Ivanenko Decision Systems and Nonstochastic Randomness

1. Auflage 2010
ISBN: 978-1-4419-5548-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 272 Seiten

'Decision Systems and Non-stochastic Randomness' presents the first mathematical formalization of the statistical regularities of non-stochastic randomness and demonstrates how these regularities extend the standard probability-based model of decision making under uncertainty, allowing for the description of uncertain mass events that do not fit standard stochastic models. The formalism of statistical regularities developed in this book will have a significant influence on decision theory and information theory as well as numerous other disciplines.

Autoren/Hrsg.

Ivanenko, V. I.

Inhaltsverzeichnis

1;Preface;10
2;Contents;12
3;Chapter 1 Introduction;14
4;Chapter 2 Decision Systems;21
4.1;2.1 Preliminaries;21
4.2;2.2 The Structure of Decision Systems;24
4.3;2.3 Decision Situations;29
4.3.1;2.3.1 The Scheme of a Decision Situation;29
4.3.2;2.3.2 Data about the Unknown;33
4.4;2.4 Experiments in Decision Systems;37
4.5;2.5 The Decision-Maker;39
4.6;2.6 Existence of Uncertainty in Decision Systems;42
4.7;2.7 Criterion Choice Rule;45
5;Chapter 3 Indifferent Uncertainty;50
5.1;3.1 Preliminaries;50
5.2;3.2 Gl -independent Sequences;56
5.3;3.3 Main Identity;63
5.4;3.4 Risk Estimates;67
5.5;3.5 Decision-Making under Gl -independence and Antagonistic Games;72
5.6;3.6 Some Examples of Gl -Independence;74
6;Chapter 4 Nonstochastic Randomness;82
6.1;4.1 Preliminaries;82
6.2;4.2 The Concept of Statistical Regularity;91
6.3;4.3 Sampling Directedness and Statistical Regularity;93
6.4;4.4 Coordinated Families of Regularities;100
6.5;4.5 Statistical Regularity of a Sequence;107
6.6;4.6 Statistical Regularity of Gl-Independence;109
6.7;4.7 Sampling Directedness as a Realization of a Random-in-a-Broad-Sense Phenomenon;113
7;Chapter 5 General Decision Problems;119
7.1;5.1 Preliminaries;119
7.2;5.2 On the Preference Relation in Mass Operations;123
7.3;5.3 Statistical Regularity and Criterion Choice;133
8;Chapter 6 Experiment in Decision Problems;140
8.1;6.1 Preliminaries;140
8.2;6.2 Informativity of Experiments;145
8.3;6.3 Optimality of Experiments;154
9;Chapter 7 Informativity of Experiment in Bayesian Decision Problems;158
9.1;7.1 Preliminaries;158
9.2;7.2 Theorem of Existence and Uniqueness of Uncertainty Functions;165
9.3;7.3 The Inverse Theorem;190
9.4;7.4 A Generalization to Countable Q;193
10;Chapter 8 Reducibility of Experiments in Multistep Decision Problems;202
10.1;8.1 Preliminaries;202
10.2;8.2 Reducibility of Bayesian Decision Problems;205
10.3;8.3 Reducibility of a General Decision Problem;209
11;Chapter 9 Concluding Remarks;213
12;Appendix A Mathematical Supplement;217
12.1;A.1 Elements of the Theory of Binary Relations;217
12.1.1;A.1.1 Basic Operations on Sets;217
12.1.2;A.1.2 Topological Spaces;220
12.1.3;A.1.3 Binary Relations;222
12.1.4;A.1.4 Operations on Binary Relations;224
12.1.5;A.1.5 Properties of Relations;226
12.1.6;A.1.6 Equivalence Relations;229
12.1.7;A.1.7 Strict Order Relations;231
12.1.8;A.1.8 Nonstrict Order Relations;233
12.1.9;A.1.9 Preference Relations;236
12.1.10;A.1.10 Mappings of Relations;239
12.2;A.2 Elements of the Theory of Real Functions;240
12.2.1;A.2.1 Rings, Semirings, and Algebras of Sets;240
12.2.2;A.2.2 Additive Functions of Set;243
12.2.3;A.2.3 Measure and Its Properties;244
12.2.4;A.2.4 Outer Measure;245
12.2.5;A.2.5 Standard Extension of Measure;247
12.2.6;A.2.6 Uniqueness of Extension of Measure;248
12.2.7;A.2.7 The Lebesgue Measure in Euclidean Space;248
12.2.8;A.2.8 Measurable Sets;250
12.2.9;A.2.9 Measurable Functions;252
12.2.10;A.2.10 Equivalent Functions, Convergence Almost Everywhere, and Convergence in Measure;255
12.2.11;A.2.11 The Lebesgue Integral of a Bounded Function;258
12.2.12;A.2.12 The Space S of Measurable Functions;262
12.2.13;A.2.13 Summable Functions;264
12.2.14;A.2.14 The Space L of Summable Functions;268
12.2.15;A.2.15 Variations of Additive Set Functions;268
12.2.16;A.2.16 The Radon Integral;270
12.2.17;A.2.17 Integration of Finitely Additive Set Functions;271
13;References;273
14;Index;278

Leseproben

"Chapter 2 Decision Systems (p. 19-20)

The correct statement of the laws of physics involves some very unfamiliar ideas which require advanced mathematics for their description. Therefore one needs a considerable amount of preparatory training even to learn what the words mean.
Richard Feynman

2.1 Preliminaries

Decision theory emerged from the requirements of diverse fields of human activity such as medicine, gambling, politics, warfare, economics and finance, and engineering. Perhaps this is the reason for the terminological diversity that sometimes impedes not only mutual understanding between specialists in different fields but also the development of decision theory itself. In this sense, control theory has been more fortunate, for its terminology turned out to be common to many spheres of its application.

There are two points of view on the relationship between decision theory and control theory. According to one such view, they have nothing in common. According to the other, these theories are gradually converging because the differences between them are not fundamental [14, 18]. The author of this book is an adherent of the latter viewpoint, and proposes the following motivation. In control theory one studies a control system that consists of a pair of objects: a plant and a controller. In decision theory one studies a pair consisting of a decision situation and a decision-maker. It is natural to call such a pair a decision system. A control system is defined similarly.

The problem of choice of a decision or a control—an action that produces some consequence—is a problem common to both systems. In both systems, one may encounter two basic difficulties in the process of making this choice: dynamics and uncertainty. The development of control theory began in engineering, and the dynamics of plants became its central problem. The development of decision theory began in economics, and uncertainty became its central problem. While this dichotomy still exists, more and more attention is now being devoted to uncertainty in control theory [53] and to dynamics in decision theory [14].

But there is still an essential difference. Whereas the choice of decision criterion is at the center of decision theory, it is still on the periphery in control theory. A systematic mathematical study of a control system becomes possible only if we define mathematical models of its components: the controlled plant, the controller, and the experiment (observation) the controller can perform over the plant. The same must be true about a decision system.

Therefore, in this chapter we introduce the notion of a decision system and mathematical models of its components: the decision situation, the decision-maker, and the experiment (observation) the decision-maker can perform over the decision situation. An attempt to define a model of the second component of a decision system (the decision-maker) may seem surprising if we do not mention that our model concerns only the sequence of specific operations any decision-maker performs in the process of decision-making."