please provide solution in hand written

2. Consider the infinitely repeated game with discount factor o E (0, 1) and the stage game A X (1,0,0) 3 R R (0,2,2) (5,0,0) (5,0,0) (2,1,1) For each of the strategy profile below, find the range of o under which the strategy profile is a subgame-perfect Nash equilibrium. (a) The play is (I, L, R) if (I, L, R) has been played at all previous days, and (X, L, L) is played otherwise. (b) On "Sundays", i.e., at t e {0, 7, 14, .. .}, (X, L, L) is played; on other days, (I, L, R) is played if (1, L, R) has been played at all previous such days and (X, L, L) is played otherwise. (c) On "Sundays", (X, L, L) is played; on other days, (I, R, R) is played if (I, R, R) has been played at all previous such days and (X, L, L) is played otherwise. 4. [Recall Problem 4 in Homework 2.] 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 cream parlors, namely 1 and 2, located at 0 and 1, respectively. Each ice cream parlor i sets a price p; 0.' A kid located in w is to pay cost clw - yl to go to a store located at y, where c E (0, p/3). Given the prices p1 and p2, 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) Assume that the above game is repeated 100 times, and find the subgame-perfect Nash equilibria. (b) Assume that the above game is repeated infinitely many times and the discount rate is 6 E (1/3, 1). For each of the following strategy profile, find the highest p' under which the strategy profile is a subgame-perfect Nash equilibrium. (Here, p* may be a function of 6. You need to choose both p* and p to make the strategy profile a subgame-perfect Nash equilibrium.) 1. At the beginning each parlor i chooses p; = p* and continues to do so until some player sets a different price; each i selects price p; = p thereafter. 2. There are two states: Collusion and War. The game starts at the state Collusion. In Collusion state, each player i chooses p; = p", and in War state each player i chooses pi = p. If both players set the price prescribed for the state, then the state in the next round is Collusion; the state in the next round is War otherwise. 3. Bonus: In part (ii), assume that p 2 0 but war can last multiple periods. State such a strategy profile formally and answer the above question for such a strategy profile