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

$\displaystyle \pi = q_i [a - (q_i + q_j) - c] $

And my quantities in monopoly are qm - after maximising profits

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

since in monopoly

$\displaystyle (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...

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.

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

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

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

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

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

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