Two competing entrepreneurs each plan to open a restaurant along the main street of a college town. Main street has nine blocks and each entrepreneur can open their new restaurant in any one of the nine blocks (they can even both choose the same block). During the school year, each block is also home to 100 college students. We will make the following assumptions: The nine blocks can be represented as regions along a line: 1 2 3 4 5 6 7 8 9 The customers for each restaurant are the college students living in each block (900 in total) and each such student will only visit the restaurant that is closest to the block where they live. If a block is equidistant from both restaurants, we assume half the students from that block visit one restaurant, and half visit the other. Example: If Restaurant A is at block 4 and Restaurant B is in block 6, then A will get all customers from blocks 1-4, B will get all customers from Regions 6-9, and each restaurant will get half the customers from block 5. We assume that the payoff to each restaurant is the number of customers. The players A and B will simultaneously choose where to build their respective restaurants. Neither knows in advance where the other is choosing to locate. Each player has the strategy set {1,2,3,4,5,6,7,8,9}, denoting their location choices. (a) Argue that for A the strategy sa = 2 is not dominated by sa = 3 (it doesn't matter that we have chosen A. The problem is symmetric and the same result will hold for B). (b) Use iterated elimination of strictly dominated strategies to show that in the dominant strategy equilibrium, each player chooses to locate at block 5. At each step be careful to show which strategies are eliminated and which strategies remain in the reduced game. (c) Before eliminating any strategies, is 5 a strictly dominant strategy for either player? Explain. Two competing entrepreneurs each plan to open a restaurant along the main street of a college town. Main street has nine blocks and each entrepreneur can open their new restaurant in any one of the nine blocks (they can even both choose the same block). During the school year, each block is also home to 100 college students. We will make the following assumptions: The nine blocks can be represented as regions along a line: 1 2 3 4 5 6 7 8 9 The customers for each restaurant are the college students living in each block (900 in total) and each such student will only visit the restaurant that is closest to the block where they live. If a block is equidistant from both restaurants, we assume half the students from that block visit one restaurant, and half visit the other. Example: If Restaurant A is at block 4 and Restaurant B is in block 6, then A will get all customers from blocks 1-4, B will get all customers from Regions 6-9, and each restaurant will get half the customers from block 5. We assume that the payoff to each restaurant is the number of customers. The players A and B will simultaneously choose where to build their respective restaurants. Neither knows in advance where the other is choosing to locate. Each player has the strategy set {1,2,3,4,5,6,7,8,9}, denoting their location choices. (a) Argue that for A the strategy sa = 2 is not dominated by sa = 3 (it doesn't matter that we have chosen A. The problem is symmetric and the same result will hold for B). (b) Use iterated elimination of strictly dominated strategies to show that in the dominant strategy equilibrium, each player chooses to locate at block 5. At each step be careful to show which strategies are eliminated and which strategies remain in the reduced game. (c) Before eliminating any strategies, is 5 a strictly dominant strategy for either player? Explain
