Consider a variant of the game described in Exercise 4. Suppose that the firms move sequentially rather than simultaneously. First, firm 1 selects its quantity q1, and this is observed by firm 2. Then, firm 2 selects its quantity q2, and the payoffs are determined as in Exercise 4, so that firm i's payoff is (12 - qi - qj)qi .

As noted in Exercise 6 of Chapter 3, this type of game is called the Stackelberg duopoly model. This exercise asks you to find some of the Nash equilibria of the game. Further analysis appears in Chapter 15. Note that firm 1's strategy in this game is a single number q1. Also note that firm 2's strategy can be expressed as a function that maps firm 1's quantity q1 into firm 2's quantity q2. That is, considering q1 , q2 ∈ [0, 12], we can write firm 2's strategy as a function s2 : [0, 12] S [0, 12]. After firm 1 selects a specific quantity q1, firm 2 would select q₂ = s₂(q₁).

(a) Draw the extensive form of this game.

That is, firm 2 selects q₂ = 5 in the event that firm 1 chooses q1 = 2; otherwise, firm 2 picks the quantity that drives the price to zero.

(b) Verify that these strategies form a Nash equilibrium of the game. Do this by describing the payoffs players would get from deviating.

(c) Show that for any x ∈ [0, 12], there is a Nash equilibrium of the game in which q₁= x and s₂(x) = (12 - x)>2. Describe the equilibrium strategy profile (fully describe s₂) and explain why it is an equilibrium.

(d) Are there any Nash equilibria   Explain why or why not.