Aggregation Procedure for Cardinal Preferences: A Formulation and Proof of Samuelson's Impossibility Conjecture

SAMUELSON MADE THE CONJECTURE stated above in his 1967 paper 7. He also formalized there the axiom of independence of irrelevant alternatives for cardinal preferences, used here. Preference are cardinal if their representation by a numerical function is invariant under, and only under, positive linear transformations. One may think that the disregard for intensity of preferences, embedded in Arrow's treatment of profiles of ordinal rankings of alternatives, leads to the impossibility result. Samuelson's conjecture points out that this is not the way to refute the conclusions of Arrow's theorem. There is also interest per se in aggregation of cardinal preferences. Such preferences are usually considered as von Neumann-Morgenstern utility, i.e., numerical representation of preferences over lotteries 11. Since uncertainty is the rule and not the exception whenever decisions are involved, it is of some importance to obtain a social N-M utility over risky outcomes. Given such a utility, the society will be able to choose a best alternative among the several feasible risky actions (i.e., lotteries). However it is not necessary to restrict the interpretation of cardinal preferences to those induced by ordinal ranking over lotteries. One can think of cardinal preferences derived from comparisons between pairs of alternatives (as in an axiomatization of a regret relation). See Alt 1 for an early work of this kind. When working with cardinal preferences a continuity assumption is needed, in addition to unanimity and independence (see the example at the end of the next section). A standard reference for Arrow's theorem is the last chapter of his book 2. For a general discussion of aggregation of cardinal preferences, see ShapleyShubik 10. Some other impossibility results involving different notions of cardinal preferences appear in the works of Sen 9, DeMeyer-Plott 4, Schwartz 8, and Fishburn 5. A model dealing with aggregation of cardinal preferences into social cardinal prefereiLces, as here, is that of Harsanyi 6. However he is interested in 1 The work of the first author was done at Northwestern University and the work of the second author began at the University of Illinois in Urbana-Champaign and it was completed at the University of Minnesota in Minneapolis. Both authors are on leave from Tel-Aviv University. The authors wish to express their thanks to E. A. Pazner, M. A. Satterthwaite, J. Kelly and the referees for helpful comments. This research was partly supported by NSF Grant # SOC-75-05317.