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LectureFeb15_MATH4321_12S

LectureFeb15_MATH4321_12S - Reduction of a Game in...

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Reduction of a Game in Extensive Form to Strategic Form. Pure strategy . A pure strategy is a player’s complete plan for playing the game. It should cover every contingency. A pure strategy for a Player is a rule that tells him exactly what move to make in each of his information sets. It should specify a particular edge leading out from each information set.
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Example: Player I has 3 information sets. The set of edges coming out from the information sets are{A,B}, {E,F}, {C,D}. Player II has two information sets. The set of edges coming out from the information sets are {a,b}, {c,d}. Set of Pure Strategies for I: {A,B}x{C,D}x{E.F} ={ACE, ACF, ADE, ADF, BCE, BCF, BDE, BDF} Set of Pure Strategies for II: {a,b}x{c,d}={ac,ad,bc,bd}
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Remark : Even for a simple game, there may be a large number of pure strategies. Example : (Two move Tic-Tac-Toe) Player I and II take turn to make a move on the Tic-Tac-Toe board. The game stops after each player has made one move. How many pure strategies are there for Player I and for Player II?
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Example : (Two move chess) There are 20 possible moves for Player I and 20 possible moves for Player II. How many pure strategies are there for Player II?
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Reduced pure strategy : Specification of choices at all information sets except those that are eliminated by the previous moves. Remark : It is not too easy to find the set of reduced pure strategies. Also we need to use full pure strategies for another important concept.
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Example: Set of Reduced Pure Strategies for I:{AE, AF, BC, BD} Set of Reduced Pure Strategies for II: {ac,ad,bc,bd}
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Now that we have set of pure strategies for each player, we need to find the payoffs to put the game in strategic form. Random payoffs . The actual outcome of the game for given pure strategies of the players depends on the chance moves selected, and is therefore a random quantity. We represent random payoffs by their average values.
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Example: Suppose Player I uses BCF and Player II uses ac.
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