Elements of auction theory. This material is not part of the course, but is included here for those who are interested

Transcription

1 Elements of auction theory This material is not part of the course, ut is included here for those who are interested

2 Overview Some connections among auctions Efficiency and revenue maimization Incentive compatiility or truth revelation 2nd price auction The generalized Vickrey auction GVA The revelation principle Independent private value versus common value Some results on revenue maimization Deriving the Nash eq. in 1 st -price auction CS206, Spring 2001 Shoham 2

3 Some connections among auctions Dutch = 1st price sealed id English ~ Japanese English ~ 2nd price sealed id oth price setting and epected revenue under IPV model; see elow CS206, Spring 2001 Shoham 3

4 Two yardsticks for good auctions Revenue: The seller should etract the highest possile price Efficiency: The uyer with the highest valuation should get the good The two are usually aligned, ut can sometimes fail e.g, 1stprice auction when uyers have different risk attitudes. CS206, Spring 2001 Shoham 4

5 Direct mechanisms and incentive compatiility In a direct mechanism you simply announce your valuation The auction is incentive compatile if it s in your est interest not to lie aout your true valuation Eample: 2nd price Vickrey auction Another eample: the generalized Vickrey auction GVA; see discussion of cominatorial auctions elow CS206, Spring 2001 Shoham 5

6 The revelation principle You can transform any auction into an equivalent one which is direct and incentive compatile Rather than lie, the mechanism will lie for you Eample: Assume two idders, with valuations drawn uniformly from a fied interval plus other assumptions. The optimal strategy is to id 1/2 your true value. But if the rule is changed so that the winner only pays half his id, it is optimal to id your true value. CS206, Spring 2001 Shoham 6

7 Various auction settings Independent Private Value IPV Common Value CV Affiliated Value These different settings define different Bayesian games; the relationships etween the private signals determine the auction settings CS206, Spring 2001 Shoham 7

8 Some results on revenue maimization IPV model CV model Risk-neutral uyers: all k-price auctions, as well as English Japanese and Dutch have identical revenues. Risk-Averse uyers: 1st + Dutch are preferale to 2nd and English; more generally, k-price is superior to k+1 price. Risk-seeking uyers: k+1 price is preferale to k price e.g., English is preferale to Dutch English and second-price are no longer revenue equivalent, although Dutch and 1st still are English is preferale to 2nd, which is preferale to 1st/Dutch, if the agents are risk-neutral. CS206, Spring 2001 Shoham 8

9 Eample of equiliirum analysis: 1 st -price auction Setting: One good, two agents The agents valuations are independently drawn from the uniform distriution on [0,1] uy=y is the utility functions of oth agents A first-price FP auction CS206, Spring 2001 Shoham 9

10 Easy part: proving a particular equilirium Assume player 1 plays z, and player 2 s strategy is y=y/2 If player 1 s valuation is his epected payoff is given y 2z v= 0 z dv = z2z = 2z 2z note: given the y/2 strategy, 1 only wins when 2 s valuation is <2z This is a quadratic equation who derivative 2 4z is equal to 0 at z = / 2 The same analysis is true of player 2 Therefore =/2 is the est response to the same strategy y the other player, and therefore the two players adopting this strategy forms an equilirium 2 CS206, Spring 2001 Shoham 10

11 Harder part: proving this equilirium unique, under certain assumptions sketch* We ll e looking for a continuous symmetric increasing equilirium. 1 s epected payoff is 1 z v= 0 z dv = Since this is a concave fn requires proof, ma is achieved when derivative is zero: 1 z z * This is not part of the class material, and is included for completeness CS206, Spring 2001 Shoham 11

12 CS206, Spring 2001 Shoham 12 Second approach cont. take this part on faith, even if you didn t see this in your intro calculus course We now look for a value for z which zeros the derivative, under the constraint that z=: Now note that =/2 is a solution; and using functional analysis, which we won t enter, you can prove it is the unique solution ' ' ' 1 1 z z + = + = ' '* 1 1 z z z z 0 ' ' 1 = + = +

13 Multi-player 1 st -price auctions More generally, with n idders and similar conditions, the symmetric equilirium is given y: = n 1 n CS206, Spring 2001 Shoham 13

14 Some of the many topics in auction theory not covered here signals and valuations: the formal model of auctions as Bayesian games Precise definition of the affiliated values model Monopoly, marginal revenue, marginal cost Risk-averse auctioneer auctioneer as trader: if idders are risk neutral auctioneer prefers 1 st >2 nd >EnglishJapanese. Constant asolute risk aversion Computational compleity of cominatorial auctions Collusion in auctions CS206, Spring 2001 Shoham 14

15 Further reading on auction theory Handook of Auction Theory P. Klemperer, Ed. Klemperer s introduction to the Handook Articles references in the Handook Many other, more recent articles CS206, Spring 2001 Shoham 15