CC BY
111
Rafiq ALIEV
Director of Joint MBA Program, professor at Georgia State University (Atlanta, U.S.A.), head of the Department of Computer-Aided Control Systems,
Azerbaijan State Oil Academy (Baku, Azerbaijan).
DECISION ANALYSIS IN FUZZY ECONOMICS
Abstract
Tracing the development of economics since the 19th century up to the present day makes it evident that at its core there is a sequence of rather precise and mathematically sophisticated axiomatic theories. At the same time, there is always a noticeable and persistent gap between economic reality and the economic predictions derived from these theories.
The main reason why economic theories have not been successful so far in modeling economic reality is the fact that these theories are formulated in terms of hard sciences characterized by their nature of preciseness.
In economics, as in any complex multi-agent humanistic system, motivations, intuition, human knowledge, and human behavior, such as perception, emotions, and norms, play dominant roles. Consequently, real economic and socioeconomic world problems are too complex to be translated into classical mathematical and bivalent logic languages, solved and interpreted in the language of the real world. The traditional modeling methodology (economics deals with models of economic reality) is perhaps not relevant or at least not powerful enough to satisfy the requirements of human reasoning and decision making activities. A new much more effective modeling language is needed to capture economic reality. According to Prof. L. Zadeh, in general, fuzzy-logic-based modeling lan-
guages have a higher power of cointension than their bivalent logic-based counterparts and present the potential for playing an essential role in modeling economic, social, and political systems.
The sheer complexity of causation in the economic arena mandates a fuzzy approach. We argue that many economic dynamical systems naturally become fuzzy due to the uncertain initial conditions and parameters.
In this study we consider an economic system as a human centric and imperfect information-based realistic multi-agent system with fuzzy-logic-based representation of the economic agents’ behavior and with imprecise constraints. We use fuzzy “if-then” language and fuzzy differential equations for modeling the economic agents.
This paper looks beyond the standard assumptions of economics that all people are similarly rational and self-interested. To be able to incorporate motivation input variables of economic agents into their models, we created behavioral model of agents by using the fuzzy and Bayes-Short-liffe approaches.
Nowadays, adding norms and motivations to utility function has become an issue for economists and business people. They also recognized that something is missing in standard utility function and even in standard economic theory. Everything is not as simple as just profit or utility maxi-
mization under budget constraint. They recognized that during all these years they have been ignoring the most important thing: the norms and motivations that actually directed the human nature of the decision makers.
It is obvious that these norms and motivations cannot be captured by statistical analysis, there are deeper and more subtle uncertain relationships between wellbeing and its determinants. We suggest the
fuzzy graph-based approach to utility function construction. The proposed approach is consistent with the behavioral and uncertainty paradigms of the decision makers. Here we put the emphasis on mathematical background rather than real case analysis.
The suggested fuzzy approaches to economic problem-solving are applied to fuzzy decision making in an oligopolistic economy.
1. Introduction
Neoclassical assumptions regarding the opportunity and efficiency of economic agents are not realistic.1 As George Akerlof argues, in New Classical macroeconomics the important element, motivation, is missing—it fails to incorporate the norms of the decision makers.2
More advanced macroeconomics interprets the behavior of economic agents through preferences that include norms, which are people’s views regarding how they, and others, should or should not behave. Although these preferences are a central feature of sociological theory, they have been ignored by most economists. Sociologists consider norms to be central to motivation; because people tend to live up to the views and principles they accept and are happy only when they can manage them. Daniel Kahneman and Amos Tversky3 have studied people’s unwillingness to take favorable odds in small bets due to loss aversion. They explain that people have a mental frame that makes them reluctant to take losses.
Unlike traditional neoclassical theory, assumptions are not fundamental to the construction of economic models in behavioral economics. The Herbert Simon tradition in behavioral economics maintains that intelligent behavior need not produce the type of optimal or efficient behaviors predicted by traditional neoclassical theory. The behavioral model makes it possible to have multiple equilibria or a fuzzy set solution to the choice set afforded to economic agents in the fundamentally important domain of production. For this reason, the behavioral model can interpret important economic facts that contradict the analytical predictions of neoclassical economic theory (the variety of complex outcomes in competitive equilibrium is indicative of the substance of economic life).
In a complex environment with a large number of heterogeneous interacting agents there is a high degree of uncertainty about their interaction and relevant information. Also agents will constantly try to find better representations of the perceived reality and will therefore experience and learn. We need a way of modeling the “mental models” of economic agents that operate within such an economic environment.
Main reasons why economic theories have not been successful in modeling economic reality is directly related to the subject of this paper, i.e. these theories are formulated in terms of classical mathematics, bivalent logic, and the classical theory of additive measures. Human reasoning and
1 See: G.A. Akerlof, “The Missing Motivation in Macroeconomics,” The American Economic Review, Vol. 97, No. 1, 2007, pp. 5-36; D. Romer, Advanced Macroeconomics, Third Edition, McGraw-Hill/Irvin, New York, 2006, p. 678; J.M. Dowling, Y. Chin-Fang, “Modern Developments in Behavioral Economics,” Social Science Perspectives on Choice and Decision Making, World Scientific Publishing Co. Pte. Ltd., Singapore, 2007, p. 446.
2 See: G.A. Akerlof, op. cit.
3 See: D. Kahneman, A. Tversky, “Prospect Theory: An Analysis of Decision under Risk,” Econometrica, No. 47, 1979, pp. 263-291.
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decision making are based on high degrees of uncertainty (usually nonstatistical) and classical mathematics is not capable of expressing this kind of uncertainty. Human preferences for complex choices are not determined, in general, by the rules of additives measures.
As stated above, well-known economists such as Akerlof, Kahneman, and Altman have advocated reformulation of economic models to include factors such as motivations and norms of agents that influence economic behavior. However, to date, no comprehensive mathematical framework has been suggested for this economic paradigm.
This research attempts to model an economic system as a human centric and imperfect information-based realistic multi-agent system with fuzzy-logic-based4 representation of the economic agents’ behavior.
2. Overview of Related Work
British economist Shackle has argued since the late 1940s that the probability theory is not useful for capturing the nature of uncertainty in economics. He has suggested that uncertainty associated with actions with unknown outcomes should be expressed in terms of degrees of possibility rather than by probabilities.5
C. Ponsard presented that fuzzy optimization makes it possible to consider that the objective or the constraint are fuzzy.6 Fuzzy optimization accounts for either the imprecision of the utility function or the imprecision of budget constraint.
C.R. Barret et al explored the problem of aggregation of ordinary fuzzy individual preferences into ordinary fuzzy social preferences.7 S. Ovchinnikov et al established the existence of fuzzy models for strict preference, indifference, and incomparability satisfying all classical properties.8 J. Kacprzyk et al presented fuzzy individual and social preference relations based on new solution concepts in group decision making and new “soft” degrees of consensus.9 B.R. Munier questioned the expected utility hypothesis as a pattern of rational decision under risk or uncertainty.10
A comparative review of the application of the fuzzy sets theory in economics was conducted by
A. Billot.11 There are also various papers (for example by J.G. Aluja and A. Kaufmann) where fuzzy microeconomics models are considered.12
4 See: L.A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353; idem, “Outline of a New Approach to the Analysis of Complex Systems and Decision Processes,” IEEE Transactions on Systems, Man, and Cybernetics, SMC3, 1973, pp. 28-44; idem, “Fuzzy Logic,” IEEE Computer, April 1988, pp. 83-93.
5 See: G.L. Shackle, Decision, Order and Time in Human Affairs, Cambridge University Press, New York and Cambridge, U.K., 1961.
6 See: C. Ponsard, “Producer’s Spatial Equilibrium with a Fuzzy Constraint,” European Journal of Operational Research, No. 10, 1982, pp. 302-313.
7 See: C.R. Barret, P.K. Pattanaik, M. Salles, “Rationality and Aggregation of Preferences in an Ordinary Fuzzy
Framework,” Fuzzy Sets and Systems, No. 49, 1992, pp. 9-13.
8 S. Ovchinnikov, M. Roubens, M. Salles, “On Fuzzy Strict Preference, Indifference and Incomparability Relations,” Fuzzy Sets and Systems, No. 49, 1992, pp. 15-20.
9 See: J. Kacprzyk, M. Fedrizzi, H. Nurmi, “Group Decision Making and Consensus under Fuzzy Preferences and
Fuzzy Majority,” Fuzzy Sets and Systems, No. 49, 1992, pp. 21-31.
10 See: B.R. Munier, “Expected Utility versus Anticipated Utility: Where do We Stand?” Fuzzy Sets and Systems, TNo. 49, 1992, pp. 55-64.
11 See: A. Billot, “From Fuzzy Set Theory to Non-Additive Probabilities: How Have Economists Reacted?” Fuzzy Sets and Systems, No. 49, 1992, pp. 75-90; idem, Economic Theory of Fuzzy Equilibria. An Axiomatic Analysis, Springer-Verlag, Berlin, Heidelberg, 1992, p. 164.
12 See: J.G. Aluja, A. Kaufmann, Introduccion de la teoria de la incertidumbre en la gestion de empresas [Introduction to the Uncertainty Theory in Enterprises Management], ed. by Milladoiro-Academy of Doctors, Vigo-Barcelona
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Various simulations of different real world problems, including the macroeconomic problems described by fuzzy differential equations, among which are the economic problems of supply and demand, the national economy model, etc., were considered by J.J. Buckley and L.J. Jowers.13
Optimal control of a complex dynamic economic system was considered in a series of works (for example, by Silberberg and Suen, Zhukovskiy and Salukvadze, etc.).14
Fuzzy control for stabilizing economic processes to implement stabilization strategies in a user-friendly way, by means of a linguistically expressed algorithm, is suggested in a paper by V. Geor-gescu.15
In recent years, a great number of papers and several books have explored the use of fuzzy logic as a tool for designing intelligent systems in business, finance, management, and economics.16 These books present recent progress in the application of constituents of Soft Computing (SC) methodologies, in particular neural networks, fuzzy logic, chaos, etc. The well-known book by R.A. Aliev,
B. Fazlollahi, and R.R. Aliev highlights some of the recent developments in the practical application of SC in business and economics.17
In this study, we consider an economic system as human centric and imperfect information-based realistic multi-agent system with fuzzy-logic-based representation of the economic agents’ behavior. We analyze different problems in its creation, mainly two concepts of a multi-agent economic system, behavioral modeling of the economic agent, fuzzy utility function construction by adding the norms and motivations of agents, fuzzy optimality in overall decision making, and generalized stability analysis and fuzzy multi-criteria optimal control of dynamical economic systems. The paper is structured as follows. In Section 3 we present an economy as a multi-agent system and propose two approaches to its optimization, as well as behavioral modeling of the economic agent using “if-then» and fuzzy differential equation languages. In Section 4 we consider the utility function construction which incorporates the decision maker’s norms and motivations and is consistent with the behavioral and uncertainty paradigms of the decision makers. Section 5 is devoted to a
(Spain), Reial Academia De Doctors, 2002 (English version Ed. Springer 2003); J.G. Aluja, A. Kaufmann, Introduccion de la teoria de los subconjuntos borrosos a la gestion de las empresas [Introduction of Fuzzy Sub-Sets Theory to Business Management], Milladoiro, Santiago de Compostela, 1986; H. Hagras, V. Callaghan, A. Lopez, “An Incremental Adaptive Life Long Learning Approach for Type-2 Fuzzy Embedded Agents in Ambient Intelligent Environments,” IEEE Transactions on Fuzzy Systems, Vol. 15, No. 1, February 2007, pp. 41-55; Fuzzy Sets in Management, Economics and Marketing, ed. by C. Zopounidis, P.M. Pardalos, G. Baourakis, World Scientific Publishing Co. Pte. Ltd., Singapore 2001, p. 269; A. Kaufmann, J.G. Aluja, Las Matematicas Del Azar Y De La Incertidumbre. Elementos Basicos Para Su Aplicacion En Economia, Editorial Centro De Estudios Ramon Areces, Madrid, 1990, p. 298.
13 See: J.J. Buckley, L.J. Jowers, Simulating Continuous Fuzzy Systems, Series: Studies in Fuzziness and Soft Computing, Vol. 188. Springer, 2006.
14 See: I. Grigorenko, Optimal Control and Forecasting of Complex Dynamical Systems, World Scientific Publishing Co. Pte. Ltd., Singapore, 2006, p. 198; E. Silberberg, W. Suen, The Structure of Economics. A Mathematical Analysis, Third Edition, Irwin McGraw-Hill, New York, 2001, p. 668; G. Heal, B. Kristrom, “A Note on National Income in a Dynamic Economy,” Economics Letters, Vol. 98, Issue 1, January 2008, pp. 2-8; S.P. Sethi, G.L.Thompson, Optimal Control Theory. Applications to Management Science and Economics, Second Edition, Springer, New York 2000, p. 504; V.I. Zhukovskiy, M.E. Salukvadze, Riski i iskhody v mnogokriterialnykh zadachakh upravlenia, Intellect Edition, Moscow, Tbilisi, 2004, 356 pp.
15 See: V. Georgescu, “Fuzzy Control Applied to Economic Stabilization Policies,” Studies in Informatics and Control, Vol. 10, No. 1, March 2001, pp. 37-60, ISSN 1220-1766.
16 See: G. Bojadziev, M. Bojadziev, Fuzzy Logic for Business, Finance, and Management, World Scientific Publishing Co. Pte. Ltd., Singapore, 1997; P. Lisboa, A. Vellido, B. Edinbury, Neural Networks. Business Applications of Neural Networks, World Scientific Publishing Co. Pte. Ltd., Singapore, 2000; S.-H. Chen, “Genetic Algorithms,” in: Evolutionary Computation in Economics and Finance, ed. by S.-H. Chen, Physica-Verlag, 2002; Soft Computing for Risk Evaluation and Management: Applications in Technology, Environment and Finance, ed. by D. Ruan, M. Fedriz-zi, J. Kacprzyk, Springer-Verlag, 2001.
17 See: R.A. Aliev, B. Fazlollahi, R.R. Aliev, Soft Computing and its Applications in Business and Economics, Springer-Verlag, Berlin, Heidelberg, 2004.
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multi-agent decision making system in an oligopolistic economy. In section 6 we give the concluding comments.
3. An Economy as a Multi-Agent System
An economy as a complex system is composed of a number of agents interacting in a distributed mode. Advances in distributed artificial intelligence, intelligent agent theory, and soft computing technology make it possible for these agents as components of a complex system to interact, cooperate, contend, and coordinate in order to form the global behavior of the economic system. Recently, there has been great interest in the development of Intelligent Agents (IA) and multi-agent systems in economics, in particular in decision analysis and control of economic systems.18
It should be noted that economic agents often deal with incomplete, contradictory, missing, and inaccurate data and knowledge.19 Furthermore, the agents have to make decisions in uncertain situations, i.e. the real world multi-agent economic systems function within an environment of uncertainty and imprecision.
In this section we consider two approaches to a multi-agent economic system: the conventional concept and the alternative concept.
In the conventional concept of multi-agent distributed intelligent systems the main idea is granulation of functions and powers from a central authority to local authorities. In these terms, the economic system is composed of several agents, such as firms, plants, and enterprises that can perform their own functions independently, and, therefore, have the information, authority, and power necessary to perform only their own function. These intelligent agents can communicate together to work, cooperate, and be coordinated in order to reach the common goal of the entire economy.20
An alternative concept of a multi-agent distributed intelligent system with cooperation and competition among its agents is distinguished from the conventional approach by the following: each intelligent agent acts fully autonomously; each intelligent agent proposes a full solution to the problem (not only to its own partial problem); each agent has access to full available input information; a total solution to the problem is determined as the proposal of one of the parallel functioning agents on the basis of a competitive procedure (not by coordinating and integrating the partial solutions of agents often performed iteratively); cooperation among the agents produces the desired behavior of the system; cooperation and competition in the system are performed simultaneously (not sequentially).21
18 See: J. Lu, G. Zhang, D. Ruan, F. Wu, Multi-Objective Group Decision Making. Methods, Software and Applications with Fuzzy Set Techniques, Imperial College Press, London, 2007, p. 390; Soft Computing Agents. A New Perspective for Dynamic Information Systems, ed. by V. Loia, IOS Press, Amsterdam, 2002, p. 254; A. Whinston, “Intelligent Agents as a Basis for Decision Support Systems,” Decision Support Systems, No. 20 (1), 1997, p. 1; Sh. Teraji, “Culture, Effort Variability, and Hierarchy,” Journal of Socio-Economics, Vol. 37, Issue 1, February 2008, pp. 157-166.
19 See: J.M. Dowling, Y. Chin-Fang, op. cit.; R.A. Aliev, R.R. Aliev, “Fuzzy Distributed Intelligent Systems for Continuous Production. Application of Fuzzy Logic,” in: Towards High Machine Intelligence Quotient Systems, ed. by M. Jamshidi, M. Titli, L. Zadeh, S. Boverie, Prentice Hall PTR, Upper Sadle River, New Jersey, USA, 1997; R.A. Aliev, M.I. Liberzon, Metody i algoritmy koordinatsii v promyshlennykh sistemakh upravlenia, Radio i svyaz, Moscow, 1987, 208 pp.; Judgment Under Uncertainty: Heuristics and Biases, ed. by D. Kahneman, P. Slovic, A. Tversky, Cambridge University Press, 1982. 544 pp.
20 See: R.A. Aliev, R.R. Aliev, op. cit.; Judgment Under Uncertainty: Heuristics and Biases; Distributed Artificial Intelligence, Vol. II, ed. by L. Gasser, M. Huhns, Morgan Koufmann, San Mateo, California, 1989, pp. 259-290.
21 See: R.A. Aliev, B. Fazlollahi, R.M. Vahidov, “Soft Computing Based Multi-Agent Marketing Decision Support Systems,” Journal of Intelligent and Fuzzy Systems, Vol. 9, 2000, pp. 1-9; B. Fazlollahi, R.M. Vahidov, R.A. Aliev, “Multi-Agent Distributed Intelligent Systems Based on Fuzzy Decision making,” International Journal of Intelligent Systems, Vol. 15, 2000, pp. 849-858.
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A similar idea of decomposition of the overall system into agents with cooperation and competition among them is implemented in several papers.22
Zhang proposed a way to synthesize final solutions in systems where different agents use different inexact reasoning models to solve a problem. In this approach a number of expert systems propose solutions to a given problem. These solutions are then synthesized using the ego-altruistic ap-
proach.23
24
3.1. The Conventional Approach
We will mainly consider systems with a so-called “fan beep” structure in which the economic system consists of N agents at the lower level and one element at the higher level (called center).
The state of i-th agent (i e [1 : N]) is characterized by vector x,. Vector x. should meet the local
constraints
Xi e Xt c Eni, (1)
where X. is a set in n. dimensional Euclidean space. A specificity of these hierarchical systems is
information aggregation at the higher level. This means that the only agent of the higher level, the center, is not concerned with individual values of variables x, but some indexes evaluating the elements’ activities (states) produced from those values. Let’s denote the vector of such indexes as:
Fi (Xi) = (fa(x,),..., fm (Xi)), i e [1: N]. (2)
The state of the center is characterized by vector F0, the components of which are the indexes of the agents of the lower level:
F0 = (Fp..., Fn), where Ft = F.(x,). (3)
Commonly, the decision making process implies the existence of a person making the final decision (Decision Maker-Agent) at the higher level.
H0(F1,., Fn) ^ max; (4)
H(Fv..., Fn) > b; (5)
F, e SF = {/{. = {. (x. ), x. e PiX (or x. e RX)} (6)
where PiX(RiX) is the set of effective (semi-effective) solutions to the problem
®i (x,) = (Pi1 (x, ),...,?>*, (x,)) ^ 'max'; (7)
x, e X. (8)
We suggest two algorithms of the solution to (4)-(6):
22 See: Y. Jong, W. Liang, L. Reza, “Multiple Fuzzy Systems for Function Approximation,” in: Proceedings of NAFIPS 97, Syracuse, New-York, 1997, pp. 154-159; T. Tsuji, A. Jazidie, M. Kaneko, “Distributed Trajectory Generation for Cooperative Multi-Arm Robots via Virtual Force Interactions,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, No. 27 (5), 1997/a, pp. 862-867; Ch. Zhang, “Cooperation Under Uncertainty in Distributed Expert Systems,” Artificial Intelligence, No. 56, 1992, pp. 21-69.
23 See: N.A. Khan, R. Jain, “Uncertainty Management in a Distributed Knowledge Base System,” in: Proceedings of International Joint Conference on Artificial Intelligence, 1985, pp. 318-320.
24 See: R.A. Aliev, M.I. Liberzon, op. cit.; R.A. Aliev, V.P. Krivosheev, M.I. Liberzon, “Optimal Decision Coordination in Hierarchical Systems,” News of Academy of Sciences of USSR, Tech. Cybernetics, No. 2, 1982, pp. 72-79.
(a) Non-iterative algorithm
Non-iterative optimization algorithms include three main phases. At the first stage, the local problems of vector optimization are dealt with. The solutions to these problems are
sets pX and sets Sf (Qf) or any approximations of these sets. The second stage implies the
implementation of the center’s task (4)-(6) as result of which we get the optimal values of
the agents’ criteria F* = (F1*,...,F*). Vector F* is then passed to the i-th agent which
implements the third stage by solving the problem:
The solution to (9) represents local variables x*. If there are several solutions, one is selected based on the preferences of the i-th agent.
Note that the center only receives information about the Fi indexes, not about vector x.. Because the dimension of F. is usually significantly less than the dimension of vector x,, it considerably reduces the amount of data circulating between the levels.
(b) Iterative algorithm
Let Q, c Eki be a subset in the space of criteria. Let’s call the elements a.e Q. from this subset coordinating signals.
Definition.25 The function F, (a,) = (f (o,),..., f im, (a, )) is called a coordinating function if the following conditions are satisfied:
(a) For Vo, eQ, there exists such an element as x,(a,)e Rf, for which F,(a,) = F,(x,(a,));
(b) Inversely, for any element x,0 of subset pX, there exists such a coordinating signal as a0 e Q,, for which F,(a0) = F(x0).
From the definition above, it follows that problem (4)-(6) is equivalent to the problem given below:
Thus, the variables of problem (10) are coordinating signals a, defined on the set of acceptable coordinating signals Q.. The rationale for such a transformation is the simpler structure of set Q. compared to the set of effective elements (points).
By choosing among different coordinating functions F, (a,) and different solution algorithms of problem (10) it is possible to construct a large number of iterative coordinating algorithms.26 It can be shown that the known decomposition algorithms, such as the Dantzig-Wolf algorithm,27 the Kor-
25 See: R.A. Aliev, M.I. Liberzon, op. cit.
26 See: R.A. Aliev, V.P. Krivosheev, M.I. Liberzon, op. cit.; R.A. Aliev, M.I. Liberzon, “Bezyterativnye algoritmy
koordinatsii v dvukhurovnevykh sistemakh,” Izvestia AN SSSR, Tekhnicheskaia kibernetika, No. 3, 1986, pp. 163-166.
27 See: G. B. Dantzig, Linear Programming and Extensions, Princeton University Press, 1998, pp. 648.
(9)
H0(F1(o1),..., Fn (a*)) ^ max;
H(F, (a,),...,Fn (a*)) > b;
O e Q,, i e [1: N].
(10)
nai-Liptak algorithm,28 or the algorithm based on the interaction prediction principle,29 are special cases of the more general suggested algorithm.30
3.2. The Alternative Approach31
3.2.1. Architecture of the Proposed Multi-Agent Economic System
Fig. 1 shows the basic structure of the proposed (alternative) multi-agent economic system.
Figure 1
Architecture of the Multi-Agent Fuzzy Decision Making System
28 See: J. Kornai, T. Liptak, “Two-Level Planning,” Econometrica, Vol. 33, No. 1, 1965, pp. 141-169.
29 See: M.D. Mesarovic, D. Macko, Y. Takahara, Theory of Hierarchical Multilevel Systems, Academic Press, New York, 1972.
30 See: R.A. Aliev, M.I. Liberzon, Metody i algoritmy koordinatsii vpromyshlennykh sistemakh upravlenia.
31 See: R.A. Aliev, B. Fazlollahi, R.R. Aliev, op. cit.; R.A. Aliev, B. Fazlollahi, R.M. Vahidov, op. cit.; B. Fazlolla-hi, R.M. Vahidov, R.A. Aliev, op. cit.; R.A. Aliev, R.R. Aliev, Soft Computing and its Application, World Scientific Publishing Co. Pte. Ltd., New Jersey, London, Singapore, Hong Kong, 2001, p. 444.
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The system is composed of N agents. All agents receive the same input information x1,x2,.,xm. The vector of input variables includes economic factors (the price of produced products, advertising, culture norm, emotional [trust, fairness]). Each agent performs inference and produces its own solution to the full problem:
Uj = [j uj2,...,ujl] , j = 1N .
The estimator uses the agents’ current solutions, U1,U2,-,U№ input information xpx2,.. ,,xm, and any other necessary information to determine the system outcomes for each agent’s proposal (e.g. profit, worker satisfaction, quality, pollution, environment, etc.). These outcomes that define alternatives are the basis for competition among agents A,...., AgN. The evaluator (Decision Maker) compares all the alternatives values and determines a “winner” (for example, j-th) agent, with the best outcome (best focus or overall goal of the problem at the top determined by the utility function). The winner’s solution is accepted as the total solution of the full system U, = [uu .,u]r.
3.2.2. Behavioral Modeling of an Economic Agent
In this study we consider an economic system as a human centric and imperfect information-based realistic multi-agent system with fuzzy-logic-based representation of the economic agents’ behavior and with imprecise constraints. We use a fuzzy “if-then” language and fuzzy differential equations for modeling the economic agents.32
This paper looks beyond the standard assumptions of economics that all people are similarly rational and self-interested. To be able to incorporate motivation input variables of the economic agents into their models, we created a behavioral model of agents using “if-then” language.
“If-Then” Modeling Language
In economics, as in a complex multi-agent humanistic system, motivations, intuition, human knowledge, and human behavior, such as perception, emotions, and norms, play dominant roles.33 Consequently, real economic and socioeconomic world problems are too complex to be translated into classical mathematical and bivalent logic languages, solved and interpreted in the language of the real world. The traditional modeling methodology (economics deals with models of economic reality) is perhaps not relevant or at least not powerful enough to satisfy the requirements of human reasoning and decision making activities. A new much more effective modeling language is needed to capture economic reality. As Prof. L. Zadeh states, in general, fuzzy-logic-based modeling languages have a higher power of cointension than their bivalent logic-based counterparts and present the potential for playing an essential role in modeling economic, social, and political systems.
In his last paper, Prof. L. Zadeh shows: “An example of a widely used fuzzy-logic-based modeling language is the language of fuzzy-if-then rules. Another example is the fuzzy-logic-based lan-
32 See: L.A. Zadeh, “Is There a Need for Fuzzy Logic?” Information Science (accepted); R.A. Aliev, “Fuzzy Dynamics in Advanced Macroeconomics,” in: Proceedings of the First International Conference on Soft Computing Technologies in Economy, ICSCTE-2007, Baku, Azerbaijan, 19-20 November, pp. 11-33; R.A. Aliev, “Modeling and Stability Analysis in Fuzzy Economics,” Applied and Computational Mathematics, Vol. 7, No. 1, 2008.
33 See: J.M. Dowling, Y. Chin-Fang, op. cit.; M. Altman, “Behavioral Economics, Power, Rational Inefficiencies, Fuzzy Sets, and Public Policy,” Journal of Economic Issues, Vol. XXXIX, No. 3, September 2005, pp. 683-706; C. Pon-sard, op. cit.
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guage of the Generalized Theory of Uncertainty (GTU)—a language which has the potential for playing an essential role in cointensive modeling of systems such as economic systems, social systems, political systems, etc., systems in which what is needed is a machinery for computation with imprecise probabilities, imprecise goals, and imprecise constraints.”34 He suggests important rationale for those of the fuzzy logic as a modeling language. For imperfect information systems fuzzy logic is the logic of choice as a basis for modeling languages. This rationale asserts that fuzzy logic of choice is the basis for modeling language.
Since the economic, socioeconomic, and behavioral input variables of economic agents in a multiagent system are usually uncertain and described by linguistic values and the outputs are perception-based outcomes, the use of the if-then rules language for modeling these agents is appropriate.
The rule base formed by a collection of fuzzy rules is the model of an agent and is described as
Rk = IF xi is Aki and x2 is Ak2,.,x„ is ^
THEN y is Bk, k = 1, m ,
where xt,i = 1,n k-th agent’s input variables, y—its input, Akj and Bk—fuzzy sets that describe the linguistic terms of the input and output variables of an agent.
Fuzzy Differential Equations as a Modeling Language
Fuzzy differential equations serve many functions in economics. They are used to determine the conditions for dynamic stability in microeconomic models of market general equilibria and to trace the time path of growth under various conditions and uncertainty in microeconomics. Given the growth rate of a function, differential equations enable the economist to find the function whose growth is described; from point elasticity, they enable the economist to estimate the demand function.
Measurements made in an economic environment are generally imprecise, ambiguous, or vague, and, therefore, uncertain. If data is uncertain, a variable is fuzzy and can be represented using fuzzy logic, which enables control of the economic system to be based on human experience. A fuzzy approach could become particularly useful in handling uncertain or imprecise economic environmental data describing dynamically changing environments. Therefore, any component of an economic system can be considered a fuzzy granule.
By representing a dynamic economic system as a fuzzy system, endogenous and exogenous parameters of the economic models could be presented as fuzzy variables. In many cases, information about the behavior of a dynamical system is uncertain. In order to obtain a more realistic (not more exact!) model, we have to take into account these uncertainties. This approach provides the background for the development of fuzzy differential equations as the basis for modeling economic systems.
Fuzzy differential equations need to be considered as a separate model instead of creating them as fuzzy analogues of crisp counterparts. We can conclude that fuzzy differential equations (FDE) are a natural way to model dynamical systems under possibilistic uncertainty.
4. Utility Function Construction
The models of economic choice used until today and based on utility maximization were created by utilitarians, beginning with Adam Smith.
[ L.A. Zadeh, “Is There a Need for Fuzzy Logic?”
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Further developments, such as expected utility, the Bayesian revision of preferences, and subjective utility, slightly relaxed the neoclassical hard assumption. Recently economists and psychologists began investigating the decision making process within the larger context of variables and situations related to the motivations of economic agents.
These investigations have demonstrated that people often violate the assumptions of the neoclassical model as they make everyday decisions. One of the most noteworthy works in this field belongs to Daniel Kahneman, with his theory of experienced utility.35 This theory is based on the belief that there is a “measurable” good that is separable from the choices people make. It is not surprising that in 2008 at the World Economic Forum in Davos, Bill Gates, founder of Microsoft, stated: “As I see it, there are two great forces of human nature: self-interest and caring for others. Capitalism harnesses self-interest in a helpful and sustainable way, but only on behalf of those who can pay.”
Modern behavioral economics needs to incorporate issues like irrationality, attitudes toward risk, and other variables that have a motivating factor on decision making, such as altruism, culture, trust, etc, into the utility function.
While economists construct modern utility functions that are consistent with the behavioral and uncertainty paradigms of the decision makers, they are frequently confronted with the uncertain constraints, goals, and state of the economic agents. Moreover, this construction and decision made on its basis can be represented in the form of human readable rules.
It is obvious that classical mathematics, bivalent logic, and statistical analysis cannot be used to incorporate the decision makers’ norms and motivations into utility (happiness) functions to achieve transparency of the latter. Linguistic variables are powerful tools used by humans to indicate choice and preference. We suggest the fuzzy graph and fuzzy implication-based approach to utility (happiness) function construction. It is essential to reformulate economics so that it can deal with fuzzy choice and preferences and model realistic economic life.
Fuzzy graphs are a very effective way to handle linguistic uncertainty if the membership functions
of variables are known.36 The decision maker uses linguistic variables with membership functions /ij,
i = 1,...,n, l = 1,...,q, where q—number of linguistic terms, n—number of variables representing the criteria of the agents’ solution findings to estimate the utility function U(x1,.,xn).37 For example, if x1 is
profit, and x2 is worker satisfaction, d could be the membership function of a “high” quantity of profit,
and the membership function for a “medium” degree of satisfaction. The membership functions of
the utility (“medium utility,” “high utility,” etc.) are denoted by jxf (xn+1 = u), i = 1,...,n. A fuzzy utility can be defined as fuzzy graph
Ijj (x1,x2,..., xn+1 = u) = max mind (xl)), i = 1,. ., n +1, l = 1,..., q
j l
Instead of a fuzzy graph we can use the implication-based models of fuzzy rules in which rules are viewed as a constraint that restricts the set of possible solutions. The result of aggregating the relations R,, i = 1,.,N rules is fuzzy relation R, in which the aggregation is carried out in the form of the same conjunction and therefore implemented through some t-norm,
N
R(x,y) = T[f (A, (x),Bj (y))],V(x,y) e Xx7 . i=1
35 See: Judgment Under Uncertainty: Heuristics and Biases, ed. by D. Kahneman, P. Slovic, A. Tversky.
36 See: C. Ponsard, op. cit.; G.E. Cantarell, V. Fedele, “Fuzzy Utility Theory for Analysing Discrete Choice Behaviour,” in: Proceedings of the Fourth International Symposium on Uncertainly Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03$17.00 2003 IEEE; P.D. Wilde, “Fuzzy Utility and Equilibria,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, Vol. 34, No. 4, August 2004, pp. 1774-1785.
37 See: P.D. Wilde, op. cit.
Utility function U, depending on one input variable x2 (worker satisfaction degree), and two input variables x1 and x2 (represented by z) are shown in Figs. 2-7. For agents with a different degree of “fairness” (x3), utility functions u depending on x1 (profit) and x2 (worker satisfaction degree) are shown in Figs. 8-13. Fairness is characterized by five terms (very low, low, medium, high, very high).
Below (on p. 65) we will give in detail the construction procedures of fuzzy utility functions for optimizing the overall goal of the problem at the top level. This is based on the multiple attributes (criteria) that define alternatives generated by agents at the middle level.
Figure 2 Figure 3
Fuzzy utility, Fuzzy utility,
Mamdani ALI-1 implication (f-norm-Aliev)
Figure 4
Fuzzy utility,
Mamdani
Figure 5
Fuzzy graph,
Mamdani
0 0,2 0,4 0,5 0,6 0,8 1
z
Figure 6
Fuzzy utility,
ALI-1 implication (f-norm-Aliev)
Figure 7
Fuzzy utility,
ALI-1 implication (f-norm-Aliev)
Figure 8
Fuzzy utility,
Zadeh implication (f-norm-min)
Figure 9
Fuzzy utility,
Zadeh implication (f-norm-min)
Pareto optimal generation of alternatives could be performed by heuristic procedures or fuzzy multi-criteria linear, nonlinear, goal programming, and other methods.38 Each alternative can be described using quality criteriaX, i = 1,.,n. Then the domain is represented by the Cartesian product
38 See: J. Lu, G. Zhang, D. Ruan, F. Wu, op. cit.; R.A. Aliev, M.I. Liberzon, Metody i algoritmy koordjnatsjj vpro-myshlennykh sistemakh upravlenia.
Figure 10 Figure 11
Fuzzy utility, Fuzzy utility,
Zadeh implication (f-norm-min) Zadeh implication (f-norm-min)
Figure 12
Fuzzy utility,
Zadeh implication (f-norm-min)
Figure 13
Fuzzy utility,
Zadeh implication (f-norm-min)
of n sets: X : X = X1 xX2 x ... xXn. A point in the domain is given by the corresponding n-tuple of values (x1, x2,.,xn) e X, where x. e X, and denoted by x(n).39
39 See: A.N. Borisov, O.A. Krumberg, I.P. Fyodorov, Priniatie reshenii na osnove nechetkikh modelei. Primery is-polzovania, Zinatne Press, Riga, 1990, 184 pp.; J. Efstathuio, V. Rajrovich, “Multi-Attribute Decision making Using a Fuzzy Heuristic Approach,” Intern. J. Man-Machine Studies, Vol. 12, No. 2, 1980, pp. 141-156.
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If the values of the criteria are determined linguistically (are implied to be fuzzy), then alternative A is fuzzy subspace X and represented as the Cartesian product of fuzzy sets on X.:
A = F(Xi) x F(X2) x ... x F(X,),
where F(X) is the fuzzy subset of X..
The membership function MA(x.) is determined in accordance with the usual determination of fuzzy composition:
^A (x(n)) = min^A (x,)
j=1,n ’
where MA(x.) is the membership function of estimation of an alternative by criterionX..
The elements u of the universal set U are chosen by DM in accordance with some simple rules, for example
U = {high, sufficiently high, medium, sufficiently low, low}.
The codebook can be extended by introducing modifiers, such as very, completely, more, or less. Utility can be interpreted as a linguistic variable, the values of which are terms of a fuzzy set defined
over the interval [0,1]. Knowledge about the utility is represented by fuzzy relation R from X = {xi(n)}
into U = {u}. The relation is a fuzzy set of Cartesian productX x R. R is defined in terms of membership function, |j.R (x", u) which assigns to each pair a grade of membership from the interval [0,1].
Considering utility as fuzzy relation R, we can characterize an alternative A e A by fuzzy subset V c U, where V = A ° R, using the formula
Vv (u) = max(min(^R (x(n), u), p.A (u))).
Construction of utility relation. The space of efficiency, X, contains a definite number of n-tuples. On the basis of the heuristics being considered we leave out non-essential tuples. As a result, we have a Pareto optimal set of solutions.
Calculation of alternatives’ utility. On the basis of the description of alternatives and tables of utility estimations of tuples values we calculate utility values V(A.).
By ranking the alternatives, we arrange them using evaluations like “BETTER,” “EQUIVALENT,” “WORSE.”
5. Simulations and Applications
5.1. Multi-Agent Decision Making System in an Oligopolistic Economy
The main idea of the proposed multi-agent economic system is based on granulation of the overall system into cooperative autonomous intelligent agents. These agents compete and cooperate with each other in order to propose a total solution to the problem and organize (combine) the individual total solutions into a final solution.
The architecture and modeling of the agent is considered in Section 3. As an example, here we consider an oligopolistic economy. In such an industry the number of firms is limited to a few, where the action of one firm has an impact on the industry’s demand. A number of firms compete for three products (x, y, and z). Each firm is managed by an agent, which pursues profitability and market share goals. The teams make marketing, production, and financial decisions each quarter using decisions incorporating data and models. Marketing decisions include decisions on mix and amounts of goods to produce, pricing, advertising expenses, and other variables. The firm’s price and advertising strat-
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egies, as well as prices and advertising expenditures of the competitors are the major factors influencing profits, market share, and other objectives.40
The existing systems are oriented toward an econometrics model of the industry’s history. In particular, as is shown by Schott and Whalen, existing econometrics models deal inadequately with information about competitors’ future behavior. The models assume that the competitor’s actions are known when the firm makes its decisions. However, information about the competitor’s future actions is at best vague. Schott and Whalen presented a procedure to model the uncertainties of competitors’ behavior as fuzzy information.41
The proposed multi-agent distributed intelligent system consists of 5 knowledge-based agents.
Input information are fuzzy variables of average price 51 (AvgPrice) and average advertising ~2
(AvgAdv) of competitors, as well as the trust degree of an agent ~3 (tr.d), and are the same for all 5 agents. Using the fuzzy inference rule each agent produces its output solutions: the firm’s own price (~i1), = 1,5 and the firm’s own advertising 2, i = 1,5. These solutions are characterized by three
objectives: profit f1), quality (f2), and worker satisfaction fj). Fuzzy rules of the knowledge base of each agent are of the following type:
For the first agent:
IF AvgPrice is HIGH and AvgAdv is LOW and tr.d is MEDIUM, THEN Price is HIGH and Advertising is MEDIUM
IF AvgPrice is LOW and AvgAdv is HIGH and tr.d is MEDIUM, THEN Price is MEDIUM and Advertising is HIGH
IF AvgPrice is MEDIUM and AvgAdv is MEDIUM and tr.d is MEDIUM, THEN Price is HIGH and Advertising is MEDIUM
For the second agent:
IF AvgPrice is HIGH and AvgAdv is LOW and tr.d is LOW, THEN Price is HIGH and advertising is LOW
IF AvgPrice is LOW and AvgAdv is HIGH and tr.d is LOW, THEN Price is MEDIUM and Advertising is HIGH
IF AvgPrice is MEDIUM and AvgAdv is MEDIUM and tr.d is LOW, THEN Price is HIGH and Advertising is LOW
For the third agent:
IF AvgPrice is HIGH and AvgAdv is LOW and tr.d is HIGH, THEN Price is LOW and Advertising is LOW
IF AvgPrice is LOW and AvgAdv is HIGH and tr.d is HIGH, THEN Price is LOW and Advertising is HIGH
IF AvgPrice is MEDIUM and AvgAdv is MEDIUM and tr.d is HIGH, THEN Price is LOW and Advertising is MEDIUM
40 See: B. Schott, T. Whalen, “Fuzzy Uncertainty in Imperfect Competition,” Information Science, No. 76, 1994, pp. 339-654.
41 See: Ibidem.
For the fourth agent:
IF AvgPrice is HIGH and AvgAdv is LOW and tr.d is LOW, THEN Price is HIGH and Advertising is LOW
IF AvgPrice is LOW and AvgAdv is HIGH and tr.d is LOW, THEN Price is LOW and Advertising is HIGH
IF AvgPrice is MEDIUM and AvgAdv is MEDIUM and tr.d is LOW, THEN Price is MEDIUM and Advertising is MEDIUM
Finally, for the fifth agent:
IF AvgPrice is HIGH and AvgAdv is LOW and tr.d is MEDIUM, THEN Price is HIGH and Advertising is MEDIUM
IF AvgPrice is LOW and AvgAdv is HIGH and tr.d is MEDIUM, THEN Price is LOW and Advertising is HIGH
IF AvgPrice is MEDIUM and AvgAdv is MEDIUM and tr.d is MEDIUM, THEN Price is MEDIUM and Advertising is HIGH
The Pareto optimal set of solutions constructed by all agents’ proposals is given in Table 1. For example, HIGH Profit is determined by fuzzy number (505,858; 195; 52,467).
The optimal solution from the set of Pareto-optimal alternatives is the proposal of the second agent.
Table 1
Alternative Agents
Agent '1 (Profit) '2 (quality) '3 (worker satisfaction degree)
1 HIGH LOW HIGH
2 MEDIUM MEDIUM HIGH
3 LOW HIGH LOW
4 HIGH MEDIUM MEDIUM
5 LOW LOW HIGH
6. Concluding Comments
In economics, as in any complex multi-agent humanistic system, motivations, intuition, human knowledge, and human behavior (such as perception, emotions, and norms) play dominant roles. Consequently, real economic and socioeconomic world problems are too complex to be translated
into classical mathematical and bivalent logic languages, solved and interpreted in the language of the real world. Two concepts of the creation of a multi-agent economic system were investigated: conventional, in which the main idea is granulation of functions and powers from a central authority to local authorities; and alternative, in which an overall consensus is achieved by cooperation and competition among agents. For both cases coordinated multi-criteria optimal decision making methods were suggested. Here we put the emphasis on mathematical background rather than real case analysis. “If-then” modeling language and fuzzy differential equations were considered as tools for the behavioral modeling of agents. As the economic, socioeconomic, behavioral input variables of economic agents in a multi-agent system are usually uncertain and described by linguistic values, and the outputs are perception-based outcomes, fuzzy graphs were used. Fuzzy utility functions were suggested that incorporate issues like irrationality and other variables that have a motivating factor on decision making, such as altruism, fairness, etc.
The suggested approach has been successfully applied to fuzzy decision making in an oligopolistic economy.
