HANDBOOK OF with Economic Applications

VOLUME I

Edited by

ROBERT J. AUMANN

The Hebrew University of Jerusalem

and

SERGIU HART The Hebrew University of Jerusalem

1992 NORTH-HOLLAND AMSTERDAM • LONDON • NEW YORK • TOKYO CONTENTS OF VOLUME I Introduction to the Series v Contents of the Handbook vii Preface xi

Chapter 1 The Game of HERBERT A. SIMON and JONATHAN SCHAEFFER 1 1. Introduction 2 2. Human chess play 3 3. Computer chess: Origins 5 4. Search versus knowledge 8 4.1. Search 8 4.2. Knowledge 11 4.3. A tale of two programs 12 5. Computer chess play 13 6. The future 14 7. Other games 15 8. Conclusion 15 References 16

Chapter 2 Games in Extensive and Strategic Forms SERGIU HART 19 0. Introduction 20 1. Games in extensive form 20 2. Pure strategies 25 3. Games in strategic form 26 4. Mixed strategies 28 5. Equilibrium points 29 6. Games of 29 7. Behavior strategies and perfect recall 32 References 40 Contents of Volume I

Chapter 3 Games with Perfect Information JAN MYCIELSKI 41 1. Introduction 42 2. Basic concepts 43 3. Open games are determined 45 4. Four classical infinite Pi-games 47 5. The game of Hex and its unsolved problem 51 6. An interplay between some finite and infinite games 52 7. Continuous Pi-games 54 8. The main results of the theory of infinite Pi-games 60 9. Proof of Theorem 8.1(ii) 62 10. The Axiom of 66 References 68

Chapter 4 Repeated Games with Complete Information SYLVAIN SORIN 71 0. Summary 72 1. .Introduction and notation 72 2. Nash equilibria 76 2.1. The infinitely G^ 77

2.2. The discounted game GA 78 2.3. The n-stage game G,, 79 3. Subgame perfect equilibria 80 3.1. G. 80

3.2. GA 81 3.3. G,, / 82 3.4. The recursive structure 84 3.5. Final comments 86 4. Correlated and communication equilibria 86 5. Partial monitoring 87 5.1. Partial monitoring and random payoffs 87 5.2. Signalling functions 91 6. Approachability and strong equilibria 94 6.1. Blackwell's theorem 95 6.2. Strong equilibria 96 7. and repetition 97 7.1. Approximate rationality 97 7.2. Restricted strategies 98 7.3. Pareto optimality and perturbed games 101 8. Concluding remarks 103 Bibliography 104 Contents of Volume I xix

Chapter 5 Repeated Games of Incomplete Information: Zero-Sum SHMUEL ZAMIR 109 1. Introduction 110 1.1. Illustrative examples 111 2. A general model 114 2.1. Classification 115 3. Incomplete information on one side 116 3.1. General properties 116 3.2. Full monitoring 120 3.3. The general case 131 4. Incomplete information on two sides 133 4.1. Minmax and maxmin 133 4.2. The asymptotic value lim,,^ V,XP) 137 4.3. Existence and uniqueness of the solution of the functional equations 138 4.4. The speed of convergence of u,,(p) 140 5. Incomplete information on two sides: The symmetric case 142 6. Games with no signals 144 7. A game with state dependent signaling 146 8. Miscellaneous results 148 8.1. •• Discounted repeated games with incomplete information 148 8.2. Sequential games 148 8.3. A game with incomplete information played by "non-Bayesian players" 150 8.4. A with signals 151 Bibliography 152

Chapter 6 Repeated Games of Incomplete Information: Non-Zero-Sum FRANCOISE FORGES ' 155 1. Introduction 156 2. Basic definitions 157 3. Nash equilibria 160 3.1. The "standard one-sided information case" 160 3.2. Games with known own payoffs 165 3.3. Existence 168 4. Communication equilibria 169 References 176

Chapter 7 Noncooperative Models of Bargaining KEN BINMORE, MARTIN J. OSBORNE and 179 1. Introduction 181 2. A sequential bargaining model 182 xx Contents of Volume I

2.1. Impatience 184 2.2. Shrinking cakes 187 2.3. Discounting 188 2.4. Fixed costs 188 2.5. Stationarity, efficiency, and uniqueness 189 2.6. Outside options 190 2.7. Risk 191 2.8. More than two players 191 2.9. Related work 192 3. The Nash program 193 3.1. Economic modeling 195 4. Commitment and concession 197 4.1. Nash's threat game 198 4.2. The Harsanyi-Zeuthen model 199 4.3. Making commitments stick 200 5. Pairwise bargaining with few agents 200 5.1. One seller and two buyers 200 5.2. Related work 202 6. Noncooperative bargaining models for coalitional games 203 7. Bargaining in markets 204 7.1. Markets in steady state 205 7.2. Unsteady states 206 7.3. Divisible goods with multiple trading 207 7.4. Related work 209 8. Bargaining with incomplete information 210 8.1. An alternating-offers model with incomplete information 211 8.2. Prolonged disagreement 213 8.3. Refinements of m bargaining models 214 8.4. Strategic delay 215 8.5. Related work 216 9. Bargaining and 217 10. Final comments 219 References 220

Chapter 8 Strategic Analysis of Auctions ROBERT WILSON 227 1. Introduction 228 2. Varieties of auctions 229 3. Auctions as games 230 4. Static single-item symmetric auctions 232 4.1. The independent private-values model 235 Contents of Volume I xxi

4.2. The common-value model 237 4.3. Auctions with many bidders 240 4.4. Superior information 241 4.5. Asymmetric payoffs 244 4.6. Attrition games 245 5. Uniqueness and existence of equilibria 246 5.1. Uniqueness 247 5.2. Existence of equilibria in distributional strategies 248 6. Share auctions 250 7. Double auctions 252 7.1. Bid-ask markets 255 8. Applications 256 9. Experimental and empirical evidence 259 9.1. Experimental studies 260 9.2. Empirical studies 261 10. Comparisons of auction rules 263 11. Optimal auctions 266 12. Research frontiers 271 Bibliography 271

Chapter 9 .- . Location JEAN J. GABSZEWICZ and JACQUES-FRANCOIS THISSE 281 1. Introduction 282 2. Location under mill price competition 284 2.1. Variable prices and parametric locations 284 2.2. Variable prices and locations 291 3. Location under discriminatory price competition 294 3.1. Variable prices and parametric locations ' 294 3.2. Variable prices and locations 297 4. Location under nonprice competition 298 4.1. Simultaneous locations 299 4.2. Sequential locations 302 5. Concluding remarks 302 References 303

Chapter 10 Strategic Models of Entry Deterrence ROBERT WILSON 305 1. Introduction 306 2. Preemption 307 3. Signaling 313 xxii Contents of Volume I

3.1. Attrition 313 3.2. Limit pricing 315 4. Predation 318 5. Concluding remarks 323 Bibliography 324

Chapter 11 Patent Licensing MORTON I. KAMIEN 331 1. Introduction 332 2. The license auction game 336 3. The fixed fee licensing game 342 4. Fixed fee licensing of a product innovation 344 5. Royalty licensing 345 6. Fixed fee plus royalty licensing 348 7. An optimal licensing mechanism: The "chutzpah" mechanism. 348 8. Licensing Bertrand competitors 352 9. Concluding remarks 353 References 353

Chapter 12 The and Balancedness YAKAR KANNAI 355 0. Introduction 356 I. Games with Transferable 358 1. Finite set of players 358 2. Countable sets of players 362 3. Uncountable sets of players / 367 4. Special classes of games 370 II. Games with Non-transferable Utility 372 5. Finite set of players 372 6. Infinite set of players 379 III. Economic Applications 381 7. Market games with a finite set of players 381 8. Approximate cores for games and markets with a large set of players 385 References 393

Chapter 13 Axiomatizations of the Core BEZALEL PELEG 397 1. Introduction 398 Contents of Volume I xxiii

2. Coalitional games with transferable utility 399 2.1. Properties of solutions of coalitional games 399 2.2. An axiomatization of the core 403 2.3. An axiomatization of the core of market games 404 2.4. Games with coalition structures 406 3. Coalitional games without side payments 407 3.1. Reduced games of NTU games 407 3.2. An axiomatization of the core of NTU games 408 3.3. A review of "An axiomatization of the core of a cooperative game" by H. Keiding 409 References 411

Chapter 14 The Core in Perfectly Competitive Economies ROBERT M. ANDERSON 413 1. Introduction 414 2. Basics 416 3. Assumptions on preferences and endowments 418 4. Types of convergence 424 5. Survey of convergence results 430 5.1..- Non-convex preferences: Demand-like theorems 433 ~5.2. Strongly convex preferences 434 5.3. «Rate'of convergence 436 5.4. Decentralization by an equilibrium price 439 5.5. Non-convex preferences: Stronger conclusions 440 5.6. Non-monotonic preferences 442 5.7. Replica and type sequences 444 5;8. -.Counterexamples 445 6. Economies with a continuum of agents •' 445 7. Non-standard exchange economies 449 8. Coalition size and the f-core 449 9. Number of improving coalitions 451 10. Infinite-dimensional commodity spaces 451 Bibliography 454

Chapter 15 The Core in Imperfectly Competitive Economies JEAN J. GABSZEWICZ and BENYAMIN SHITOVITZ 459 1. Introduction 460 2. The mathematical model 463 3. Budgetary exploitation: A general price property of core allocations in mixed markets 464 xxiv Contents of Volume I

3.1. The "budgetary exploitation" theorem 464 3.2. An example of a monopolistic market with no equivalence 465 4. Competitive allocations and the core of mixed markets 467 4.1. The core when large traders are similar 467 4.2. The core when to large traders correspond similar small traders 468 5. Restricted competitive allocations and the core of mixed markets 470 6. Budgetary exploitation versus utility exploitation 473 6.1. "Advantageous" and "disadvantageous" monopolies 473 6.2. When budgetary exploitation implies utility exploitation 474 6.3. Disadvantageous monopolies and disadvantageous endowments 476 7. Syndicates 476 8. Conclusions 479 References 482

Chapter 16 Two-Sided Matching ALVIN E. ROTH and MARILDA SOTOMAYOR 485 1. Introduction 486 2. Some empirical motivation 486 2.1. The case of American physicians 486 2.2. Bidder rings in auctions 490 3. Several simple models: Stability, and the polarization of interests in the core 491 3.1. The marriage model 492 3.2. The reformulated college admissions model 494 3.3. Complex preferences over groups 498 3.4. The assignment model 502 4. The structure of the set of stable matchings 507 4.1. Size of the core 511 4.2. The linear structure of the set of stable matchings in the marriage model 512 4.3. Comparative statics: New entrants 514 5. Strategic results 515 5.1. Strategic behavior in models of one-to-one matching 518 5.2. Many-to-one matching: The college admissions model 525 5.3. Incomplete information 527 6. Empirical overview 530 6.1. Some further remarks on empirical matters 532 Bibliography 535

Chapter 17 Von Neumann-Morgenstern Stable Sets WILLIAM F. LUCAS 543 Contents of Volume I xxv

1. Introduction 544 2. Abstract games and stable sets 544 3. The classical model 549 4. Stable sets for three-person games 554 5. Properties of stable sets 562 6. Special classes of games 566 6.1. Simple games 567 6.2. Symmetric games 569 6.3. Simple and symmetric games 569 7. Symmetric stable sets 572 8. Discriminatory stable sets 575 9. Finite stable sets 578 10. Some conclusions 585 Bibliography 587

Chapter 18 The Bargaining Set, Kernel, and Nucleolus MICHAEL MASCHLER 591 1. Introduction 592 2. Basic definitions and notations for games with side payments 594 3. The bargaining set 595 4. The kernel 603 5. The nucleolus 610 6. The reduced game property and consistency 616 7. The dynamic theory 621 8. Related solutions 625 9. Classes of games 628 10. Applications / 631 11. Experiments and empirical data 637 11.1. Laboratory experiments 637 11.2. Evidence from empirical data 641 12. Games without side payments 642 13. Games with an infinite number of players 645 13.1. Games with a countable number of players 645 13.2. Limit behavior 645 13.3. Games with a continuum of players 646 Bibliography 647

Chapter 19 Game and Decision Theoretic Models in Ethics JOHN C. HARSANYI 669 I. Social Utility 671 xxvi Contents of Volume I

1. Ethics as a branch of the general theory of rational behavior 671 2. The axioms of Bayesian decision theory 672 3. An equi-probability model for moral value judgments 675 4. Axioms for rational choice among alternative social policies 677 5. Interpersonal utility comparisons 679 6. Use of von Neumann-Morgenstern in ethics 682 6.1. Outcome utilities and process utilities 682 6.2. Gambling-oriented vs. outcome-oriented attitudes 683 6.3. Von Neumann-Morgenstern utility functions and outcome utilities 683 6.4. Von Neumann-Morgenstern utilities as cardinal utilities 684 II. Rule Utilitarianism, Act Utilitarianism, Rawls' and Brock's Nonutilitarian Theories of Justice 685 7. The two versions of utilitarian theory 685 8. The effects of a socially accepted moral code 687 9. The negative implementation effects of act utilitarian morality 687 10. The value of free individual choice 688 11. Morally protected rights and obligations, and their expectation effects 689 12. The advantages of the rule utilitarian approach 691 13. A game-theoretic model for a rule utilitarian society 692 14. Rawls' theory of justice 694 15. Brock's theory of social justice based on the Nash solution and on the 696 15.1. Nature of the theory 696 15.2. Brock's theory of "need justice" 698 15.3. Brock's theory of "merit justice" 701 III. Reassessing Individual Utilities / 702 16. Mistaken preferences vs. informed preferences 702 17. Exclusion of malevolent preferences 703 18. Exclusion of other-oriented preferences 704 19. Conclusion 706 References 706

Index 709