Two firms produce and sell identical products. The market demand for the same product is given by the equation Q = 3 − 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 four possible values: 0, 1, 2 , or 3 (other price are not allowed). Let p1 be firm 1’s price choice and p2 be Firm 2’s price choice. The market price equals the lowest value of the two price choices. 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:

• If p1 < p2 then Firm 1 receives a payoff equal to (3 − p1)p1 and Firm 2 receives payoff zero. Here is the reason: Firm 1, who has the lowest price, sells a quantity equal to market demand, 3− p1, and gets the profit of (3− p1)p1 , which is quantity 3− p1 times price p1; Firm 2 on the other hand sells zero quantity and receives zero profit because its price is higher than Firm 1’s and the products are the same, consumers only buy from 1 Firm1. Forexample, if(p1 =2,p2 =3)thenFirm1’sprofitequals(3−2)×2=2 and Firm 2’s profit is zero because its price is higher and will sell zero quantity.

• If p1 > p2 then Firm 1 receives payoff zero and Firm 2 receives a payoff equal to (3 − p2)p2. This is just the opposite of the above case.

• If p1 = p2 = p then both firms receive the same payoff (3−p)p . In this case, the two 2 prices are equal, and each firm sells a quantity equal to half of market demand (3 − p)/2, namely they split the total demand equally.

Answer the following questions.

(a) Constructthe4×4payoffmatrix.

(b) Is there any weakly or strictly dominant strategy for either player? If so, find them all.

(c) Is there any weakly or strictly dominated strategy for either player? If so, find them all.

(d) Find all of the Nash equilibria using best responses.