written on ‘learning by doing’ - so it might be that he assumes standard economic rationality. If that would be perfect foresight, then I(~a) is implied. However, it is better to hold that Arrow in that period discussed constitutional choice for agents and not by agents. The choice for people then is made by some algorithm or calculating machine. His axioms do not describe educated people involved in constitutional choice. Alternatively put, another new result in this chapter is the widening of the scopes of utility and rationality to the inclusion of knowledge about the constitutional process itself. In that sense the original Arrowian axioms can be called incomplete. Alternatively, if the idea is that these axioms concern educated people, then there is a hidden inconsistency, in that reasonable agents are assumed to regard inconsistent axioms as reasonable. [107]
Hence:
ARC = ARe & ARa
Restatement of Arrow’s Theorem
It appears very useful to discuss the example given by the Marquis de Condorcet 1785. Sen (1970) gives a simple example that appears to be presented first by Nanson 1882. A similar example is reproduced in Table 12, and I will refer to it as “the Condorcet case”. There are three parties and three topics A, B and C on ballot, and the numbers of seats and the preferences are such that, with pairwise voting and a majority rule, a cycle results: A < B < C < A.