Game Theory

Game Theory - Economics 146 Fall, 2007 2-Person 0-Sum Game...

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Unformatted text preview: Economics 146 Fall, 2007 2-Person 0-Sum Game Theory Last update: 11/23/07. Introduction. Game Theory is the study of conflict and/or cooperation in situations where 2 or more players by their choices of action attempt to (at least partially but not completely) determine the outcome of the interaction. There is a very strong link between linear programs and games of a type called 2-person 0-sum games. In particular, the solu- tions to a game of this type can be completely computed as the solutions to a corresponding linear program. Structure of the Games. We start with an example called Two Fingers. Example. (Two Fingers) There are two players, called I and II. Both players acting at the same time put out 1 or 2 fingers. Each makes his or her choice independently of the other player. If the number of fingers put out by both players are the same ( match ) then II wins $1 from I; if not, then I wins $1 from II. 2 Note that in the game Two Fingers there are 2 players and that, in each outcome of the game, what I wins is equal to the negative of what II wins, viz. the sum of Is winnings and IIs winnings in each play of the game is 0. Hence the terminology 2-person 0-sum game . Because there are only 2 players and because what one player wins the other must lose, the objectives of the players are diametrically opposed. No cooperation is possible. A course of action for a player is called a strategy. More precisely, we define a strategy for a player in a game to be a comprehensive, prior set of instructions which prescribes a course of action for a player under all possible contingencies that may arise in the play of the game. At the end of each play of a game, an outcome occurs and a payoff is made to each player (payoffs can be negative). In the example of Two Fingers, each player has two strategies put out 1 finger or put out 2 fingers. We emphasized that each player chooses a strategy for the play of game independently of each other. (In this example, each player has the same set of strategies; this need not be the case for other games.) We name the strategies for player I 1 : put out 1 finger 2 : put out 2 fingers and for player II 1 : put out 1 finger 2 : put out 2 fingers 1 We write out the matrix for the payoffs in Two Finger as 1 2 1- 1 +1 2 +1- 1 It is traditional that player I is the row player and that player II is the column player. The outcome of the game is known at the end of each play in our case, match or not match and payoffs are paid to the players in our case, the match outcome pays off +1 to player II (and, hence, -1 to player I) and the not match outcome pays off +1 to player I (and, hence,-1 to player II). Since the sum of the payoffs to the players is 0, we write in the matrix only the payoff to player I....
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This note was uploaded on 09/23/2008 for the course ECON 146 taught by Professor Farmer during the Fall '07 term at UCLA.

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Game Theory - Economics 146 Fall, 2007 2-Person 0-Sum Game...

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