Introduction to Game Theory / Edition 1

Introduction to Game Theory / Edition 1

by Peter Morris
     
 

This is a textbook for a course in the theory of games. It is intended for advanced undergraduates and graduate students in mathematics and other quantitative disciplines, e.g., statistics, operations research, etc. It treats the central topics in game theory and is meant to give students a basis from which they can go on to more advanced topics. The subject matter

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Overview

This is a textbook for a course in the theory of games. It is intended for advanced undergraduates and graduate students in mathematics and other quantitative disciplines, e.g., statistics, operations research, etc. It treats the central topics in game theory and is meant to give students a basis from which they can go on to more advanced topics. The subject matter is approached in a mathematically rigorous way, but , within this constraint, an effort is made to keep it interesting and lively. New definitions and topics are motivated as thoroughly as possible. The mathematical prerequisites for understanding the book are modest: basic probability together with a little calculus and linear algebra.

Among others, two topics of great current interest are discussed in this book. The idea of iterated Prisoner's Dilemma (super games) is considered. It is specially of great interest to biologists, sociologists and others who use it in studying the evolution of cooperative behavior both in nature and in human society. Also covered are challenging game-playing computer programs.

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Product Details

ISBN-13:
9780387942841
Publisher:
Springer New York
Publication date:
07/28/1994
Series:
Universitext Series
Edition description:
1994
Pages:
252
Product dimensions:
9.21(w) x 6.14(h) x 0.53(d)

Table of Contents

1. Games in Extensive Form.- 1.1. Trees.- 1.2. Game Trees.- 1.2.1. Information Sets.- 1.3. Choice Functions and Strategies.- 1.3.1. Choice Subtrees.- 1.4. Games with Chance Moves.- 1.4.1. A Theorem on Payoffs.- 1.5. Equilibrium N-tuples of Strategies.- 1.6. Normal Forms.- 2. Two-Person Zero-Sum Games.- 2.1. Saddle Points.- 2.2. Mixed Strategies.- 2.2.1. Row Values and Column Values.- 2.2.2. Dominated Rows and Columns.- 2.3. Small Games.- 2.3.1. 2 × n and m × 2 Games.- 2.4. Symmetric Games.- 2.4.1. Solving Symmetric Games.- 3. Linear Programming.- 3.1. Primal and Dual Problems.- 3.1.1. Primal Problems and Their Duals.- 3.2. Basic Forms and Pivots.- 3.2.1. Pivots.- 3.2.2. Dual Basic Forms.- 3.3. The Simplex Algorithm.- 3.3.1. Tableaus.- 3.3.2. The Simplex Algorithm.- 3.4. Avoiding Cycles and Achieving Feasibility.- 3.4.1. Degeneracy and Cycles.- 3.4.2. The Initial Feasible Tableau.- 3.5. Duality.- 3.5.1. The Dual Simplex Algorithm.- 3.5.2. The Duality Theorem.- 4. Solving Matrix Games.- 4.1. The Minimax Theorem.- 4.2. Some Examples.- 4.2.1. Scissors-Paper-Stone.- 4.2.2. Three-Finger Morra.- 4.2.3. Colonel Blotto’s Game.- 4.2.4. Simple Poker.- 5. Non-Zero-Sum Games.- 5.1. Noncooperative Games.- 5.1.1. Mixed Strategies.- 5.1.2. Maximin Values.- 5.1.3. Equilibrium N-tuples of Mixed Strategies.- 5.1.4. A Graphical Method for Computing Equilibrium Pairs.- 5.2. Solution Concepts for Noncooperative Games.- 5.2.1. Battle of the Buddies.- 5.2.2. Prisoner’s Dilemma.- 5.2.3. Another Game.- 5.2.4. Supergames.- 5.3. Cooperative Games.- 5.3.1. Nash Bargaining Axioms.- 5.3.2. Convex Sets.- 5.3.3. Nash’s Theorem.- 5.3.4. Computing Arbitration Pairs.- 5.3.5. Remarks.- 6. N-Person Cooperative Games.- 6.1. Coalitions.- 6.1.1. The Characteristic Function.- 6.1.2. Essential and Inessential Games.- 6.2. Imputations.- 6.2.1. Dominance of Imputations.- 6.2.2. The Core.- 6.2.3. Constant-Sum Games.- 6.2.4. A Voting Game.- 6.3. Strategic Equivalence.- 6.3.1. Equivalence and Imputations.- 6.3.2. (0,1)-Reduced Form.- 6.3.3. Classification of Small Games.- 6.4. Two Solution Concepts.- 6.4.1. Stable Sets of Imputations.- 6.4.2. Shapley Values.- 7. Game-Playing Programs.- 7.1. Three Algorithms.- 7.1.1. The Naive Algorithm.- 7.1.2. The Branch and Bound Algorithm.- 7.1.3. The Alpha-Beta Pruning Algorithm.- 7.2. Evaluation Functions.- 7.2.1. Depth-Limited Subgames.- 7.2.2. Mancala.- 7.2.3. Nine-Men’s Morris.- Appendix. Solutions.

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