A Course in Game Theory

Citation

Osborne, M. J., & Rubinstein, A. (1994). A Course in Game Theory. The MIT Press.

Chapter Summary

The book has fifteen chapters in four parts, preceded by a preface and followed by a list of results, references and an index. Its length is about 350 pages, and each chapter closes with a section of notes. The authors describe it as aimed at graduate students and advanced undergraduates, with precise definitions and full proofs, accepting less generality and a narrower scope in return.

Chapter 1: Introduction

The authors present game theory as a set of analytical tools for studying decision-makers who interact, and explain that they treat it as a social science rather than as a branch of mathematics. The chapter distinguishes noncooperative and cooperative models, strategic and extensive games, and games with perfect and imperfect information, which is how the four parts are organised. It also contrasts game theory with competitive equilibrium, then discusses rational behavior, the steady-state and deductive interpretations of solutions, bounded rationality, and terminology and notation.

Part I: Strategic Games (Chapters 2–5)

Chapter 2, Nash Equilibrium, defines strategic games and Nash equilibrium, gives examples and an existence result, and covers strictly competitive games and Bayesian games. Chapter 3, Mixed, Correlated, and Evolutionary Equilibrium, treats mixed-strategy equilibrium and its interpretations, correlated equilibrium and evolutionary equilibrium. Chapter 4, Rationalizability and Iterated Elimination of Dominated Actions, covers rationalizability and the elimination of strictly and weakly dominated actions. Chapter 5, Knowledge and Equilibrium, models knowledge and common knowledge, asks whether people can agree to disagree, links knowledge to solution concepts, and ends with the electronic mail game.

Part II: Extensive Games with Perfect Information (Chapters 6–10)

Chapter 6 defines extensive games with perfect information and subgame perfect equilibrium, discusses the interpretation of a strategy, two notable finite-horizon games, and iterated elimination of weakly dominated strategies. Chapter 7, Bargaining Games, studies alternating-offers bargaining and its subgame perfect equilibrium, with variations. Chapter 8, Repeated Games, distinguishes infinitely and finitely repeated games and presents strategies as machines and a series of folk theorems under different criteria (limit of means, overtaking, discounting). Chapter 9, Complexity Considerations in Repeated Games, introduces machine games and the structure of their equilibria. Chapter 10, Implementation Theory, covers the implementation problem and implementation in dominant strategies, in Nash equilibrium and in subgame perfect equilibrium.

Part III: Extensive Games with Imperfect Information (Chapters 11–12)

Chapter 11 defines extensive games with imperfect information, principles for the equivalence of extensive games, framing effects, mixed and behavioral strategies, and Nash equilibrium. Chapter 12, Sequential Equilibrium, develops strategies and beliefs, sequential equilibrium, perfect Bayesian equilibrium for games with observable actions, refinements, and trembling hand perfect equilibrium.

Part IV: Coalitional Games (Chapters 13–15)

Chapter 13, The Core, defines coalitional games with transferable payoff, the core and conditions for its nonemptiness, markets with transferable payoff, games without transferable payoff and exchange economies. Chapter 14, Stable Sets, the Bargaining Set, and the Shapley Value, presents these four solution concepts. Chapter 15, The Nash Solution, treats bargaining problems, the axiomatic definition, its relation to the alternating-offers game and an exact implementation.

Key Concepts

Chapters 1–5

• Strategic game and Nash equilibrium: a simultaneous-move model and the profile in which no player gains by deviating.
• Mixed and correlated equilibrium: equilibria with randomization, and the interpretations the authors give of it.
• Rationalizability: the actions that survive iterated reasoning about others' beliefs.
• Common knowledge: what the players know, know that others know, and so on, and its role in solution concepts.

Chapters 6–10

• Subgame perfect equilibrium: equilibrium that remains optimal after every history.
• Folk theorems: the sets of outcomes sustainable in infinitely repeated games.
• Machine games: repeated games in which strategy complexity matters.
• Implementation: designing rules so that equilibrium outcomes match a desired social choice.

Chapters 11–15

• Sequential equilibrium and perfect Bayesian equilibrium: equilibrium with beliefs at information sets.
• Trembling hand perfection: robustness to small mistakes.
• The core, stable sets, bargaining set, Shapley value: solutions for coalitional games.
• Nash solution: the axiomatic solution to a two-person bargaining problem.

Critical Analysis

Strengths

• Foundations and interpretation. The authors say they emphasise foundations and the interpretation of basic concepts, and Chapter 1 gives an unusually explicit discussion of what a solution means, including the steady-state and deductive readings and bounded rationality. Sections 3.2 and 6.4 are devoted to interpreting mixed strategies and strategies in extensive games.
• Precise and compact. Definitions and results are stated exactly, with full proofs; the whole four-part theory fits in about 350 pages, with a list of results as a reference aid.
• Broad coverage of the theory. The book covers strategic, extensive and coalitional games, repeated games, implementation, knowledge and bounded-complexity strategies in one book.
• Support from the authors. The authors' site maintains corrections to the English edition and offers a solution manual.

Limitations

• Pure theory. The authors say they stay almost entirely in the territory of pure theory; the art of applying a model to a real situation is left to another book.
• Concise. The authors accept less generality for the sake of rigour and scope, and the text is not a gentle on-ramp for readers who have never seen formal game theory.
• Dated emphasis. The authors put less emphasis on imperfect-information games because those models were developed intensively only in the 1980s and were less mature; the text, published in 1994, does not cover later developments.

Where It Fits in the Course

The other books in this course start from easier ground. Dixit, Skeath and McAdams's Games of Strategy and Dixit and Nalebuff's The Art of Strategy teach strategic thinking through examples with little mathematics, and Binmore's Game Theory: A Very Short Introduction is a brief, lively introduction with very little formalism. Gibbons's Game Theory for Applied Economists, titled for applied economists, is the other advanced book in this list.

Osborne and Rubinstein is the book to move to when you want the definitions and proofs behind those introductions and a critical sense of what each solution concept means. Read it after an applied or intuitive course, and use it as the reference for the theory.

How to Use This Book

The book is for graduate students and advanced undergraduates with enough mathematical maturity to read definitions and proofs. An applied or intuitive course beforehand will make the abstraction easier to follow.

A practical plan:

• Core: Chapter 1; Chapters 2–4 (strategic games); Chapter 6 (extensive games and subgame perfection); Chapters 11–12 (imperfect information and sequential equilibrium).
• Next: Chapter 7 (bargaining), Chapter 8 (repeated games) and Chapter 13 (the core).
• Optional: Chapter 5 (knowledge), Chapters 9–10 (complexity and implementation) and Chapters 14–15 (further coalitional solutions).

Read the notes at the end of each chapter for references, use the list of results to look up statements, and consult the authors' corrections file.