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Math Help - Zero Sum Games 4

  1. #1
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    This question illustrates the following theorem.
    If one player of a 2-person zero-sum game employs a fixed strategy, then the opponent has an optimal counter strategy that is pure. In other words, if Player I knows that Player II is using a particular mixed strategy y, then Player I can maximize their expected payoff by using a pure strategy, and vice versa.

    a. Work out the following.
    i. Find the expected payoff, E(x,y), to Player I if Player I
    uses mixed strategy (x1, 1-x1), 0 ≤ x1 ≤ 1, and Player II uses
    mixed strategy (y1, 1-y1), 0 ≤ y1 ≤ 1, in the 2-person zerosum
    game with the following payoff matrix.

    6 , 1
    2 , 3

    (this is a 2x2 matrix)


    ii. Now, think of y1 is being fixed to y1=0.3, find the value(s)
    of x1 that maximizes E(x,y) for fixed y1. Make your case
    that, in general, if Player I can use a pure strategy to get
    the maximum expected payoff, given that Player I knows
    Player II’s strategy.

    b. Suppose that Player II knew that Player I is to use a mixed
    strategy (0.3, 0.1, 0.6) in the game with the following payoff
    matrix. Work out the three expected payoffs if Player II uses a
    pure strategy and hence deduce the best strategy for Player II.

    ⎡ 0 1 −2 ⎤
    1 0 1
    ⎣ 2 −1 0

    (this is a 3x3 matix, sorry for the formatting)

    Using the same game given in Question b:

    Write down the linear programming problems induced by the
    game.
    Last edited by mr fantastic; May 12th 2009 at 02:52 AM. Reason: Merged posts
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