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ISBN 10: 9813227346
ISBN 13: 978-9813227347
Author: Thomas Ferguson
Game theory is a fascinating subject. We all know many entertaining games, such as chess, poker, tic-tac-toe, bridge, baseball, computer games — the list is quite varied and almost endless. In addition, there is a vast area of economic games, discussed in Myerson (1991) and Kreps (1990), and the related political games [Ordeshook (1986), Shubik (1982), and Taylor (1995)]. The competition between firms, the conflict between management and labor, the fight to get bills through congress, the power of the judiciary, war and peace negotiations between countries, and so on, all provide examples of games in action. There are also psychological games played on a personal level, where the weapons are words, and the payoffs are good or bad feelings [Berne (1964)]. There are biological games, the competition between species, where natural selection can be modeled as a game played between genes [Smith (1982)]. There is a connection between game theory and the mathematical areas of logic and computer science. One may view theoretical statistics as a two-person game in which nature takes the role of one of the players, as in Blackwell and Girshick (1954) and Ferguson (1968). Games are characterized by a number of players or decision makers who interact, possibly threaten each other and form coalitions, take actions under uncertain conditions, and finally receive some benefit or reward or possibly some punishment or monetary loss. In this text, we present various mathematical models of games and study the phenomena that arise. In some cases, we will be able to suggest what courses of action should be taken by the players. In others, we hope simply to be able to understand what is happening in order to make better predictions about the future.
A Course in Game Thoery 1st Table of contents:
Introduction
Contents
Part I. Impartial Combinatorial Games
1. Take-Away Games
1.1 A Simple Take-Away Game
1.2 What is a Combinatorial Game?
1.3 P-positions, N-positions
1.4 Subtraction Games
1.5 Exercises
2. The Game of Nim
2.1 Preliminary Analysis
2.2 Nim-Sum
2.3 Nim with a Larger Number of Piles
2.4 Proof of Bouton’s Theorem
2.5 Misère Nim
2.6 Exercises
3. Graph Games
3.1 Games Played on Directed Graphs
3.2 The Sprague-Grundy Function
3.3 Examples
3.4 The Sprague-Grundy Function on More General Graphs
3.5 Exercises
4. Sums of Combinatorial Games
4.1 The Sum of n Graph Games
4.2 The Sprague-Grundy Theorem
4.3 Applications
4.4 Take-and-Break Games
4.5 Exercises
5. Coin Turning Games
5.1 Examples
5.2 Two-dimensional Coin-Turning Games
5.3 Nim Multiplication
5.4 Tartan Games
5.5 Exercises
6. Green Hackenbush
6.1 Bamboo Stalks
6.2 Green Hackenbush on Trees
6.3 Green Hackenbush on General Rooted Graphs
6.4 Exercise
Part II. Two-Person Zero-Sum Games
7. The Strategic Form of a Game
7.1 Strategic Form
7.2 Example: Odd or Even
7.3 Pure Strategies and Mixed Strategies
7.4 The Minimax Theorem
7.5 Exercises
8. Matrix Games — Domination
8.1 Saddle Points
8.2 Solution of All 2 × 2 Matrix Games
8.3 Removing Dominated Strategies
8.4 Solving 2 × n and m × 2 Games
8.5 Latin Square Games
8.6 Exercises
9. The Principle of Indifference
9.1 The Equilibrium Theorem
9.2 Nonsingular Game Matrices
9.3 Diagonal Games
9.4 Triangular Games
9.5 Symmetric Games
9.6 Invariance
9.7 Exercises
10. Solving Finite Games
10.1 Best Responses
10.2 Upper and Lower Values of a Game
10.3 Invariance under Change of Location and Scale
10.4 Reduction to a Linear Programming Problem
10.5 Description of the Pivot Method for Solving Games
10.6 A Numerical Example
10.7 Approximating the Solution: Fictitious Play
10.8 Exercises
11. The Extensive Form of a Game
11.1 The Game Tree
11.2 Basic Endgame in Poker
11.3 The Kuhn Tree
11.4 The Representation of a Strategic Form Game in Extensive Form
11.5 Reduction of a Game in Extensive Form to Strategic Form
11.6 Example
11.7 Games of Perfect Information
11.8 Behavioral Strategies
11.9 Exercises
12. Recursive and Stochastic Games
12.1 Matrix Games with Games as Components
12.2 Multistage Games
12.3 Recursive Games. ϵ-Optimal Strategies
12.4 Stochastic Movement Among Games
12.5 Stochastic Games
12.6 Approximating the Solution
12.7 Exercises
13. Infinite Games
13.1 The Minimax Theorem for Semi-Finite Games
13.2 Continuous Games
13.3 Concave Games and Convex Games
13.4 Solving Games
13.5 Uniform [0,1] Poker Models
13.6 Exercises
Part III. Two-Person General-Sum Games
14. Bimatrix Games — Safety Levels
14.1 General-Sum Strategic Form Games
14.2 General-Sum Extensive Form Games
14.3 Reducing Extensive Form to Strategic Form
14.4 Overview
14.5 Safety Levels
14.6 Exercises
15. Noncooperative Games
15.1 Strategic Equilibria
15.2 Examples
15.3 Finding All PSE’s
15.4 Iterated Elimination of Strictly Dominated Strategies
15.5 Exercises
16. Models of Duopoly
16.1 The Cournot Model of Duopoly
16.2 The Bertrand Model of Duopoly
16.3 The Stackelberg Model of Duopoly
16.4 Entry Deterrence
16.5 Exercises
17. Cooperative Games
17.1 Feasible Sets of Payoff Vectors
17.2 Cooperative Games with Transferable Utility
17.3 Cooperative Games with Non-Transferable Utility
17.4 End-Game with an All-In Player
17.5 Exercises
Part IV. Games in Coalitional Form
18. Many-Person TU Games
18.1 Coalitional Form. Characteristic Functions
18.2 Relation to Strategic Form
18.3 Constant-Sum Games
18.4 Example
18.5 Exercises
19. Imputations and the Core
19.1 Imputations
19.2 Essential Games
19.3 The Core
19.4 Examples
19.5 Exercises
20. The Shapley Value
20.1 Value Functions — The Shapley Axioms
20.2 Computation of the Shapley Value
20.3 An Alternative Form of the Shapley Value
20.4 Simple Games. The Shapley–Shubik Power Index
20.5 Exercises
21. The Nucleolus
21.1 Definition of the Nucleolus
21.2 Properties of the Nucleolus
21.3 Computation of the Nucleolus
21.4 Exercises
Appendix 1 Utility Theory
Appendix 2 Owen’s Proof of the Minimax Theorem
Appendix 3 Contraction Maps and Fixed Points
Appendix 4 Existence of Equilibria in Finite Games
Solutions to Exercises of Part I
Solutions to Chap. 1
Solutions to Chap. 2
Solutions to Chap. 3
Solutions to Chap. 4
Solutions to Chap. 5
Solution to Chap. 6
Solutions to Exercises of Part II
Solutions to Chap. 7
Solutions to Chap. 8
Solutions to Chap. 9
Solutions to Chap. 10
Solutions to Chap. 11
Solutions to Chap. 12
Solutions to Chap. 13
Solutions to Exercises of Part III
Solutions to Chap. 14
Solutions to Chap. 15
Solutions to Chap. 16
Solutions to Chap. 17
Solutions to Exercises of Part IV
Solutions to Chap. 18
Solutions to Chap. 19
Solutions to Chap. 20
Solutions to Chap. 21
Solutions to Exercises of Appendix 1
References
Index
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