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Game Theory

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

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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 I’s winnings and II’s 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 • σ 1 : put out 1 finger • σ 2 : put out 2 fingers and for player II • τ 1 : put out 1 finger • τ 2 : put out 2 fingers 1 We write out the matrix for the payoffs in Two Finger as τ 1 τ 2 σ 1- 1 +1 σ 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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Game Theory - Economics 146 – Fall 2007 2-Person 0-Sum...

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