A strong (Ross) characterization of multivariate risk aversion

Using the addition of uncorrelated noise as a natural definition of increasing risk for multivariate lotteries, I interpret risk aversion as the willingness to pay a (possibly random) vector premium in exchange for a reduction in multivariate risk. If no restriction is placed on the sign of any co-ordinate of the vector premium then (as was the case in Kihlstrom and Mirman's (1974) analysis) only pairs of expected utility maximizers with thesame ordinal preferences for outcomes can be ranked in terms of their aversion to increasing risk. However, if we restrict the premium to be a non-negative random variable then comparisons of aversion to increasing risk may be possible between expected utility maximizers withdistinct ordinal preferences for outcomes. The relationship between their utility functions is precisely the multi-dimensional analog of Ross's (1981)global condition forstrongly more risk averse.