To parametric decision problems with money income

Abstract

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 .

Two criteria in multiple decision making

“…Therefore, according to that proved for the functional u ¥ , conditions V1 and V2 are satisfied on B 0 . Moreover, condition V3¢ is satisfied for the functional u ¥ (see [5]), which can be proved similarly to the respective statement in Theorem 1 from [4].…”

“…Therefore, according to Theorem 1 from [5], there exists a statistical regularity P P Î ( ) Q such that for any f B Î 0…”

“…Therefore, the continuation of u onto B also has the form (14) since conditions V1, V2, and V3 ' are satisfied on B for such a functional in view of Theorem 1 [5]. Therefore, for any u U We denote by w the contraction of the utility function w P X 0 ( ) onto the set X Y P X Ì Ì 0 ( )and in view of the affinity of the function w P X 0 ( ) we obtain from the last ratio …”

“…In view of the condition of positive homogeneity, we denote the extension of the functional u onto all B 0 by u as well. The property V1 of the monotonicity of the functional u on B 0 remains, and property V 2 holds for it (see [5]). …”

“…In view of Theorem 2 from [5], there exists a unique continuation of the functional u on B, for which conditions V1, V2, and V3 ' are satisfied. Therefore, the continuation of u onto B also has the form (14) since conditions V1, V2, and V3 ' are satisfied on B for such a functional in view of Theorem 1 [5].…”