Question

Auctioning k identical items:

A particularly simple case of combinatorial auctions is when there are k identical items to be sold and n > k bidders who are interested in acquiring one of them. That is, each bidder has their own value for getting an item, but that value does not depend on which item and there is no additional value for acquiring more than one. In this setting VCG has a particularly simple and natural form. Explain what VCG does in this setting. That is, who does it allocate items to and how much does it charge them?

Answer 1

The VCG mechanism allocates identical items to bidders with the highest individual values and charges them based on the harm they cause to other bidders. In the case of k identical items, the top k bidders receive one item each.

Step-by-step explanation:

In the setting where there are k identical items to be sold and n > k bidders who are interested in acquiring one of them, the VCG (Vickrey-Clarke-Groves) mechanism works by allocating the items to the bidders with the highest individual values for them. The mechanism charges each winning bidder an amount equal to the total harm they cause to all other bidders by winning the item.

For example, let's say there are 4 identical items and 8 bidders. Each bidder submits a bid indicating their value for getting one item. The VCG mechanism would allocate one item to each of the 4 bidders with the highest bids and charge each of them the sum of the differences between their bid and the next highest bid.

Key points:

• VCG mechanism allocates items to bidders with highest individual values for them.
• Winning bidders are charged the sum of the differences between their bid and the next highest bid.
• In the case of k identical items, the top k bidders will receive one item each.