game_partII

# game_partII - GAME THEORY Thomas S Ferguson Part II...

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GAME THEORY Thomas S. Ferguson Part II. Two-Person Zero-Sum Games 1. The Strategic Form of a Game. 1.1 Strategic Form. 1.2 Example: Odd or Even. 1.3 Pure Strategies and Mixed Strategies. 1.4 The Minimax Theorem. 1.5 Exercises. 2. Matrix Games. Domination. 2.1 Saddle Points. 2.2 Solution of All 2 by 2 Matrix Games. 2.3 Removing Dominated Strategies. 2.4 Solving 2 × n and m × 2Games . 2.5 Latin Square Games. 2.6 Exercises. 3. The Principle of Indi±erence. 3.1 The Equilibrium Theorem. 3.2 Nonsingular Game Matrices. 3.3 Diagonal Games. 3.4 Triangular Games. 3.5 Symmetric Games. 3.6 Invariance. 3.7 Exercises. II – 1

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4. Solving Finite Games. 4.1 Best Responses. 4.2 Upper and Lower Values of a Game. 4.3 Invariance Under Change of Location and Scale. 4.4 Reduction to a Linear Programming Problem. 4.5 Description of the Pivot Method for Solving Games. 4.6 A Numerical Example. 4.7 Exercises. 5. The Extensive Form of a Game. 5.1 The Game Tree. 5.2 Basic Endgame in Poker. 5.3 The Kuhn Tree. 5.4 The Representation of a Strategic Form Game in Extensive Form. 5.5 Reduction of a Game in Extensive Form to Strategic Form. 5.6 Example. 5.7 Games of Perfect Information. 5.8 Behavioral Strategies. 5.9 Exercises. 6. Recursive and Stochastic Games. 6.1 Matrix Games with Games as Components. 6.2 Multistage Games. 6.3 Recursive Games. ² -Optimal Strategies. 6.4 Stochastic Movement Among Games. 6.5 Stochastic Games. 6.6 Approximating the Solution. 6.7 Exercises. 7. Continuous Poker Models. 7.1 La Relance. 7.2 The von Neumann Model. 7.3 Other Models. 7.4 Exercises. References. II – 2
Part II. Two-Person Zero-Sum Games 1. The Strategic Form of a Game. The individual most closely associated with the creation of the theory of games is John von Neumann, one of the greatest mathematicians of the 20th century. Although others preceded him in formulating a theory of games - notably ´ Emile Borel - it was von Neumann who published in 1928 the paper that laid the foundation for the theory of two-person zero-sum games. Von Neumann’s work culminated in a fundamental book on game theory written in collaboration with Oskar Morgenstern entitled Theory of Games and Economic Behavior , 1944. Other discussions of the theory of games relevant for our present purposes may be found in the text book, Game Theory by Guillermo Owen, 2nd edition, Academic Press, 1982, and the expository book, Game Theory and Strategy by Philip D. Straﬃn, published by the Mathematical Association of America, 1993. The theory of von Neumann and Morgenstern is most complete for the class of games called two-person zero-sum games, i.e. games with only two players in which one player wins what the other player loses. In Part II, we restrict attention to such games. We will refer to the players as Player I and Player II. 1.1 Strategic Form. The simplest mathematical description of a game is the strate- gic form, mentioned in the introduction. For a two-person zero-sum game, the payo± function of Player II is the negative of the payo± of Player I, so we may restrict attention to the single payo± function of Player I, which we call here L .

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## This note was uploaded on 02/02/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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game_partII - GAME THEORY Thomas S Ferguson Part II...

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