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ECON20005: Competition and Strategy

Semester 2, 2021

Assignment 2

1. Consider a game of two players. The following information is common knowledge between the two players. First, Lisa chooses between actions u and d. Then, Sam observes which action Lisa has chosen with probability 1/3. With probability 2/3, he does not observe the action Lisa has chosen. (Hence, at the time Lisa makes her decision, she does not know whether Sam will observe or will not observe her choice. She believes that with probability 1/3, her choice will be revealed to Sam and with probability 2/3, her choice will not be revealed to Sam.)

Regardless of whether Sam has observed that Lisa chose u, or he has observed that Lisa chose d, or he has not observed any action, Sam chooses between actions l and r. The payoff of each player is 1 after (u, l) and (d, r), and 0 otherwise.

(a) (5 marks) Write the above game in extensive form.

(b) (5 marks) Write the above game in normal form.

2. Consider the following asymmetric variation of the Rock-Paper-Scissors game.

(a) (7 marks) Suppose x > 1. Find a mixed-strategy Nash Equilibrium of this game. (Hint: Before you start to solve for the mixed strategies, it is always a good idea to check whether you can simplify the game.)

(b) (3 marks) What is the row player's expected payoff in the equilibrium you found in part (a)?

3. Two firms (Firm 1 and Firm 2) are in a market where demand is given by P = 120 - (q1 + q2). The firms'marginal cost of production is 60.

(a) (3 marks) Suppose both firms compete by setting their quantities simultaneously and that the game is played only once. Find the best response functions of the firms and draw them.

(b) (3 marks) Find the Nash Equilibrium of the game in part (a). What are the equilibrium profits of the firms?

(c) (6 marks) Now suppose that instead of setting quantities simultaneously, the firms set their quantities sequentially. Specifically, Firm 1 moves first and chooses its quantity (q1). After observing the quantity choice of Firm 1, Firm 2 moves second and chooses its quantity (q2). Find the Subgame Perfect Nash equilibrium of this game. Explain why the quantities in the sequential-move game differ from those in the simultaneous-move game.

(d) (3 marks) Consider the sequential-move game specified in part (c). Suppose that the firms play the Nash Equilibrium strategies you identified in part (b). That is, Firm 1 produces , and Firm 2 produces  no matter what Firm 1 produces, where  and  are the Nash Equilibrium quantities you found in part (b). Show that although these strategies constitute a Nash Equilibrium of the game in part (c), they are not subgame perfect.

(e) (6 marks) Now suppose that the two firms play the following game. In stage 1, they simultaneously decide whether to enter or not. Each firm that decides to enter has to pay a fixed cost of entry, F. In stage 2, if both firms enter, they play the simultaneous-move game in part (a). That is, they choose simultaneously how many units to produce. If only one firm enters, it becomes a monopolist. Find the Subgame Perfect Nash equilibrium of this game if F ≤ 400.

(f) (3 marks) Now suppose 400 < F ≤ 900. Find the Subgame Perfect Nash Equilibrium of the game in part (e).

4. Consider the following game.

(a) (3 marks) Find all pure-strategy Nash Equilibria of this game.

(b) (5 marks) Suppose the players play the game repeatedly for 75 periods. Assume the countries do not discount future payoffs (i.e., the discount factor is = 1). Describe the two players' choices along the Subgame Perfect Nash Equilibrium path. Briefly explain your solution method.

(c) (5 marks) Now consider the infinitely repeated version of this game. The perperiod payoffs are discounted by where 0 < < 1. Suppose that both players follow the following grim strategy: "Play c as long as no one has ever played d; otherwise play d:" Find the minimum value of such that this is a Subgame Perfect Nash Equilibrium. That is, find , the critical value of such that for all > , the two players would cooperate and play c in all periods.

(d) (3 marks) Which player determines the critical value for supporting cooperation? Briefly discuss the intuition for the result.