1. Suppose there are n 2 people contemplating whether to purchase a Mac operating system or a Windows operating system, so that the strategy set of each player is (Mac, Windows). The payoff to buying a Mac is 100 + 10m, where m = the number of people who choose Mac. The payoff to buying Windows is 100w, where w = the number of people who choose Windows, and that everyone buys one or the other (so m + w = n). Note that these payoffs incorporate the assumption that Mac is better than Windows, but that there are network effects: each products value to the consumers depends on how many other consumers are using that product.

a. Are there any symmetric equilibria in this game? (HINT: your answer for m=n may depend on n!) Be very systematic about showing your work.

b. If n = 100, can it be an equilibrium for 91 people to choose Mac and 9 people to choose Windows? Show your answer carefully.

2. Suppose there are n firms, and each can choose a price p from the set {10, 20, 30}. Each firm i has the identical profit function 100 + 5*max[p1, p2, ..., pn] 25 pi.

a. Is this game symmetric? Explain.

b. Characterize all Nash equilibria of this game.

3. Recall that in a Bertrand duopoly, firms compete on price --- that is, each firm chooses the optimal price to charge for its output, always thinking of what the competing firm will do. It will then produce the quantity dictated by the demand curve for its product. Suppose we have two firms (A and B) competing in a market. The quantity demanded for each firms product will fall as that firm increases its price, but will rise as the other firm increases its price (as some of the second firms customers will instead buy the first firms product). Firm As demand is given by qA = 30 2PA + PB, while firm Bs demand is qB = 20 PB + 2PA.1 Firm As total cost function is CA = 3qA, and firm Bs total cost function is CB = 10qB.

a. What is the profit function for each firm?

b. What is the best reply function of each firm? Show your work!

c. Draw a large graph of the firms best reply functions. (Be sure to make your graph large --- take up a whole page --- as the areas should get small quickly, and show each iteration in a different color as we did in class. There is free graph paper available on the Web, but you dont have to draw to scale if you label carefully. For the quiz, be prepared to show either firm on the vertical axis.)

d. In equilibrium, what quantity should each of these firms choose to produce, and what profit will each firm earn?

e. Use a numerical example to show that the firms can do better by colluding. (NOTE: this does not necessarily mean PA = P B!)

4. Consider the BR functions you graphed in the previous question. Show three rounds of rationalizability (i.e. three steps for each player) to approach the Nash equilibrium. Be sure to explicitly state your reasoning for each step.