1. There are three locations A, B, and C, in a city. The first player, Alice, has an opportunity to either open a production facility in location A or C (but not both), but she cannot open a production facility in location B. The second player, Bob, has an opportunity to either open a production facility in location B or C (but not both), but he cannot open a production facility in location A. Two products are on offer in the city: product X and product Y. The cost of pro- ducing cach product at a particular location varies and is specified in the following table. product Y product X location A cost 10 cost 30 location B cost 30 cost 30 location C cost 30 cost 10 There is only one customer for product X in the city and he is willing to pay a maximum price of 30 for product X and there is only one customer for product Y in the city and she is willing to pay a maximum price of 30 for product Y. Alice and Bob pick their location and then set a price for each product. The customers will buy the product from whoever offers it more cheaply as long as that cheaper price does not exceed the maximum price they are willing to pay. If both players set a price exceeding the 30 maximum, the customer will not buy anything. If Alice and Bob offer the product for the same price, customers prefer to buy from Alice. Note that Alice and Bob are allowed to produce both products at the production facility at their respective location and it is allowed for both players to have a production facility in the same location. The profit of a player consists of the amount of money they receive from the cus- tomers (if any) minus the cost they incur (based on the table above) for producing the product. (a) Model this as a two-player game between Alice and Bob in which both play- ers have two pure strategies corresponding to the possible location of their respective production facility. You should assume that the prices players set will depend on the strategies chosen by both players and will be chosen optimally by both players for each of the four outcomes of the game. [10] (b) Find all Nash equilibria of the game. [10] (c) For each customer that successfully buys their product, we define the value surplus as 30 minus the cost of producing the product (the latter will only depend on the location from which the product is supplied). Customers that do not obtain their product have a value surplus of 0. The social surplus is the sum of the value surpluses of each customer. Calculate the social surplus [5] for all pure strategy profiles in the game.