Chapter summary

In this chapter we present the theory of auctions, which is considered to be one of the most successful applications of game theory, and in particular of games with incomplete information. We mainly study symmetric auctions with independent private values and risk-neutral buyers. An auction is presented as a game with incomplete information and the main interest is in the (Bayesian) equilibrium of this game, that is, in the bidding strategies of the buyers and in the expected revenue of the seller. A hallmark of this theory is the Revenue Equivalence Theorem, which states that in any equilibrium of an auction method in which (a) the winner is the buyer with the highest valuation for the auctioned item, and (b) any buyer who assigns private value 0 to the auctioned item pays nothing, the expected revenue of the seller is independent of the auction method. This theorem implies that a wide range of auction methods yield the seller the same expected revenue. We also prove that the expected revenue to the seller increases if all buyers are risk averse, and it decreases if all buyers are risk seeking.

The theory is then extended to selling mechanisms. These are abstract mechanisms to sell items to buyers that include, e.g., post-auction bargaining between the seller and the buyers who placed the highest bids. We prove the revelation principle for selling mechanisms, which allows us to consider only a simple class of mechanisms, called incentive-compatible direct selling mechanisms.