ANSWER NO 4

2. (15 marks] Anna and Bob are twins. This year, their parents only got them one cake for their birthday. And they must decide how to split it. That is, they must decide two numbers a and b, that represent the fraction of the that goes to Anna and the fraction of the cake that goes to Bob, respectively. If a+b is less than one, the rest of the cake is thrown away. Anna's utility equals the share of the cake that she gets. Bob's utility equals the share of the cake that he gets. Problems 3-5 ask you to analize different mechanisms that Anna and Bob could use to choose a split. a (a) Find the set of all feasible splits. That is, what properties do a and b have to satisfy? (b) Find the set of all Pareto efficient splits. 4. (15 marks] The second mechanism they are considering is as follows. First, Anna proposes a split. Then, Bob decides whether to accept or reject Annas proposal. If Bob accepts, then they split the cake according to Anna's proposal. Otherwise, they throw away the entire cake. (a) Use backward induction to predict the outcome of this mechanism. (b) Is the outcome of this mechanism always Pareto efficient? (c) Do you think this mechanism is fair? 2. (15 marks] Anna and Bob are twins. This year, their parents only got them one cake for their birthday. And they must decide how to split it. That is, they must decide two numbers a and b, that represent the fraction of the that goes to Anna and the fraction of the cake that goes to Bob, respectively. If a+b is less than one, the rest of the cake is thrown away. Anna's utility equals the share of the cake that she gets. Bob's utility equals the share of the cake that he gets. Problems 3-5 ask you to analize different mechanisms that Anna and Bob could use to choose a split. a (a) Find the set of all feasible splits. That is, what properties do a and b have to satisfy? (b) Find the set of all Pareto efficient splits. 4. (15 marks] The second mechanism they are considering is as follows. First, Anna proposes a split. Then, Bob decides whether to accept or reject Annas proposal. If Bob accepts, then they split the cake according to Anna's proposal. Otherwise, they throw away the entire cake. (a) Use backward induction to predict the outcome of this mechanism. (b) Is the outcome of this mechanism always Pareto efficient? (c) Do you think this mechanism is fair