Mechanism Design
There are three kids: Alice, Bob and Clara. Their mother wants to split a cake of size 1 into three pieces and distribute them among the kids. Every kid wants to eat as much cake as possible.
a) (10 rp) Consider the following mechanism. Alice cuts the cake in three pieces the way she likes; then Bob takes any piece he likes, and then Clara takes any piece she likes (of the two that are left), so Alice is left with another remaining piece. How will Alice cut the cake?
She should divide the cake in 3 equal parts. Otherwise, hers sibling will take the larger pieces and she will be left with the smallest one.
b) (20 rp) Now consider a more complicated situation. A kid is unhappy if he or she gets less than a certain share of a cake. In particular, Alice will be happy if she gets at least a piece of size a, Bob needs at least b, Clara needs at least c. For every kid, getting a piece of the minimum required size is better than getting no cake, which is, in turn, better than getting a piece of less than the minimum required size. If one is already happy, he or she nevertheless prefer getting more cake to less cake. The mother knows that 0<a,b,c<1 but does not know a,b or c. All three children know all three numbers. Under which a,b,c does a mechanism that ensures each of the kids is happy exist? Suggest such a mechanism.
Such mechanism cannot exist if a+b+c>1. In this case, the cake is too small to make everyone happy. We will show that the mechanism exists for any a,b,c such that a+b+c \leq 1.
For example, the following mechanism can be applied. Alice offers some distribution of the cake to her siblings. Each of the siblings then votes in favour or against this distribution. If both votes are in favour, the distribution is accepted. Otherwise, no one gets anything.
In such a mechanism, Alice has an incentive to offer the distribution that makes both siblings and herself happy (which is always possible as long as a+b+c \leq 1 ). Bob and Clara have incentives to accept such distribution because they will get nothing otherwise. Should Alice offer any distribution which makes Bob or Clara unhappy, the unhappy sibling(s) will vote against because they prefer to get nothing.
The crucial features of any proposed mechanism are the following:
1. The distribution of a cake among the kids must be done one of them (because the mother is unaware of a,b,c values).
2. If suggested distribution doesn't satisfy kids' minimal requirements, everyone gets nothing.