Managerial Logic

Hervé Raynaud, a former student of the école Polytechnique and an honorary emeritus professor of Mathematics, has worked with about 20 academic organizations around the world. A former psychoanalyst, researcher, and international consultant, he is now teaching at the Sigmund Freud University of Vienna, in its Paris branch, and at the Conservatoire National de la Villette in France.

Kenneth J. Arrow is an American economist and a joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. His most significant works deal with social choice theory, notably "Arrow's impossibility theorem" and his research on general equilibrium analysis. He is currently the Joan Kenney Professor of Economics and Professor of Operations Research, Emeritus at Stanford University.

General Introduction xi

PART 1. A PARADOXICAL RESEARCH FIELD 1

Chapter 1. The Initial Problem 5

1.1. Introduction 5

1.2. The decision makers and their consultants’ usual work11

1.3. Toward a paradigm for managerial decision-making 21

1.4. Exercises 28

1.5. Corrected exercises 32

Chapter 2. Paradoxes 35

2.1. Arrow’s axiomatic system 36

2.2. May’s axiomatic system 43

2.3. Strategic majority voting 44

2.4. Exercises 47

2.5. Corrected exercises 53

PART 2. A CENTRAL CASE: THE MAJORITY METHOD 57

Chapter 3. Majority Method and Limited Domain 61

3.1. Sen’s lemma [SEN 66] 62

3.2. Coombs’ condition 63

3.3. Black’s unimodality condition [BLA 48, BLA 58] 66

3.4. Romero’s arboricity 67

3.5. Romero’s quasi-unimodality 69

3.6. Arrow–Black’s single-peakedness 72

3.7. The Cij’s 74

3.8. Exercises 78

3.9. Corrected exercises 80

Chapter 4. Intuition Can Easily Suggest Errors 87

4.1. Inada’s conditions 87

4.2. Is the bipartition the same as the NITM condition? 88

4.3. Diversity of the NIMT condition 92

4.4. Exercises 94

4.5. Corrected exercises 94

Chapter 5. Would Transitivity be a Prohibitive Luxury? 97

5.1. Star-shapedness 98

5.2. Ward’s condition 101

5.3. The failure of the majority method 104

5.4. Exercises 106

5.5. Corrected exercises 106

Conclusion of the Second Part 109

PART 3. AXIOMATIZING CHOICE FUNCTIONS 111

Chapter 6. Helpful Tools for the Sensible Decision Maker 117

6.1. The “habitual” decision maker and his/her traditional means 117

6.2. The habitual decision maker 124

6.3. A “sensible” decision maker confronted with a difficult decision 137

6.4. The urgency of raising the moral standard of the market 138

6.5. Conclusion 141

6.6. Exercises 146

6.7. Corrected exercises 149

Chapter 7. An Important Class of Choice Functions 153

7.1. Introduction 153

7.2. The problem: various definitions 154

7.3. Natural properties of the E-matrices and B-F-matrices 156

7.4. Choice functions that depend only on the E-matrix or on the B-F-matrix 158

7.5. Characterization of the choice functions that depend only on the E-matrix (respectively, B-F-matrix) 161

7.6. Conclusion 163

7.7. Exercises 165

7.8. Corrected exercises 167

Chapter 8. Prudent Choice Functions 171

8.1. Introduction 171

8.2. Toward the prudence axiom 172

8.3. Properties related to prudence for choice functions 179

8.4. Exercises 182

8.5. Corrected exercises 186

Chapter 9. Often Implicit Axioms: Sovereignty, Homogeneity, Decision by Rejection or Selection, Prudence and Violence 191

9.1. Introduction 191

9.2. Sovereignty 193

9.3. Homogeneous choice 195

9.4. Choice by selection and choice by rejection 198

9.5. Violent choice and prudent choice 202

9.6. Exercises 205

9.7. Corrected exercises 207

Chapter 10. Coherent Choice Functions 211

10.1. Introduction 211

10.2. Characterization of the Borda method 211

10.3. Coherence and the other axioms 218

10.4. Exercises 223

10.5. Corrected exercises 224

Chapter 11. Rationality and Independence 227

11.1. Introduction 227

11.2. Rationalities 228

11.3. Axioms of independence 237

11.4. The inclusive iteration principle 242

11.5. Conclusion 243

11.6. Exercises 245

11.7. Corrected exercises 246

Chapter 12. Monotonic Choice Functions 251

12.1. Introduction 251

12.2. Monotonicity defined 252

12.3. Prudence and monotonicity 258

12.4. Prudence and binary monotonic independence 260

12.5. Strong monotonicity 262

12.6. Exercises 263

12.7. Corrected exercises 264

PART 4. MULTICRITERION RANKING FUNCTIONS 267

Chapter 13. Sequentially Independent Rankings 275

13.1. Introduction 275

13.2. The sequential independence axioms 277

13.3. Sequential independence with current choice and rejection functions 281

13.4. Exercises 287

13.5. Corrected exercises 290

Chapter 14. Prudent Rankings 293

14.1. Introduction 293

14.2. Some unexpected theorems 294

14.3. Prudent rankings 297

14.4. Prudence in preorders and iterated prudent choice 300

14.5. Exercises 305

14.6. Corrected exercises 307

Chapter 15. Coherent Condorcet Rankings 313

15.1. Introduction 313

15.2. What does one call Kemeny’s method or second Condorcet method? 313

15.3. Young and Levenglick’s theorem 319

15.4. Exercises 322

15.5. Corrected exercises 326

Chapter 16. Monotonic Rankings 333

16.1. Definitions of monotonicity for ranking functions 333

16.2. Monotonicity of the most ordinary non-sequential multicriterion ranking function 339

16.3. Various remarks 346

16.4. Exercises 348

16.5. Corrected exercises 350

Concluding Remarks 355

Bibliography 367

APPENDICES 377

Appendix 1. Benjamin Franklin’s Letter 379

Appendix 2. Pyramids and Snakes: Romero’s Algorithm 381

Appendix 3. A Few Widespread Commercial Multicriterion Decision Techniques 387

Index 405

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