Game theory

2. Consider the standard Stackelberg model of quantity competition but now with three firms. Each firm i c {1, 2,3} must choose how much quantity to produce, q, but firm 1 makes their choice first, firm 2 observes the choice q1 before making their choice 92, and firm 3 observes both q1 and 92 before making their choice 93. The market price is determined by an inverse demand which depends on the total quantity produced by the firms Q = q1 + 92 + 93, where the price per unit is given by P(Q) = max {A - Q,0}. Firm 3 has a per unit cost of c3 = c > 0, firm 2 has a per unit cost of c2 = . c of producing each unit, while firm I has a cost of ci = 2 . c per unit . The total profit is given by: Ti(91, 92, 93) = P(q1 + 92 + 93) . qi - Ci . qi. (a) Draw the extensive-form for the Stackelberg game above, making sure to label the generic histories q, and (91,92) and terminal nodes (91, 92, 93), the players that make choices at those histories, and the terminal history payoffs (v1, v2, v3). (b) Draw the 2 generic proper sub-games of the above game, (T(qi) and I(q1, 92)). (c) For the sub-game at the generic history (q1, 92) find the best response of firm 3, q5(91, 92). (d) Given 95(91,92), solve for the sub-game perfect best response q?(q) at the generic sub game (q1). (c) Provide the complete sub-game perfect strategy for this 3-player Stackelberg game assuming A > 4c. (f) How would your answer change if we 4 . c > A > 2 . c