Decision Theory with Imperfect Information

Abstract

ForewordProfessor Aliev's magnum opus, Decision Theory with Imperfect Information (with a Foreword by Prof. Lotfi Zadeh), or DTII for short, is a path-breaking contribution to a better understanding of how to make decisions in an environment of imperfect information-information which is uncertain, imprecise, incomplete or partially true. In real-world settings, such information is the norm rather than exception. Professor Aliev's work builds on existing theories of decision-making, but leaves them far behind.Decision theory as we know it today, is rooted in the pioneering work of von Neumann and Morgenstern, Theory of Games and Economic Behavior, 1944. von Neumann and Morgenstern's brilliant work spawned a huge literature. A notable development in the evolution of decision theory was the development of Prospect Theory by Kahneman and Tversky in 1979-a theory for which they were awarded the Nobel Prize in economics in 2002.Prospect theory was an important step in the direction of enhancing the capability of decision theory to deal with realworld problems. Professor Aliev's magnum opus contains a very insightful and very detailed critical analysis of existing theories of decision-making. Existing theories contain much that is deep, rigorous and elegant. However, there is a basic problem. Existing theories are based on the classical, Aristotelian, bivalent logic. Bivalent logic is intolerant of imprecision and partiality of truth. Fundamentally, bivalent logic is not the right logic to deal with the pervasive imprecision of the real world. Professor Aliev's work shifts the foundation of decision theory from bivalent logic to fuzzy logic. Informally, fuzzy logic is a system of reasoning and computation in which the objects of reasoning viii Decision Theory with Imperfect Informationand computation are classes with unsharp (fuzzy) boundaries. In the real world, such classes are the norm rather than exception. It is important to note that almost all words in a natural language are labels of classes with unsharp boundaries. Examples. Tall, cheap, fast, usually, most, likely, etc. Representation of the meaning of such words as probability distributions, is not effective. In the realm of decision theory, what is of particular importance is representation of imprecise probabilities and imprecise rewards as fuzzy probabilities and fuzzy rewards, respectively. In Professor Aliev's work these and other concepts are allowed to be described in a natural language. One of the major contributions of Professor Aliev's work is the development of a conceptual framework for decision-making and computation with information which is described in a natural language. A concept which plays an important role in computation with natural language is that of precisiation of meaning. Informally, if p is a proposition drawn from a natural language, then its meaning is precisiated by representing it in a mathematically well-defined form. More concretely, p may be viewed as a carrier of information about a variable, X, which in general is implici...

The general theory of decisions

“…Definition 13 (A Z-number valued probability measure [6]). The probability measure is a functionP : F → Z [0,1] satisfying: 472 (1)P(V) ≥ (0, 1) for any V ∈ F; 473 (2) For any sequence…”

“…475 Definition 14 (A Z-number valued random variable [6]). Given a probability space ( , F, P), a mapping X : → Z is said to be 476 a Z-valued random variable if it is D-Borel measurable, that is, it is a measurable mapping w.r.t.…”

“…, (A n , B n )). Also Definition 12 (Supremum metrics for Z-numbers [6]). The supremum metrics in Z n is defined as…”

“…553 4.2. Discussion of some special cases [6] 554 We would like to briefly discuss here application of the suggested theory to special cases in the existing literature. Let us 555 show that the existing decision theories like EU, CEU, PT and CPT which are the basic existing utility models applied on space 556 S are special cases of the suggested general theory of decisions.…”

“…A Z-number valued function[6]). In general, a Z-number valued function is a mappingf : → Z, Z-number valued functionf : → Z, a fuzzy number valued function ϕ : → D 1 is called its A-valued function 481 whenever for any ω ∈ one has ϕ(ω)= A ∈ D 1 ifff (ω) = (A, B) ∈ Z.…”