Reserve prices in online auctions 1

Transcription

1 Reserve prices i olie auctios 1 Susaa Cabrera Yeto 2, Rosario Gómez 3, Nadège Marchad 4 Jauary 2007 Abstract: I this paper, we ivestigate the effect of miimum bids i electroic auctios. The extesive use of auctios i electroic markets is explaied by efficiecy. I practice, auctio houses ad olie auctioeers allow sellers to specify a umber of differet parameters whe listig a item for auctio. Amog these are the level of the opeig bid ad the amout of a "reserve price" below which the seller will ot sell the item. O the oe had, the reserve price restores the seller cotrol o the auctio results. O the other had, it elimiates all the potetial buyers whose offer would have bee lower tha this level. If all buyers valuatios are below the reserve price, the item is t attributed. The itroductio of reservatio price i the auctio itroduces dramatic chages i the auctioeer behaviors. Our paper aalyzes the effects of public ad secret reserve prices, ad the selectio of reserve prices itself, i Eglish auctios usig laboratory experimets. I our experimetal desig, the seller decides to sell the item at a miimum price (Riley ad Samuelso 1981.) which is either commuicated to potetial buyers (public reserve price) or ot (secret reserve price). The, the three bidders bid i a auctio for a item. The itroductio of a reserve price reduces the efficiecy level of the auctio, by geeratig coflicts, as well as the expected reveues of the seller ad the buyers. 1 We wish to thak Marie-Claire Villeval, Lauret Flochel for their useful commets. We also thak Rudy Saboadiere for skillful research assistace. Fiacial support from the CNRS/CGP ad PAI/Picasso program is gratefully ackowledged (this paper was part of the project "Société de l'iformatio"). Ay errors are our ow. 2 Departameto de Estructura Ecoómica, Uiversidad de Málaga, El Ejidos/. yeto@uma.es 3 Departameto de Teoría e Historia Ecoómica, Uiversidad de Málaga, El Ejidos/. rosgomez@uma.es 4 GATE, 93, chemi des Mouilles, B. P. 167, Ecully. marchad@gate.crs.fr 1

2 1. Itroductio Olie auctios are oe of the most successful forms of electroic commerce. Web sites as ebay couts some 3.8 millio registered members to take part i auctios. 5 Olie auctios are oe of the leadig iovatios i electroic market. I practice, olie auctioeers allow some freedom to the sellers i choosig a umber of parameters whe listig a item for auctio (see the seller guide of Yahoo, QXL, ebay.) Thus, sellers determie importat elemets of the auctio rules, which dramatically ifluece the result. Amog these are the level of the opeig bid ad the amout of a "reserve price" below which the seller will ot sell the item. Both optios restore the cotrol of the seller o the occurrece of a sell. Ideed, the seller decides the miimum price he wats to receive for the item, which elimiates all the potetial buyers whose offer would have bee lower tha this level. If the buyers valuatios of the object are below the reserve price, the item is ot sold. Note that the extesive use of auctios i electroic markets is explaied by efficiecy. I a auctio without reserve price, the good is always attributed to a buyer. As a cosequet, auctio geerates higher expected reveue tha other bilateral sell mechaism such as egotiatios i which a coflict is likely (Bulow, Klemperer, 1996.) As emphasis by the theoretical literature (see Klemperer 1999 for a survey) ad olie seller guide, the determiatio of the reservatio price is a critical decisio. Despite the widespread use of reserve prices i auctio, there is, to the best of our kowledge, few empirical works o that topic. Accordig to Bajari ad Hortaçsu (2000), the miimum bid or a public reserve price is oe of the most importat determiats of etry i auctio. Public reserve price cesors the distributio of bidder valuatios by discouragig some bibbers. Katkar, Luckig-Reiley (2000) ad Luckig-Reiley (2000) ru field experimets o secret vs. public reserve price effects o bids by sellig pokémo card ad magic card o the ebay. Their mai coclusio is that secret reserve prices reduce sigificatly the probability of sell ad, therefore, the earig of the seller. Nevertheless, those studies disregard importat factors, which may ifluece behaviours such as buyers private valuatio of the item, risk aversio. Laboratory experimets have bee extesively used to ivestigate a wide variety of hypotheses from auctio theory (see Kagel, J.H, 1995 for a survey), but, to the best of our kowledge, 5 See The ecoomist, July 22, 1999, The Heyday of Auctio. 2

3 effects of a secret or public reserve price are ot amog the topics covered, eve whe it is a usual practice of auctio houses. Istitutio desig: Roth details matters. Our paper aalyzes the effects of public ad secret reserve prices i Eglish auctios usig laboratory experimets. Accordig to Riley ad Samuelso (1981), the reserve price value is idepedet of the umber of bibbers. Ideed, we form groups of four participats composed of three buyers ad a seller. I our experimetal desig, three bidders bid i a auctio for a item, which have a radom value at the momet the seller decides to sell the item at a miimum price. How do bidders value items they are biddig o? Oe aswer to this questio is to say that each bidder idexed i gets utility v i from wiig the item. If bidder i kows v i but ot the v s of other bidders, the valuatio of the item is private. This situatio makes sese if the bidder is buyig the item without itetio to resale it. I the remaiig, we assume that buyers have private ad idepedet valuatios of the item, such that, they ca t resale it. The reserve price is either public (treatmet 1) or secret, the bidders bid without kowig the miimum price accepted by the seller (treatmet 2.) The itroductio of a reserve price reduces the efficiecy level of the auctio, by geeratig coflicts, as well as the expected reveues of the seller ad the buyers. Sellers select the reserve prices optimally. Sellers behaviour is affected by the itroductio of a reserve price; we ote a importat umber of overbiddig behaviours. We obtai a sigificat treatmet effects due to the iformatio o the reserve price. Bidders are sigificatly less aggressive whe they kow the value of the reserve price. Therefore, the probability to sell the good i auctio with reserve price is sigificatly higher for secret reserve price. The remaider of the paper is orgaized as follows. Sectio 2 presets the game theoretical aalysis of reserve price auctio derived from auctio theory. This aalysis provides the basis to uderstad the buyers ad seller s behaviour. Sectio 3 itroduces the experimetal desig ad describes the theoretical hypothesis. Sectio 4 aalyzes the experimetal data ad discusses the results. Fially, sectio cocludes. 3

4 2. Theoretical backgroud Cosider a sigle seller of a idivisible good with reservatio value v 0, who faces risk-eutral potetial buyers. Each buyer i holds a reservatio value v i, with i=1, 2,.., ad v i [ v, v] the followig, we assume that private valuatios of parties are idetically distributed ad idepedetly draw from the uiform distributio with F ( v) = 0, F ( v) = 1. Note that the aalysis for commo distributio, F(v), is preseted i the appedix I I the Eglish auctio, the good is awarded to the buyer who makes the fial ad highest bid. The buyer placig the highest valuatio o the good pays approximately the maximum of the reservatio values of the other -1 buyers. As Vickrey oted, this is equivalet to a secod-price sealed-bid auctio i which each buyer submits a bid ad the high bidder pays the secod highest rather tha the highest bid. 6 The outcomes of the Eglish ad secod-price auctios satisfy a strog criterio of equilibrium: they are domiat equilibria; that is, each bidder has a well-defied best bid regardless of how high he believes his rivals will bid. I a secod-price auctio, the domiat strategy is to bid true valuatio; i the Eglish auctio, the domiat strategy is to remai i the biddig util the price reaches the bidder s ow valuatio. Ideed, these domiat strategies are idepedet of buyers risk posture. The seller ca either, take part to a simple auctio (subsectio 2.1) or, determie a reserve price (subsectio 2.2). I former case, the buyer who submits the highest bid wi the auctio ad a sell always occurs. I the latter situatio, the seller determied a reserve price, which correspods to the smallest bid he will accept. If all bids are lower tha the reserve price of the seller, the item is t sold. The ext subsectios preset the equilibrium behaviours of the buyers ad sellers as well as their expected reveues i both cases. 6 I geeral, with the assumptios that uderlie the model, ay mechaism which always gives the good to the highest-value bidder i equilibrium, ad a bidder with the lowest feasible valuatio has o chace of ay surplus, yield the same expected paymet by each bidder ad the same expected reveue for the seller. This is the Reveue Equivalece Theorem. See, for example, Vickrey (1961). However, equivalet auctios to the Eglish auctio ad to the secod-price sealed-bid auctio do ot share the feature of domiat equilibrium ecessarily. 4

5 2.1 Auctios with o reserve price Give that each bidder will aouce his true valuatio i the secod-price sealed-bid auctio. 7 A idividual with valuatio v will wi with probability F 1 ( v ), which is the probability that all others will have valuatios ad bids below v. The expected paymet to this bidder is: 8 ( 1) v + v P ( v) = F( v) So the expected payoff to a buyer with value v to makig a bid is: e F( v) ( v v) π =. The expected paymet by the buyer with value v, cotiget o wiig, will be: 9 ( 1) v + v H ( v) = I the case of the uiform distributio, the expected k th highest value amog values idepedetly draw from the distributio o [ v; v] is: k v + ( v v) + 1 I a secod-price (or ascedig) auctio, the seller s expected reveue is oted Λ with: 11 1 Λ = v + ( v v) + 1 That is the expected secod-highest valuatio of the valuatios. 12 Note that, a bidder with a lower valuatio tha the seller s valuatio ca purchase the item. I this case, the outcome is ot efficiet. Note also that the expected reveue of the seller, Λ, is a strictly icreasig fuctio of the umber of buyers,. Thus, the more bidders there are, the larger is the seller s expected reveue Reserve prices If the seller aouces a reserve price R, his expected reveue icreases util: 7 I a ascedig auctio, each bidder will remai i the biddig util the price reaches his ow valuatio. 8 The expected paymet correspods to the expected price paid to the seller by this buyer. 9 The geeral expressios of P(v) ad H(v) ca be foud, for example, i Bulow ad Roberts (1989). 10 See, for example, Klemperer (1999). 11 A proof of the geeral expressio of Λ is give i Riley ad Samuelso (1981). 12 Note that i the case of the uiform distributio Λ=H(v) if v = v + ( v v) ; that is, if the wier s + 1 valuatio is the highest expected valuatio. 5

6 Ω = v + [ v( + 1) 2( R + v) ] 1 ( R v) ( v v) ( + 1)( v v) Note that if o reserve price was aouced, or v is greater tha the reserve price, R, the the expected reveue equals Ω. The, the reserve price, which maximizes the expected reveue of the seller, is: 13 R * v0 + v Note that this reserve price, R *, is idepedet of the umber of buyers. Furthermore, if the reserve price is higher or equals to the seller private valuatio, R * v 0, the item is sold ad a bidder with a valuatio greater tha the seller s valuatio purchased the item. Thus, the seller receives a positive payoff. Ideed, the equilibrium reserve price, R *, is icreasig i his private valuatio, v 0, ad the highest value of the buyers support distributio, v. = 2 Propositio: Whether the optimal reserve price selected by the seller is aouced publicly before biddig or ot, does o affect either the seller s expected reveue or bids. As i Eglish auctios ad secod-price auctios, the strategy of biddig oe s true valuatio is a domiat strategy. The seller caot ifluece bids by cocealig his reserve price. The, the optimal silet reserve price is the same as the optimal aouced reserve price, ad the expected seller reveue is idetical. 14 Substitutig i Ω for equilibrium reserve price, R *, the total expected retur to the seller is oted Γ ad equals: Γ = v 0 F * ( R ) + Ω = v ( v + v0 2v) ( v v) ( + 1) ( v v) * 0 R The term v F ( ) idicates that if o sell occurs (i.e. all bidders have values below the reserve price) the, the seller keeps the item ad receives his ow valuatio. 13 See Riley ad Samuelso (1981). 14 As Riley ad Samuelso (1981) ote, the argumet is more complex i the case of Dutch auctios ad first-price sealed-bid auctios, but oce agai it ca be show that there is o advatage i usig a silet reserve price. However, the argumet does ot apply out of the idepedet-private-values-model. Vicet (1995) shows that i a commo-value auctio, a silet reserve price icreases participatio of buyers, which may icrease the expected reveue of the seller. 6

7 The aalysis of the differeces betwee the expected reveues of the seller i auctios with ad without optimal reserve price, Γ-Λ, allows us to compare the two auctio mechaisms from the seller s poit of view. The sig of the derivative of Γ-Λ with respect to is egative. Hece, the more bidders there are, the less is the differece betwee the seller s expected reveue with ad without a optimal reserve price. Ideed, this differece is icreasig i the seller private valuatio, v 0. Whe the seller optimally sets a bidig reserve price, a iefficiet outcome may occur. If all bidders valuatios are below the optimal reserve price, R *, the good is ot sell. I this case, the seller may keep the good despite the presece of some bidders with a higher valuatio tha the seller s ow valuatio. The itroductio of a optimal reserve price iduces iefficiecy i auctio. Ideed, the secodprice sealed-bid ad Eglish auctios are ot optimal sellig mechaism if they are supplemeted by the optimally set reserve price. 15 The, the seller could, for example, have several rouds of biddig, or charge bidders etry fees, or allow oly a limited time for the submissio of bids. Noe of these more complicated strategies would icrease the expected price. 16 However, while this result is theoretically appealig, we have o idea about whether it characterizes biddig behaviours i real auctios. Therefore, our mai cocer is ow to test the empirical properties of reserve prices i auctio ad examie whether the behaviour of real participats complies with the theory. The ext sectio presets the experimetal procedures ad theoretical predictios before focusig o the experimetal results. 3. Experimetal desig ad theoretical predictios Sectio 3.1 explais the parameters, theoretical predictios ad hypothesis i details, sectio 3.2 provides a geeral descriptio of the experimetal procedure. 3.1 Experimetal parameters ad theoretical predictios At the begiig of each period, we form groups of four players, oe seller ad three buyers. Each buyer i (with i = 1,, 3) is assiged a private reservatio value v i idepedetly draw 15 The result exteds to Dutch auctios ad first-price auctios. 16 See, for example, McAfee ad McMilla (1987). 7

8 from a uiform distributio with support {0, 1, 2,, 59, 60}. The participats kow exclusively their ow reservatio values, but ot the values of the other subjects. The private valuatio of the seller equals 20. We let kow to the buyers that this value was also draw from a uiform distributio with support {0, 1, 2,, 59, 60}. The i each group, the seller determies his reserve price, R, ad the three buyers choose simultaeously a biddig price. Buyers are iformed whether the reserve price was reached or ot. First, if there are oe or more bidders with bids greater tha the reserve price, the buyer with the highest bid wi the item. If at least oe bid is greater tha the reserve price, the bidder who submitted the highest bid wis the auctio ad pays the maximum of the reserve price ad the secod highest bid submitted. I case of a tie i the high bid submitted, the high bidders are iformed of which of them wis radomly the auctio. The selected bidder pays his bid. Secod, if o bidder submitted a bid greater tha the reserve price, o sell occur ad the seller keeps the item. Whe the item is sold by auctio, the bidder who wis the auctio receives the differece betwee his valuatio ad the amout paid; the other bidders ear 0. The seller gets the amout paid by the wiig bidder mius his ow valuatio. If the item is ot sold by auctio, bidders get 0 ad the seller receive his ow valuatio. The experimet is based o a factorial 2 2 desig: There are four treatmets which exclusively differ with respect to the iformatio coditio o the reserve price value ad the possibility for buyers to overbid. Reserve price Overbiddig Iformatio Allowed Not allowed Private Treatmet 1 Treatmet 2 Public Treatmet 3 Treatmet 4 Table 1: Presetatio of the treatmets The mai questio of our study is whether private or public iformatio about reserve price affects biddig behaviours, reserve price values ad the efficiecy of the auctio mechaism. Therefore, i some of the treatmets, participats take part to a secret reserve price auctio i which they submit their bibs without kowig the value of the sellers reserve price. I other treatmets, subjects iteract i a public reserve price auctio i which they are iformed about the value of the sellers reserve price before submittig their bids. However, experimetal studies o auctios poit out overbiddig behaviours (Kagel, 1989). I order to evaluate the impact of this behavioural regularity, we itroduce treatmets allowig or ot buyers to overbid. 8

9 Whe overbiddig is ot allowed, buyers ca t submit a bid higher tha their ow valuatios of the item. At the ed of each period, the subjects were iformed whether or ot a sell occurs, for the buyers if they wi the auctio ad the sell price, as well as their ow payoff i the curret period ad their total profit up to this time. To iduce risk eutral prefereces for sellers ad buyers, we use a lottery procedure after each auctio. 17 Subjects earigs i the auctio are measured i poits. After the auctio, each subject who wo a positive umber of poits, h, face a lottery with two prizes, A=100 UME ad B=0 UME. The probability of wiig A is h/n, ad the probability of wiig othig is 1-(h/N), where N is the maximum umber of poits that a subject ca obtai i the auctio. I the settig, the maximum umber of poits that ay seller or buyer ca obtai i the auctio each period is N=40. O the oe had, if the object is sold by auctio, the buyer who wis the auctio ca obtai a maximum umber of 40 poits whe she has the maximum valuatio of 60, the others have miimum valuatios of 0, ad the reserve price is the lowest feasible, v 0 = 20. The, the buyer pays p=20 ad obtais v-p=60-20=40 poits. O the other had, the seller ca also obtai a maximum umber of 40 poits whe at least two buyers valuatios are 60 ad the reserve price is R 60. The, the wier buyer pays p=60 ad the seller receives p- v 0 =60-20=40 poits. So the probability of wiig the high prize, A=100 UME, for a subject who has eared h poits is h/40. Note that iducig risk eutrality for buyers is iocuous theoretically, because bidders have a domiat strategy, which is to bid true valuatio. I table 2, we use the theoretical backgroud developed i sectio 2 to compute the predictios tested i our experimet. Seller Buyers Equilibrium biddig R * = 40 v i = b i Efficiecy R e = 20 v i = b i Table 2: theoretical predictios Assumig participats are risk eutral, we derive the followig hypotheses from our theoretical backgroud. Hypothesis 1: Bidders bid their true valuatios. Thus, oe should observe o differece i biddig betwee treatmets. 17 See, for example, Roth ad Malouf (1979), Berg et al. (1986) ad Berg et al. (2003) 9

10 If bidders have valuatios greater tha the publicly kow reserve price, they bid their true valuatio; Whe bidders do ot kow the reserve price, all of them bid their true valuatio. Hypothesis 2: Reserve prices are optimal. Whether the reserve price is public or secret does ot have ay effect o reserve prices selected by sellers. Hece, for each sub-treatmet, the earigs to the seller are the same o average i all treatmets. Hypothesis 3: The itroductio of private or public reserve prices lowers the efficiecy of auctio mechaism. Auctios are efficiet mechaisms. I case of a reserve price, coflict may appear betwee the wier of the auctio ad the seller. Thus, all sells will ot be implemeted because of too low price offer. 3.2 Experimetal procedure I all experimetal coditios described below, subjects participated as a seller or a buyer i a secod price sealed bid auctio, oe seller ad three buyers formig a group. Role assigmet remaied the same throughout the etire sessio. The experimets were ru i the GATE experimetal laboratory with 160 participats ad cosisted i 10 sessios, with each sessio comprisig 40 periods. The participats were radomly recruited from a subject pool of studets of several uiversities ad the graduate school of maagemet (Lyo). All of them were iexperieced i auctio experimets ad o subject participated i more tha oe of the sessios. I each of the 40 periods, oe seller ad three buyers were re-matched such that the same group did ever iteract i two cosecutive periods. Therefore, i our setup, all the theoretical results hold for all periods: Sice iteractio is aoymous ad oe-shot the 40 periods are repetitios of static games ad ot a dyamic game givig rise to further equilibria All together, we collected 6400 observatios, which provide us 3, respectively 2, idepedet observatios for treatmets 1 ad 3 ad for treatmets 2 ad 4. 10

11 Upo arrival, participats were radomly assiged to a specific computer termial. 19 I the begiig of each sessio, istructios were distributed ad read aloud (see Appedices 6.2 for a Eglish traslatio). Clarifyig questios were asked ad aswered privately. The, we asked the participats to fill i a cotrol questioaire i order to check for uderstadig. Oly after all questios had bee correctly aswered, the experimet started. The experimet was computerized usig the REGATE software (Zeiliger 2000). O average, each sessio lasted oe hour, excludig paymet of subjects. All amouts were give i ECU (Experimetal Currecy Uits), with coversio ito Euros at a rate of 1 for 100 ECU for buyers ad 150 ECU for sellers upo completio of the sessio. The fial paymet was the sum of the sigle payoffs of the 40 periods plus a 2 show-up fee. The average payoffs per roud (i ECU) ad the stadard deviatios (i brackets) are reported i the followig table: Treatmets Sellers Buyers Reserve price Overbiddig Mea Std. Dev. Mea Std Dev. Private Allowed Not allowed Public Allowed Not allowed Table 3: Average earigs per roud. For each treatmet the seller ears sigificatly more poits tha the buyers. 4. Experimetal Results Mai differeces arise from a compariso of biddig behaviours i public vs. secret reserve price treatmet. Furthermore, the objective is the to compare behavioural differeces related to iformatio o the reserve prices (public vs. secret) ad/or restrictio o submitted bids. The ecoometric aalysis evaluates the impact of treatmet variables o idividual proposals Reserve prices Accordig to theoretical predictio, we obtai o differeces i meas by comparig public ad secret reserve price distributios (see table 4 ad figure 1.) Ideed, meas ad media values are close to the predicted reserve price. 19 The GATE experimetal laboratory has privacy coditios sufficiet to assure that participats could ot observe each other s decisios. 11

12 With overbiddig Without over biddig Total Secret Public Secret Public Secret Public Mea 41,1 41,7 37, ,56 40,12 Media Std. Dev. 8,31 7,33 6,36 7,17 7,81 7,52 Table 4 : Reserve prices The distributio of reserve prices seems to be more flat i secret price treatmets, which is cosistet with other studies (Luckig Riley 2000.) Frequecies 0,35 0,3 0,25 0,2 0,15 0,1 Secret Public 0, Reserve price values We ru a radom effect liear regressio o reserve prices to ivestigate the effect of our strategic variables. I order to ivestigate the effect of our strategic ad treatmet variables o the idividual behaviours, we ru the followig radom effects liear regressio (for each party): y t = X t β+ε t = 1,...,N ad t = 1,...,T ε t = u +v t +w t where X t is the vector of the idepedet variables ad β the vector of the estimated coefficiets. The umber of idividuals equals 120 (N=120) ad umber of periods equals 40 (T=40). I our experimet, the variables, which characterize the model, are the explaatory variables 1 ad 2 are the two treatmet variables: These dummies study separately the impact of the iformatio o the reserve price (variable public) ad overbiddig opportuity (variable overbid) o the buyers' bids. The cross variable (variable public*overbid) aalyzes the joit ifluece of these two variables o biddig behaviour. By cosiderig the variable begi (resp. ed), we attempt to test the presece of a start (resp. ed) game effect. 12

13 The results from model 1 relate to icetives. I model (2), we added characteristics of the participats as cotrolled variables. We have geder ad we costructed dummies for graduate studets ad studets i ecoomics or i related fields (busiess, math, computer sciece), which are expected to perform better i experimet ivolvig some aspects of game theory. The variable Previous participat iforms us whether the subject has participated i a laboratory experimet i the past.. Depedat var. Model 1 Model 2 Reserve prices Coefficiet Std. Dev. Coefficiet Std. Dev. Cste (1.637) *** (7.181) Public (1.728) (1.602) Overbid ** (1.764) ** (1.591) Pub*Overbid (0.005) (0.004) Begi ** (0.495) ** (0.545) Ed (0.495) (0.545) Male (1.740) Studet * (5.104) Graduate (1.211) Eco. or related field studet *** (0.563) Previous part (1.662) R² 24.5 % 6.2% Log-likelihood Nb. of obs Table 20 Accordig to theoretical predictio, the reserve prices are idepedet o the iformatio coditios ad sigificatly icrease over time. Result 1 : Sellers set similar reserve price uder secret or public iformatio coditios. The absece of restrictio o biddig behaviour icreases sigificatly the reserve prices regardless the iformatio coditio o public reserve price. 20 *** Statistically sigificat at 1%, **statistically sigificat at 5%, *statistically sigificat at 10%. Stadards errors are reported i parethesis. 13

14 4.2. Impact of a reserve prices o biddig behaviours Followig the experimetal results, if o restrictio applied o bids, the buyers bids deviate strogly from equilibrium ad efficiecy. Ideed, their strategies cosist of offerig higher bids tha their reservatio values, such that both proportios are high, 48% or 86%, depedig o the reserve price iformatio. These results are illustrated i figure 3: 1 0,8 0,6 0,4 0,2 0-0,2 w ith overbiddig w ithout overbiddig Secret reserve price Public reserve price Figure 3 O the idividual level, figures 1 ad 2 report the bids made by participats depedig o their private valuatios. All poits located o the straight-lie correspod to the equilibrium ad efficiet bids (i.e. the player bids his reservatio value b i = v i ). Biddig behaviours of buyers i secret reserve price treatmets Bids Private valuatio Bids Private valuatio Figure 1A. Overbiddig ot allowed Figure 1B. Overbiddig allowed 14

15 Biddig behaviours of buyers i public reserve price treatmets Bids Private valuatio Figure 2A. Overbiddig ot allowed Bids Private valuatio Figure 2B. Overbiddig allowed Result 2. Whe o restrictio applies o bids, the buyers' behaviours deviate from equilibrium ad efficiecy by overbiddig. I public reserve price treatmets, the deviatios are more importat. Whe the buyers submit a bid, they kow the reserve price value demaded by the seller. Therefore, buyers, whose private valuatio is lower tha the reserve price set by the seller, have o opportuity of profit. The theory predicts that they should also bid their private valuatio of the item. I fact, real buyers are discouraged ad set either very low bids or overbids. Result 3. Public reserve price discouraged buyers with private valuatio lower tha the miimum price demaded by the seller. I order to ivestigate the effect of our strategic ad treatmet variables o the idividual behaviours, we ru the followig radom effects liear regressio (for each party): y t = X t β+ε t =1,...,N ad t=1,...,t ε t = u +v t +w t where X t is the vector of the idepedet variables ad β the vector of the estimated coefficiets. The umber of idividuals equals 120 (N = 120) ad umber of periods equals 40 (T = 40). I our experimet, the variables, which characterize the model, are the explaatory variables public ad overbid. The cross variable (public*overbid) aalyzes the joit ifluece of these two variables o biddig behaviour. Variable reserve price capture the ifluece of the 15

16 reserve price value i public reserve price treatmets. Variable value correspod to the private valuatio of buyer i. Variable serious bidder is a dummy which idicate if the private valuatio of buyer I is higher tha the public reserve price. By cosiderig the variable begi (resp. ed), we attempt to test the presece of a start (resp. ed) game effect. The results from model 1 relate to icetives. I model (2), we added characteristics of the participats as cotrolled variables. We have geder ad we costructed dummies for graduate studets ad studets i ecoomics or i related fields (busiess, math, computer sciece), which are expected to perform better i experimet ivolvig some aspects of game theory. The variable Previous participat iforms us whether the subject has participated i a laboratory experimet i the past. Depedat var. Model 1 Model 2 Bids Coefficiet Std. Dev. Coefficiet Std. Dev. Cste ** (0.925) Val *** (0.008) *** Public *** (1.651) *** Public*reserve price *** (0.027) *** Overbid *** (1.129) *** Pub*Overbid (1.588) Serious bidder *** (0.953) *** Serious bidder*val *** (0.021) *** Begi ** (0.329) * Ed (0.329) Male Studet Graduate Eco. or related field studet Previous part R² % % Log-likelihood Nb. of obs Table 21 Whe the seller kows that three bidders will take part to the auctio, he uses the reserve price as a sigal to eforce competitio ad overbiddig behaviours. 21 *** Statistically sigificat at 1%, **statistically sigificat at 5%, *statistically sigificat at 10%. Stadards errors are reported i parethesis. 16

17 Bids icrease sigificatly with the private valuatio of the item ad the overbiddig opportuity. Cotrary to theoretical predictio, bids are sigificatly lower whe the reserve price value is kow. Result 4. I public reserve price treatmets, the bids deped sigificatly o the reserve price. O the oe had, bids sigificatly icrease with the reserve price value. O the other had, buyers with private valuatio higher or equal to the reserve price set by the seller set sigificatly higher bids but they are decreasig i their private valuatio. The latter result is obvious because serious bidder have high private valuatio of the object. Thus, they set higher bids tha others. The former results poits out that serious bidders do ot overbid. Serious bidders are more likely to wi the auctio, which meas that overbiddig may be costly for them Probability of a sell I order to ivestigate the effect of our strategic ad treatmet variables o coflict resolutio, we ru the followig radom effects Probit model: y t = X t β+ε t =1,...,N ad t=1,...,t ε t = u +v t +w t y t = 1 if y t 0 y t = 0 if y t <0 where X t is the vector of the idepedet variables ad β the vector of the estimated coefficiets. Furthermore, y t equals 1 if a agreemet is reached ad 0 otherwise. We cosider the variables public, overbid ad public overbid to capture the effect of our strategic variable o the probability of a sell, as well as the private valuatio of the three buyers ad the reserve prices set by the seller. Variables begi ad ed capture respectively the possibility of a start ad/or ed game effects. 17

18 Coefficiet Std. Dev. Margial effects Std. Dev. Cste *** *** Public *** ** Overbid Pub*Overbid Reserve price *** *** Bid *** *** Bid *** *** Bid *** *** Begi Ed ** ρ ** Log-likelihood Restricted Log-likelihood Chi-squared % of predicted observatios 87.35% Nb. of obs Table 22 Despite the strog overbiddig behaviours of the buyers, the absece of restrictio o bids does t icrease sigificatly the occurrece of a sell. Ideed, overbiddig is more severe for o serious bidders. The probability of a sell icreases with the private value of the three buyers. Ad, as oted i previous studies (Kathar, Luckig Rieley, 2000 ad Reiley 2000 for fields experimets ad Bajari, Hortaçsu, 2000, for ecoometric study), it decreases sigificatly with the value of the reserve price. Result 5. The probability of a sell is a decreasig fuctio of the reserve price value. Public iformatio o the reserve price ifluece bidders behaviour ad icrease sigificatly the probability of a sell. This result is also reported i Katar ad Luckig- Reiley (2000). Result 6. Public reserve price icreases the probability of a sell. 22 *** Statistically sigificat at 1%, **statistically sigificat at 5%, *statistically sigificat at 10%. Stadards errors are reported i parethesis. 18

19 6. Appedix 6.1. Expected results for a commo distributio. Cosider a commo distributio, (v) ad differetiable over the iterval [ v; v]. F, with F ( v) = 0, F ( v) = 1, ad F(v) strictly icreasig Auctio with o reserve price: I the secod-price sealed-bid auctio, bidders aouce their true valuatios, v, ad wi with probability F 1 ( v ). The expected paymet to the buyer with value v is: His expected payoff is: v P ( v) = wdf ( w) = vf ( v) F ( w) dw v e 1 π ( v) = vf ( v) P( v). His expected paymet, cotiget o wiig, will be: P( v) H ( v) = = v 1 F ( v) v v F v v 1 F 1 ( w) dw + 1 k v + ( v v) + 1 For the geeral expressios of P(v) ad H(v) see Bulow ad Roberts (1989). I a secod-price (or ascedig) auctio, the seller s expected reveue Λ, is (see Riley ad Samuelso (1981) for the geeral expressio of Λ): v [ vf'( v) + F( v) 1 ] + 1 Λ= F ( v) dv v Reserve prices If the seller aouces a reserve price R, the expected reveue to the seller icreases util Ω Ω = v [ vf' ( v) + F( v) 1 ] R F( v) The reserve price, which maximizes the expected reveue of the seller, is (Riley ad Samuelso (1981): ( v) 1 dv * R = v 0 * 1 F( R ) + * F' ( R ) 19

20 6.2. Istructios (private vs public reserve price) You take part i a experimet about decisio makig i ecoomics i which you will have to make decisios. Durig that experimet, you will make moey. Your earigs will deped o you decisios ad the decisios of other persos. Each of you will make idividually her decisio i frot of her computer. Please do ot try to commuicate with other participats. I the experimet, oe seller ad four buyers form a aoymous group. You will play 40 idepedet rouds of game. At the begiig of each roud, the groups are rematch radomly with oe seller ad four buyers. You will either be a seller or a buyer ad remaied i that positio of play durig the all experimet. Each group of participat has to agree o the exchage price of a good. Oe roud of play : - Private valuatio of the good - At the begiig of each roud, the buyers ad the seller get a private valuatio for the good. For the buyers: their valuatios correspod to the higher price that they agree to pay to buy the good. The private valuatio are radomly ad idepedetly draw i the iterval: [0,1,..,59, 60]. Each iteger i this iterval has the same chace to be selected. For the seller: her valuatio correspods to the lower price that they agree to accept to sell the good. The private valuatio are radomly ad idepedetly draw i the iterval: [0,1,..,59, 60]. Each iteger i this iterval has the same chace to be selected. - Reserve price - Oce iformed about her private valuatio, the seller makes a price offer. This price correspods to his reserve price, i.e. the lowest bid he will accept. (- Iformatio of the buyers o the reserve price value - Before to make a decisio, the three buyers are iformed o the reserve price selected by the seller.) - Bid submissio - Oce iformed o her private valuatio (ad the reserve price ad the) each of the four buyers submits simultaeously a bid. - The selectio of the buyer- Cas 1 : If all the bids are lower tha the reserve price of the seller, the sell is cacelled. Cas 2 : If at least oe of the four bids is higher tha the reserve price of the seller, the sell occurs ad the good is attributed to the buyer with the highest bid. - Determiatio of the sell price - If a sell occurs, the computer determies the sell price pay by the buyer to the seller. This price depeds o the reserve price of the seller ad the secod highest bid of the four buyers. If the secod highest bid is higher tha the reserve price, the sell price is equal to the secod highest bid. 20

21 If the secod highest bid is lower tha the reserve price, the sell price is equal to the reserve price of the seller. I poits : - Computatio of the earigs If the sell occurs, the buyers ad the seller received a umber of poits equals to : For the selected buyer, her private valuatio mius the sell price. For the three other buyers, o poit. For the seller, the sell price mius her private valuatio. If the sell is cacelled, the umber of poits of each participats (the seller ad the three buyers) is equal ad correspods to zero. I ECU: How the computer determie the actual earigs of the seller ad the selected buyer? The computer radomly draw a umber i the iterval [0,1,..,39,40], each umber has the same chace to be selected. The, it compares this umber with the umber of poit of the each participat. If this umber is lower tha the umber of poits collected i the roud, the participat (buyer or seller) eared the 100 ECU. Otherwise, his earig is ull. Feedback iformatio at the ed of each roud: At the ed of each roud, accordig to your decisios, you will be iformed about the followig elemets: - Your private valuatio - Your bid (for the buyer) ; - Your reserve price (for the seller / for all) - If the sell occurs ; - If you are the selected buyer (i.e. the buyer who submitted the highest bid; for the buyer oly) - The sell price (if the sell occurs) ; - Your earig for the roud, 21

22 The earig computatio for the experimet: Your total earig for the all sessio is determied by the sum of your earigs i each roud. The value of your accout will be coverted i euros with a exchage rate of 1 for 150 ECU for the sellers ad 3 for 100 ECU for the buyers. Before to start the experimet, we will ask to fill a uderstadig questioaire about these istructios. To go further, all participats have to aswer correctly to all the questios. At the ed of the experimet, we will ask you to give us iformatio about your age, sex, level ad field of study, uiversity or school ad either or ot you had already take part i a experimet. Please, take some additioal time to read agai these istructios. If you have ay questio, please, raise had up; we will come to aswer your questios. Durig the all sessio, we kidly ask you to ot ask questio or speak loudly. Thaks for you cooperatio. 22

23 Refereces. Bajari, P. ad Hortaçsu, A., 2000, Wier s Curse, Reserve Prices ad Edogeous Etry: Empirical Isights from ebay Auctios, Staford Uiversity Workig Paper, Bulow, J. ad Roberts, J., 1989, The Simple Ecoomics of Optimal Auctios. Joural of Political Ecoomy 97 (5): Burguet, R. ad Sákovics, J., Reserve Prices without Commitmet. Games ad Ecoomic Behavior 15: Kagel, J.H., 1995, Auctios: A Survey of Experimetal Research, i the Hadbook of Experimetal Ecoomics, J. Kagel ad Roth A., eds Priceto: Priceto Uiversity Press, Klemperer, P., 1999, Auctio Theory: A Guide to the Literature. Joural of Ecoomic Surveys 13 (3): Katkar, R., ad Luckig-Reiley, D., 2000, Public versus Secret Reserve Prices i ebay Auctios: Results of a Pokémo Field Experimet, Workig Paper, Vaderbilt Uiversity. Levi, D. ad Smith, J., 1996, Optimal Reservatio Prices i Auctios, Ecoomic Joural, 106: Luckig-Reiley, D., 2000, Field Experimets o the Effects of Reserve Prices i Auctios: More Magic o the Iteret, Workig Paper, Vaderbilt Uiversity. Luckig-Reiley, D., 2000, Auctios o the Iteret: What s Beig Auctioed, ad How? Joural of Idustrial Ecoomics 48: McAfee, R.P. ad McMilla, J., 1987, Auctios ad Biddig. Joural of Ecoomic Literature 25: McAfee, R.P. ad Vicet, D.R., 1997, Sequetially Optimal Auctios. Games ad Ecoomic Behavior 18: Riley, J.G. ad Samuelso W.F. 1981, Optimal Auctios. America Ecoomic Review 71 (3): Samuelso W. F., 1985, Competitive Biddig with Etry Costs. Ecoomics Letters, 17: Vickrey, W., 1961, Couterspeculatio, Auctios, ad Competitive Sealed Teders. Joural of Fiace 16 (1): Vicet, D.R., 1995, Biddig Off the Wall: Why Reserve Prices May be Kept Secret. Joural of Ecoomic Theory 65: