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Math Help - Zero Sum Games 3

  1. #1
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    Zero Sum Games 3

    Find the expected payoff to each player, if Player I uses mixed strategy 0.2, 0.3, 0.5) and Player II uses mixed strategy (0.1, 0.5, 0.2, 0.2), in the two-person zero-sum game with the following payoff matrix.

    (It's a 3x4 matrix, I couldn't type it any other way)

    -1 0 1 1
    2 1 -2 -1
    1 3 -1 0
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  2. #2
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    Hello, Victor!

    Find the expected payoff to each player,
    if Player 1 uses mixed strategy (0.2, 0.3, 0.5)
    and Player 2 uses mixed strategy (0.1, 0.5, 0.2, 0.2)
    in the two-person zero-sum game
    with the following payoff matrix: . \begin{pmatrix}\text{-}1& 0& 1& 1 \\ 2 &1 &\text{-}2& \text{-}1 \\ 1& 3& \text{-}1& 0 \end{pmatrix}
    We have: . \begin{array}{c||c|c|c|c|}<br />
& 0.1 & 0.5 & 0.2 & 0.2 \\ \hline \hline<br />
0.2 & \text{-}1 & 0 & 1 & 1 \\ \hline<br />
0.3 & 2 & 1 & \text{-}2 & \text{-}1 \\ \hline<br />
0.5 & 1 & 3 & \text{-}1 & 0 \\ \hline \end{array}


    Player 1's expectation: . \begin{array}{c}<br />
+(0.2)(0.1)(\text{-}1) + (0.2)(0.5)(0) + (0.2)(0.2)(1) + (0.2)(0.2)(1) \\ \\[-4mm]<br />
+(0.3)(0.1)(2) + (0.3)(0.5)(1) + (0.3)(0.2)(\text{-}2) + (0.3)(0.2)(\text{-}1) \\ \\[-4mm]<br />
+ (0.5)(0.1)(1) + (0.5)(0.5)(3) + (0.5)(0.2)(\text{-}1) + (0.5)(0.2)(0)\end{array}

    . . . . . . . = \;-0.02 + 0 + 0.04 + 0.04 + 0.06 + 0.15 - 0.12 - 0.06 + 0.05 + 0.75 - 0.10 + 0

    . . . . . . . = \;0.79


    If the given units are measured in dollars,

    . . Player 1 can expect to win an average of 79 cents per game,

    . . Player 2 can expect to lose an average of 79 cents per game.

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