Modeling Of Decision Maker under Imperfect Information

In real-life, imperfect information is commonly present in all the components of the decision making problem. In decision making problems a DM is almost never provided with perfect, that is ideal decision relevant information to determine states of nature, outcomes, probabilities, etc. We are known that, relevant information almost always comes imperfect. Imperfect information is information which in one or more respects is imprecise, uncertain, incomplete, unreliable, vague or partially true [1]. Imprecision is one of the widest concepts including variety of cases. We will discuss uncertainty concepts of imperfect information and it application to problem modeling of decision maker. In the first stage of the modeling the identification determinants of a decision maker was implemented using Delphi method. The aim of the second stage consists of the linguistic evaluation of the factors. At the final stages decision makers model was realised by using possibilityprobability based method and Dempster-Shafer theory based model.


I. INTRODUCTION
Two main concepts of imperfect information are imprecision and uncertainty [1,2]. For purposes of differentiation between imprecision and uncertainty, Prof. L.A. Zadeh suggested the following example: "For purposes of differentiation it is convenient to use an example which involves ethnicity. Assume that Robert's father is German and his mother's parents were German and French. Thus, Robert is 3/4 German and 1/4 French. Suppose that someone asks me: What is Robert's ethnicity. If my answer is: Robert is German, my answer is imprecise or, equivalently, partially true. More specifically, the truth value of my answer is 3/4. No uncertainty is involved. Next, assume that Robert is either German or French, and that I am uncertain about his ethnicity. Based on whatever information I have, my perception of the likelihood that Robert is German is 3/4. In this case, 3/4 is my subjective probability that Robert is German. No partiality of truth is involved. No partiality of truth is involved. A proposition is a carrier of information. In the above example, call it the Robert example, the information carried by the proposition "Robert is German" is precise but not entirely correct, that is, is partially true. When imprecision is related to partiality of truth with no uncertainty involved, it will be referred to as strict imprecision, or s-imprecision for short. As was noted already, there is no connection between s-imprecision and uncertainty." In this work we will discuss modelling of decision maker under imperfect information. Making decisions is certainly the most important task of a manager and it is often a very difficult one. It depends on two factors: the statement of the decision making problem and the determinants of a decision maker. Decisions are an inevitable part of human activities. It requires the right attitude. Every problem properly perceived, becomes an opportunity. In most cases the decision maker must view the problems as opportunities rather than solving problems. A pessimist sees the difficulty in every opportunity, an optimist sees the opportunity in every difficulty. It all depends on the decision-maker's attitude. Decision maker looks at problems using reaction and emotion. Decision making depends on a character of a decision maker. This requires including different behavioral characteristics of decision maker into decision making model. The analysis of the existing works [3][4][5][6][7][8][9][10][11][12][13][14][15] of the field modeling of decision maker shows that emotion, altruism, reciprocity, fairness, social responsibility and etc. are basic attributes of human behavior. Authors of works [3][4][5][6][7] developed theory of reciprocal altruism for games behavior. Decision maker modeling under second-order uncertainty using the method based on the possibility-probability measure is discussed in work [16,19.20].
[21] is devoted to the problems of decision making in fuzzy environment for management systems of oil refinery enterprise (ORE), namely, to the problems of development of new methods and tools which allow to take into account uncertainty of environment and a decision maker's behavior (DM). Two main factors characterized by uncertainty and influencing decision making in management systems of ORE are determined and necessity of development of new methods of decision making under second-order uncertainty is argued. It is conducted an analysis of psychological determinants of a DM which influence decisions made under uncertainty with the latter intrinsic for both a DM's behavior and a decision making environment. On the base of the suggested analysis, DM's behavior has been modeled by using of fuzzy measure, possibility measure and belief measure. Human behavioral modeling based on the Dempster -Shafer theory of belief and fuzzy logic is suggested by Yager in [17]. In [17] authors considers the appropriateness of fuzzy sets and fuzzy production rules for representing human centered cognitive concepts. It is noted that production systems is one of the oldest techniques of knowledge representation. Human behavioral modeling requires an ability to formally represent experienced informative or cognitive concepts that are often at best described in imprecise linguistic terms. It is shown in [17] how probabilistic uncertainty can be included into the output of a fuzzy rule by using Dempster-Shafer paradigm. This methodology that combines fuzzy and probabilistic uncertainty provides a framework for creation of models that can include both the concepts and unpredictability needed to model human behavior. A tractable model of reciprocity and fairness is discussed in [6]. The income distribution and the kindness or unkindness of others' choices systematically affect a person's emotional state. The emotional state systematically affects the marginal rate of substitution between own and others' payoffs, and thus the person's subsequent choices. The proposed model is applied to two sets of laboratory data: simple binary choice mini-ultimatum games, and Stackelberg duopoly games with a range of choices. The results confirm that other-regarding preferences respond to others' intentions as well as to the income distribution. Different approaches to decision making problems to introducing trust, reciprocity, altruism, fairness etc were proposed by James C.Cox and his colleagues and other authors [22][23][24][25][26][27][28][29][30]. The existing approaches don't deal with possibilistic and probabilistic uncertainty which is characterizing decision makers behavior. In this paper we try to do modeling of decision maker by using emotion and altruism factors. The rest of the paper is organized as follows: In section 2 the process of determining decision makers attributes and a statement of the problem are given. In section 3 modeling process is shortly described under second-order uncertainty using the possibilityprobability measure based method. In section 4 we create the decision makers model by using obtained data . Decision maker behavioral modelling using fuzzy and Dempster-Shafer theories suggested in 5. Section 6 is conclusion.

II. STATEMENT OF THE PROBLEM
The basic problem is to evaluate personal quality of a decision maker by using psychological determinants. For determining psychological determinants as basic factors influencing a choice of a decision maker we use the Delphi method. For determining basic factors of a decision maker the following questionnaire is created:

Query 1.
Please indicate by "+" which of the following should be considered as determinants of a decision maker (see in table 1).

Query 2. Identification of total index of a DM.
Please indicate what term should be used for a total index (resulting dimension) of a DM as an overall evaluation to be determined on the base of the determinants indicated in the previous query.

A) personal quality B) power of decision C) other(please indicate)
These surveys have been sent the Internet to experts. The answers received from experts(see Fig.1) are operated on the basis of Delphi method. Altruism, emotion, trust, reciprocity, risk, social responsibility, tolerance to ambiguity etc. are obtained as the basic determinants. Therefore in this work two psychological determinants are chosen for modelling of the decision-maker [21]. As the index decision-making of the decision-maker personal quality is accepted. The following type model is offered on the basis of received answers: IF U_1 is A_i1 and U_2 is A_i2 and U_r is A_ir THEN V is Di and CFi where CFiis the confidence degree of the rule that is defined by expert. It expresses the belief degree of the expert to the truth degree of the rule. A_i1, A_i2, A_ir, Di are linguistic value of the linguistic variable U_1,U_2 , U_r, V.

III. MODELING OF A DECISION MAKER UNDER SECOND-ORDER UNCERTAINTY USING THE POSSIBILITY-PROBABILITY MEASURE BASED METHOD
Knowledge in a production systems can be described in different ways. Some of the post-modern techniques for representation of knowledge include logical calculus, production systems and structured model. This work is devoted to the production system based approaches of knowledge representation. The production systems is the simplest. A production systems consist of three items:1) a set of production rules,2) dynamic database, called the working memory, 3) control structure or interpreter, which interprets the database using the set of production rules. The production system has large applications in decision making problems, in oil refinery problem, in psychology, in business problems, in technical problems, in social sciences [3,21,31].
The structure of a production rule can be formally stated as follows. Before presenting the technique for knowledge representation by product systems, we define the term knowledge, which the widely used in this paper.
The production description of knowledge in the knowledge base of decision maker is based on fuzzy interpretation of antecedents and consequents in production rules [3]. 1 1 2 2 The basic steps of the method are given below:

R IF x is A and x is A and and x is A THEN u is B and u is B and and u is
1. The truth degree of the rule is computed as: where j is a number of a rule. After all these rules have been executed (with different truth degrees) the next rule (rules) ought to be executed:  Dear professor Arkadiy Borisov, Our department conducts research on decision theory with imperfect information with University of California, Berkeley, during 4 years. One of the branches of our research area is to combine state of nature and state of a decision maker. Consequently, we need to model power of decision maker and features of a decision maker. For the purpose of this we use Delphi Method to collect and to process opinion of experts in decision making area. I would like to ask you as an expert in decision area to express your opinion on main features of a decision maker and a resulting dimension characterizing power of a decision maker (for example, power of a decision maker, personal quality of a decision maker) by participating in our survey. We are sending you 1 st questionnaire and I ask you to send back your answers, if possible, within a 3 week. Kind regards, R.A.Aliev and L.A. Gardashova Query 1. Please indicate by "+" which of the following should be considered as determinants of a decision maker (see in Please indicate what term should be used for a total index (resulting dimension) of a DM as an overall evaluation to be determined on the base of the determinants indicated in the previous query.
A) personal quality B) power of decision C) other(please indicate)

Fig.1. The answers received from experts
The above described model is realized by using the ESPLAN expert system shell and different tests are performed . Example.
Let us describe the model taking into account the characteristic features of DM:

IF trust level of a DM is about 70 and altruism level of a DM is about 70 THEN personal quality of a DM (V) is equal?
Below is given computer simulation results by ESPLAN expert system shell(see Fig. 2.). -arithmetic operations with fuzzy numbers; -realization of simple question-ask dialogue by using special functions; -set a confidence degree for any rule (in per cent); -call of external programs; -data interchange using file system. All above mentioned abilities are supported by ESPLAN knowledge representation language based on production rules. The inference engine of ESPLAN allows : -forward-chaining width-first inference with truth degree calculation on the continuous scale [0,100]; -set of a truth threshold during run-time in order to cut a rules with current truth degree less than the threshold; -tracing inference to the screen; -tracing inference to disk for further generation of the explanation; The shell of ESPLAN has own WORDSTAR compatible text editor. The shell of ESPLAN is represented to a user like the multi-window interface.

V.MODELLING OF DECISION MAKER ON THE BASIS OF FUZZY AND DEMPSTER-SHAFER THEORY
Now we consider modeling on the basis of Dempster -Shafer theory. Human behavioral modeling requires an ability to represent and manipulate imprecise cognitive concepts. It also needs to include the uncertainty and unpredictability of human action [17]. Human behavioral modeling requires an ability to formally represent sophisticated cognitive concepts that are often at best described in imprecise linguistic terms. Fuzzy sets provide a powerful tool for enabling the semantical modeling of these imprecise concepts within computer based systems [17]. With the aid of a fuzzy set we can formally represent sophisticated imprecise linguistic concepts in a manner that allows for the types of computational manipulation needed for reasoning in behavioral models based on human cognition and conceptualization. Now we consider a DM behavioral modeling using fuzzy and Dempster-Shafer theories suggested in [17].
The Dempster-Shafer approach fits nicely into the fuzzy logic since both techniques use sets as their primary data structure and are important components of the emerging field of granular computing. In [17] the behavioral model is represented by partitioning the input space. We can represent relationship between input and output variables by a collection of n "IF-THEN" rules of the form:    [17] in this example the empty set takes the value 0.09. But in accordance with Dempster-Shafer theory m-value of the empty set should be zero. In order to achieve this, m values of the focal elements should be normalized and m value of the empty set made equal to zero. The normalization process is as follows: performed, which demonstrated the approximately same results. Both methods are efficient, but the preference of the second method is the possibility to evaluate the confidence degree of the result.

VI. CONCLUSION
In this paper, the decision maker behavioral modelling under imperfect information or secondorder uncertainty is proposed. By using Delphi method psychological determinants of a decision maker were determined. The described models are realized by using the expert system shell, the language of technical computing Matlab and different tests are performed. The obtained results proved validity of the suggested approach. .