Some classes of preference choice rules for decision-making problems
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SOME CLASSES OF PREFERENCE CHOICE RULES FOR DECISION-MAKING PROBLEMS UDC 519.81
V. M. Mikhalevich
Abstract. 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. Keywords: statistical regularity, situation of decision problem, unobservable parameter, preference choice rule, model, guaranteed result principle.
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 = 2Q . 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 = 2Q
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