Utilities of the finite family of random variables

In this paper, we first represent addition of independent random variables by the symmetric product on the symmetric algebra S(V), and then develop two utilities of random vectors: 1) an additive utility of the sum of independent random variables (Proposition 1) deduced by a utility over vector space V (Theorem 2); and 2) a utility over a tensor space T/sup k/ (Theorem 3). The distinction between two utilities over V and over T/sup k/(V) is dependent on two (weaker and stronger) solvability conditions. The weaker one deduces a utility of random variables which are connected by a non-commutative operation. Finally, we give a "holistic" order to such random vectors that they are decomposed into two types of components represented by the convolution and the non-commutative operation.