Set Functions, Games and Capacities in Decision Making

Description

Foreword......Page 3 Preface......Page 4 Contents......Page 7 1.1 Notation......Page 13 1.2 General Technical Results......Page 15 1.3.1 Binary Relations and Orders......Page 19 1.3.2 Partially Ordered Sets and Lattices......Page 20 1.3.4 Linear Inequalities and Polyhedra......Page 25 1.3.5 Linear Programming......Page 27 1.3.6 Cone Duality......Page 29 1.3.7 Support Functions of Convex Sets......Page 30 1.3.8 Convex Optimization and Quadratic Programming......Page 31 1.3.9 Totally Unimodular Matrices and PolyhedronIntegrality......Page 32 1.3.10 Riesz Spaces......Page 33 1.3.11 Laplace and Fourier Transforms......Page 34 2 Set Functions, Capacities & Games......Page 36 2.1 Set Functions and Games......Page 37 2.3 Capacities......Page 38 2.4.1 In Decision and Game Theory......Page 39 2.4.2 In Operations Research......Page 41 Reliability......Page 42 2.5 Derivative of a Set Function......Page 43 2.6 Monotone Cover of a Game......Page 44 2.7 Properties......Page 45 2.8.2 Unanimity Games......Page 53 2.8.3 Possibility and Necessity Measures......Page 54 2.8.5 Decomposable Measures......Page 55 2.8.6 λ-Measures......Page 57 2.10 The Möbius Transform......Page 60 2.10.1 Properties......Page 63 0-1-Capacities......Page 65 Belief and Plausibility Measures......Page 66 Possibility and Necessity Measures......Page 67 λ-Measures......Page 68 2.11 Other Transforms......Page 69 2.12.1 Definitions and Examples......Page 71 2.12.2 Generator Functions, Cardinality Functions......Page 73 2.12.3 Inverse of Cardinality Operators......Page 74 2.12.4 The Co-Möbius Operator......Page 75 2.12.5 The Interaction Operator......Page 76 2.12.6 The Banzhaf Interaction Operator......Page 80 2.12.7 Transforms of Conjugate Set Functions......Page 82 2.13 k-Additive Games......Page 84 2.14 p-Symmetric Games......Page 85 2.15.1 The Vector Space of Games......Page 86 2.15.2 The Cone of Capacities......Page 88 2.15.3 The Cone of Supermodular Games......Page 89 2.15.4 The Cone of Totally Monotone Nonnegative Games......Page 90 2.15.5 The Riesz Space of Games......Page 91 2.15.6 The Polytope of Normalized Capacities......Page 92 2.15.7 The Polytope of Belief Measures......Page 99 2.15.8 The Polytope of At Most k-Additive Normalized Capacities......Page 100 2.16 Polynomial Representations......Page 102 2.16.1 Bases of PB(n)......Page 103 2.16.2 The Fourier Transform......Page 107 2.16.3 Approximations of a Fixed Degree......Page 112 Approximation of Degree 1: Approximation of a Game by an Additive Game......Page 115 2.16.4 Extensions of Pseudo-Boolean Functions......Page 119 The Owen Extension......Page 120 The Lovász Extension......Page 127 2.17.1 Transforms and Bases......Page 128 2.17.2 The Inverse Problem......Page 134 2.18 Inclusion-Exclusion Coverings......Page 135 2.19 Games on Set Systems......Page 140 2.19.1 Case Where X Is Arbitrary......Page 141 Null Sets......Page 143 Supermodular and Convex Games......Page 144 2.19.2 Case Where X Is Finite......Page 145 Weakly Union-Closed Set Systems......Page 146 Comparisons and Further Remarks......Page 147 Supermodular and Convex Games......Page 149 Modular and Additive Games......Page 151 k-Monotone and Totally Monotone Games......Page 152 3 Core & Selectope of Games......Page 156 3.1 Definition and Interpretations of the Core......Page 157 3.2.1 Nonemptiness of the Core......Page 159 3.2.2 Extreme Points of the Core......Page 165 3.2.3 Additivity Properties......Page 167 3.3.1 Nonemptiness of the Core......Page 168 3.3.2 Boundedness......Page 169 3.3.3 Extremal Rays......Page 172 3.3.4 Extreme Points......Page 173 3.3.6 Bounded Faces......Page 180 3.4 Exact Games, Totally Balanced Games, Large Cores and Stable Sets......Page 185 3.5 The Selectope......Page 192 4 Integrals......Page 199 4.1 Simple Functions......Page 200 4.2 The Choquet and Sugeno Integrals for Nonnegative Functions......Page 201 4.3 The Case of Real-Valued Functions......Page 206 4.3.1 The Choquet Integral......Page 207 4.3.2 The Sugeno Integral......Page 210 4.4.1 The Choquet Integral of Nonnegative Functions......Page 212 4.4.2 The Sugeno Integral of Nonnegative Functions......Page 214 4.4.3 The Case of Real-Valued Functions......Page 216 4.5.1 The Case of Nonnegative Functions......Page 217 4.5.2 The Case of Real-Valued Integrands......Page 220 4.6 Properties......Page 221 Elementary Properties......Page 222 Comonotonic Additivity......Page 225 Horizontal Additivity......Page 226 Concavity......Page 231 4.6.2 The Sugeno Integral......Page 237 4.7.1 The Choquet Integral......Page 244 4.7.2 The Sugeno Integral......Page 247 4.8.1 The Choquet Integral......Page 249 4.8.2 The Sugeno Integral......Page 254 4.9.1 The Choquet Integral......Page 256 4.9.2 The Sugeno Integral......Page 261 4.10.1 Computation of the Choquet Integral......Page 264 4.10.2 Equimeasurable Rearrangement......Page 268 4.11.1 The Shilkret Integral......Page 269 4.11.2 The Concave Integral......Page 270 4.11.3 The Decomposition Integral......Page 275 4.11.4 Pseudo-Additive Integrals, Universal Integrals......Page 280 4.12 The Choquet Integral for Nonmeasurable Functions......Page 282 5 Decision under Risk & Uncertainty......Page 290 5.1.1 The Components of a Decision Problem......Page 291 5.1.2 Introduction of Probabilities......Page 293 5.1.3 Introduction of Utility Functions......Page 294 5.2 Decision Under Risk......Page 295 5.2.1 The Expected Utility Criterion......Page 296 5.2.2 Stochastic Dominance......Page 298 5.2.3 Risk Aversion......Page 300 5.2.4 The Allais Paradox......Page 301 5.2.5 Transforming Probabilities......Page 302 5.2.6 Rank Dependent Utility......Page 303 5.2.7 Prospect Theory......Page 309 5.3.1 The Expected Value Criterion and the Dutch Book Argument......Page 312 5.3.2 The Expected Utility Criterion......Page 315 5.3.3 The Ellsberg Paradox......Page 317 5.3.4 Choquet Expected Utility......Page 318 5.3.5 Ambiguity and Multiple Priors......Page 323 5.4 Qualitative Decision Making......Page 326 5.4.1 Decision Under Risk......Page 327 5.4.2 Decision Under Uncertainty......Page 330 6 Decision with Multiple Criteria......Page 333 6.1 The Framework......Page 334 6.2.1 The Fundamental Problem of Measurement......Page 336 6.2.2 Main Types of Scales......Page 337 6.2.3 Ordinal Measurement......Page 338 6.2.4 Difference Measurement......Page 342 6.3 Affect, Bipolarity and Reference Levels......Page 344 6.3.1 Bipolarity......Page 345 6.3.2 Reference Levels......Page 346 6.3.3 Bipolar and Unipolar Scales......Page 347 6.4.1 The MACBETH Method......Page 349 6.4.2 Determination of the Value Functions......Page 350 6.6 The Weighted Arithmetic Mean as an Aggregation Function......Page 352 6.7 Towards a More General Model of Aggregation......Page 354 6.7.1 The Unipolar Case......Page 355 6.7.2 The Bipolar Case......Page 357 6.8 The Multilinear Model......Page 362 6.9 Summary on the Construction of the Aggregation Function......Page 365 6.10.1 Importance and Interaction Indices for a Capacity......Page 366 6.10.2 Importance and Interaction Indices for an Aggregation Function......Page 368 6.10.3 A Statistical Approach: The Sobol' Indices......Page 371 6.10.4 The 2-Additive Model......Page 373 6.11 The Case of Ordinal Measurement......Page 375 6.11.1 The Emergence of the Sugeno Integral Model......Page 376 6.11.2 Monotonicity Properties of the Sugeno Integral Model......Page 378 6.11.3 Lexicographic Refinement......Page 380 7 Dempster-Shafer & Possibility Theory......Page 384 7.1.1 Dempster's Upper and Lower Probabilities......Page 385 7.2 Shafer's Evidence Theory......Page 386 7.2.1 The Case Where m()>0......Page 391 7.2.2 Kramosil's Probabilistic Approach......Page 392 7.2.3 Random Sets......Page 393 7.3 Dempster's Rule of Combination......Page 398 7.3.1 The Rule of Combination in the Framework of Evidence Theory......Page 399 To Normalize or Not to Normalize?......Page 401 The m()>0 Issue Again and the Nonnormalized Rule......Page 402 The Combination Rule in the Framework of Random Sets......Page 403 7.3.3 Decomposition of Belief Functions into Simple Belief Functions......Page 404 7.4 Compatible Probability Measures......Page 405 7.5 Conditioning......Page 406 7.5.1 The General Conditioning Rule......Page 407 7.5.2 The Bayes' and Dempster-Shafer Conditioning Rules......Page 413 7.6 The Transferable Belief Model......Page 418 7.7 Possibility Theory......Page 420 7.7.1 The Framework......Page 421 7.7.2 Link with Dempster-Shafer Theory......Page 425 7.7.3 Links Between Possibility Measures and Probability Measures......Page 426 7.7.4 The Possibilistic Core and Totally Monotone Anticore......Page 430 Belief Functions......Page 434 k-Monotone and Totally Monotone Functions......Page 435 Properties of Belief Functions......Page 436 Probability Measures on Distributive Lattices......Page 439 Possibility and Necessity Measures......Page 441 7.8.2 Infinite Spaces......Page 444 A.1 Bases and Transforms of Set Functions......Page 445 A.2 Conversion Formulae Between Transforms......Page 446 Symbols......Page 448 Refs......Page 455 Index......Page 469