Parameterization of the Lottery Model of Nonparametric Decision-Making Situation
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PARAMETERIZATION OF THE LOTTERY MODEL OF NONPARAMETRIC DECISION-MAKING SITUATION V. I. Ivanenko,a† O. V. Kuts,a‡ and I. O. Pasichnichenkob
UDC 519.71:330.46
Abstract. The paper focuses on the parametric description of a nonparametric decision-making situation, i.e., where it is impossible to reveal the objective parameter determining the consequences of decisions. For the case of strict uncertainty, the classes of matrix schemes containing those and only those schemes that can be used to model certain nonparametric situation are described and the formula for class cardinality is proved. The cases are established where there are grounds to choose the matrix scheme with the smallest, in its class, cardinality of the set of values of the parameter. Keywords: strict uncertainty, decision making, lottery scheme, matrix scheme, parameterization.
By a decision system (similarly to the term “control system” in control theory) we will mean the structure that occurs always where decision-maker (DM) appears in a situation that requires decision-making (SDM)1, i.e., choosing a unique decision from a setU. Any decision u ÎU in decision system yields a unique indefinite consequence c from a set of possible ones C u Í C. Two types of SDM are distinguished: parametric and nonparametric. In the case of a parametric situation, it is possible to determine an “objective” parameter q Î Q that stipulates the consequence together with the decision; in the case of a nonparametric situation, this is impossible. Noncooperative game is an example of a parametric SDM; choice among lotteries is a nonparametric SDM. Two types of SDM correspond to two types of models of these situations: matrix and lottery. Lottery model has the form S l = ( Z l , I l ), where Z l is lottery scheme of the SDM. By a lottery scheme we mean triple Z l = (U , C , y ( . )) , where y : U ® 2C \ {Æ }, y( u ) = C u , is the structure of cause-and-effect relationships in the SDM in the form of a many-valued mapping, I l is information about the cause-and-effect mechanism of the generation of consequences according to the lottery model. Matrix model has the form S m = ( Z m , I m ), where Z m is matrix scheme of the SDM. By a matrix scheme we mean the quadruple Z m = ( Q, U , C , g (. , . )) , where g : Q´ U ® C is the structure of cause-and-effect relationships in the SDM in the form of a single-valued mapping, I m is information about the cause-and-effect mechanism of the generation of consequences according to the matrix model. It is natural to use lottery model for the analysis of a nonparametric situation and matrix model for the analysis of a parametric one. The construction of a matrix model equivalent to a lottery model of a nonparametric situation, and of a lottery model equivalent to a matrix model of a parametric situation, is described in [1, 3]. Though nonparametric situations occur more often in practice, matrix model is mainly considered in theoretical studies after the publication by Savage [4] (for example, [5–8]), especially in connection with subject
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