Infinite Regress and Arrow's Theorem

Abstract

Alfred MacKay has presented an instructive argument for relationship between traditional problem of infinite regress and Arrow's Impossibility Theorem.' In order show relationship he has constructed his own proof of theorem. In this proof, problem of infinite regress is made explicit and pervasive, whereas in Arrow's revised proo f2 problem of infinite regress does not appear explicitly-indeed, show that it plays role at all requires derivation. Both proofs, although somewhat different in scope, essentially show that handful of apparently unobjectionable conditions on aggregations from individual preference orderings social preference orderings cannot simultaneously be met. That they cannot is not intuitive, is like (conceptual) action at distance.3 MacKay's task is to find way of understanding four conditions, way of thinking about them, which will help us see why they conflict. He seems think he has found the ... reason why.4 I think he is largely wrong, although he is interestingly wrong. MacKay's and Arrow's proofs are genuinely different, because Arrow's depends far less on problem of infinite regress. Indeed, it requires an infinite regress essentially rule out simple majority decision as an acceptable social choice rule in populations of odd numbers, and not generally otherwise, whereas MacKay's proof turns fundamentally on an infinite regress. Hence, although in MacKay's proof of (a slight variant of) Arrow's theorem theorem seems be a generalization of long-ago-recognized Voter's Paradox,5 in Arrow's own proof it does not. It is only Voter's Paradox, and not more abstruse and more general General Possibility Theorem (now commonly known as Arrow's Impossibility Theorem), which is an analogue of traditional problem of infinite regress. To make this point clear, it should suffice show how socially cyclic preferences enter into MacKay's and Arrow's proofs.