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Math Help - Help with Game Theory question

  1. #1
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    Help with Game Theory question

    Okay kind of a basic one here, just going through the Gibbons book question 1.5,

    I'm trying to calculate the both payoffs in the monopoly game


    I have my profit function
     \pi = q_i [a - (q_i + q_j) - c]

    And my quantities in monopoly are qm - after maximising profits

     q_i * = (a + q_j* - c )/2     \textnormal{and}       q_j * = (a + q_i* - c )/2

    since in monopoly

     (q_m / 2) = q_i = q_j = [((a-c)/2) / 2] = [(a-c)/4]

    my profits are (a-c)^2 / 4

    which is wrong and should be divided by 8 - not sure where I have gone wrong...
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  2. #2
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    Re: Help with Game Theory question

    I cant tell where you went wrong because you jumped from the expression for qi and qj to the final profit. Here's the solution.

    q_i=q_j=\frac{a-c}{4}

    \pi=q_i(a-q_i-q_j-c)

    \pi=q_i(a-\frac{a-c}{4}-\frac{a-c}{4}-c)

    \pi=q_i(a-\frac{a-c}{2}-c)

    \pi=q_i(\frac{a-c}{2})

    \pi=\frac{a-c}{4}(\frac{a-c}{2})
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