This paper studies a decision-making system in which a situation has its numerical consequences with the natural order as the preference relation of a decision-maker. A rather wide class of situations is considered in which the decision-maker can use the criterion of the mentioned type under some rather natural conditions based on the principle of guaranteed result, which depends only on a regularity that describes randomness in a general sense, i.e., the regularity of a mass phenomenon representing a state of nature. Keywords: statistical regularity, scheme of a situation, preference choice rule.The present article continues the investigations performed in [1,2]. The objective of these investigations is a generalization of the results of analysis of the "general decision problem" that are obtained in [3].In a context Q , we denote by B 0 ( ) Q or simply by B 0 the set of all finite-valued S-measurable functions on Q , i.e., B f B C a r df 0 ( ) ( ): ( ) Q Q Q = Î <¥ def { } , and by B a b 0 ( , ), where a b , Î R and ( ), -> a b 0, the set of all finite-valued S-measurable functions on Q with their values in the interval ( , ) a b , B a b f f B f a b 0 0 ( , ) : : ( ) ( , )Let L be an arbitrary convex set of bounded S-measurable functions on Q that are denoted by B and are such that some a b , Î R can be found for which the set B a b 0 ( , ) is contained in L,(1)Next we denote by V L ( )the class of all functionals u on L, i.e., u : L ® R, and by V L V L 0 ( ) ( ) Ì its subclass satisfying the conditions formulated below for any f f L 1 2 , Î .

A decision-making system consisting of a decision maker and a situation of decision making is considered. A parametric decision problem with financial incomes on which preferences obey a natural numerical order is investigated. A decision preference criterion that is the maximum of expected losses among distributions that constitute a statistical law describing the random nature of natural states is obtained. By this criterion, the class of decision makers is specified axiomatically.

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The topic under discussion is modeling the subject of a decision making system, a decision maker (DM), up to the information, which it relies upon while making a concrete decision, directing towards the purpose before it, i.e., choosing the best possible action. Clearly, the indicated model of a DM substantially depends on the object of decision making, the situation of decision making (SDM), which this DM roughly presents in the form of the so-called situation of decision problem (SDP) through discarding the decisions impossible (or not interesting) for itself as well as impossible (according to its view) consequences from the initial situation, thus obtaining the so-called scheme of the situation of decision problem (SSDP). The interchangeability of the obtained models is studied.A decision-making system under study [2] is considered as a pair: a decision maker (DM) and a situation of decision-making (SDM), which is the object of the decision-making system. As a result of a DM's action (of any nature), a consequence appears in the SDM (of any nature as well). By a decision we will mean the DM's choice in order to reveal an element from a set of possible actions.H. Raiffa told: "We shall find ourselves partially concerned with situations in which the cosequences of any action you may take are not certain, because events may intervene that you cannot control or predict with certainty and whose outcomes will inevitably affect your final condition." Nevertheless, "... an individual who is faced with a problem of choice under uncertainty should go about choosing a course of action that is consistent with his personal basic judgment and preferences" [4].In what follows, we will call the set of "events that you cannot control or predict with certainty" the set of values of unobservable parameter and denote it by Q, and if the set Q is known, the situation will be called parametric. The set Q is considered together with a fixed (arbitrary) algebra S of its subsets; we will call elements of this algebra random events for the space of the unobservable parameter Q. If the algebra S is not specified, it is assumed by default that S Q = 2 .Moreover, the set X of consequences possible for the DM is also considered together with a fixed (arbitrary) algebra X of its subsets. The algebra X either always appear in the context or by default X = 2 X . To attain the objective, the DM uses modeling of the decision-making system considered here. Let us introduce necessary definitions and concepts, beginning with the major concept of statistical regularity given in [1], which generalizes the concept of probability distribution. Definition 1. A statistical regularity on Q, where Q is an arbitrary set with a given algebra of subset S (if S is not specified, then by default S Q = 2 ), is any nonempty closed set P in the topology t( ) Q of the space PF p p ( ): ([ , ]) : ( ) Q Q S = Î = { 01 1, p C D p C p C D C D ( ) ( ) ( \ ) , È = + " ÎS} (1) of all additive probability measures on Q, being the trace of *-weak topology in B S Q ( ) con...