(For students of economics only) Suppose a duopoly where two firms, 1 and 2, select their output q1 and q2 once and independently of one another (i.e. non-co-operatively) in order to maximise their...

(For students of economics only) Suppose a duopoly where two firms, 1 and 2, select their output q1 and q2 once and independently of one another (i.e. non-co-operatively) in order to maximise their respective profit. Assuming that (a) the price is set automatically by the market according to (the inverse demand) function
, and (b) each firm has the same fixed cost (F) and constant marginal cost equal to 10, then show that there exist a pair of quantity strategies
 which constitute this game’s a unique Nash equilibrium in pure (quantity) strategies. Moreover, show that these strategies are also the game’s unique rationalisable strategies.