Cancellation Conditions for Multiattribute Preferences on Finite Sets

Applications of decision theory for multiple criteria or multiple attributes often assume a Utility representation for preferences that is additive over criteria or attributes. Axiomatic theories for additive Utilities are well developed but are not without gaps. A case in point arises with finite sets of alternatives, where two preference axioms are necessary and sufficient for additive Utilities. One is weak ordering. The other is a cancellation ax-iom that consists of an infinite scheme of cancellation conditions, one for each positive integer K ≥ 2. It is known that the infinite scheme can be truncated to a finite scheme for K ≤ K* that depends on the size of the set of alternatives, but very little is known about the value of K* which ensures additivity for all finite sets of that size. The present paper contributes to the determination of K*. A fundamental result is that if there are m attributes, the jth of which has n_j elements in its attribute set, then Σn_j - (m - 1) is an upper bound on K*. Lower bounds on K* that are near to this upper bound are obtained for special cases of (n₁, n₂, ..., n_m.