The initial goal of this paper was to study how to characterize the behavior of an EU maximiser decision maler according to the sign of u[p], p-th derivative of his utility function. The answer is well-known for u', u'', has been studied for u''' (Menezes, Geiss ant Tressler, Kimball), u'''' (Kimball). In a general case, the answer to this question led us to introduce two categories of properties concerning the preferences of the decision maker. These properties that we call p-order downside-mindedness (increasing or decreasing) consist in a particular attention paid by the decision maker to lower values of prizes (in risk theory) or poorer individuals (in inequality theory). Roughly speaking, a p-downside-increasing-minded (pDIM) decision maker prefers, after a defined process that the lower value concerned be increased. Thus these properties are defined in a very general framework and are not restricted to utilitarian model. However we show that in utilitarian model, the sign of the p-th derivative of utility function is given if and only if the decision maker is p-downside-increasing-minded or p-downside-decreasing-minded for every p=1,2,....
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