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# Introduction to Mathematical Economics ECON C103

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This 28 page Class Notes was uploaded by Dr. Janiya Bernier on Thursday October 22, 2015. The Class Notes belongs to ECON C103 at University of California - Berkeley taught by D. Sraer in Fall. Since its upload, it has received 31 views. For similar materials see /class/226706/econ-c103-university-of-california-berkeley in Economcs at University of California - Berkeley.

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Date Created: 10/22/15

Econ 103C Lecture 6 An introduction to Auction Theory David Sraer Berkeley U n iversity April 21 2008 Some common auction forms a Open ascending price auction or English auction 0 Open descending price auction or Dutch auction not commonly used in practice 0 Sealed bid first price auction 0 Sealed bid second price auction a ebay 3 7 i I Aprii 2L 2008 2 30 Valuations o Auctions are used because the seller ignores the values that bidders attach to the object a Question design a mechanism to overcome this informational issue when only 1 item is sold 9 Two situations 0 The bidder knows the value of the object to himself at the time of bidding A private values EX a CD on ebay 9 Each bidder only have an imperfect value of the true common value of the object A common values EX bidding for an oil field April 21 2008 3 30 Evaluating auctions Two grounds on which to evaluate auctions 0 Revenue How much utility does a particular auction provide the seller of the good with o Efficiency does the object fall in the hand of the agent with the highest utility for it 3 s ADVll at 2008 4 30 What is an auction Two important general characteristics of auctions o Auctions are mechanisms that elicit information in the form of bids from potential buyers of a good regarding their willingness to pay and the outcome is determined solely on the basis of the received information a universality o In an auction the identity of the bidder plays no role in determining who wins the object and who pays how much a anonymity April 21 2008 5 30 The formal framework 1 0 There is a single object for sale 0 There are N potential buyersbidders 0 Each bidder i E 17 N is willing to pay X for the object at most 0 Each X corresponds to the realization of a random variable X which is iid on some interval 07w according to the increasing distribution function F continuous and with full support 0 EX lt 00 April 2L 2008 6 3o The formal framework 2 a Bidder 139 knows only the realization X of his own valuation and simply knows that others39 valuation are drawn iid from F o Bidders are risk neutral o All components of the model other than the realized values are commonly known to all bidders 0 There are no liquiditybudget constraints each bidder can pay up to hisher value X April21 2008 1 3o Analysis of second price auctions 1 0 Call 3 07w HR the equilibrium bid function of bidder i 0 An equilibrium is defined by an n tuple of bidding functions 3 73 such that for all bidder i with value X it is optimal to bid 3Xi if other bidders use the bid functions B 0 We are interested in symmetric equilibria ie in equilibria where all bidders use the same bid function BA 0 Consider a second price auction Call b the bid of player 139 His payoff is given by X 7 max by if b gt max by 7 J79 JI I 0 if b lt max by h I ll April 21 2008 s 30 Analysis of second price auctions 2 Proposition In a second price sealed bid auction it is a weakly dominant strategy to bid according to 3X X 0 Assume the other player 7i bid according to 3 Call p1 maxht bj maxhtixj the highest competing bid 0 Assume that bidder i decides to bid 2 lt X If p1 lt z or p1 gt Xi then bidding z or bidding X is equivalent However if X gt p1 gt 2 then bidding X strictly dominates bidding z 0 Assume similarly that bidder i decides to bid 2 gt X If p1 lt X or p1 gt z then bidding z or bidding X is equivalent However if z gt p1 gt X then bidding X strictly dominates bidding z Remark The previous proof does not rely on the iid assumption 39 quot 39 7W4 39ll April 21 2008 9 30 Expected payment in a second price auction 0 In a second price auction 0 A bidder i with valuation x wins the auction with probability Cx where G is the distribution function of YllNill defines as the highest value among N 71 bidders 9 Conditional on winning the auction bidder i will pay on expectation EMN UMN lt x a Note that G can be easily computed PY1quot 1lt x PX1 lt x x1 lt x FXN1 0 Overall a bidder with valuation X expected payment is given by FXN 1EY1N 1lY1N 1 lt X April 21 2008 3910 3o First Price Auctions 0 In such an auction the payoff for bidder i submitting a sealed bid b is given by X 7 b b gt Hi J79 0 if b lt maxbj J79 0 Clearly the previous equilibrium strategy is no longer optimal as it would deliver 0 profit to the bidder for sure 0 Clear tradeoff for bidder 139 increasing the bid 1 increases the probability of winning but 2 decreases the expected gain conditional on winning April 21 2008 3911 30 First Price Auctions 2 0 Assume there is a symmetric increasing and differentiable equilibrium strategy B 0 Consider bidder i with valuation X and assume the other bidders play the equilibrium strategy 3 First notice that it is never optimal for bidder i to choose a bid b gt w as reducing the bid would not affect the probability of winning the auction while increasing profit conditional on winning 0 A bidder with value 0 will bid 0 ie 0 0 0 Bidder 139 wins the auction whenever b gt maxiigl BX gt Since 3 is increasingmaX1 BX max1Xi y1 v 1 where Y1N71 is defined as before I ll April 21 2008 12 30 Firs Price Auctions 3 0 Thus bidder 1 wins the auction whenever WM lt b gt YE lt W 1b 0 His expected payoff by bidding b is therefore G lt 71 X7 b where G is again the distribution of VIM71 0 Take the FOC with respect to b g 3 71 b lt 71 gt xi b 7 G WWW 0 W W bi 0 Under a symmetric equilibrium the optimal bid b must be BMW ie 500 WW X gX X XgX o Equivalently d a CX x XgX 0 Using 30 0 we have mx foxygydy EMN IMN lt 1 I H Aprii 21 2008 13 30 First Price Auctions 4 o The equilibrium bidding strategy B can be rewritten using integration by parts G BWX WM 7 Kandy xi f3dy 0 Obviously the equilibrium bids are lower than the value X s m 1 0 As GM 7 RX the degree of shading depends on the number of competing bidders as N increases the equilibrium bid approaches X I April 21 2008 3914 30 Two exa m ples 0 Assume that the values are uniformly distributed on 01 0 If FX X then CX X V 1 and BMW X 7 f ll71 dy x a the bidder bids a constant fraction of his value a j 39 Anni 211008 3915 no Revenue comparison Proposition With i i d private values the expected seller39s revenue in a first price auction is the same as in a second price auction 0 A bidder with value X in a first price auction will win with probability Cx a Conditional on winning the auction he will pay EYfquot 1l Yf V l lt x o The expected payment by a bidder with value X is thus the same in a first price and in a second price auction 0 The ex ante payment of a particular bidder is given in both auctions by EmX fsd mXfxdx where W CXEY1N 1lY1N 1 lt X I ll April 21 2008 16 30 Revenue comparison EmX o The ex ante expected revenue generated by the seller is R N x fowy i Fygydy 0 Call YZW the second highest of N values We have 1PY2N k 1PY1N k NPY1N gt kamp Y1quot 1 lt k FkN N 1 7 Fk FkN 1 NFkN 1 7N71Fkquot I April 21 2008 11 30 Revenue comparison 0 Thus the density associated with YQN is awn NmeivMVMW4ueFM 0 But gk N 71fkFkN 2 so that QWMMMWHW 0 Thus R A f2NydyEY2Ni o The seller39s revenue is equivalent in the second and the first price auction Revenue equivalence resut I Aprii 21 2008 3918 30 Revenue comparison variance Proposition With iid private values the distribution ofequilibrium prices in a second price auction is riskier in the sense ofa mean preserving spread than the distribution of equilibrium prices in a first price auction 0 Let39s first define a mean preserving spread Let X be a random variable with distribution F 0 Let Z be a random variable such that for all potential values X of X EZlX X 0 and call H the distribution of Z a Call Y X l Z the random variable obtained from drawing from X and then drawing Z from the conditional distribution on X ie HlX X Call G the distribution of Y 0 G is a mean preserving spread of F a Y and X have the same mean but Y is more spread out than X as it isjust X l some noise 39l April 21 2008 19 30 I Revenue Comparison Variance 0 Remember that 1 in a second price auction the seller39s revenue is directly the random variable R2 YZW and 2 in a first price auction it is given by the random variable R1 BY1N o ElelR1 p EY2Nl Y1N p ElelNllYfN W p 0 But quite simply for all y m M y EiYEN WYfN lt y o So finally ER2lR1 p EY1N 1lY1N 1 lt ylt1p w 1p p 0 Thus if one call Z the random variable R2 7 R1 we have 1 R2 R1 l Z and 2 EZlR1 p 0 Thus R2 is a mean preserving spread of R1 I ll April 21 2008 20 3o Reserve price 0 O O 0 The seller often uses the possibility to set a reserve price r gt 0 ie a price that the winner of the auction will have to pay if the price set in the auction is lower than r If no bid exceeds the reserve price then the good is left unsold Consider the case of a second price auction The equilibrium strategy for a bidder is still to bid its value the reserve price excludes bidder with value lt r but the proof derived above can be applied to bidders with value 2 r The expected payment of a bidder with value equal to r is simply rGr The expected payment of a bidder with value X 2 r is 60 ffygmdy ill April 21 2008 2130 Reserve price first price auction 0 In a first price auction the reserve price still excludes bidder with value X lt r a A bidder with value r has to bid BU r 0 Now we can apply the exact same proof as before to find the following differential equation d a CX x XgX 0 So that 6X frxngMy o The expected payment of a bidder with value X 2 r is thus given by X mx7 r Gmmx ram ygwy gt The revenue equivalence still holds when there is a reserve price April 21 2008 22 3o Reserve price effect on revenue 0 We can compute the ex ante expected payment of a bidder in both the second and the first price auction as mm w ram Dandy fxdx rGr1e Fr Wendy fxdx m rGr1e Frl1 PM wow V r 0 xgx1eFxdx a Call X0 the seller39s utility if the good is left unsold The seller39s utility is thus I39lo Nmr l FrNX0 0 Utility maximizing reserve price dr N1 PM 7 rl r Cr Crfrxo I a 77 April 21 2008 23 3o Reserve price effect on revenue 2 9 Introducing the hazard rate X 1220 dl39l Tr N e 0 7x0 m 1 e Fm cm 0 In r X0 the derivative is positive a the optimal reserve price is more than the seller39s value for the good 0 Why Consider X0 O and consider setting r gt O assume only 2 bidders Tradeoff 0 Possibility that the good is left unsold Expected probability Fr2 Magnitude of loss r 9 Increase in the price paid for the good when Y2 lt r Expected probability 2Fr17 Fr Magnitude of gain r a When r is close to 0 first order gain vs second order loss gt strict gain by imposing a gt O reserve price It is optimal to exclude some bidders with value larger than X0 in order to raise the expected price exclusion principle I W ill Apl ll 21 2008 24 3o Reserve price effect on revenue 3 a The relevant FOC r 7 X0 r 1 o In particular note that the optimal reserve price does not depend on the number of bidders Intuition the reserve price is useful only when there is one bidder with value above r a With F uniform on 01 and X0 0 then r so that L lir r i a Without reserve price the expected revenue with two bidders is and it goes up to 157 with the optimal reserve price a 25 increase in seller39s profit I April 21 2008 25 3o Risk Averse Bidders Assume now that each bidder has a utility function u R a R such that u0 0 u is strictly increasing and concave Proposition With symmetric independent private values the expected revenue in a first price auction is greater than that in a second price auction 0 First notice that the proof for the equilibrium strategy in a second price auction still applies with risk aversion gt still optimal for the risk averse bidder to bid its own value o In a first price auction consider 39y 07w gt R the equilibrium bidding strategy with 39y0 0 Assume that 39y is increasing and differentiable I ill April 21 2008 2 6 30 Risk Averse Bidders 2 o Nash equilibrium all other bidders are playing 39y It must be optimal for bidder 1 to bid 39yx o If bidder 1 bids 2 proba of winning is G y 1z and utility conditional on winning is uxi z o Equilibrium condition 39yx argmaxZ G 4 H X 7 z where G FA 1 0 Using FOC mgxux 7 39yx GXLlX7 39yx or 7 Xi YlXD V X mew ax I April 21 2008 21 3o Risk Averse Bidders 3 0 With risk neutrality B X X7 x 469 a Note that 39y0 O 30 0 Because u is strictly concave and u0 0 uy 2 u yy so that W 2 x7 x 0 Thus 3X 2 39yx implies 39y x 2 x This combined with 39y0 O 30 implies that 3X lt 39yx for X gt 0 gt Risk Aversion causes an increase in equilibrium bids Risk averse bidders wants to hedge themselves against the possibility of losing the auction I April 21 2008 28 3o

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