These points will be clarified below.
Note that people have in practice rejected some of Arrow’s properties. Even those scholars who seem to accept the general claim AGV, accept, a fortiori, the implied inconsistency, and thus in practice drop some assumptions to cope with the real world. Unfortunately, however, the literature has not converged to some agreement on which properties are best to drop. The position of this paper will be to forward the proposition that the Arrow axiom of Pairwise Decision Making (formerly known as the Independence of Irrelevant Alternatives) is the culprit to kill. It is a bad axiom for rational collective decision making, since it appears to be incongruent with that very notion itself.
In the following we develop the concepts, give a short proof and discussion of Arrow’s Theorem, construct the argument against the claims, reappraise the literature, and conclude.
Basic concepts
Please note that we will have to redefine some symbols for this chapter only.
Let X be the commodity domain. An element in the commodity domain can be called an item or a candidate. An agent is a compound of various properties such as utility, wealth etcetera. Let S be the set of possible compounds on X. With n agents, our interest concerns the function c: Sn
S. which maps the society into an aggregate compound. This is generally called the ‘Arrow type of social welfare function’ or simply a constitution.
A constitution differs from the ‘Bergson-Samuelson type of social welfare function’ (SWF) - and the latter is defined directly over X as SWF: X