A unit mass of kids are uniformly located on a street, denoted by the [0; 1] interval. There are two ice cream parlors, one located in x and the other is located in 1 ? x, where x 0. Given the prices p and q for the ice cream in stores located at x and 1 ? x, respectively, each kid buys one unit of ice cream from the store with the lowest total cost, which is the sum of the price and the cost to go to the store. (If the total cost is the same, she flips a coin to choose the store to buy.)

(a) Compute the revenue for each firm, as a function of price vector (p; q). The revenue is price times the total mass of the kids who buy from the given store.

(b) Assume that each store set their own price simultaneously and try to maximize the expected value of its own revenue, .

(c) Compute the set of Nash equilibria.

(d) Compute the set of rationalizable strategies.

U U 0-1113 Blim (a) Find a subgaJne-perfect Nash equilibrium in pure strategies under which the average payo of each player is in between 1.1 and 1.2. Verify that your strategy prole is indeed a subgameperfeet Nash equilibrium. (b) Find a subgame perfect Nash equilibrium in pure strategies under which the average payo of player 1 is at least 5.7. Verify that your strategy prole is indeed a subgame- perfect Nash equilibrium. (c) Can you nd a subgameperfeet Nash equilibrium under which the average payoff of player 1 is more than 5.8? Consider the repeated game with the following stage game. A unit mass of kids are uniformly located on a street, denoted by the [0, 1} interval. There are two ice creme parlors, namely 1 and 2, located at [l and 1, respectively. Each ice cream parlor 12 sets a. price p,- 5 :5 for its own ice cream, simultaneously, where f! > 0.1 A kid located in w is to pay cost c |w y| to go to a store located at y, where c E (0.15/3). Given the prices In and pg, each kid buys one unit of ice cream from the store with the lowest total oust, which is the sum of the price and the cost to go to the store. (If the total cost is the seine, she ips a coin to chase the store to buy.) (a) Assume that the above game is repeated 100 times, and nd the subgame-perfeot Nash equilibria. (b) Assume that the above game is repeated innitely many times and the discount rate is 5 E (1/3, 1]. For each of the following strategy prole, nd the highest if under which the strategy prole is a snbgame-perfect Nash equilibrium. (Here, 31\" may be a. mction of 6. You need to choose both 31'\" and ii to make the strategy prole a suhgame-perfect Nash equilibrium.) 1. At the beginning eatdi parlor :5 chooses p; = p' and continues to do so until some player sets a different price; each 1' selects price p,- = 33 thereafter. 2. There are two states: Collusion and War. The game starts at the state Collusirni. In Challesion state, each player 1' chases p.- = p'. and in War state each player 1' ohm p,- = :3. If both players set the price premrihed for the state, then the state in the need round is Collusion; the state in the next round is War otherwise. 3. Brains: In part (ii), ai-sunm that f: 3 0 but war can last multiple periods. State such a strategy prole formally and answer the show. question for such a strategy prole