Notes20

# Notes20 - CS 573 Algorithmic Game Theory Instructor Chandra...

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CS 573: Algorithmic Game Theory Lecture date: 28th March, 2008 Instructor: Chandra Chekuri Scribe: Rodolfo Pellizzoni 1 Revenue Equivalence Theorem During last class we studied first and second price auctions in the symmetric Bayesian setting, and we showed that the revenue and expected payments are identical in both auctions at the equilibrium. In fact, this is not just a coincidence but is instead a consequence of the revenue equivalence principle that we will state below. In general, consider a direct revelation auction A for a single item and suppose that A assigns the item to the highest bidder. Then it can be shown that revenue at equilibrium is the same in all such auctions. Let us consider two further examples. Third price auction: in third price auction, the item is allocated to the highest bidder who is then charged third highest bid (it is left to the reader as an exercise to prove that such auction is not truthful). Although this auction is not used in practice, it is an interesting exercise to study it. All-pay auction : again, the item is awarded to the highest bidder. However, each bidder pays his/her bid even if he/she does not win the item. This type of auction is used to model lobbying kind of activities where costs/bids are sunk costs. Theorem 1.1 (Revenue Equivalence Theorem) Suppose bidders have independent and iden- tically distributed valuations and are risk neutral. Then any symmetric and increasing equilibrium of a direct revelation auction A that assigns the item to the highest bidder such that the expected payment of bidder with value 0 is 0, yields the same expected revenue. The key idea of the theorem is that at equilibrium the revenue depends only on the allocation rule used by the mechanism. As a matter of fact, a more general revenue equivalence theorem states that for any social choice function f implementable by two mechanisms in a Bayesian Nash equilibrium, the payments/revenue will be the same with some normalization assuming losers pay 0 (see the textbook for more details). Proof: The proof proceeds by deriving properties on the expected payments that do not depend on the specific mechanism being used. Let s : [0 , w ] R be a symmetric equilibrium strategy for auction A . We will characterize m A ( x ), the expected payment of a fixed bidder (suppose bidder 1 without loss of generality) over v - 1 , assuming that all other bidders play according to s . By assumption m A (0) = 0. In equilibrium bidder 1 should bid s ( x ). Suppose he bids some value s ( z ). We can then derive m A ( x ) solving a differential equation as follows.

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