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1、Lecture XVI: AuctionsMarkus M. M obiusApril 20, 2006 Osborne, chapter 9 Gibbons, chapter 3 Paul Klemperers website at http:/www.paulklemperer.org/ has fan- tastic online material on auctions and related topics. I particularly rec- ommend his beautiful online book “Auctions: Theory and Practice” for
2、a supremely readable and non-technical discussion of auctions and the survey chapter “A Survey of Auction Theory” for a concise overview of the main theory.1IntroductionWe already introduced a private-value auction with two bidders last time as an example for a game of imperfect information. In this
3、 lecture we expandthis definition a little bit.In all our auctions there are n participants and each participant has a valuation viand submits a bid bi(his action).The rules of the auction determine the probability qi(b1,.,bn) that agent i wins the auction and the expected price pi(b1,.,bn) which he
4、 pays. His utility is simple ui= qivi pi.aaThe agent is always assumed to be risk-neutral and just maximize his expected profit - new issues arise if the bidders are risk-averse.1There are two important dimensions to classify auctions which are based on this template:1. How are values vidrawn? In th
5、e private value environment each bidder has a private value vifor the item to be auctioned off. In the important independent private value case the types are drawn independently from some distribution Fifor each player i with support v,v.For our purposes we normally assume that all bidders valuation
6、s are not just private but also drawn independently and from an identical distribution F (we say private values are i.i.d.).1In the correlated private value environment the viare not independent - for example if I have a large vithe other players are likely to have a large value as well. Somewhat co
7、nfusingly the special case where all bidders have the same valuation vi= vjis called the pure common value environment.22. What are the rules? In the first price auction the highest bid winsand the bidder pays his bid. In the case of a tie a fair coin is flipped to determine the winner.3In the secon
8、d price auction the highest bidder wins but pays the second-highest bid. In the all-pay auction all bidders pay and the highest bidder wins. All these three auctions are static games. The most famous dynamic auction is the English auction where the price starts at zero and starts to rise.Bidders dro
9、p out until the last remaining bidder gets the auction.4The Dutch auction is the opposite of the English - the price starts high and decreasesuntil the first bidder jumps in to buy the object. The Dutch auctionis strategically equivalent to the first-price auction. Note, that in May 2004 Google deci
10、ded to use a Dutch auction for its IPO.1We also make the technical assumption that the distribution is continuous and has noatoms. 2There are much more general environments. A very general formulation which en-compasses both private and common value auctions is due to Wilson (1977). Each bidder gets
11、 a signal tiabout her valuation which is drawn from some distribution gi(ti,s) where s is a common noise term for all players. The players value is then a function v(ti,s). If vi= tiwe get back the private value environment. Similarly, if vi= v(s) we have the pure common value environment. 3In most
12、equilibria ties are no problem because they occur with zero probability.4The second price auction with private values is very similar to the English auctionwhere the price starts at zero and increases continuously until the last bidder drops out (he pays essentially the second highest bid)22Solving
13、Important Private Value AuctionsUnless otherwise stated we assume symmetric private value i.i.d. environ- ments.2.1First-Price AuctionWe focus on monotonic equilibria bi= fi(vi) such that fiis strictly increasing (one can show this always holds).It will be simplest to work just with two bidders but
14、the method can easily be extended to many bidders. We also assume for simplicity that values are drawn i.i.d.from the uniformdistribution over 0,1 (this can also be generalized - the differential equation becomes more complicated then). The probability of player i winning the auction with bidding b
15、isProb(fj(vj) b) = Prob(vj f1j(b) = F(f1j(b) = f1j(b)(1)The last equation follows because F is the uniform distribution. The expected utility from bidding b is therefore:Prob(fj(vj) b)(vi b) = f1j(b)(vi b)(2)The agent will choose b to maximize this utility. We differentiate with respectto b and use
16、the first-order condition:1 f?j(f1 j(b)(vi b) f1j(b) = 0(3)From now on we focus on symmetric equilibria such that fi= fj. We know that in equilibrium b = fi(vi) such that:1 f?(vi)(vi f(vi) vi= 0(4)This is a differential equation and can be rewritten as follows:vi= vif?(vi) + f(vi)(5)We can integrate both sides and get:1 2v2 i+ K = vif(vi)(6)3where K is a constant. This gives us finally:f(vi) =1 2vi+K vi(7)You can check that f(0) = 0 - the player with a zero v
