1. Two firms produce and sell identical products. The market demand for the same product is given by the equation Q = 4 P where Q is the total quantity demanded and P is the market price (same for all consumers and the two firms). Each firm's strategy is the price it chooses. Each firm must choose a price from five possible values: 0, 1, 2 , 3, or 4 (other price are not allowed). Let p1 be firm 1's price and p2 be Firm 2's price. The market price P equals the lowest of the two prices. For both firms, production cost equals zero, so profits equal to the revenue they get from selling the product. Specifically, the payoffs (profits) of the two firms are calculated using the following rules:
2. If p1 p2 then Firm 1 receives a payoff equal to (4 p1)p1 and Firm 2's payoff equals zero. Here is the reason: Firm 1, who has the lowest price, sells a quantity equal to

1

market demand, 4 p1, and gets the profit of (4 p1)p1 , which is quantity 4 p1 times price p1; On the other hand, because Firm 2's price is higher than Firm 1's price and the products are identical Firm 2 cannot sell anything so receives zero profit. For example, if (p1 =2,p2 =3)thenFirm1'sprofitequals(42)2=4andFirm2'sprofitiszero.

If p1 > p2 then Firm 1 receives payoff zero and Firm 2 receives a payoff equal to (4 p2)p2. This is just the opposite of the case in the above.

(4p)p Ifp1=p2=pthenbothfirmsreceivethesamepayoff 2 . Inthiscase,thetwo

prices are equal, and each firm sells a quantity equal to half of market demand (4 p)/2, namely they split the total demand equally.

Answer the following questions.

1. (a)(4points)Constructthe55payoffmatrix.
2. (b)(2 points) Is there any weakly or strictly dominant strategy for either player? If so, find all of them.
3. (c)(4 points) Is there any weakly or strictly dominated strategy for either player? If so, find all of them.
4. (d)(2 points) Find all of the Nash equilibria using best responses.

3. (12 points) Two rms produce and sell identical products. The market demand for the same product is given by the equation Q = 4 P where Q is the total quantity demanded and P is the market price (same for all consumers and the two rms). Each rm's strategy is the price it chooses. Each rm must choose a price from ve possible values: 0, 1, 2 , 3, or 4 (other price are not allowed). Let 191 be rm 1's price and 192 be Firm 2's price. The market price P equals the lowest of the two prices. For both rms, production cost equals zero, so prots equal to the revenue they get from selling the product. Specically, the payos (prots) of the two rms are calculated using the following rules: 0 If p1 p2 then Firm 1 receives payoff zero and Firm 2 receives a payoff equal to (4 p2)p2. This is just the opposite of the case in the above. a If p1 2 p2 = p then both rms receive the same payoff w. In this case, the two prices are equal, and each rm sells a quantity equal to half of market demand (4 p) / 2, namely they split the total demand equally. Answer the following questions. (a) (4 points) Construct the 5 x 5 payo matrix. (b) (2 points) Is there any weakly or strictly dominant strategy for either player? If so, nd all of them. (c) (4 points) Is there any weakly or strictly dominated strategy for either player? If so, nd all of them. (d) (2 points) Find all of the Nash equilibria using best responses