Say we have k firms in a market where each of the firms will choose a quantity to produce [0,100]. Let the total quantity be defined as Q = q1 + ... + qk where each of them have the same payout
p(q1,...,qk) = (100-Q)*qi
if Q<= 100
It is easy to find the Nash equilibrium for 2 firms using best responses etc. but I want to find a nash equlibrium (pure) for each k (in a market with k firms and not only 2). I guess a start is to write each payoff and take the derivative w.r.t. each firm, right?
p1 = (100-Q)*q1
pk = (100-Q)*qk
then partial derivatives of each one and then I get the best choice for each firm. Then what do I do?