Question

Suppose that a seller has one item, which we will call item x. There are three buyers, whom we will call A, B, and C. The values that these buyers (A, B, and C) have for the item are 6, 3, and 1, respectively.

(a) Suppose that the seller runs a second price auction for the item. Which buyer will win the auction and how much will this buyer pay? Make it clear what are the bids.

(b) Now let us suppose that the seller uses the VCG procedure to allocate the item. Remember that the first step in running the VCG procedure when there are more buyers than items is to create fictional items, which each buyer values at 0, so that the number of items to be allocated is the same as the number of bidders. Let us call these additional (fictional) items y and z.

i. Find the allocation that results from running the VCG procedure.

ii. What are the prices charged to each buyer for the item that they receive? iii. Explain why the price that buyer A pays is the harm that he causes to the remaining bidders by taking the item he is assigned.

Answer 1

a)

Bids of each person:

A - 6

B - 3

C - 1

Person A wins as bids the highest and he has to pay 3.01 ( 0.01 more than the second highest bidder i.e., B)

b)

Bid of each person for item x:

A - 6 B - 3 C - 1

Bid of each person for item y:

A - 0 B - 0 C - 0

Bid of each person for item z:

A - 0 B - 0 C - 0

(i) The outcome of the auction is bidder A getting item x, and bidder B getting y and bidder C getting z.

Hence, value achieved by A = 6

value achieved by B = 0

value achieved by C = 0

(ii) C pays 0, B pays 0, A pays 3

(iii)

If A was not in the auction, the item would be assigned to person B. In that case person B would pay 3 but he is currently paying 0. Hence A has to pay that harm of 3.

If B was not in the auction, the item would be assigned to person A. In that case person A would pay 10 but he is currently also paying 10. Hence B has to pay for no harm

If C was not in the auction, the item would be assigned to person A. In that case person A would pay 10 but he is currently also paying 10. Hence C has to pay for no harm