Answer the following questions regarding the matrix below, which represents the strategic interaction between the two largest movie production studios, Universal Pictures and 20th Century Fox. They are each deciding the release dates for their own summer blockbuster action movies. They can release their movies in June, July, or August. Because of limited demand for going to the movies, each firm would generally rather release their movies earlier than later. Similarly, they would rather not release their movies at the same time, as it dilutes the demand for their movies.
The matrix form of the game is as follows:

Fox
June July August
June 5,5 6,4 8,6
Universal July 4,6 5,5 6,5
August 7,8 4,6 5,4

(a) Suppose Universal and Fox make their decisions simultaneously.
Are there pure strategy Nash equilibria? If so, what are they? Show your work. (b) Suppose they move sequentially, and that Universal moves first. First set up the game tree. What will be the equilibrium path of the subgame perfect Nash equilibrium outcome? Explain intuitively why or why not. Would Universal want this to happen?

a. A pure strategy Nash equilibrium of a game is a strategy profile in which both players are best responding to each other's strategies. For the given payoffs, we have 2 pure strategy Nash equilibrium (June, August) and (August. June). The arguments are as follows:

- Universal playing June and Fox playing August is equilibrium. This is because given that Universal in releasing in June. FOX maximizes its payoff by releasing in August and the same reasoning applies for Universal as well.

• Universal playing August and Fox playing June is equilibrium. This is because given that Universal is releasing in August, Fox maximizes its payoff by releasing in June and the same reasoning applies for universal as well.

b. The extensive form diagram is given below. The sub game perfect  equilibrium is computed by backward induction. Fox chooses its action given what universal has decided. These actions are marked by the red arrows. Universal therefore faces the following payoffs:

- By playing June, Universal gets 8 and Fox gets 6

- By playing July, Universal gets 4 and Foxs gets 6

- By playing August, Universal gets 7 and Fox gets *

Thus, given the above information by backward induction, Universal will choose to release in June and then Fox will release in August leading to payoffs (8,6).

Universal would want this to happen because the subgame perfect equilibrium gets rid of the Nash equilibrium that generated lower payof for them. That is , the only SPE is giving payoffs (8,6) and (7,8). Since Universal prefers 8 over 7, it would want the game to be sequential

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