Practice game theory problems 10/15

1. Suppose that there are two firms, A and B, in a duopoly market. Each firm can charge one of three prices – high, medium and low. Suppose that whichever firm charges the lower price gets the entire market. If the two firms charge the same price, they share the market equally. Given these assumptions, we have the following normal form game, where the entries are the profits.

			Firm B
		High	Medium	Low
Firm A	High	(6, 6)	(0, 10)	(0, 8)
	Medium	(10, 0)	(5, 5)	(0, 8)
	Low	(8, 0)	(8, 0)	(4, 4)

Solve for the Nash equilibrium in pure strategies. Solve for the iterated dominant equilibrium. Explain your reasoning for each equilibrium obtained. (This is an example of the model examined by Bertrand (French economist in the 19^th century) in contrast to Cournot. See ECON 303.)

2. This is the classic Cournot duopoly problem. Suppose that there are two firms (firm 1 and firm 2) in the industry, each producing a homogeneous product. The demand function is given by

Q = Q1 + Q2 = 10 – P,

Where Q1 is firm 1’s output and Q2 is firm 2’s output. Note that the price in the market depends on the aggregate production (duopoly). Each firm must estimate the production decision of the other firm. Suppose further that each firm has a constant marginal cost of $1 per unit of output. (See ECON 303.) The game is a simultaneous move game.

a. Determine each firm’s reaction function. A reaction function is a best response function (optimal Qi) for any given output by the other firm. This requires using calculus. Set up the profit maximizing problem and solve for each Qi.

b. Graph each firm’s reaction function in the plane (Q1, Q2). What is the Nash equilibrium? Is it stable?

c. What is the market equilibrium price?

d. What is each firm’s output in the equilibrium?

3. Suppose that we have the matching pennies game given below.

		B
		Heads	Tails
A	Heads	(1, -1)	(-1, 1)
	Tails	(-1, 1)	(1, -1)

a. Solve the game for the Nash equilibria in pure strategies.

b. Solve the game for the mixed strategy equilibrium.

4. Suppose that Coke is considering entering the Eastern European market where Pepsi is already established. Pepsi can respond to Coke’s entry by playing tough (T) or by accommodating (A). Coke can also make similar choices – T or A. The post entry decisions are made simultaneously. The payoffs for the post entry part of the game are the following:

		Pepsi
		Tough	Accommodate
Coke	Tough	(-2, -1)	(0, -3)
	Accommodate	(-3, 1)	(1, 2)

If Coke stays out, then Pepsi makes profits of 5 and Coke has profits of 0 in the Eastern European market.

a. Set up the game in terms of the normal form. Solve for all of the Nash equilibria in pure strategies.

b. Set up the extensive form of the game. Solve for the sub game perfect equilibria.

c. Discuss the equilibria. Are the two sets the same? If not why not?