Stochastic Dominance Rules for Multi-attribute Utility Functions

Abstract

Several attempts have been made recently to extend the general framework of stochastic dominance from the single variable to the multivariate case. Levy (1973) developed sufficient rules for firstand second-degree dominance when utility functions are defined on terminal wealth and there is independence among outcomes in different periods. Levy and Paroush (1974a, 1974b) extended these results for rules of first degree dominance omitting the requirement of inter-period independence of outcomes. In addition they developed necessary and sufficient rules of first-degree dominance for additive utility functions. Huang et al. (1978) extended all the results in Levy (1973) and the results obtained for additive utility functions in Levy and Paroush (1974b) to all important classes of stochastic dominance. Levhari et al. (1975) provide sufficient and necessary conditions for the general case of monotone increasing (first-degree dominance) and quasi-concave utility functions. Unfortunately, the conditions developed are difficult to verify. In this paper we investigate rules of stochastic dominance for multi-attribute utility functions. After establishing (by means of Theorem 1) the equivalence of rules for multivariate utility functions u = u(x1, ..., x) and univariate utility functions defined on multivariate outcome space u = us(P(xl, ..., x)), we focus upon the latter.