econ game theory questions

Problem 1: [20 pts]

Part a: [10 pts] Consider the following normal form game, which we call game G1:

D E

A 1, 1 0, 0

B 2, 1 −1,−1 Consider also a different normal form game in which player 1 has the additional option of playing strategy C, which we call game G2:

D E

A 1, 1 0, 0

B 2, 1 −1,−1 C x, y a, b

Are there some possible values of x, y, a, b for which all of the Nash equilibria of game G2 give strictly worse payoff to player 1 (row player) than the Nash equilibrium of game G1? If so, provide an example of possible values of x, y, a, b. If not, justify why not.

Part b: [10 pts] Consider the following normal form game.

L M R

U 1, 0 1, 2 0, 1

D 0, 3 0, 1 2, 0

K −1, 0 2, 0 3,−1

Find all the rationalizable strategies for players 1 (row player) and 2 (column player). Justify your answer, showing each step of elimination.

Part c: [10 pts, Extra Credit] Suppose that in a two player normal form game player 1 has strategies A1, B1, C1, D1 and player 2 has strategies A2, B2, C2, D2. Suppose we know that in this game,

1. A1 is a best response to A2, B1 is a best response to B2;

2. and A2 is a best response to B1, B2 is a best response to A1.

Can we be sure that A1, B1, A2, B2 are rationalizable? If yes, justify. If no, provide an explicit example.

Problem 2: [20 pts] Consider the following n player game. Each player has two pure strategies, {0, 1}. The utilities of each player is as follows. Player 1 obtains 1 utility if he chooses a strategy different from that chosen by player 2 and otherwise obtains a utility of 0.1 Similarly, any player i < n obtains a utility of 1 if he chooses a strategy different from that chosen by player i + 1, and otherwise obtains a utility of 0. Finally player n obtains a utility of 1 if he chooses a strategy different from that chosen by player 1, and otherwise obtains a utility of 0.

Part a: [10 points] Find all Nash equilibria in pure strategies of this game when n = 3. Justify both why the Nash equilibria you found are Nash equilibria and why there are no other Nash equilibria.

Part b: [10 points] Find all Nash equilibria in pure strategies of this game when n = 4. Justify both why the Nash equilibria you found are Nash equilibria and why there are no other Nash equilibria.

1Note that the strategy of player 3 does not affect player 1’s utility.

Problem 3: [30 pts] Consider two firms that compete via Bertrand competition. The demand of consumers is given by:

Q(p) = 150−p.

Firms 1 and 2 both have marginal costs of production of 15. Each firm 1 and 2 choose prices p1 ≥ 0 and p2 ≥ 0 simultaneously. Consumers buy from the firm with the lowest price. If the firms choose the same price, then the firms share the consumers equally.

For example, if both firms choose a price of p1 = p2 = 20, then each firm obtains profits of

150−20 2

(20−15).

Part a: [5 pts] Compute the best response functions of each firm in this game.

Part b: [5 pts] What are all of the Nash equilibria of this game? Justify your answer.

Part c: [5 pts] Are any of the Nash equilibria that you found in part b Pareto efficient? If yes, explain why. If not, provide an example of a strategy profile that Pareto dominates each such Nash equilibria.

Part d: [5 pts] Now suppose that each firm can only set prices in increments of one dollar.2 What are the Nash equilibria of this game?

Part e: [10 pts] Go back to the original game where firms can set any price that is a positive real number. Now suppose that another firm, firm 3, with marginal cost 20 enters the market. All three firms compete via Bertrand competition. Give an example of a Nash equilibrium, and justify your answer.

Problem 4: [30 pts] Two companies compete via Cournot competition. Each firm simultaneously chooses a quantity q1, q2 ≥ 0 to produce. Given quantities of q1, q2 chosen by the two firms, each firm can sell each of the units of the good at a price (in dollars) given by the inverse demand function:

P(q1 + q2) = 150−q1 −q2.

Suppose that firm 1 has a marginal cost of production of 10 dollars and firm 2 has a marginal cost of production of 20 dollars. As a result, each firm’s utility (profit) is given by:

u1(q1, q2) = (150−q1 −q2)q1 −10q1 u2(q1, q2) = (150−q1 −q2)q2 −20q2.

Part a: [10 pts] Compute and write the best response function for each firm. Show your work.

Part b: [10 pts] Solve for all of the Nash equilibria of this game. What is the price that is charged by the firms in this Nash equilibrium? Show your work.

Part c: [10 pts] Suppose now that the government thinks that prices are too high because the firms are producing too little. To incentivize more production, the government implements a subsidy. For each unit of the good sold, the government promises to pay each firm an additional 5 dollars. What are the Nash equilibria of this new game? Is the government successful in lowering the prevailing price?

2Prices must be some positive integer.