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Auction theory can be viewed as a branch of game theory, especially bayesian games, but it can also be viewed as a separate field of economics. Auction theory develops mathematical models of auctions. What distinguishes it from the rest of game theory is that its development has been strongly motivated by a familiar real-world institution, and this has given the theory a special flavor.
Auction theory originated in the important work of a Nobel-prize winning economist William Vickrey, Counterspeculation, Auctions, and Competitive Sealed Tenders.
There are many real-world examples of auctions. Think of eBay or Amazon. Or think about Google that prices online advertising through auction. Non-internet examples include offshore oil auctions administered by US Department of Interior, highway procurement auctions (e.g. the Indiana Department of Transportation INDOT), or timber auctions administered by the US Forest Service.
Auction theory distinguishes between open auctions, in which bidders bid with at least some knowledge of other bids already submitted, and sealed bid auctions, in which bidders have no such information. eBay is an example of an open auction, as are auctions at Christie's and Sotheby's. Most of procurement in the US is conducted through sealed-bid auctions. These are also known as sealed tenders. Off-shore oil is also leased through sealed-bid auctions. The US Forest Service sells timber through both sealed-bid and open auctions.
Auction theory also distinguishes between private value auctions and common value auctions. This distinction is made according to the nature of the items sold. If the item has a price (valuation in the terminology of auction theory) that is more or less perfectly known to a buyer, then this is a private value auction. For example, think of buying a digital camera on eBay. Before you enter a bid, you must determine the maximum amount that you are willing to pay. This amount will depend on several factors, among them your level of desire for a new camera as well as the prices at which it is available for purchase elsewhere. But the main thing is that you know more or less perfectly how much you are prepared to pay, and how much other bidders value the camera is irrelevant for you. (Of course, it is relevant in the sense that you may not win the auction if others value it more, but it is irrelevant for the maximum amount you yourself are prepared to bid.)
You may be in a quite different situation however if you are bidding for a painting on eBay. Chances are you won't be allowed to examine the painting before you bid, so some of its important aspects will remain unknown before you win. For example, you most likely won't know if the painting is authentic or not. But the bidders each would receive some signal about the authenticity , perhaps by unconsciously comparing in their minds the painting's features to other paintings by the same artist. Even simpler, some of the bidders may be experts. You as a bidder would clearly benefit from the information available to other bidders. This is an example of what auction theorist call a common value auction.