Auctions with Interdependent Valuations. Theoretical and Empirical Analysis, in particular of Internet Auctions

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2 Auctios with Iterdepedet Valuatios Theoretical ad Empirical Aalysis, i particular of Iteret Auctios Julia Schidler Viea Uiversity of Ecoomics ad Busiess Admiistratio Jauary 003

3 Abstract The thesis ivestigates a umber of auctio formats both theoretically ad empirically. The effect of differet auctio rules o the fial price ad o bidder valuatios is aalysed. Results from a experimetal sale of real goods, testig reveue equivalece of the ope ad sealedbid secod-price auctio do ot coform to theoretical predictios: the ope auctio leadig to sigificatly lower prices tha the sealed-bid auctio. It turs out that the ope auctio format allows bidders to satisfy a tedecy to stick together with their valuatios. The empirical results motivate a dyamic biddig model of iterdepedet valuatios, bidders beig ucertai about their valuatios ad learig from the exit-prices of their rivals. Furthermore, biddig behaviour o the Iteret is ivestigated i the hard close ad the automatically exteded auctio. Late biddig is show to be a ratioal strategy i the hard close auctio, but ot i the automatically exteded auctio. Theoretical results show that the expected fial price ad seller reveue is lower i the hard close auctio tha i the automatically exteded auctio, where prestige-cocers ca lead to a explosive fial price. Moreover, Yahoo auctio data cofirms the strog presece of late biddig i the hard-close auctio ad the seller s preferece for the automatically exteded auctio.

4 Itroductio... 8 PART ONE: Auctio Theory The Four Stadard Auctio Formats Eglish Auctio Dutch Auctio First-Price Sealed-Bid Auctio....4 Vickrey Auctio (Secod-Price Sealed-Bid Auctio).... The Idepedet Private Values Model (IPV).... Biddig strategies i the IPV Model..... First-Price Sealed-Bid Auctio..... Dutch Auctio Secod-Price Sealed-Bid Auctio Eglish Auctio Extesio: Sealed-Bid Higher k th -Price Auctios Results of the Idepedet Private Values Model Extesio: Ucertaity About the Number of Bidders Effect of the Number of Bidders o the Price Effect o the Biddig Strategy Beyod Stadard Assumptios Risk Aversio Asymmetries Betwee Bidders Iterdepedet Values The Commo Value Model The Symmetric Model, the Milgrom-Weber Model Affiliatio Biddig Strategies i the Milgrom-Weber Model Secod-Price Sealed-Bid Auctio Eglish Auctio Results of the Milgrom-Weber Model Rakig of Expected Prices Likage priciple Reveue Rakig Accordig to Theoretical Predictios

5 5. Experimetal Tests of Biddig Behaviour ad Auctio Reveue Field Experimets Field Experimets o the Iteret Laboratory Experimets Theoretical Predictios ad Empirical Results PART TWO: Dyamic Price Formatio i the Japaese Auctio Itroductio Experimet Experimetal Set-Up The Goods Results No Reveue-Equivalece Lower Bid-Variace i Japaese Auctio Average Bid Not Sigificatly Differet Reasos for Lower Bid-Variace i Japaese Auctio Learig Effects Iterpretatio The Model The First Roud The First Exit Commo Value Estimatio ad the Updatig Procedure The Geeral Procedure Estimatio Procedure The Expected Fial Price Results of the Model Coclusio Appedix PART THREE: Experimetal Test of Reveue Equivalece Motivatio Experimetal Set-Up

6 3. Reveue Equivalece Betwee the Secod-Price ad the Japaese Auctio Breakdow of Reveue Equivalece Testig the Effect of Revealig Public Iformatio Results Eglish Outcry versus Secod-Price Sealed-Bid Auctio Results Reveue Equivalece: First-Price Sealed-Bid ad Dutch Auctio Results Compariso: Secod-Price Sealed-Bid, First-Price Sealed-Bid ad Japaese Auctio Results Iterpretatio Six Results of the Experimetal Ivestigatio of Reveue Equivalece Coclusio PART FOUR: Iteret Auctios ad their Framework Itroducig Iteret Auctios Three Busiess Models: Ebay, Amazo, ad Yahoo Reveue Network Effects Loyalty The Sellig Mechaism Auctios ad Posted Prices The Goods Suitable for Auctios Goods Sold The Auctio Formats Used The Choice of Auctio Format by the Auctio House Iteret-Specific Characteristics Iteret Specific Advatages Iteret Specific Problems Outlook

7 3. The Iteret Auctio Rules Bid Submissio ad Procedure Bidder ad Seller Registratio Auctio-Legth Auctio Fees Additioal Features Some Implicatios of the Auctio Rules PART FIVE: Late Biddig Ivestigatio Itroductio Theoretical Ivestigatio Theoretical Ivestigatio of Late biddig The Moral Hazard Icetive Iterdepedet Values Geeral Biddig Model Model of the Hard Close Auctio Reveue Compariso: Hard Close ad Automatically Exteded Auctio Prestige Value Model Symmetrical Case Expert-Amateur Case Result of Prestige Value Auctios Payoffs i the Automatically Exteded Auctio Payoff Compariso: Hard Close ad Automatically Exteded Auctio Milgrom-Weber Model Biddig i the hard close auctio Geeral Predictio for Iterdepedet Valuatios Late biddig with Respect to the Edig-Rule Late Biddig Accordig to Good Type.... Seller s Choice of Edig-Rule Empirical Ivestigatio Late Biddig Existece of Late biddig Complete Auctio Duratio Last Twelve Hours

8 3.. Late biddig: Depedecy o Edig-Rule Complete Biddig-Path Last Twelve Hours Reasos for Late Biddig i Automatically Exteded Auctios Late biddig: Accordig To Type of Good Art: Strogest Late biddig Computers: Late biddig Similar For Both Edig Rules Late biddig I Car Auctios: Strogly Depedet O Edig-Rule Operatioal Ivestigatio of Late Biddig Time-Ivariace Wier s Biddig Behaviour Etry Time of Wier Wiig Bid: Sigle Bid or Proxy Bid? Seller s Choice of Edig-Rule The Preferred Edig-Rule Successful Matchigs Average Number of Bids Buy-Price Results of the Empirical Aalysis The Four Mai Hypotheses ad the Empirical Evidece Further Importat Results Coclusio PART SIX: Coclusio Literature

9 Itroductio What is a auctio? A mechaism that determies the price ad allocatio of goods by comparig competig bids. Auctios have bee used for sellig goods i aciet cultures such as early Chia, Greece ad the Roma Empire. Herodotus reports auctios of wome o the aual marriage market as early as 500 B.C. i Babylo. Nowadays, auctios are a widely used sellig device for diverse items, such as govermet bods, state-owed firms ad mieral rights for oil ad other atural resources. Sotheby s ad Christies, fouded i the 8 th Cetury i Great Britai, represet a brach of traditioal auctio houses kow for sellig exquisite items such as art, atiquities ad jewellery, or collectibles, such as cois ad stamps to the wider public. Aother set of goods frequetly sold i auctios are perishable products, such as flowers (i Hollad) ad fish (i Japa). Due to the Iteret ad the cosequetly low trasactio costs, auctios have boomed. Ebay, Amazo ad Yahoo auctios eable cosumers to buy ad sell items o a virtual platform ope to bidders aroud the world. Supplier cotracts are auctioed-off olie. Whether traditioal or olie, the seller wats to receive the highest possible price for his good. The questio of how buyers form their bids ad which auctio format realises the highest auctio reveue for the seller eeds to be aswered give the iformatio techology era ad its ew empirical isights. The dissertatio is set-up as follows: I part oe I preset a overview of auctio theory. I part two, a experimetal test of reveue equivalece betwee the secod-price sealed-bid ad the Eglish auctio (also kow as the Eglish ascedig bid auctio) is coducted. The empirical results do ot coform to theoretical predictios: the ope Eglish auctio yields sigificatly higher reveue tha the secod-price sealed-bid auctio. Furthermore, bids are 8

10 far more arrowly dispersed i the Eglish tha i the secod-price sealed-bid auctio. The empirical observatios call forth a dyamic biddig model of iterdepedet valuatios. Bidders are ucertai about their valuatios ad follow a boudedly ratioal-learig rule to update their valuatios i the course of the auctio. I part three, further tests of reveue-equivalece are coducted. The sealed-bid format is compared to the ope format for the two pairs of strategically equivalet auctios: First-price sealed-bid ad Dutch auctio, Eglish outcry ad secod-price sealed-bid auctio. The results are tested uder the effect of revealig public iformatio. I part four the rules ad framework of the curretly existig Iteret auctios are preseted. I part five biddig behaviour i Iteret auctios is aalysed usig two models of iterdepedet valuatios: a geeral ad a prestige value model. Late biddig is foud to be a ratioal biddig strategy i the hard-closig auctio, lowerig the price ad seller reveue. O the other had prestige-effects ca lead to exorbitat seller reveue i the automatically exteded auctio. The theoretical predictios are tested usig Yahoo auctio data with respect to two edig rules (hard close) ad three categories of goods (cars, computers ad paitigs). Late biddig is foud to be strogly preset both i terms of the dyamic ad operatioal biddig path as well as the wiig bidder s etry time. The proposed seller s preferece for automatically exteded auctios is empirically cofirmed. I part six I coclude. 9

11 PART ONE: Auctio Theory Auctio rules ca be chose with respect to two goals. Oe goal is maximizatio of the seller reveue; the secod goal is efficiecy; efficiecy meaig that the good is allocated to the highest-valuig bidder. Efficiecy ad reveue-maximisatio do ot ecessarily coflict. Here we focus o private goods, where a seller is usually cocered with fidig a auctio mechaism to maximise his reveue. I the best case, the seller could charge the highest valuig bidder a price exactly equal to this valuatio. But, the seller does ot kow the bidders valuatios. The goal of the bidder is to maximise his utility, which is the differece betwee the valuatio of the good ad the price he has to pay. Thus, the bidder has o iterest i revealig his valuatio to the seller. No auctio mechaism ca determie prices directly i terms of bidder prefereces ad iformatio. The seller must choose auctio rules that reveal iformatio about the bidders prefereces. There are a large umber of rules a seller could choose whe desigig his persoal auctio sellig device. Auctio theory works with the followig four auctio formats:. The Four Stadard Auctio Formats. Eglish Auctio The Eglish auctio is a ope, ascedig bid auctio. The price is raised sequetially util oly oe active bidder remais. The good is allocated to the highest bidder, who has to pay a price equal to the secod-highest bid.. Dutch Auctio The Dutch auctio is a ope, descedig bid auctio. A couter showig the curret price is lowered cotiuously util the first bidder cries: halt. The Dutch auctio is for example used i Hollad for sellig flowers. 0

12 .3 First-Price Sealed-Bid Auctio The first-price sealed-bid auctio is a closed auctio. Every bidder eters a private bid. The good is awarded to the highest bidder at the price of his ow bid..4 Vickrey Auctio (Secod-Price Sealed-Bid Auctio) The secod-price sealed-bid auctio is a closed auctio. The good is awarded to the highest bidder at the price of the secod-highest bid.. The Idepedet Private Values Model (IPV) I order to aalyse which auctio format is the reveue maximisig choice, we eed to make some assumptios about the way bidders form their valuatio of the good. Oe frequetly chose set of assumptios is the idepedet private values model: Assumptios: - Risk-eutrality: All bidders are risk-eutral, maximisig their expected profits. - Idepedece: The bidders values are private ad idepedetly distributed. - Symmetry: The values of the bidders are distributed accordig to the same distributio fuctio. - No budget costrait: Bidders have the ability to pay up to their respective values. A risk-eutral seller wats to sell a idivisible object that he himself values with zero. There are bidders. Bidder i (i =,.., ) draws his valuatio x i from the distributio fuctio F i (x i ) idepedetly ad idetically distributed o the iterval [ x, x] with the desity fuctio f i (x i ) f i (x) = f(x) for all x [ x, x]. This is equivalet to assumig the good has already bee produced ad the seller s utility from usig it is zero. Jehle ad Rey (00), p.374.

13 Every bidder kows his valuatio, but caot observe the private valuatios of the other bidders. The seller does ot kow the bidder valuatios, but he ad all bidders kow the distributio of the bidder valuatios ad the umber of bidders. A bidder s valuatio is idepedet ad private, deotig differeces i taste. The value of the good depeds oly upo persoal prefereces; a bidder s value is uaffected by the valuatios of the other bidders (eve if he kew them, his valuatio for the good would remai uchaged). The wiig bidder receives the good ad has to pay a price p. His payoff is give by: x i p. If he does ot wi the good, his payoff is zero.. Biddig strategies i the IPV Model Biddig behaviour i a IPV auctio is a o-cooperative game. Bidders devise a strategy; i.e. a biddig fuctio β :[0, ω] R that maps every possible value bidder i could draw ito a o-egative bid. Bidders search for the biddig fuctio that leads to the most desirable outcome, give that all other bidders also form their bid accordig to that same biddig fuctio... First-Price Sealed-Bid Auctio I a first-price sealed-bid auctio the bidder with the highest bid wis ad pays a price equal to his bid. Π i xi bi = 0 ( xi bi ) / k, k : = {arg maxb j} j if if if b i b b i i < max < max = max j i j i j i b b b j j j I the first-price sealed-bid auctio bidders shade their bids; i.e. bid less tha their valuatio. If a bidder bid his true valuatio, he would have to pay a price equal to his valuatio i case of wiig ad receive a payoff of zero.

14 Bidders face a trade-off; shadig the bid dowwards meas lowerig the probability of wiig but also meas icreasig the expected gai (whe beig the wiig bidder). G distributio fuctio of Y, where Y is the secod-highest private sigal. g the desity of Y, g = G The expected payoff of the wiig bidder is give by: G ( β ( b))( x b) Maximisig this with respect to b, yields the followig first-order coditio: g β ( b)) ( x b) G( β β '( β ( b)) ( ( b)) = 0 I a symmetric equilibrium b = β (x) ad yields the followig differetial equatio: ' G ( x) β ( x) g( x) β ( x) = xg( x) or equivaletly, d dx ( G( x) β ( x)) = xg( x) sice β ( 0) = 0, β ( x) = yg( y) dy G( x) 0 = E[ Y Y x < x] F The symmetric equilibrium strategy i a first-price auctio is: β x) = E[ Y Y < ] ( x Proof: Oly strictly icreasig biddig fuctios are cosidered; it is assumed that bidders with higher valuatios make higher bids. z deotes the value for which b is the equilibrium bid, z = β ( b), so that β ( z ) = b. 3

15 Bidder s payoff from biddig ) (z β whe his value is x is give by: = = = < = = Π z z z dy y G z x z G dy y G z z G x z G dy y yg x z G z Y E Y z G x z G z x z G x b ) ( ) )( ( ) ( ) ( ) ( ) ( ) ( ] [ ) ( ) ( )] ( )[ ( ), ( β It follows that: 0 ) ( ) )( ( ) ), ( ( ) ), ( ( = Π Π z x dy y G x z z G x z x x β β If all bidders follow the strategy β, a bidder with a value of x will be best off biddig ) (x β ; thus β is a symmetric equilibrium strategy. The equilibrium bid ca be writte as: = x F dy x G y G x x 0 ) ( ) ( ) ( β This is the symmetric Nash equilibrium of a first-price sealed-bid auctio. The biddig fuctio is strictly icreasig i x ad offers a uique solutio. As ca be see from the expressio above, bidders i a first-price auctio bid less tha their valuatio. The degree of bid shadig depeds o the umber of rival bidders, because ) ( ) ( ) ( ) ( = N x F y F x G y G As the umber of bidders icreases, the equilibrium bid approaches x. (x) F β I the case of uiformly distributed valuatios o [0,]: F(x) = x, the G(x)=x N- ad x N N x F ) ( = β The expected seller reveue, i.e. expected price is: ] [ = N N R E F The expected utility of the wiig bidder with sigal x N is: N x N 4

16 .. Dutch Auctio I the Dutch auctio a bidder eeds to decide at what price to cry halt. The wiig bidder, has to pay a price equal to his bid. The Dutch auctio is strategically equivalet to the firstprice auctio...3 Secod-Price Sealed-Bid Auctio I the secod-price sealed-bid auctio, the price the wier has to pay is determied by the secod-highest bid ad is thus idepedet of the wier s bid. Π i xi bi = 0 ( xi bi ) / k, k : = {arg maxb j} j if if if b b i b i i < max < max = max j i j i j i b b b j j j It is a uique weakly domiat strategy to bid oe s ow valuatio: S β ( x) = x Proof: By biddig above his valuatio, a bidder rus the risk of wiig the auctio, i cases where he would make a loss. Assume that bidder has a valuatio x ad that the highest competig bid is: p = max b. By biddig b = x, bidder will wi if b p ad does ot wi if b > p <. j i j I case bidder bids a amout higher tha his valuatio: b > x. If b > x p, the bidder wis with a payoff: x. This is the same payoff he would have received from biddig a p amout equal to his valuatio. If p > b, bidder loses. If > p >, bidder wis but makes a loss equal to x p x x, whereas by biddig a amout equal to his valuatio he would ot have made a loss. It follows that it is ever profitable for bidder to bid above his valuatio, as this ever icreases his profit, but may actually decrease his profit. b By biddig below his valuatio, a bidder lowers his chaces of wiig: he does ot wi i cases where he could have received a positive payoff. Thus, the pay-off maximisig strategy for bidder i is to bid his valuatio. 5

17 Assume bidder has a valuatio of x ad that the highest competig bid is p = max b. By biddig b = x, bidder will wi if b p ad does ot wi if b > p <. j i j I case bidder bids a amout smaller tha his valuatio: b < x. If x > b p, the bidder wis with a payoff: x p. This is the same payoff he would have received from biddig a amout equal to his valuatio. If p > x b, bidder loses. If x > p > b, bidder loses, but could have wo by biddig b =. It follows that it is ever profitable for x bidder to bid below his valuatio, because this may decrease his profit...4 Eglish Auctio I a Eglish auctio a bidder has to decide whe to drop out of the auctio. The Eglish auctio differs from the sealed-bid auctios i that bidders observe the exit-prices of the others ad have the possibility to revise their valuatio as log as they are active participats. I a Eglish auctio truthful biddig is a weakly domiat strategy. If weakly domiated strategies are elimiated, the bidder with the highest valuatio wis ad pays a price equal to the secod highest valuatio...5 Extesio: Sealed-Bid Higher k th -Price Auctios Theoretically it is possible to coduct a auctio, where it is either the highest bid that determies the price the wier has to pay, or the secod-highest bid, but istead the thirdhighest or fourth-highest bid. Usig higher k- th price auctios leads to the followig results:.) Bids are higher tha valuatios..) Equilibrium bids icrease as k icreases. 3.) Equilibrium bids decrease as the umber of bidders is icreased. The reaso why third-price auctios ad higher are geerally ot foud i practice, is because they expose the seller to higher risk tha the stadard auctio formats. Higher k-th price auctios, meaig higher tha secod-price auctios. 6

18 . Results of the Idepedet Private Values Model 3 Result : The Dutch auctio is strategically equivalet to the first-price sealed-bid auctio. Two auctio formats are strategically equivalet, whe the expected seller reveue is equal ad a idetical bidder would choose the same strategy uder both auctio formats. Result : The Eglish auctio is strategically equivalet to the secod-price sealed-bid auctio, but i a weaker form tha the strategic equivalece of the Dutch ad first-price sealed-bid auctio - the latter holdig eve whe bidders are ucertai about their valuatio. Result 3: The secod price ad the Eglish auctio lead to a efficiet allocatio, a Paretooptimal outcome. The Dutch ad first-price sealed-bid auctio also lead to a efficiet outcome as log as the bidders valuatios are draw from a symmetric distributio. Result 4: The expected seller reveue is equal to the expected value of the secod highest bidder. Result 5: The seller s expected reveue is equally high i all four auctio formats. This is the famous reveue equivalece theorem by Vickrey (96). Result 6: The four stadard auctio forms ca be desiged so as to produce a optimal outcome by usig etry fees or reserve prices. This result is true for may commo sample distributios, icludig the ormal, expoetial, ad uiform distributio. Result 7: Whe the seller, the bidders or both are risk-averse, the seller strictly prefers the Dutch or first-price sealed-bid auctio to the Eglish or secod-price auctio..3 Extesio: Ucertaity About the Number of Bidders 3 As oted i Milgrom ad Weber (98). 7

19 It is geerally assumed i auctio theory that the umber of participatig bidders is kow. I reality bidders ofte face ucertaity with respect to the umber of bidders, for example i Iteret auctios, where bidders are allowed to eter util the very last momet..3. Effect of the Number of Bidders o the Price I the idepedet private values model, as the umber of bidders icreases, the secod highest valuatio approaches the upper limit of the distributio of valuatios, ad thus the price teds to the highest possible valuatio (Holt 979). As log as the umber of bidders is fiite, the price the wiig bidder has to pay is smaller tha his valuatio. A higher umber of bidders raises the seller reveue ad lowers the bidder reveue i all four auctio formats..3. Effect o the Biddig Strategy The umber of bidders does ot ifluece the biddig strategy i the secod-price auctio, but does ifluece it i the first-price auctio. The biddig strategy i the secod-price auctio is give by: β ( x ) = x. Ucertaity about the umber of participatig bidders, ad cosequetly uder- or overestimatig the umber of participatig bidders has o effect i the secod-price auctio. Whe values are uiformly distributed x ~U [0,] the biddig strategy i the first-price auctio is determied by: β ( x) = N x. The expected seller reveue i the first-price N auctio is equal to N N ad the expected payoff of the wiig bidder (with the private sigal x N ) is equal to x N. N Uder- or overestimatig the umber of participatig bidders i the first-price auctio affects a bidder s probability of wiig ad his expected reveue. Uderestimatig the umber of participatig bidders reduces the idividual bidder s probability of wiig. If all bidders uderestimate the umber of participatig bidders, the seller s expected reveue falls. Overestimatig the umber of bidders icreases the idividual bidder s probability of wiig, but lowers his expected reveue. The seller s expected reveue rises whe all bidders overestimate the umber of participatig bidders. 8

20 3. Beyod Stadard Assumptios The Reveue Equivalece Theorem does ot always hold whe assumptios of the idepedet private value model are relaxed. 3. Risk Aversio Whe either the seller or the buyers are risk-averse, the first-price auctios lead to higher seller reveue tha the secod-price auctios. Uder risk-aversio the equilibrium biddig strategy i the secod-price auctio remais uchaged, but chages i the first-price auctio. I a first-price auctio bidders shade their bid whether they are risk-eutral or risk-averse. Risk-averse bidders i a first-price auctio shade their reservatio price more heavily tha whe they are risk-eutral. Risk-eutral bidders shade their bid less, because the icrease i expected-paymet due to a margial icrease of the bid, is less costly tha the reduced probability of ot wiig the auctio due to the lower bid. This raises the seller s expected reveue ad lowers the bidder s expected payoff 4. With costat absolute risk-aversio the first-price auctio produces higher expected reveues tha the secod-price auctio. 3. Asymmetries Betwee Bidders If the assumptio of symmetrical bidder valuatios is removed, the first-price auctio does ot always create a efficiet outcome (the good is ot always awarded to the highest valuatio bidder). Roughly speakig, the sealed-bid auctio geerates more reveue tha the ope auctio whe bidders have distributios with the same shape (but differet supports). I cotrast the ope 4 Riley ad Samuelso (98). 9

21 auctio geerates more reveue tha the sealed-bid auctio whe distributios have differet shapes but approximately the same support. Ex ate asymmetries ca discourage participatio by lower valuig bidders. Small asymmetries ca lead to highly asymmetric equilibria that result i low seller reveues Iterdepedet Values The private value model is ofte urealistic, because there are may goods where bidders are ucertai about their valuatio ad are iflueced by the valuatios of the other bidders. I the followig sectio the private values assumptio is relaxed ad istead bidders are assumed to have iterdepedet values. Iterdepedet values imply that every bidder has some private iformatio i form of a sigal, but a bidder does ot perfectly kow his valuatio for the object. It may ow be the case that other bidders possess iformatio that would - if kow to the bidder - affect his valuatio. This ca be due to resale or prestige cosideratios: a buyer of a old-timer might wat to resell the car after some time, thus he will let his valuatio be somewhat depedet o the other bidder s valuatios. Iterdepedet values do ot imply aythig about the distributio of the bidders sigals: sigals ca be idepedetly distributed or correlated. The best-kow model of iterdepedet values is that of Milgrom ad Weber (98). They assume that bidders sigals are affiliated, which is a special form of positive correlatio (see Part Two Chapter.3. below). Milgrom ad Weber s geeral symmetric model of iterdepedet valuatios ca accout for the case of strictly private valuatios (see above for the IPV model), for the itermediate cases ad for strictly commo valuatios (see below) The Commo Value Model A good havig a sigle objective value is offered for sale. The bidders do ot kow the value of the object, but every bidder has access to some iformatio o its value, each bidder 5 See Klemperer (998), p

22 makig a differet estimate of the good s value. V is the true value of the good, draw radomly from a probability distributio (here: a uiform distributio) o the iterval: [ x, x]. Each bidder receives a private sigal x i, i =,..,N. The private sigals are idepedet draws from the uiform distributio o [V-ε, Vε]. A first-price auctio is cosidered here: Bidders do ot kow the true value V ad try to estimate the correct expected value. The expected value of the item coditioal o sigal x i is: E [ V X i ] = x. I this case every bidder i would take his private sigal to be the best estimate of the good s value, kowig that the expected mea sigal is equal to the site s true value. But if every bidder bids his private sigal, the wiig bidder will be the oe with the highest private sigal. He will have overbid the true value V most highly, makig a loss i tur. This is kow as the wier s curse. Foreseeig that a bidder will oly wi whe his sigal is the highest sigal, he bids the expected value coditioal o beig the high bidder: E V X [ max = x ] = i x i N ε N The expected value coditioal o beig the high bidder is lower tha the expected value coditioal o the private sigal: E V X = x ] = x > E[ V X = x ] for N > [ i i i max i The wier s curse ca be measured as the differece i the two expected values. Avoidig the wier s curse requires cosiderable discoutig of bids relative to the sigal values. The size of the discout is a icreasig fuctio of the umber of bidders N ad the dispersio of the sigals aroud the true value ε. Raisig the umber of bidders or lowerig the precisio of the sigals leads to a higher wier s curse (whe bidders igore their judgemetal failure). The symmetric equilibrium bid fuctio is equal to: b( xi ) = xi ε Y where = ε N Y exp ( x ( x ε ) N ε ) i

23 Expected profits for the high bidder are equal to: ε Y N Y dimiishes rapidly as x i moves beyod x ε. Igorig Y, the biddig fuctio is approximately equal to b ( x i ) x ε ad the high bidder s profit equal to = i ε. N If bidders igore the wier s curse, the bid fuctio uder risk eutrality is: s ε Y b ( xi ) = xi N N The model predicts that the high sigal holder always wis the auctio. This is because all bidders use the same bid fuctio, their oly differece beig their private iformatio x i regardig the value of the item The Symmetric Model, the Milgrom-Weber Model As itroduced i Sectio 3.3 the most promiet model of iterdepedet valuatios is the Milgrom-Weber model (98). Bidders have some ucertaity about their valuatio, due to resale or prestige cosideratios. Beig a geeral model it ca take accout of the various degrees of ucertaity ragig from the purely private value model to the purely commo value model. For all itermediate cases they assume that private sigals are positively correlated by affiliatio: There are bidders. Bidder i s value of the object is V i = u i (S, X). The bidder s valuatio does ot oly deped upo his private sigal, but also upo the other bidders private sigals. S = (S,..., S m ) is a vector of variables measurig the good s quality, which ifluece the value of the object to the bidders. The bidders caot observe S, but the seller ca observe some or all compoets of S. X = (X,...,X ) is a vector of value sigals observed by the idividual bidders. Let Y,,Y - represet the largest to the smallest estimates from amog X,,X. Every bidder observes a private sigal about the value of the good. Bidder i, i =,.., observes the private sigal X i about the value of the good. Bidder s value is: V = u (S, S m, X, Y, Y )

24 Assumptio : There is a fuctio u such that for all i, u i (S,X) = u(s,x i,{x j} j i ). Thus, all of the bidders valuatios deped o S i the same way, ad each bidder s valuatio is a symmetric fuctio of the other bidders sigals. Assumptio : u is oegative, cotiuous ad o-decreasig i its variables. Assumptio 3: For each i, E [V i ] < Whe V i = X i for all i, the model is reduced to the idepedet private value model. Whe V i = S for all i, the model is reduced to the commo value model. Bidders are risk-eutral ad their valuatios are i moetary uits, so that whe bidder i receives the object ad has to pay p, his payoff is V i p. f(s,x) is the joit probability desity of the radom elemets of the model. Two assumptios are made about the joit distributio of S ad X: Assumptio 4: f is symmetric i its last argumets. Assumptio 5: The variables S,..., S m, X,..., X are affiliated. E [V X = X, Y = Y, Y - = Y - ] is o-decreasig i x Affiliatio Every bidder has some private iformatio about the value of the good. This private iformatio is expressed i the sigal he draws. I case of affiliatio it is assumed that the bidders sigals X, X,..., X correlatio ad meas that if a subset of the are positively affiliated. Affiliatio is a strog form of positive take o large values, the the remaiig also take o large values. Variables are affiliated if large values for some of the variables make the other variables more likely to be large tha small. A high value of oe bidder s estimate makes high values of the other bidders estimates more likely. X i X j For variables with desities, affiliatio ca be defied as such: Let z z deote the compoet-wise maximum of m dimesioal vectors z ad z ad let z z deote the compoet-wise miimum. Variables are affiliated if, for all z ad z, f(z z )f(z z ) f(z)f(z ). 3

25 Three implicatios of affiliatio 6 :.) Y,,Y - are the largest to the smallest estimates from amog X,, X. If the variables X,...,, X X are affiliated, the the variables X, Y,..., Y are also affiliated..) G( x) deotes the distributio of Y coditioal o X = x. ad Y beig affiliated X implies that if x >x, the G( x' ) domiates G( x) i terms of the reverse hazard rate, that is, for all y, g( y x') g( y x) G( y x') G( y x) 3.) If γ is ay icreasig fuctio, the x >x implies that: E [ γ ( Y ) X = x'] E[ γ ( Y ) X = x ] Biddig Strategies i the Milgrom-Weber Model Secod-Price Sealed-Bid Auctio Bidder s decisio problem i the secod-price sealed-bid auctio is to choose a bid b that maximises the expected value mius the price coditioal o bidder s sigal (whe this is the highest sigal). S The equilibrium strategy of every bidder is to bid β ( x) = v( x, x) v x, y) : = E[ V X = x, Y = ]. v is o-decreasig. ( y I the case of private values (where v(x,x)=x) the equilibrium strategy is weakly domiat. With geeral iterdepedet values however S β is ot a domiat strategy Eglish Auctio The Eglish auctio i the Milgrom-Weber Model is modelled as a Japaese Auctio. The auctio begis at a price of zero, at which all bidders are active. The auctioeer raises the price ad bidders ca quit the auctio by depressig a butto. Bidders who have quit the auctio caot retur at a later poit i time. Every time a bidder quits the auctio, the exitprice is revealed to all remaiig bidders. 6 Take from Krisha (00), p.86. 4

26 A biddig strategy i the Eglish Auctio must specify for each of his possible valuatios, whether he will be active at ay give price level, as a fuctio of the biddig activity observed util the. The exit-prices ca be ordered as follows: p p k Bidder i s strategy ca be described by a fuctio b ik (x i p,, p k ), specifyig the price at which bidder i will quit if, at that poit, k other bidders have left at the prices p,, p k. Because the price ca oly rise, b ik (x i p,, p k ) has to be greater or equal to p k. The symmetric equilibrium is: * β ( x) = E V 0 * β ( x p,..., p β k * k ( Y * β ( Y 0 k ) = [ X = Y =... = Y = x] p,..., p p ], k ) = E[ V k X ) = p k = Y,..., k =,..., =... = Y = x, * β 0 ( x) deotes the optimal bid whe all bidders are active, * β ( x p,..., ) the equilibrium bid after k bidders have quit the auctio. k p k First-Price ad Dutch Auctio The first-price sealed-bid ad the Dutch auctio are strategically equivalet ad ca be treated equally. The equilibrium strategy for a bidder is: F β ( x) = x 0 v( y, y) dl( y x) x g( t t) where, L( y x) = exp dt G( t t) y G( x) is the distributio fuctio of Y, i.e. the secod highest sigal; uder the coditio that the highest sigal is equal to x. g( x) is the coditioal desity fuctio of Y. Milgrom ad Weber further prove that the expected seller reveue of the secod-price sealedbid auctio is greater or equal to that of the first-price sealed-bid auctio. 5

27 Results of the Milgrom-Weber Model.) The Dutch ad the first-price sealed-bid auctio are strategically equivalet..) Whe bidders are ucertai about their value estimates, the Eglish ad the secod-price sealed-bid auctio are ot equivalet. The Eglish auctio leads to higher expected prices due to the likage priciple (see below). 3.) Whe bidders value estimates are statistically depedet, the secod-price sealed-bid auctio geerates higher average prices tha does the first-price sealed-bid auctio. 4.) If the seller has access to a private source of iformatio, his best policy is to commit himself to hoesty (always reportig all iformatio completely). This is true for the firstprice sealed-bid, Dutch, secod-price sealed-bid ad the Eglish auctio Rakig of Expected Prices Eglish > Secod-price sealed-bid > Dutch = First-price sealed-bid Likage priciple The likage priciple explais why the Eglish auctio yields higher reveue tha the secodprice sealed-bid auctio uder affiliatio ad why revealig public iformatio raises the price. Every bidder receives idirect iformatio about the valuatios of the other bidders from their publicised exit-prices. Observig biddig behaviour by others makes bidders more cofidet ad lets them bid higher o average. All measures that icrease the iformatio of bidders, for example quality guaratees, are price icreasig ad advatageous to the seller. The price is liked to the valuatios of the o-wiig bidders ad the wiig bidder. The auctio prices deped o the reports the bidders make ad o the seller s iformatio. The Dutch ad first-price sealed-bid auctio with o likages to the other bidders estimates, yield the lowest expected price. The Eglish auctio with likages to ay of the estimates of the o-wiig bidders yields the highest expected price. Revealig public iformatio raises the price i all three auctios, by addig a likage. 4. Reveue Rakig Accordig to Theoretical Predictios The table below summarises the theoretical predictios about seller-reveue for the idepedet private values ad the affiliated values model (for private ad iterdepedet 6

28 valuatios). As ca be see the first-price sealed-bid ad Dutch auctio always yield equal reveue. The secod-price sealed-bid ad the Eglish auctio yield equal reveue uder the private values assumptio, but ot whe values are iterdepedet: Model IPV ad risk eutral bidders IPV ad risk averse bidders Affiliated, privately kow values ad risk eutral bidders Affiliated, privately ukow values 7 ad risk eutral bidders Table : take from Luckig-Reiley (999), p.065 Reveue Rakig All equal Dutch = st Price > d Price = Eglish Dutch = st Price < d Price = Eglish Dutch = st Price < d Price < Eglish Returig to our origial questio of fidig the auctio format that leads to the highest seller reveue out of the four stadard formats, we arrive at the result that whe bidders are riskeutral the reveue of the Eglish auctio is greater (affiliated values) or equal (private values) to that of the other auctio formats. Uder risk-aversio the first-price auctios yield highest seller-reveue. If the theoretical predictios were trustworthy ad the seller were able to kow what riskattitude the buyers have ad how the bidder valuatios are distributed, the he could choose the auctio format accordigly. I the ext chapter empirical tests of the theoretical predictios are preseted. 5. Experimetal Tests of Biddig Behaviour ad Auctio Reveue Experimetal tests of auctio reveue ca either be coducted through cotrolled laboratory experimets or field studies. There are far more laboratory tests comparig auctio-reveue tha field-studies. Oe disadvatage of usig field-studies for reveue comparisos is that the 7 The bidder is ucertai about his valuatio oly havig received a oisy sigal about his value. 7

29 theoretical reveue predictios rely o assumptios about bidder valuatios, however it is difficult (impossible) to cotrol field-data for the type of bidder valuatios. 5. Field Experimets There is very little field data comparig auctio formats, because real auctios ted to be coducted accordig to oe pre-determied mechaism. Oe example of data makig a empirical compariso of two auctio formats possible, is the U.S. Forest Service auctio for timber harvestig rights i the Pacific Northwest. Due to a chage i federal law, the U.S. Forest Service coducted some of its auctios by a first-price sealed-bid auctio ad the others by a Eglish auctio. Mead (967) ad Johso (979) used this data i a empirical study ad foud that the first-price sealed-bid auctio raises sigificatly higher reveue tha the Eglish auctio. However, Hase (985, 986) fids that after correctig the data for a bias i the selectio method for the timber lots, the lower reveue of the Eglish auctio is o loger statistically sigificat. The strikig part of the results is that timber sales are likely to have strog commo value or at least correlated private value elemets, which i theory should lead to higher prices i the Eglish auctio. Teorio (993) studies multi-uit auctios by usig data from Zambia currecy auctios. Teorio fids that multi-uit auctios yield higher reveue whe the price is determied by a discrimiatory rule tha whe it is determied by a uiform-pricig rule. 5.. Field Experimets o the Iteret Luckig-Reiley (999) coducted auctios of Magic Cards o Ebay to empirically test reveue- equivalece o the Iteret. He fids that the Dutch auctio leads to thirty percet higher reveue tha the Eglish auctio ad that the secod-price sealed-bid ad the Eglish auctio are roughly reveue-equivalet. 5. Laboratory Experimets Most empirical reveue-comparisos are carried out by cotrolled laboratory experimets. Bidders are assiged valuatios distributed accordig to the assumptios of the theoretical 8

30 model tested. Uder the assumptio of private valuatios, the participats i the experimet are told that their valuatio for the good is exactly x moetary uits. The experimet tests whether the participats ca guess the ratioal biddig strategy, i the case of idepedet private values, whether they realise that they are supposed to bid their valuatio i the secodprice auctios ad are supposed to shade their bid i the first-price auctios. Laboratory experimets that test the existece of the wier s curse i the commo value model, test whether the participats realise that they are ot supposed to bid their private sigal, but are supposed to shade their bid to discout for the strategic error of overbiddig due to the wier s curse. Laboratory experimets testig the private-values assumptio show that bids ted to be higher i the sealed-bid tha i the ope auctios: Coppiger et al (980) ad Cox et al (98,983) fid that reveue i the first-price sealedbid auctio is sigificatly higher tha theoretically predicted by the risk eutral Nash equilibrium strategy (RNNE); reveue i the Dutch auctio is approximately equal to or slightly below the RNNE predictio. Kagel et al (987) ad Kagel ad Levi (993) fid that reveue i the secod-price sealedbid auctio is higher tha i the Eglish auctio format: bidders bid their valuatios i the Eglish auctio but bid above their valuatio i the secod-price format. Results were tested with respect to bidder experiece, but the breakdow of reveue-equivalece remais. Experimet Coppiger et al (980) Cox et al (98, 983) Kagel et al (987) Kagel ad Levi (993) Table : take from Luckig-Reiley (999), p.066 Results st price > Dutch st price > Dutch d price > Eglish d price above theoretical predictios 9

31 6. Theoretical Predictios ad Empirical Results The theoretical predictios of auctio theory strogly rely o assumptios about the distributio of bidder valuatios. Auctio theory expects reveue-equivalece i the case of private values ad expects the Eglish auctio to yield highest auctio reveue i the case of affiliated values. Cotrary to theoretical predictios, experimetal laboratory results show that the sealed-bid format leads to higher reveue tha the ope auctio format. Possible explaatios for this discrepacy may iclude importat aspects beig eglected i auctiomodels or experimets beig carried accordig to usuitable methods. 30

32 PART TWO: Dyamic Price Formatio i the Japaese Auctio Overview A dyamic biddig model is preseted i which bidders are ucertai about their ow valuatio. Bidders lear about their private valuatio from the exit prices observed. As a result, the secod-price sealed-bid auctio produces sigificatly higher reveue tha the Japaese auctio: moreover, bids i the Japaese auctio are far more arrowly spread tha i the secod-price sealed-bid auctio. The model explais this result by showig that bidders are able to satisfy a tedecy to stick together i the ope Japaese auctio, whereas the secret secod-price sealed-bid auctio offers o such opportuity. Furthermore, the model ca explai the results of a experimetal sale of real goods. 3

33 . Itroductio Sellers wat to use the auctio mechaism that maximises their expected reveue. I this sectio we compare two auctio formats: the Japaese 8 ad the secod-price sealed-bid auctio 9. Turig to auctio theory, the seller has to make a assumptio about the bidders valuatios, whether bidders have purely private, purely commo, or iterdepedet valuatios. Auctio theory predicts that whe bidders have private values, a good yields equivalet expected reveue whether sold by a Eglish or secod-price sealed-bid auctio. This is a result of William Vickrey s fudametal Reveue Equivalece Theorem 0. The secod-price sealed-bid auctio ad the Eglish auctio are ot oly reveue equivalet, but are also strategically equivalet. The domiat strategy of both auctio formats is to bid a amout equal to oe s private valuatio. The assumptios uderlyig the private value model are striget ad maybe urealistic, as they impose that bidders value the good idepedetly of the valuatios of all other bidders. I may istaces bidders are iflueced by the values that their rivals assig to a good. Imagie for example art, secod-had objects or collector items where bidders are ofte subject to reputatioal cocers. Bidders partly base their valuatio o other bidders value judgemets, believig the good to be more precious whe others value the good highly ad less valuable whe others do ot care much for the good. A importat istace whe bidders do ot act accordig to the predictios of the private value model ca be observed i Iteret auctios. Goods - loosely classifiable as privatevalue goods - hardly receive bids for days util oly some hours or miutes before the plaed auctio ed, whe all of a sudde biddig activity rises icomparably. Late biddig 8 The Japaese auctio is a sub-variat of the Eglish auctio ad is also called ascedig-clock auctio. The Eglish auctio is a ope, ascedig-bid auctio. 9 The high-bidder wis, but pays oly the secod highest bid. 0 It states that all four stadard auctio formats (first-price, secod-price, Eglish ad Dutch auctio) lead to equally high expected seller-reveue uder the assumptio of idepedet private valuatios. Furthermore, the domiat strategy is uaffected whe bidders have private affiliated values, see Kagel ad Roth (995), p

34 occurs despite bidders havig the possibility to use a proxy-biddig aget. Whe bidders have private values, they are expected to have o icetive to hold back their valuatio. A possible explaatio could be that the high-valuig bidder believes that by biddig early ad publicisig his high bid, he will cause low valuig bidders to revise their valuatio upwards, raisig the price the wier has to pay. Prestige cosideratios ad ucertaity about the true quality of the good could be causes of this behaviour. At the other extreme of idepedet private values lie purely commo values: the good havig a ukow but commo value to all bidders. Oly few goods are pure commo value goods, such as for example oil fields or gold uggets. May goods, however, have some ucertaity surroudig their true quality, makig them irrecocilable with both the purely idepedet private value model ad the purely commo value model. For most goods it is realistic to relax the private values assumptio ad istead to assume that values are iterdepedet 3. Iterdepedet values ca be of may a kid, but auctio theory focuses almost exclusively o the Milgrom ad Weber model of iterdepedet values. I their geeral model of symmetric iterdepedet values they assume that the bidders private sigals are affiliated 4, i.e. positively correlated, ad predict that the Japaese auctio yields higher expected reveue tha the secod-price sealed-bid auctio. Laboratory experimets test theoretical predictios based o the models assumptios. Theoretical predictios cocerig auctio reveue strogly rely o assumptios about the distributio of the bidders valuatios. There are may laboratory experimets testig the private values predictios 5 (both for affiliated ad idepedet private values), experimets A bidder ca submit his maximum-willigess-to-pay to the proxy-biddig aget, who will the bid o his part, raisig the curret high-bid by a miimum-icremet util he appears as the high-bidder. The proxybiddig aget will stop biddig oce the maximum-willigess to pay is reached. 3 See Part Oe Chapter 3.3. Iterdepedet values: Bidders have some ucertaity about their values, their value partly beig iflueced by private iformatio held by the other bidders. 4 Affiliatio: Bidders kow the value of the item to themselves with certaity, but a higher value of the item for oe bidder makes higher values for the other bidders more likely (private values are positively correlated relative to the set of possible valuatios). Kagel ad Roth (995), p Kagel, Harstad ad Levi (987) test reveue equivalece for affiliated private values. Empirical results show failure of the theoretically predicted strategic-equivalece betwee the secod-price sealed-bid ad the Japaese auctio. 33

35 testig the existece of the wier s curse for commo values, but there is a lack of experimets comparig reveue for iterdepedet valuatios 6. Laboratory experimets are coducted by assigig a private value or value estimate to every bidder ad observig whether bids ad reveue correspod to domiat strategy predictios. This method has the drawback that viewed critically it is merely a test of a bidder s cogitive ability of guessig the domiat biddig strategy. A seller watig to kow which of the two auctio formats yields higher expected reveue, might prefer a experimet that is less cotrolled but has a set-up that makes its results more meaigful to the practical udertakig. We desiged a experimet to test reveue equivalece of the Japaese ad secod-price sealed-bid auctio i a realistic settig: i the experimetal sale of real cosumptio goods. The experimetal results i Chapter Three show that biddig behaviour differs i the two auctio formats examied, specifically bids i the ope auctio beig far more clustered tha uder the sealed-bid format. As a result, the fial price i the secod-price sealed-bid is higher tha i the Japaese auctio. The results suggest that bidders do ot solely base their valuatio o their private value estimate, but istead partly base their reservatio price o the other bidders valuatio of the good. Motivated by the experimetal observatios, a biddig model is preseted i Chapter Five. This model differs from Milgrom ad Weber s geeral model i a umber of respects. Milgrom ad Weber assume valuatios are exogeous ad affiliated. I our boudedly ratioal model bidders are ucertai about their ow valuatio ad partly base their ow valuatio o other bidders private iformatio, i.e. idepedet sigals. Bidders update their valuatio usig the iformatio revealed through the exit prices of the other bidders. The fial price is reached i a dyamic process, bidders formig their valuatio adaptively. 6 A exceptio beig Kirchkamp ad Moldovau (00), who coduct laboratory experimets for a simple model of iterdepedet values testig efficiecy of the Japaese ad secod-price sealed-bid auctio. Their empirical results with respect to reveue are cosistet with the theoretical predictios: they fid that seller reveue is equal uder both formats ad bidder-payoff higher i the Japaese-auctio. 34