Cournot Duopoly firms guaranteed to produce the same amount

If a market is given to be a Cuornot duopoly and Q = q1 + q2. What assumptions can you make about q1 and q2 being equal? Is it possible for q1 > q2 and thus for company 1 to have greater market share in an equilibrium?

Edit: I assume they have the same cost and indistinguishable products. I guess my question could be more simply boiled down to: Are they guaranteed to have inverted reaction functions?

• Hint: Try solving the problem with $MC_1<MC_2$, namely, firm 1 is more efficient / has a lower marginal cost than firm 2. Oct 19, 2022 at 2:11
• @HerrK. yes, oops meant to assume, that. I think I need to focus on the reaction function I am just not sure how to prove they have the same reaction function Oct 19, 2022 at 2:23
• Hi, what do you mean by an inverted reaction function? Oct 19, 2022 at 19:03
• @Citrus I mean would one firms reaction function necessarily be the inverse of the other firms reaction function Oct 21, 2022 at 23:25

Further restrictions on the reaction functions are necessary to guarantee a symmetric equilibrium. A sufficient condition is that they are identical and linear in $$y_i$$.