Amy produces lemonade. She can choose to produce high quality lemonade, which is more costly to her, or can produce low quality lemonade, which is less costly. Brian decides whether to buy a bottle of lemonade from Amy or not. The price of the lemonade is 3, cost to Ann of high quality lemonade is 2, and low quality tea is 1. High quality lemonade is worth 5 to Brian, and low quality lemonade is worth 4 to him. Brian chooses whether to buy or not buy. (Amy produces lemonade even if Brian doesn't buy. The payoff of Brian from not buying is 0, and players are expected payoff maximizers.)

(i). Suppose Amy and Brian make their choices simultaneously. Draw the payoff matrix of this game. List any strictly dominated strategies for either of the players? What is the Nash Equilibrium? Draw the tree diagram of this game.

(ii). Now Assume that first Brian chooses whether to buy or not, and then observing Brian's choice Amy chooses the quality level. Assume that regardless of whether Brian buys or not buys, Amy pays the cost of any production she makes. Draw the tree diagram of this game. Find the solution using backward induction.

(iii). Now assume that if Brian doesn't buy Amy has 3 strategies, produce high quality, produce low quality, or not produce. As before if Brian chooses to buy then Amy has only the choice between produc ing high or low quality. For rest of the question, assume that a high quality lemonade is worth 4 and a low quality lemonade is worth 2 to Brian. Draw the tree diagram and solve using backward induction.

(iv). Suppose now that there are 2 periods. In the first period, Brian chooses whether to buy, and Amy chooses whether to produce and if so what quality (as in part c). At the end of period 1, Brian observes the choice Amy has made, and the game moves to second period. In period 2, Brian makes a new choice of buying or not. If Brian buys, then Amy has no choice, but to produce the same type of lemonade she produced in the first period, if she did produce lemonade in the first period. If she didn't produce tea in the first period, then she will have the option to choose the quality of the tea she will produce. If in period 2 Brian doesn't buy, then Amy has her options as described in the previous sentence, and also the option not to produce. Players maximize the sum of the payoffs that they get in the 2 periods. Draw the game tree, and solve using backward induction.