Game Theory: A Very Short Introduction

Citation

Binmore, K. (2007). Game Theory: A Very Short Introduction. Oxford University Press.

Chapter Summary

The book has a preface and ten chapters in about 175 pages. The publisher's description presents it as a non-mathematical tour of game theory, the study of how to play games rationally, that starts from everyday games such as driving in traffic, bidding on eBay and setting supermarket prices, and includes short biographies of the subject's founders, including John Nash. The first three chapters build the basic toolkit (Nash equilibrium, mixed strategies, backward induction); the remaining seven apply and extend it to conventions, repeated games, information, auctions, biology and bargaining, and close with a chapter on common fallacies.

Chapter 1: The name of the game

Introduces game theory and its assumption that players behave rationally. It presents von Neumann's minimax theorem, the idea of utility as a numerical ranking of outcomes, and the Nash equilibrium, illustrated by the Prisoner's Dilemma.

Chapter 2: Chance

Explains mixed strategies, in which a player randomises to keep an opponent guessing, and the mixed Nash equilibria they produce. The chapter notes that players randomise only between strategies that give them equal payoffs, that equilibrium is usually reached through trial-and-error learning, and returns to the minimax theorem as a worst-case approach.

Chapter 3: Time

Shows how the timing of moves matters in games such as chess and poker, representing them as game trees rather than payoff tables. Backward induction and the elimination of dominated strategies lead, in games of perfect information, to equilibria in the whole game and in every subgame.

Chapter 4: Conventions

As the title indicates, this chapter turns to conventions; the closing chapter later argues that conventions are established by cultural evolution rather than handed down by some outside authority.

Chapter 5: Reciprocity

Examines repeated games. Backward induction rules out cooperation when a game is repeated a known, finite number of times, but open-ended relationships make it feasible. The chapter discusses the folk theorem (cooperation can be sustained without an external enforcer, provided deviations are visible), the limits of tit-for-tat in groups, and game-theoretic accounts of authority, duty and emotions.

Chapter 6: Information

Covers games of imperfect information, using poker and bluffing as the main example. Drawing on Hobbes, it introduces a player's "type", the assumption that types are common knowledge, and signalling, which usually requires a costly action.

Chapter 7: Auctions

Introduces mechanism design: how a principal can use rules and incentives to make agents behave in a desired way. Auctions of public assets are contrasted with the earlier "beauty contest" approach; the chapter describes English, Dutch and sealed-bid auctions and stresses that auction design must be tailored to the situation.

Chapter 8: Evolutionary biology

Applies game theory to evolution, treating behaviour as strategy and fitness as utility. It introduces replicators and evolutionarily stable outcomes, revisits the Hawk-Dove game, the Prisoner's Dilemma and Chicken, and discusses cooperation within families and cultural replicators, or memes.

Chapter 9: Bargaining and coalitions

As its title indicates, this chapter deals with bargaining between players and with the coalitions they form.

Chapter 10: Puzzles and paradoxes

Dismantles popular fallacies, especially about the Prisoner's Dilemma: arguments from the categorical imperative, the "fallacy of the twins", the wasted-vote myth and the transparent-disposition fallacy. It also discusses Newcomb's paradox and the surprise test paradox.

Key Concepts

Chapters 1–3: The Basic Toolkit

• Nash equilibrium: a combination of strategies in which no player can do better by changing strategy alone.
• Utility: a numerical scale that ranks the outcomes of a game for a player.
• Mixed strategy: choosing among pure strategies at random to keep an opponent guessing.
• Backward induction: solving a game tree by working back from its final moves.

Chapters 4–6: Social Interaction and Information

• Folk theorem: in indefinitely repeated games, many cooperative outcomes can be sustained by the threat of punishment.
• Common knowledge: what all players know, know that the others know, and so on; assumed for player types in standard analysis.
• Signalling: costly actions that reveal a player's type.

Chapters 7–10: Applications and Fallacies

• Mechanism design: designing the rules of a game, such as an auction, to achieve a goal.
• Replicator: anything, such as a gene or a meme, that copies itself and determines behaviour in a game, the basis of evolutionary game theory.
• Prisoner's Dilemma fallacies: mistaken arguments that rational players should cooperate in a one-shot dilemma.

Critical Analysis

Strengths

• Written by a leading game theorist. Binmore is presented by the publisher as a renowned game theorist, and despite its length the book covers mechanism design and auctions, evolutionary game theory and the logic of conventions as well as the standard solution concepts.
• No equations needed. The publisher stresses that the subject is explained without mathematical equations; payoff tables and game trees carry the arguments.
• Breadth across disciplines. Applications run from poker and auctions to the sex ratio among bees, cultural evolution and moral philosophy, which suits readers from economics, biology, politics and philosophy.
• Corrects common misreadings. The final chapter's treatment of fallacies about the Prisoner's Dilemma is a useful corrective for students who meet the game in popular writing.

Limitations

• Very compressed. Ten chapters in about 175 pages means that each concept gets a few pages; readers wanting worked exercises or formal definitions will need a textbook.
• A distinctive point of view. The closing chapter sets out to discredit a series of arguments as fallacies, and the reciprocity chapter offers game-theoretic explanations of authority, duty and emotions; readers should recognise these as the author's positions and compare them with other treatments.
• No exercises. The book is designed for reading, not for practising technique.

Where It Fits in the Course

Within this course list, Binmore's book is the shortest overview and the quickest way to see what game theory covers. Dixit and Nalebuff's The Art of Strategy is a longer, story-driven non-technical introduction that asks readers to work through cases; the two pair well, with Binmore more conceptual and Dixit and Nalebuff more practical. Dixit, Skeath and McAdams's Games of Strategy is the undergraduate textbook that turns the same ideas into a course with exercises.

For more formal study, Gibbons's Game Theory for Applied Economists gives a compact, more formal treatment of equilibrium, repeated games and incomplete information for economics students, and Osborne and Rubinstein's A Course in Game Theory is a graduate text with precise definitions and proofs. Binmore's own full-length textbook, Playing for Real (Oxford University Press, 2007), is a further option for readers who like his approach.

How to Use This Book

The book suits beginners from any discipline, students about to start a game theory course, and readers who want a map of the field before tackling a textbook. School mathematics is enough.

• Read Chapters 1–3 first and slowly: they introduce the solution concepts that the rest of the book uses.
• Choose by interest after that: economics students should prioritise Chapters 5–7 and 9; biology and philosophy students Chapters 4, 8 and 10.
• Work with pencil and paper: redraw the payoff tables and game trees and check the equilibria yourself.

As a companion to a course, read the relevant chapter before each lecture block (for example Chapter 3 before extensive-form games, Chapter 6 before incomplete information, Chapter 7 before auctions), then do exercises from Games of Strategy or Gibbons.