2 Allocating Bundles of Goods Suppose we must allocate three items {a,b,c} and three agents. Suppose that it is possible to allocate more than one item to one agent, and that each agent's valuation over bundles of goods is common knowledge: a b cabac bc abc P11 21 6 4 5 11 P2 3 5 3 5 5 3 5 P3 4 5 0 7 5 7 9 For example, for P1, item a is worth $1 and item b is worth $2, while having both a and b is worth $6. That is, a and b are complements for P1 since their value together is more than the sum of the value of owning each on its own. On the other hand, a and b are substitutes for P2 since their value together is worth less than the sum of the value of owning each on its own. a. What is the efficient allocation of the items a, b, and c to the agents? That is, what is the allocation that maximizes the sum of the three agents' utilities from the items? Remember, each agent may receive zero item, one item, or more than one item, but no item can be given to two different agents. b. Suppose that the VCG mechanism is used to allocate these items. (This may seem a bit strange since the VCG mechanism is usually used when the players have private information, but in principle it could also be used when every player just has one type.) Compute the payment of each agent. 2 Allocating Bundles of Goods Suppose we must allocate three items {a,b,c} and three agents. Suppose that it is possible to allocate more than one item to one agent, and that each agent's valuation over bundles of goods is common knowledge: a b cabac bc abc P11 21 6 4 5 11 P2 3 5 3 5 5 3 5 P3 4 5 0 7 5 7 9 For example, for P1, item a is worth $1 and item b is worth $2, while having both a and b is worth $6. That is, a and b are complements for P1 since their value together is more than the sum of the value of owning each on its own. On the other hand, a and b are substitutes for P2 since their value together is worth less than the sum of the value of owning each on its own. a. What is the efficient allocation of the items a, b, and c to the agents? That is, what is the allocation that maximizes the sum of the three agents' utilities from the items? Remember, each agent may receive zero item, one item, or more than one item, but no item can be given to two different agents. b. Suppose that the VCG mechanism is used to allocate these items. (This may seem a bit strange since the VCG mechanism is usually used when the players have private information, but in principle it could also be used when every player just has one type.) Compute the payment of each agent