Suppose two players sell a product to three possible buyers. Each buyer wants to buy one unit of the product. Buyers A and C have access to one seller only, namely 1 and 2, respectively. However, buyer B can buy the product from any of the two sellers. All three buyers have a maximum value 1 for the item, i.e., will not buy the product if the price is above 1. The sellers play a pricing game they each name a price pi in the interval [0, 1]. Buyers A and C buy from sellers 1 and 2, respectively. On the other hand, B buys from the cheaper seller. To fully specify the game, we assume that if both sellers have the same price, B buys from seller 1. For simplicity, we assume no production costs, so the income of a seller is the sum of the prices at which they sold goods.

(a) Show that there is no pure Nash equilibrium. (b) Which condition of the fixed-point theorem is not satisfied?