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.

3. The first mechanism they are considering is as follows. Anna and Bob simultaneously write on a piece of paper what fraction of the cake they would like to have. Then, their parents will open both envelopes. If the sum of the numbers that Bob and Anna wrote are less or equal than 1, then each of them will get the fraction of cake that they wrote down and throw away the rest. Otherwise,if the sum is greater than one, their parents will throw away the entire cake.

(a)Assuming common knowledge of rationality, what outcomes are possible in this mechanism?

(b)Is the outcome of the mechanism always Pareto efficient?(c)Do you think this mechanism is fair?

4. The second mechanism they are considering is as follows. First, Anna proposes a split. Then, Bob decides whether to accept or reject Anna's 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? 1

5.The third mechanism they are considering is as follows. First, Anna cuts the cake into two different pieces. Then, Bob chooses the piece that he would like to eat. Anna gets the other piece.

(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?