game_part3 - GAME THEORY Thomas S. Ferguson Part III....

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GAME THEORY Thomas S. Ferguson Part III. Two-Person General-Sum Games 1. Bimatrix Games — Safety Levels. 1.1 General-Sum Strategic Form Games. 1.2 General-Sum Extensive Form Games. 1.3 Reducing Extensive Form to Strategic Form. 1.4 Overview. 1.5 Safety Levels. 1.6 Exercises. 2. Noncooperative Games. 2.1 Strategic Equilibria. 2.2 Examples. 2.3 Finding All PSE’s. 2.4 Iterated Elimination of Strictly Dominated Strategies. 2.5 Exercises. 3. Models of Duopoly. 3.1 The Cournot Model of Duopoly. 3.2 The Bertrand Model of Duopoly. 3.3 The Stackelberg Model of Duopoly. 3.4 Entry Deterrence. 3.5 Exercises. 4. Cooperative Games. 4.1 Feasible Sets of Payo± Vectors. 4.2 Cooperative Games with Transferable Utility. 4.3 Cooperative Games with Non-Transferable Utility. 4.4 End-Game with an All-In Player. 4.5 Exercises. III – 1
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PART III. Two-Person General-Sum Games 1. Bimatrix Games — Safety Levels The simplest case to consider beyond two-person zero-sum games are the two-person non-zero-sum games. Examples abound in Economics: the struggle between labor and management, the competition between two producers of a single good, the negotiations between buyer and seller, and so on. Good reference material may be found in books of Owen and Straffin already cited. The material treated in Part III is much more oriented to economic theory. For a couple of good references with emphasis on applications in economics, consult the books, Game Theory for Applied Economists by Robert Gibbons (1992), Princeton University Press, and Game Theory with Economic Applications by H. Scott Bierman and Luis Fernandez (1993), Addison-Wesley Publishing Co. Inc. 1.1 General-Sum Strategic Form Games. Two-person general-sum games may be de±ned in extensive form or in strategic form. The normal or strategic form of a two- person game is given by two sets X and Y of pure strategies of the players, and two real-valued functions u 1 ( x, y )and u 2 ( x, y ) de±ned on X × Y , representing the payo²s to the two players. If I chooses x X and II chooses y Y , then I receives u 1 ( x, y )andII receives u 2 ( x, y ). A ±nite two-person game in strategic form can be represented as a matrix of ordered pairs, sometimes called a bimatrix. The ±rst component of the pair represents Player I’s payo² and the second component represents Player II’s payo². The matrix has as many rows as Player I has pure strategies and as many columns as Player II has pure strategies. For example, the bimatrix (1 , 4) (2 , 0) ( 1 , 1) (0 , 0) (3 , 1) (5 , 3) (3 , 2) (4 , 4) (0 , 5) ( 2 , 3) (4 , 1) (2 , 2) (1) represents the game in which Player I has three pure strategies, the rows, and Player II has four pure strategies, the columns. If Player I chooses row 3 and Player II column 2, then I receives 2 (i.e. he loses 2) and Player II receives 3. An alternative way of describing a ±nite two person game is as a pair of matrices. If m and n representing the number of pure strategies of the two players, the game may be represented by two m × n matrices A and B . The interpretation here is that if Player I chooses row i and Player II chooses column j , then I wins a ij
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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_part3 - GAME THEORY Thomas S. Ferguson Part III....

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