To parametric decision problems with money income
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TO PARAMETRIC DECISION PROBLEMS WITH MONEY INCOME UDC 519.81
V. M. Mikhalevich
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., def
B 0 ( Q ) = { f Î B ( Q ): Card f ( Q ) < ¥ }, and by B 0 ( a, b ), 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 0 ( a , b ) : = { f Î R Q : f Î B 0 , f ( Q ) = ( a , b )}. 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 0 ( a, b ) is contained in L, B 0 ( a, b ) Í L = co L Í B .
(1)
Next we denote by V ( L ) the class of all functionals u on L, i.e., u : L ® R, and by V0 ( L ) Ì V ( L ) its subclass satisfying the conditions formulated below for any f1 , f 2 Î L . V1. If f1 ( q ) £ f 2 ( q ) "q Î Q , then u( f1 ) £ u( f 2 ) . V2. If a¢ , b¢ Î R , a¢ ³ 0 and f1 ( q ) = a¢f 2 ( q ) + b¢ " q Î Q, then u( f1 ) = a¢u( f 2 ) + b¢. 1 ö æ1 V3. The following inequality is fulfilled u( f1 ) + u( f 2 ) ³ 2u ç f1 + f 2 ÷ . 2 2 ø è LEMMA 1. Condition V2 is equivalent to the following condition. æ b ö V2'. If a, b1 Î R , a Î[ 0, 1) , ç 1 ÷ Î B 0 ( a, b ) , and f1 ( q ) = af 2 ( q ) + b1 " q Î Q, then u( f1 ) = a( f 2 ) + b1 . è 1- a ø Q Proof. Condition V2' trivially follows from condition V2. National University of Kyiv-Mohyla Academy, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 163–169, September–October 2011. Original article submitted September 7, 2010. 812
1060-0396/11/4705-0812
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2011 Springer Science+Business Media, Inc.
Let us show that condition V2' implies condition V2. In fact, the condition f1 ( q ) = af 2 ( q ) + b1 implies that b 1 f 2 ( q ) = f1 ( q ) - 1 . a a Then, by virtue of condition V2', we have b 1 u( f 2 ) = u( f1 ) - 1 . a a In view of the arbitrariness of a Î[ 0, 1) and b1 Î R , we have obtained that condition V2 is fulfilled for a¢ Î [ 0, 1) È (1, + ¥ ) . Let us make sure that condition V2 is also fulfilled when a¢ = 1. 1 1 1 1 Let f1 ( q ) = f 2 ( q ) + b¢; then we have f1 ( q ) = 2 éê f1 ( q )ùú + b¢ since if f Î L , then f =
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