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Information-gap decision theory. Decisions under severe uncertainty. (English) Zbl 0985.91013
San Diego, CA: Academic Press. xii, 330 p. (2001).
Information gap decision theory models uncertainty as an information gap rather than in a probabilistic (probability theory) or possibilistic (fuzzy set theory) way. Assuming no knowledge about uncertainty in the form of distribution functions, information gap models are well suited for decision making under severe uncertainty. This book is a comprehensive text on the subject. It is of interest for decision analysts, researchers, students, and teachers.
Some info-gap models of uncertainty are introduced in Chapter 2. An info-gap model is a family of nested sets, determined by the level of uncertainty $$\alpha$$ and a nominal function $$\widetilde u(t)$$. Chapters 3 to 6 are the main part of the book, where the essential concepts of info-gap models are discussed. The centerpiece of the theory are the immunity functions introduced in Chapter 3. The robustness function $$\widehat \alpha(q,r_c)$$ measures the maximal level of uncertainty that is acceptable to guarantee critical reward $$r_c$$ resulting from decision $$q$$. The opportunity function $$\widehat \beta(q,r_w)$$ measures the minimal level of uncertainty, so that windfall reward $$r_w$$ is possible. The concepts are extensively illustrated with examples.
Chapter 4 covers value judgments, in particular normalization, reasoning by analogy, rationality, and preference. The relation between robustness and opportunity functions and the behavior of these functions as depending on the decision vector $$q$$ is considered in Chapter 5. The immunities can be sympathetic or antagonitisc, depending on whether the opportunity function at the robust optimal decision decreases or increases with increasing critical reward.
In the context of info-gap models risk appears as low immunity to failure or low opportunity for sweeping success. Chapter 6 discusses risk sensitivity and gambling. Topics covered are robustness, reward, and opportunity premiums. These can assist the decision maker regarding his options of choice among actions, taking into account risk-aversity or risk-proclivity.
Chapters 7 to 9 deal with some more advanced topics: quantification of the value of information based on the robustness function, learning as fundamental revision of structures for information-processing, and coherent uncertainties and consensus. Following the quantitative study of info-gap theory is a qualitative essay in the context of project managemnet. The final part of the book consists of two chapters. Hybrid uncertainties, i.e. the combination of info-gap and probabilistic uncertainty and implications of (info-gap) uncertainty, with some practical and philosophical issues. Examples in the book are drawn from a variety of fields reaching from engineering design to portfolio selection. Problems are given for each of the chapters.

##### MSC:
 91B06 Decision theory 91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance 62C99 Statistical decision theory