Signalling in Auctions: Experimental Evidence

7261 2018

September 2018

Signalling in Auctions: Experimental Evidence Olivier Bos, Francisco Gomez-Martinez, Sander Onderstal, Tom Truyts

Impressum:

CESifo Working Papers ISSN 2364‐1428 (electronic version) Publisher and distributor: Munich Society for the Promotion of Economic Research ‐ CESifo GmbH The international platform of Ludwigs‐Maximilians University’s Center for Economic Studies and the ifo Institute Poschingerstr. 5, 81679 Munich, Germany Telephone +49 (0)89 2180‐2740, Telefax +49 (0)89 2180‐17845, email [email protected] Editors: Clemens Fuest, Oliver Falck, Jasmin Gröschl www.cesifo‐group.org/wp An electronic version of the paper may be downloaded ∙ from the SSRN website: www.SSRN.com ∙ from the RePEc website: www.RePEc.org ∙ from the CESifo website: www.CESifo‐group.org/wp

CESifo Working Paper No. 7261 Category 13: Behavioural Economics

Signalling in Auctions: Experimental Evidence

Abstract We study the relative performance of the first‐price sealed‐bid auction and the second-price sealed‐bid auction in a laboratory experiment where bidders can signal information through their bidding behaviour to an outside observer. We consider two different information settings: the auctioneer reveals either the identity of the winning bidder only, or she also reveals the winner’s payment to an outside observer. We find that the first‐price sealed‐bid auction in which the winner’s payment is revealed outperforms the other mechanisms in terms of revenue and efficiency. Our findings may have implications for the design of charity auctions, art auctions, and spectrum auctions.

JEL-Codes: C920, D440, D920.

Keywords: auctions, signalling, experiments.

Olivier Bos LEMMA, Department of Economics

Panthéon-Assas University Paris / France

[email protected]

Francisco Gomez-Martinez Department of Economics

BI Norwegian Business School Oslo / Norway

[email protected]

Sander Onderstal University of Amsterdam

Amsterdam / The Netherlands [email protected]

Tom Truyts CEREC, Saint-Louis University

Brussels / Belgium [email protected]

August 16, 2018 We are grateful to Francis Bloch, David Ettinger, Alex Possajennikov, Jean‐Marc Tallon, Ted Turocy and seminar participants at the University of Amsterdam, GATE Lyon, Orléans, Panthéon‐Assas, the Tinbergen Institute Amsterdam, the CEREC Workshop in Economics in Brussels, and EARIE Athens for their helpful comments. This research has been conducted as part of the project Labex MMEDII (ANR11‐LBX‐0023‐01). We also thank the ANR DCPG and the University of Amsterdam Research Priority Area in Behavioral Economics for their financial support. Tom Truyts gratefully acknowledges financial support from the French‐speaking Belgian community ARC project n° 15/20‐072, “Social and Economic Network Formation under Limited Farsightedness: Theory and Applications”, Université Saint‐Louis ‐ Bruxelles, October 2015 ‐ September 2020.

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1.Introduction

In many auction settings, bidders care about how their behaviour in the auction isinterpreted by others. Market analysts can consider the performance of a firm in anauction, winning or losing, as a signal of the firm’s management quality, financialposition,orconfidenceinitstechnologicaledgeonthecompetition.1Signallinghasalsobeen shown to be an important motivator for bidders in charity and art auctions:winningaVanGoghpaintingcomeswithagreatdealofprestige,whereasfailingtowina charity auctionmay leave some wondering about the losing bidder’s true financialpositionormagnanimity.2Insuchsettings,signallingconcernsconstituteanadditionalcomponent in a bidder’s bidding strategy. In the past two decades, the theoreticalliterature has devoted ample attention to signalling in auctions.3 In this paper, weaddress the question how the information revealed in auctions affects signallingincentivesand,inturn,revenueandefficiencyusingalaboratoryexperiment.

A key finding from the theoretical literature is that an auction’s equilibrium revenuedepends on both the auction format used and the kind of information that theauctioneerrevealsabouttheoutcomeoftheauction.Insettingswherebiddershaveanincentive tooverstate theirprivate information, the first‐price (FP) sealed‐bidauctionand second‐price (SP) sealed‐bid auction yield the same expected revenue in aseparatingequilibriumiftheauctioneerrevealsonlythewinner’sidentity(GiovannoniandMakris,2014)orthewinner’sidentityandbid(Goeree,2003;Haile,2003;KatzmanandRhodes‐Kropf,2008;GiovannoniandMakris,2014).4GiovannoniandMakris(2014)tie these revenue‐equivalence results togetherbyelicitingconditionswhichguaranteethat an auction’s expected revenue only depends on the information revealed,independently of the auction format used. In contrast, if the winner’s payment isrevealed (rather thanherbid), revenueequivalencebreaksdown. In that case, theSPauctiondominatestheFPauctionintermsofexpectedrevenue(GiovannoniandMakris,2014;BosandTruyts,2017).Finally,revealingeitherthewinner’sbidorthewinner’spayment increases revenue in both the first‐price and the second‐price sealed‐bidauctioncomparedtothecasewhereonlythewinner’sidentityisrevealed(GiovannoniandMakris,2014;BosandTruyts,2017).

1Liu (2012)argues that signalling incentives couldarise inbiddingcontestswhere thewinningbidderissuesequityordebtforfinancingherpayment.2Mandel (2009) distinguishes threemainmotives for buying art: investment, direct consumption, andsignalling, and suggests that the latter two explain the old puzzle as to why art systematicallyunderperformsasaninvestmentcomparedtobondsandequity.Charitiesoftenraisefundsbyauctioningobjectsprovidedtothembycelebrities(SchramandOnderstal,2009).Abroadtheoreticalandempiricalliteraturesuggeststhatsignallingandstatusareimportantmotivesforcontributionstocharities.Glazerand Konrad (1996) and Harbaugh (1998a,b) show that signalling is an important factor to explainpatternsindonationstouniversities.3 See Goeree (2003); Das Varma (2003); Haile (2003); Katzman andRhodes‐Kropf (2008);Monar andVirag(2008);Liu(2012);GiovannoniandMakris(2014);Marinovic(2016);andBosandTruyts(2017).4 Goeree (2003) and Das Varma (2003) show that in settingswhere bidderswant to understate theirprivateinformation,separatingequilibriamayfailtoexist.

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We experimentally test these results using Bos and Truyts’ (2017) framework. Weconsider a symmetric independent private values setting in which the bidders careaboutthebeliefofanoutsideobserverabouttheirvalues.Theoutsideobserverispartlyinformed about the auction outcome and uses this information to update her beliefsabout the bidders’ values. We consider two different information settings: theauctioneerrevealseithertheidentityofthewinningbiddertotheoutsideobserveronly,orshealsorevealsthewinner’spayment.

AsTurocy(2009)notes,signallinggamesarehardforhumanstoplay.Thismayexplainwhyexperimentsregardingsignallinggamesarenotverycommon.Moreover,mostofthese experiments focus on equilibrium selection, given the usual equilibriummultiplicity in signalling games.5 Auctions with signalling opportunities to outsideobservershavehardlybeenanalysedinthelab.6Tothebestofourknowledge,Fonsecaetal.(2016)istheonlyexception.Theyconsiderasettingwherebidderscansignaltheirproductivitytofirmsthatarehiringonalabourmarket.Fonsecaetal.focusonseveralinformationdisclosure policieswithin the same auction format: the first‐price sealed‐bidauction.Whiletheyfindthatsignallingopportunities leadtomoreaggressivebids,they observe consistent underbidding compared to equilibrium. Our experimentalresultscomplementtheirsinthatourdesignfacilitatesbetween‐auctioncomparisons.

Ourmainresultisthatthefirst‐pricesealed‐bidauctioninwhichthewinner’spaymentis revealed outperforms the othermechanisms in terms of revenue and efficiency. Inbothauctions,weobservemoreaggressivebiddingcomparedtocontrol treatments inwhich the outside observer’s estimates do not affect the bidders’ payoffs. Thisunderlines the importance of revealing information to outsiders when bidders careabouthowtheirbehaviourisinterpretedbyothers.However,likeFonsecaetal.(2016),we find underbidding relative to the equilibrium prediction. Underbidding isparticularly striking for the second‐price sealed‐bid auction where the winner’spaymentisrevealedtotheoutsideobserver.Asaresult,wedonotfindsupportforthetheoreticalpredictionthatthesecond‐priceauctionyieldsmorerevenuethanthefirst‐price sealed‐bid auction in this information regime.Moreover, revealing thewinner’spayment only boosts revenue in the first‐price sealed‐bid auction, not in the second‐pricesealed‐bidauction.Bothfindingsarequalitativelyinlinewithriskaversebiddingandvariationsintheoutsideobserver’saccuracyacrosstreatments.

The remainder of this paper is structured as follows. In Section 2, we describe ourexperimental design and protocol. Section 3 includes the theoretical results and thehypothesestested.Section4containsourexperimentalfindings.Section5concludes.

5 Brandts andHolt (1993), de Haan et al. (2011), Drouvelis et al. (2012), and Jeitschko andNormann(2012). 6 Previousexperimentalworkonauctionsstudiestheeffectsofdisclosingpreviousbids(see,e.g.,Casonetal., 2011; Dufwenberg and Gneezy, 2002; and Neugebauer and Selten, 2006), bidders’ types (see, e.g.,Andreonietal.,2007),andinformationabouttheobject(see,e.g.,GoereeandOfferman,2002)tobidders..

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2.Experimentaldesignandprotocol

Theexperimentwascomputerized7andrunattheCREEDlaboratoryoftheUniversityofAmsterdam.Weemployeda full2x2x2factorialdesignvarying(betweensubjects) theauction type (FP and SP), the information about the auction outcome that iscommunicatedtotheoutsideobserver(withorwithoutinformationaboutthewinner’spayment),andwhetherthebidders’payoffsdependontheoutsideobserver’sestimate(inthemaintreatments,itdid,inthecontroltreatments,itdidnot).Table1summarizesthe resulting eight treatments of the experiment. The control treatments serve as abenchmarkwhencomparingresultsbetweenauctiontypes.

Table1:Experimentaldesign

Treatment AuctionInformationtotheoutside

observer

Dobidders’payoffsdependonoutsideobserver’s

estimate?

FPW FP Thewinner Yes

FPWP FP Thewinnerandherpayment Yes

SPW SP Thewinner Yes

SPWP SP Thewinnerandherpayment Yes

FPWcontrol FP Thewinner No

FPWPcontrol FP Thewinnerandherpayment No

SPWcontrol SP Thewinner No

SPWPcontrol SP Thewinnerandherpayment No

Eachtreatmentwascomprisedofsevengroupsoffourparticipants.All224participants,recruitedbypublicannouncementfromtheundergraduatepopulationoftheUniversity,tookpart inonlyonesessioneach.Atthestartofeachsession,werandomlyallocatedparticipantsoverthecomputerssotheycouldnotinferwhichotherparticipantswereinthe same group. We provided computerized instructions to the participants. Theinstructions for treatment FPWP can be found in Online Appendix D.8 Before theexperiment started, participants answered test questions to make sure that theyunderstood the experimental protocol.9 Sessions lasted between 45 and 75 minutes.Paymentconsistedofashow‐upfeeof7euros,plusapayoffrelatedtothetotalprofitsearnedinthe30rounds.Theexchangeratewas1eurofor50experimentalpoints.Onaverage,participantsearned16.70euros(includingtheshow‐upfee).

7TheprogramwaswrittenusingPHPandmySQL.8Theinstructionsoftheothertreatmentsareavailablefromtheauthorsuponrequest.9Thesequestionsareavailablefromtheauthorsuponrequest.

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Inallsessions,participantsinteractedinfixedgroupsoffour(norematching).Ineachofthe 30 rounds of a session, a fictitious goodwasauctioned. In each round, one groupmemberwasrandomlychosenbythecomputertoplaytheroleoftheoutsideobserver.The remaining three group members were bidders in an auction. We let subjectsinteract in fixed groups, and have them take turns playing the role of the outsideobserverintheirgroup.ThiswasdonetofosterthelearningneededtoreachaperfectBayesian equilibrium, which crucially requires a coordination between the bidders’strategies and the outside observer’s beliefs. Role switching also renders biddercollusionmoredifficultasthatrequirescoordinationamongmoreplayers.

Atthestartofeachround,allbidderswereprivatelyinformedabouttheirvalueforthegood.Valueswereindependentlydrawnaccordingtoauniformdistributionontheset{1,2,3,…,100}. For the sake of comparability between treatments, we kept the valuedraws constant across treatments. In the auction, each of the three biddersindependently submitted a bid for the fictitious good from the set {0,1,2,…,200}. Thebidderwiththehighestbidwonthegood.IntheFPauction,thewinnerpaidhisownbid,while in the SP auction, the winner paid the second highest bid. Ties were resolvedrandomly.Inthemaintreatments,thebidderpayoffsdependedonboththeoutcomeoftheauction,andtheestimateoftheoutsideobserver.Thewinningbidderobtainedthedifferencebetweenhisvalueandpayment.Aftertheauction,theoutsideobserverwasaskedtoguessthevaluesofeachofthethreebiddersafterobtaininginformationaboutthe outcome of the auction. Each bidder, win or lose, received half of the outsideobserver’sestimateofhisvalue.Theresultingpayoffforbidder isgivenby

, , ,/2 if

/2 if ,

where denotestheauctionwinner, thewinner’spayment, bidder ’svalue,and theoutsideobserver’sestimateforbidder ’svalue.Thisisareduced‐formwaytomodelbidders’benefitingfromoutsidersbelievingtheyattachahighvaluethegood,e.g.,asitsignals their generosity, wealth, or productivity. For instance, a telecommunicationsfirm’svalueforradiospectrummightbecorrelatedwiththequalityofitsmanagement.Ahighbidintheauctionservesasapositivesignaltooutsideinvestorssothatthefirmmaybeabletoattractfinancialresourcesunderfavourableconditionsinthefuture.

In all treatments, the outside observer was informed about which bidder won theauction before reporting her estimates. In the WP treatments, she also obtainedinformationregardinghowmuchthewinnerpaid.Thepayoffsof theoutsideobserverdependedontheaccuracyofherestimates,alsointhecontroltreatments.Onceshehadentered value estimates for all bidders, the computer drew one of the three bidders’

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estimatesat random.When theoutsideobserver’sestimate for thisbidderdeviated pointsfromtheactualvalue,herpayoffwasequalto40 .10

One interpretationof themodel is thatafter theauction, theoutsideobserverdecideshowmuchtoinvestineachbidder,winorlose.Theoutsideobserveroptimallyinvestsmorethehigherthebidder’svalue.Bidder ’sexpectedsurplusoftheinteractionequals/2whiletheoutsideobservermovesfurtherawayfromitsoptimalinvestmentlevelin

thebidderthelessaccurateisherestimate.

3.Theoreticalpredictions

In this section, we describe the theoretical predictions. The analysis followsstraightforwardly fromBosandTruyts (2017).Like them,werestrictourattention torisk‐neutralbiddersandperfectBayesianNashequilibriathatsurviveBanksandSobel’s(1987)D1criterion(referredtoas“equilibrium”intheremainderofthispaper).Table2contains equilibrium predictions for all treatments. The formal derivations are inAppendixA.

Table2:Equilibriumpredictionspertreatment

Treatment AuctionInformationtotheoutsideobserver

EquilibriumbidsExpectedrevenue

FPW FP Thewinner23

22 72

FPWP FPThewinnerandher

payment 75

SPW SP Thewinner 22 72

SPWP SPThewinnerandher

payment 262.5 87.5

FPWcontrol FP Thewinner23

50

FPWPcontrol FPThewinnerandher

payment23

50

SPWcontrol SP Thewinner 50

SPWPcontrol SPThewinnerandher

payment 50

Notes:EquilibriumbidsintheuniquesymmetricD1equilibriumforbiddervalues beingindependentlydrawnfrom 0,100 ,andtheexpectedrevenueoftheauctioninthisequilibrium.

10 Negative payoffs were subtracted from the participants’ balances. In principle, the participants’balances might become negative. However, in the experiment, all participants accumulated a positiveamountofmoneyoverthe30rounds,withaminimumof€5.20.

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Inalltreatments,auniquestrictlyincreasingandsymmetricequilibriumbiddingcurveexists.Forthecontroltreatments,theexistenceofanoutsideobserverhasnoeffectonthe bidders’ payoffs. Therefore, the predictions are standard, and imply revenueequivalenceacrosstreatments(see,e.g.,Vickrey,1961).Ifonlytheidentityofthewinnerisrevealedtotheoutsideobserver,bidders’payoffswhenwinningareincreasedbyhalfthedifferencebetween theoutsideobserver’s value estimates forwinners and losers.Equilibrium bids are inflated by this number compared to the control treatments. InAppendixA,we show that a bidder’s payoff fromwinning the auction (andhence hisequilibriumbid)isincreasedbyabout22.

If thewinner’s payment is also revealed, then the bidderswill also take into accounthowtheoutsideobserverupdatesherbeliefsaboutthebidders’valuesinfunctionoftheobservedpayment.Inequilibrium,thewinner’svalueisexactlyrevealedtotheoutsideobserver in the FP auction, since the equilibrium bidding curve is strictly increasing.Therefore, the outside observer will perfectly predict the winner’s value. Moreover,bidders will take into account thatwhen losing, the outside observer estimates theirvaluetobeequaltohalfthewinner’svalue.Alow‐valuebidderisbetteroff,intermsoftheoutsideobserver’s equilibriumestimate,by losingagainsta sufficientlyhigh‐valuebidder,ratherthanbywinningtheauction.Intheoppositecase,thedifferencebetweenwinningandlosingislargewhenviewedintermsoftheoutsideobserver’sequilibriumestimate for high‐value bidders. As a result, the equilibrium bids of low [high] valuebidders are lower [higher] if the outside observer sees the winner’s payment ascomparedtoasituationwhereonlythewinnerisrevealed.

InSPWP,thewinner’spaymentrevealsthevaluationofthesecondhighestbidderinthefully separating equilibrium, but the outside observer cannot deduce which losingbiddermadethesecondhighestbid.Thedifferencebetweenwinningandlosingis–intermsoftheoutsideobserver’sestimate–verylargeforalow‐valuebidder.Ifhewins[loses],theoutsideobserveroptimallyestimateshisvaluetobetheaveragebetweenthesecondhighestvalueand100[3/4ofthesecondhighestvalue].Thisleadsalow‐valuebidder to submit a considerably higher bid when the outside observer obtainsinformationaboutthewinner’spayment.

Thetheoreticalpredictionsregardingtheauction’srevenueandbiddingbehaviouryieldthefollowinghypotheseswhichwewilltestusingourexperimentaldesign:

Hypothesis 1 In the FP auction, revealing both the winner and his payment to theoutsideobserverincreasestheaverageauctionrevenueascomparedtoasettingwhereonlytheauctionwinnerisrevealed.

Hypothesis 2 In the SP auction, revealing both the winner and his payment to theoutsideobserverincreasestheaverageauctionrevenueascomparedtoasettingwhereonlytheauctionwinnerisrevealed.

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Hypothesis3Inthesettingwhereboththewinnerandhispaymentarerevealedtotheoutsideobserver,theaverageauctionrevenueishigherintheSPauctionthanintheFPauction.

Hypothesis4Inthesettingwhereonlythewinnerisrevealedtotheoutsideobserver,theFPauctionandtheSPauctionyieldthesamerevenue,onaverage.

4.Results

In this section, we present our experimental results. We start in subsection 4.1 bycomparing the auction revenue between treatments. In subsection 4.2, we analysebiddingbehaviour. In subsection4.3,wediscuss the outsideobserver’s estimates andtheir effect on bids. Finally, in subsection 4.4, we present an efficiency comparisonbetweenauctions.Concerningthestatisticalanalysis, two‐sidedMann‐WhitneyUtestsare employed in the case of non‐parametric analysis, using groups as singleobservations.Theparametricanalysesarebasedonordinary least‐squareregressions,wherestandarderrorsareclusteredbygroup.Unlessstatedotherwise,theresultsreferto the main treatments, i.e., where the outside observers’ estimates affect bidders’payoffs.

Figure1:Averageauctionrevenuebytreatment

4.1.Auctionrevenue

Inthissubsection,weexploretheeffectoftheauctiontypeandtheinformationrevealedtotheoutsideobserveronauctionrevenue.Figure1showstheaverageauctionrevenue

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foreachofthetreatments.TheFPauctionwhereboththewinnerandhispaymentarecommunicatedto theoutsideobserveryields thehighestrevenueonaverage;averageauctionrevenueissignificantlyhigherinFPWPthanintheothernon‐controltreatments(p=0.08,p=0.00,andp=0.04forFPW,SPWP,andSPW,respectively).Inparticular,intheFP auction, revealing thewinnerandhispayment increases theauction’s revenue, onaverage,byalmost7unitsascomparedtothecasewhereonlythewinnerisrevealed.The increase in revenues is even greater when comparing FPWP with both SPtreatments.Aparametricanalysis,wherethehighestvaluationis includedasacontrolvariable, confirms this result (seeTable3). Figure2 represents the regression resultsgraphically. The revenue estimates are higher in FPWP than in any of the othertreatmentsforallhighestvaluesabove20(i.e.,for98%oftherealizationsofthehighestvalues).

DuetothefactthattheFPauctiongenerallyyieldsmorerevenueinthelabthantheSPauctiondoes,11wecheckfortherobustnessofourfindingsinadifference‐in‐differenceanalysis where we compare the revenues in the main treatments, correcting for therevenuesobtainedinthecontroltreatments.Wedosobyrunningalinearregressionofrevenueontreatmentdummies.Table4reportstheestimateddifferencesbetweenthetreatmentsandthecorrespondingp‐values.

Table3:Auctionrevenuepertreatmentcontrollingforthehighestvalue

Revenue(1) Revenue(2)Intercept ‐5.8736(3.3687) 10.0810(2.4631)***HighestValue 0.9891(0.0499)*** 0.7847(0.0268)***FPWP 7.1048(2.2286)** ‐5.2341(8.7926)SPWP ‐8.3524(2.0827)*** ‐27.8704(7.5080)***SPW ‐4.4095(3.3687) ‐36,3713(6,8149)***HighestValue*FPWP 0.1543(0.1318)HighestValue*SPWP 0.2442(0.0923)**HighestValue*SPW 0.3999(2.4631)***N 840 840

Notes:Clustered standarderrors inparentheses. FPW is the reference treatment.HighestValuedenotesthehighestvalueamongthethreebidders.FPWP,SPWP,andSPWaredummyvariableswhichareequalto1ifandonlyiftheobservationinvolvestreatmentsFPWP,SPWP,andSPWrespectively.FPWPraisessignificantlyhigherrevenuethanSPWandSPWP(p<0.01).SPWPandSPWdonotdiffersignificantlyfromeachotherintermsofrevenueraisedatthe90%level.*p<0.1,**p<0.05,***p<0.01.

11SeeKagel(1995)foranoverview.

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Table 4: Difference in auction revenue between treatments relative to thecorrespondingcontroltreatments

FPWP SPWP SPWFPW ‐7.29***(p=0.01) 4.60(p=0.23) ‐3.50(p=0.39)FPWP ‐ 11.89**(p=0.02) 3.79(p=0.26)SPWP ‐ ‐8.00*(p=0.06)

Notes:Thenumbers reported are thedifferencesbetween the treatment in the rowand theone in thecolumn in terms of the revenue differences between main treatment and control treatment. *p<0.1,**p<0.05,***p<0.01

Figure2:Revenueestimates as a function of the highest value in a bidder grouppertreatment

Note:ThecurvesarebasedonthelinearregressionestimatesreportedinTable3.

Result 1: In the FP auction, revealing both thewinner and his payment significantlyincreases revenue for the auctioneeras compared to the casewhere only thewinner isrevealed.

Thisresultconfirmshypothesis1.TheFPWPtreatmentyieldshigherrevenues for theauctioneer than the FPW treatment does. This means that in the FP auction, theauctioneercanincreaseherrevenuebypublishingtheauction’swinningbid,ratherthanpublishingthewinneronly.

RegardingtheSPauction, therearenosignificantdifferencesbetweenSPWPandSPWtreatments(p=0.41).

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Result 2: In the SP auction, revealing both the winner and his payment does notsignificantly increaseauctionrevenueascompared to thecasewhereonly thewinner isrevealed. In a difference‐in‐difference analysis, revealing the winner’s payment yieldssignificantlylowerrevenue.

This result contradicts hypothesis 2. As we show in Appendix B, risk aversion maycontributetoexplainingtheobservedrevenue‐rankingreversal fortheSPauction.ForSPWP,equilibriumrevenueforCARAriskpreferencesislowerthanintherisk‐neutralcase.Intuitively,inSPWP,winningisrelativelyunattractivebecausethewinnerfacesarandomoutside observer’s estimate as it depends on the secondhighest bid. In SPW,equilibriumbids are unaffected because bidding value plus the (deterministic) payofffromtheoutsideobserverprovidingahigherestimateforthewinnerthanforthelosers.We find that for sufficiently risk aversebidders, SPWPyields lowerexpected revenuethatSPW.

Result 3:When both thewinner and his payment are revealed, the FP auction raisessignificantlymoremoneythantheSPauctiondoes.

Result 3 is inconsistent with hypothesis 3. As shown in Appendix B, risk aversion isagain a potential candidate to explain the discrepancy between the data and thetheoretical predictions. As said, for SPWP, CARA bidders submit lower bids than riskneutralbidders. In contrast, riskaversionproducesan increase inequilibriumbids inFPWPcomparedtotherisk‐neutralitycase.Theintuitionisstraightforward.Firstofall,likeinthestandardcase,abiddermitigatestheriskoflosingtheauctionbysubmittingahighbid.Second,inoursetting,ariskaversebidderhasanadditionalincentivetowinbecausethewinnerobtainsasurepayofffromtheoutsideobserver’sestimate,whichisdeterministicforthewinningbid,whilealoserfacesastochasticestimatethatdependsonthewinner’sbid.Asaconsequence,forsufficientlyriskaversebidders,therevenuerankingbetweenFPWPandSPWPreverses.

ComparingFPandSPauctionswhereonlythewinnerisrevealed,theaveragerevenueis3.53pointshigherinFPWthaninSPW,butnotsignificantlydifferent(p=0.18).

Result4:Whenonlythewinnerisrevealed,FPandSPauctionsdonotdiffersignificantlyintermsofaverageauctionrevenue.

This result is in linewith hypothesis 4. The observed average revenues for FPW andSPWarealsonotfarfromthetheoreticalpredictions.

4.2.Biddingbehaviour

Inthissubsection,weanalysethesubjects’biddingbehaviourtodiscovertheextenttowhichit is inlinewiththetheoreticalpredictionsand, ifnot,howitcontributestothe

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rejectionofsomeofourhypothesesintheprevioussubsection.FigureC.1inAppendixCincludesscatterplotswhichcontrastbidssubmittedwiththetheoreticalpredictions.

Westartbyexploringhowbiddingstrategiesdependonwhethertheoutsideobserverinfluencesbidders’payoffs.Figure3showsaveragebidspertreatment.ThescatterplotsinAppendix C indicate that the bid distributions in the control treatments are in linewithwhatiscommonlyobservedinFPandSPauctions(see,e.g.,Kagel,1995):Bidsinthe FP are typically in between the risk‐neutral equilibrium bid and value; In the SPauction, quite somebids are above value, perhaps evenmore thanwhat is commonlyobserved.Forallauctions,andforallvaluesgreaterthanzero,equilibriumbidsare,onaverage, higher if the outside observer’s estimates affect bidders’ payoffs. In the FPauction, when outside observers’ estimates affect bidders’ payoffs, bidders bid moreaggressivelywhencomparedtothetreatmentswheretheirpayoffsonlydependsontheoutcomeoftheauction(p=0.00andp=0.07forFPWPandFPW,respectively).IntheSPauction,averagebidsarehigherthaninthecontroltreatments,althoughthedifferenceis only statistically significant in SPW and not in SPWP (p=0.06 and p=0.57,respectively).12

In themain treatments, revealing thewinner andhis payment increases averagebidssignificantlyintheFPauctionascomparedtothecasewhenonlythewinnerisrevealed(p=0.04).Incontrast,nosignificantdifferencesbetweenbidsarefoundintheSPauctionwhen the information revealed to the outside observer ismodified (p=0.95). Perhapsnotsurprisingly,theseobservationsareinlinewiththeobservedtreatmentdifferencesintermsofrevenue.

Figure3:Averagebids

12ApotentialexplanationofthelatterobservationisthehighfrequencyofextremeoverbiddingcomparedtotheweaklydominantstrategyofbiddingvalueinSPWPcontrol(seeFigureC.1inAppendixC).

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Wenow zoom in on bidding behaviour in the FP auction. Table 5 presents estimatedbidding functions,andFigure4contrasts thesewiththeequilibriumbiddingcurves inTable 2. Bids in FPWP diverge from the theoretical predictions to some extent. Theintercept issignificantlygreater(p<0.01)andtheslope is lesssteep(p=0.03)thanthetheoreticalprediction.Asaconsequence, low‐valuebidderssubmithigherbidsthaninequilibrium,whilehigh‐valuebidderssubmitslightlylowerbids.ForFPW,theestimatedbidding curve lies below the theoretical prediction (p=0.03 and p=0.02 for thedifferences in slope and intercept, respectively, between the observed bids and thetheoreticalprediction)forlowandintermediatevalues.AsthescatterplotinFigureC.1indicates, it ismainlysubjectswith lowandintermediatevalueswhounderbid.Noticethat equilibrium bidding entails bids above value for values below 66 while in theexperiment,70%ofthebidsareactuallybelowvalueinthecasethatvalueislowerthan66.Incontrastwiththemaintreatments,subjectsbidmoreaggressivelyinthecontroltreatmentsthanintheequilibriumprediction,notless.Theobservedoverbiddinginthecontrol treatments is in line with what is generally observed in the FP auction instandard independent private values settingswithout an outside observer.13 Result 5summarizesthemainfindingsforbiddingbehaviourintheFPauction.

Result 5: In the FP auction, compared to the control treatments, bidders bid moreaggressivelywhen the outside observer’s estimates affect their payoffs.Bidders in FPWtend to underbid relative to the equilibrium prediction, particularly for low andintermediatevalues.InFPWP,biddersoverbidforlowvaluesandunderbidforhighvalues.

Table5:EstimatedbiddingfunctionfortheFPauction

Bid TheoreticalpredictionIntercept 6.2476*(2.8823) 22Value 0.8048***(0.0481) 2/3FPWP 2.1628*(1.0733) ‐22FPWPcontrol ‐6.0022**(2.7355) ‐22FPWcontrol ‐5.9411*(2.6659) ‐22Value*FPWP 0.0224(0.0444) 1/3Value*FPWPcontrol 0.0118(0.0687) 0Value*FPWcontrol 0.0441(0.0485) 0N 2518

Notes: Clustered standard errors in parentheses. FPWP, FPWPcontrol, and FPWcontrol are dummyvariableswhichareequalto1ifandonlyiftheobservationinvolvestreatmentsFPWP,FPWPcontrol,andFPWcontrol,respectively.FPWisthereferencetreatment.*p<0.1,**p<0.05,***p<0.01.

13ThisobservationisinlinewithwhatisgenerallyobservedintheFPauctionsettingswithoutanoutsideobserver.See,e.g.,Coxetal.(1988);Harrison(1989);Kagel(1995);andGoereeetal.(2002).

14

Figure4:Estimatedbiddingfunctionsvs.theoreticalpredictionsfortheFPauction

Note:The solid lines are estimates based on the linear regression estimates reported in Table 4. ThedashedlinesrefertothetheoreticalpredictionsinTable2.

Figure5:Estimatedbiddingfunctionsvs.theoreticalpredictionsfortheSPauction

WenowturntotheSPauction.Figure5displaysestimatedbiddingfunctionsbasedonthe regressions reported in Table 6. In both treatments, bidders tend to significantlyunderbid relative to the equilibrium predictions. In SPWP, low‐value bidders inparticular bid substantially lower than the equilibrium prediction (p<0.01 for thedifferencesbetweentheobservedbidsandthetheoreticalpredictionforboththeslopeand the intercept). In SPW, bidders underbid on average over the entire value range(p=0.45andp=0.01forthedifferencesinslopeandinterceptrespectivelybetweentheobservedbidsand the theoreticalprediction).Biddersbid22pointsabovetheirvalueaccording to the equilibrium prediction. Again, subjects do not tend to submit bidssignificantlyabovetheirvalues,asthescatterplotinFigureC.1shows.Only63%ofthebidsareabovevalue,andthemajorityoftheseareinbetweenvalueandvalueplus22.

Result6: InSPW, theaveragebid is significantlyhigher than in the control treatment.Bidderstendtounderbidascomparedtotheequilibriumprediction.

15

Result 7: In SPWP, the average bid is not significantly higher than in the controltreatment. Bidders tend to underbid as compared to the equilibrium prediction,particularlybidderswithlowvalues.

Table6:EstimatedbiddingfunctionsfortheSPauction

Bid TheoreticalpredictionIntercept 11.4418***(2.6629) 22Value 1.0526***(0.06521) 1SPWP 8.0788**(3.1726) 40.5SPWPcontrol ‐3.3336(4.8040) ‐22SPWcontrol ‐9.2593**(2.8356) ‐22Value*SPWP ‐0.1781**(0.0664) ‐1/2Value*SPWPcontrol 0.0251(0.0553) 0Value*SPWcontrol 0.0122(0.0966) 0N 2509

Notes: Clustered standard errors in parentheses. SPWP, SPWPcontrol, and SPWcontrol are dummyvariableswhichareequalto1ifandonlyiftheobservationinvolvestreatmentsSPWP,SPWPcontrol,andSPWcontrol,respectively.SPWisthereferencetreatment.*p<0.1,**p<0.05,***p<0.01.Inalltreatments,underbiddingrelativetoequilibriumismostprominentinSPWP.Riskaversion may partly drive this observation. The equilibrium analysis in Appendix BshowsthatriskaversebiddersbidlessaggressivelythanriskneutralbiddersinSPWP,incontrasttoFPWP,FPW,andSPW.Moreover,thehighbidsoflow‐valuebiddersintheequilibrium prediction crucially depend on the outside observer making the correctinferences. In the next subsection, we shall see that the observed outside observer’sestimatesaresystematicallybiased,andthatthisisparticularlythecaseforSPWPinthecaseoflowpaymentsbythewinner.Asaresult,revenueinSPWPisnotgreaterthaninFPWPorSPW,incontrasttohypotheses2and3.

Finally,wecomparebiddingfunctionsinSPWPandSPW.Eventhoughaveragebidsdonotdifferbetweenbothtreatments,thebiddingfunctionsdiffersignificantlyfromeachother. In particular, low‐value bidders place higher bids in SPWP than in SPW (theinterceptissignificantlyhigherinSPWP).Thisresultisreversedforhigh‐valuebidders(bidding function is significantly steeper in SPW). As such, these findings arequalitativelyinlinewiththetheoreticalpredictions.

4.3.Outsideobservers’estimatesandtheireffectonbids

All inall,theavailabilityofmoreinformationincreasestheauctioneer’srevenueintheFPauction,butnotintheSPauction.Anadditionaldrivingforcebehindthisresultmaybe that theaccuracyof theoutsideobservers’estimatesofbidders’values ishigher inFPWP compared to FPW, but not in SPWP compared to SPW, as we will show in

16

subsection 4.4. A second factor that may contribute to explaining this result are thedifferencesintheoutsideobservers’accuracy,andhowbiddersrespondtodifferencesin value estimates for winners and losers. The analysis of bidding behaviour insubsection4.2 shedsmore lighton thisdiscrepancybetween theexperimental resultsandthetheoreticalpredictions.

Intheprevioussubsection,weobservedthatbidsinFPWPareclosetoequilibrium,onaverage,whileintheothertreatments,weobserveconsistentunderbidding.Muchoftheunderbidding is explained by bidders relying on bidswhich are close to their values,while equilibrium sometimes requires substantial overbidding. In this subsection, weexplore the extent to which the outside observers’ behaviour drives these biddingpatterns. Bidding above value is optimal in equilibrium because winning the auctionimplies an additional reward in terms of the outside observer’s inference. In theexperiment,thisparticularrewarddependsontheactualbehaviourofthesubjectintheroleoftheoutsideobserver.Wefirstanalysehowthetypesofauctions,andthedifferentelementsofinformationprovidedtotheoutsideobserver,influenceherestimatesofthebidders’ values. We then analyse how these estimates affect bidding behaviour. Inparticular, we conjecture two mechanisms through which the outside observer canaffect bidding strategies: first, the difference in the outside observers’ estimates forwinners and losers may differ between treatments or among groups, and this mayinfluence bidding behaviour. Second, the accuracy of the outside observer’s estimatesmayalsohaveaneffectonthebids.

FiguresC.2andC.3inAppendixCcontrastthevalueestimatesoftheoutsideobserversand the actual values, forwinners, for losers, and for thedifferencebetweenwinnersand losers, considering different winners’ payments and all main treatments. Whenoutside observers are only informed about the identity of the winner, their guessescannot depend on the winner’s payment. Therefore, by construction, estimates forwinners and losers do not depend on the winner’s payment in FPW and SPW. Inaddition,inbothtypesofauctions,outsideobserversunderestimatewinners’valuesandoverestimate losers’ values on average. As a consequence, in both FPWand SPW, thedifferences between the value of the winners and the value of the losers areunderestimated,suchthattheadditionalbenefitforthewinningbidderissmallerthaninthetheoreticalprediction.Inparticular,onaverage,theestimateddifferencebetweenthevalueofthewinnerandthevalueoftheloseris20.63and23.62forFPWandSPW,respectively.Thesedonotdiffersignificantly(p=0.95).Thebidders’bestresponseistoinflatetheirbidsrelativetothecontroltreatmentsbyhalfthatdifference,i.e.,by10.32and11.81points inFPWandSPW, respectively.According to thedata,bidders inflatetheirbidswithrespecttothecontrolsby3.66and8.61pointsonaverageinFPWandSPW,respectively (seeFigure3).Overbidding is slightly lower thanexpected,but it iscloseto,andqualitativelyinlinewith,thetheoreticalpredictions.

InFPWPandSPWP, theoutsideobservers canadjust their estimates forwinners andlosers depending on the information received regarding the winner’s payment (the

17

highest bid in FPWP and the second highest bid in SPWP). Figures C.2 and C.3 inAppendixCshowthat theestimates for thevaluesofboth thewinnersand the losersincrease with the winners’ payment in both FPWP and SPWP (p<0.02). Similarly, inauctions where only the winner is revealed, the outside observers generallyunderestimatethewinners’values,andoverestimatethelosers’values.Onaverage,thedifferencebetweentheestimatedvalueofthewinnerandthevalueoftheloseris23.21and18.42forFPWPandSPWP,respectively.Thesenumbersdonotsignificantlydifferbetween both auctions, and in fact, they do not differ significantly between the fourtreatments.

Figure6:Differences in the outside observer’s value estimates betweenwinners andlosers

Notes:Thefigureplotstheresultsoflinearregressionsoftheobserveddifferencesbetweentheoutsideobserver’svalueestimatesforthewinnersandthelosers(dashedline)aswellastheactualdifferencesbetweenwinners’andlosers’valuesonthewinner’spayment.

Thedifferenceinestimates,conditionalonthewinner’spayment,doesdiffersignificantlybetweenauctions.Theempiricaldifferencesbetweentheoutsideobservers’estimatesofthewinners’andlosers’valueareplottedinTablesC.2andC.3inAppendixC.Figure6includesresultsfromlinearregressionsoftheactualdifferencesinthewinner’spayment.Intheory,thedifferencebetweentheoutsideobserver’sestimatesofthewinners’andlosers’valueincreaseswiththewinner’spaymentintheFPWPanddecreaseswiththewinner’spaymentintheSPWP.Inlinewiththetheory,thedifferencesignificantlyincreaseswithrespecttothewinner’spaymentintheFPWP(p=0.02),albeittoalesserextentthantheoreticallypredicted.Incontrast,thedifferencebetweenestimatesforwinnersandlosersdoesnotdependinastatisticallymeaningfulwayonthewinner’spaymentsintheSPWP(p=0.95).Thissuggestsapossibleexplanationforthefactthatthelow‐valuebidders,especially,underbidincomparisontothetheoreticalpredictionintheSPWP,andthattheobservedbehaviourdoesnotcorroboratehypothesis3.

18

Result 8: The outside observers generally underestimate the value of thewinners andoverestimatethevalueofthelosers.Asaconsequence,thedifferencebetweenthewinner’sandthelosers’valueisunderestimatedinalltreatments.

4.4.Efficiencyandearnings

Inthissubsection,weundertakeanefficiencyanalysistodeterminewhichcombinationofauctionandinformationscenarioyieldsthehighestaverageaggregatepayoffsforallpartiesthatparticipateinoursetting.Foragivengroupandauctiontype,theefficiencyinperiodtcomprisesthesumofthreeterms:

12

where isthevalueofthewinneroftheauctionatperiod , are the earnings of the outside observer at period , and is the outsideobserver’s estimate of bidder at period . measures the auction’sefficiency in terms of allocating the object. The second term in the expression abovemeasures thepayoffs of the outside observer. The last termmeasures the sumof thepayoffsobtainedbythethreebiddersthroughtheestimatesfromtheoutsideobserver.Efficiencydoesnotdependontheauction’srevenue(itisawelfare‐neutraltransactionbetween a bidder and the auctioneer). Figure 11 compares the average value of eachtermandtheoverallaverageefficiencybetweentreatments.

The average value of the winner in the auction is the highest in the FPW treatment(p=0.06, p=0.06 andp=0.08with respect to FPWP, SPWP and SPW respectively). Theaveragevalueof thewinner isnot significantlydifferentwhen compared to theothertreatments.ThismeansthattheFPauctionwhereonlythewinnerisrevealedallocatesthe good in the most efficient way among the bidders. In other words, even thoughrevealing the winner’s payment in the FP auction increases average earnings for theauctioneer,itreducestheallocativeefficiencyoftheauction.

Table7presents twoothermeasuresofallocativeefficiency.The firstmeasureshowsthepercentageofauctionsinwhichthebidderwiththehighestvaluationwinsforeachtreatment.Thesecondmeasure is theaverageratioofthevalueofthewinningbidderover the value of the bidder with the highest value. According to both measures,allocative efficiency varies significantly across treatments. In particular, efficiency inFPWPissignificantlylowerthaninFPW.NotethattheallocativeefficiencyinbothWPtreatments is also clearly lower than the usual efficiency levels reported in theliterature.14

14See,forexample,Coxetal.(1982)andKagelandLevin(1993).

19

Figure11:Efficiencycomparisonbetweentreatments.

Table7:Allocativeefficiency

Notes:*p<0.1,**p<0.05,***p<0.01.

Theoutsideobserver’saverageearningsaresignificantlyhigher inFPWPascomparedto FPW, SPWP, and SPW (p=0.04, p=0.02, p=0.02, respectively). No significantdifferences in outside observer’s earnings are found between the other threetreatments.Thismeansthattheoutsideobserverhasthehighestaccuracyinestimatingbidders’valuesintheFPauctionwhereboththewinnerandhispaymentarerevealed.RevealingthewinningpaymenthelpsoutsideobserverstoincreasetheaccuracyoftheirestimatesintheFPauction,butitdoesnothelpintheSPauction.Apossibleexplanation

75.46

78.25

74.5576.27

Average winner's value

FPWP FPW SPWP SPW

18.09

14.11

15.83

13.14

Average outside observer earnings

FPWP FPW SPWP SPW

84.93

76.47

87.78

80.35

Average bidders' payoffs for estimates

FPWP FPW SPWP SPW

178.47

168.83

178.16

169.76

Average overall efficiency

FPWP FPW SPWP SPW

Treatment%highestvaluewins

DifferencewithFPW

Valuewinner/HighestvalueDifferencewithFPW

FPW 83.3% 97.5% FPWP 71.9% ‐11.43%*** 93.9% ‐3.68%**SPW 76.7% ‐6.67% 95.2% ‐2.34%SPWP 76.2% ‐7.14%* 92.7% ‐4.84%***

20

forthisisthatrevealingthewinner’spaymentgivesdifferentinformationtotheoutsideobserversintheFPauctionthanintheSPauction.ThewinningpaymentrepresentstheownbidofthewinnerintheFPauction,butitonlyrepresentsthesecondhighestbidintheSPauction.Hence,thewinner’spaymentallowstheoutsideobservertotheoreticallypinpointthevalueofatleastonebidderwithcertainlyintheFPauction.While,ontheotherhand,nobiddercanbefullyidentifiedintheSPauction.

We find that in FPWPandSPWP, bidders obtain significantlyhigherpayoffs from theoutside observer’s estimates than in FPW and SPW, respectively (p=0.01 and p=0.06respectively). Therefore, revealing more information induces outside observers toincrease their value estimates so that, in turn, bidders benefit more from signalling.NeitherthedifferencesbetweenFPWPandSPWP,northosebetweenFPWandSPW,arestatisticallysignificant(p=0.53andp=0.25,respectively).

All in all, none of the treatments outperform the others in all three performancemeasures.Inparticular,thereisatrade‐offbetweenallocativeefficiencyandthepayoffsbiddersearnfromtheoutsideobserver’sestimates.TotalaverageefficiencyishigherinFPWP and SPWP as compared to FPW and SPW (p=0.03 and p=0.03, respectively).Therefore, more information increases overall average efficiency in both FP and SPauctions.However,weshouldbecautiouswith this resultbecause theearningsof theoutsideobserverand thebiddersarenotnecessarilyexpressed in thesamemonetaryunits.

Result 9: Allocative efficiency is significantly higher in FPW than in the other threetreatments which, in turn, do not differ significantly between each other in terms ofallocative efficiency. FPWP yields significantly higher bidder payoffs from the outsideobserver’sestimates than theother treatmentswhich, in turn,donotdiffer significantlyregardingthisoutcomemeasure.Bothbidderpayoffsfromtheoutsideobserver’sestimatesandoverallefficiencyaresignificantlyhigher inFPWPandSPWPthan inSPWandFPW,respectively.Whenkeepingtheinformationrevelationpolicyfixed,thetwoauctionsdonotdiffersignificantlyalongthoseoutcomemeasures.

5.Conclusion

In many auction settings, bidders have the opportunity to signal their generosity,wealth, or productivity to outside observers. Applications range from local charityauctionstomulti‐milliondollarartauctions,andmulti‐billiondollarspectrumauctions.Aprimary‐schoolpupil’smothermaysubmitahighbidintheschool’sfundraisingeventtosignalhergenerositytootherparents.Abidderinanartauctionmaywanttosignalhis wealth to third parties by submitting high bids. A telecommunications firm’sbehaviour in a spectrum auction contains information about the quality of itsmanagement whichmay be a valuable signal by the firm’smanagement to investors.Signallinginauctionshasreceivedampleattentioninrecentliterature.Still,ourpaperis

21

the first study which examines the relative performance of commonly used auctionformats in an experimental setting. In the experiment, we compared the first‐pricesealed‐bidauctionandsecond‐pricesealed‐bidauctionsundertwoinformationregimes:inone,theauctioneeronlyrevealstheidentityofthewinner,andintheother,shealsopublishesthewinner’spayment.

Ourkeyfindingisthatthefirst‐pricesealed‐bidauctioninwhichthewinner’spaymentisrevealedperformsthebestamongthemechanismsstudied intermsofrevenueandoverall efficiency. Moreover, revealing the winner’s payment inflates the bids in thefirst‐priceauction,butitdoesnotdosointhesecond‐priceauction.Thesefindingsarerobust in that we obtain qualitatively the same results in a difference‐in‐differenceanalysis where we compare the revenues in the main treatments correcting for therevenues obtained in control treatments. Our efficiency analysis reveals that both theoutsideobserverandthebiddersbenefit fromrevealingthewinner’spayment inbothauctions,albeitatthecostofallocativeefficiencyinthefirst‐priceauction.Riskaversionandvariationsintheoutsideobserver’saccuracyacrosstreatmentspotentiallyexplainthewaytheobservedbidsdeviatefromtherisk‐neutralNashequilibrium.

Overall, our experimental results suggest that in a context where bidders care abouthowtheirbehaviour intheauction is interpretedbyothers,boththeauctiontypeandthe amount of information revealed can have a significant impact on the auctionperformance. The average revenue in the first‐price sealed‐bid auction where theauctioneer reveals the winner’s payment is 10‐25% higher than in the three othermechanisms studied. This finding suggests that it might be in the best interest oforganizers of charity auctions, art auctions, and spectrum auctions to use first‐priceauctions, rather than second‐price auctions, and, moreover, to reveal how much thewinnerpays.

A natural follow‐up question concerns the extent to which our results can beextrapolatedinthefield.Fieldexperimentsmayrevealthecircumstancesunderwhichfirst‐priceauctionsactuallyperformbetterthansecond‐priceauctionsinsettingswherebidders have signalling opportunities. Carpenter et al. (2008) provide suggestiveevidenceinthisdirection.Inafieldexperimentconductedduringfundraisingcampaignsatpreschools, they findthat the first‐pricesealed‐bidauctionraisesmoremoneythanthesecond‐pricesealed‐bidauction.Moregenerally,futureexperiments,bothinthelaband in the field,might identifyauctiontypesaswellas informationrevelationpoliciesthatperformevenbetterthantheonesexaminedinourlabexperiment.

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24

AppendixA:DerivationofequilibriumbiddingcurvesIn this appendix, we derive the equilibrium bidding curves. Consider a setting with

2bidders,indexed 1,… , ,biddingforasingle,indivisibleobject.Bidders’values

fortheobjectarei.i.d.accordingtoasmoothdistributionfunction on 0, ̅ , ̅ 0.The

auctionoutcome ispartly revealed to anoutsideobserver.Weassume thatabidder’s

payoffs are increasedby ( 0), if the outside observer’s estimate of thebidder’s

value equals . For the analysis,we presume that bidders bid according to the same,

strictly increasing, bidding curves. In equilibrium, the outside observer updates her

beliefsaboutthebidders’valuesaccordingly.

The structure of this appendix is as follows: in sections A.1 and A.2, we derive

equilibriumbiddingcurvesforsettingsinwhichtheoutsideobserverisinformedabout

whowinstheauctionsandhowmuchthewinnerpaysinafirst‐pricesealed‐bidauction

andsecond‐price sealed‐bidauction, respectively. In sectionA.3,weconsider the case

wheretheoutsideobserverisonlyinformedaboutthewinneroftheauction.

A.1First‐pricesealed‐bidauctionwinnerpayment

Assume that bidders bid according to a strictly increasing bidding curve . Now,

consider a bidder with a value bidding as if his valuewere ∈ 0, ̅ . If the other

bidderssticktotheequilibriumbiddingcurve,thisbidder’sexpectedpayoffsequal

, ,

where denotes the distribution of the highest‐order statistic of 1 i.i.d. draws

from .ThefirsttermontheRHSreferstothecaseinwhichthebidderwinsandthen

theoutsideobserver induces that thebidder’svalueequals .Thesecond term is the

bidder’s payoff when losing the auction, where denotes the outside observer’s

optimalvalueestimateforlosingbidders.TheequilibriumFOCisgivenby

,0

⇔ ,

25

where is the density function corresponding to . Taking into account the

boundary condition 0, we find the following solution for the resulting

differentialequation:

.

For the parameters used in the experiment ( 1/2, 3, 0,1 ), we have

/2and fromwhichitfollowsthat .

A.2Second‐pricesealed‐bidauctionwinnerpayment

The analysis for the second‐price sealed‐bid auction is analogous to the first. Let

| denote the distribution of the second highest value among the 1 other

biddersifabidderpretendstohavevalue ,conditionalonthehighestvalueamongthe

other bidders being . Moreover, we will let represent the outside observer’s

estimateofthewinner’svalue,conditionalonthesecond‐highestvaluebeing .Wethen

have:

, | .

where | | | represents the expected value of the

outsideobserver’svalueestimateofabidderwhobidsasifhavingvalue ,conditional

onthehighestvalueamongthe 1otherbiddersbeing .Noticethat ↑ |

.

TheequilibriumFOC:

↑ ||

0 ⇔

1|

/ .

Fortheparametersusedintheexperiment, /2 5/8.

26

A.3Winner‐only,bothauctiontypes

The predictions for the winner‐only treatments are straightforward: bidders’

equilibriumbidsarethestandardequilibriumbidsinasettingwithoutoutsideobserver

inflatedby timesthedifferencebetweentheoutsideobserver’svalueestimatesforthe

winnerandthe losingbidders.Whenestimatingabidder’svalue, theoutsideobserver

minimizesw.r.t. :

| | ,

where is the outside observer’s belief, i.e., the distribution function of the bidder’s

value.TheFOC:

0 ⇔ 2 1 0.

Forthethree‐biddercase,withvaluesuniformlydistributedon 0,100 , /10

forthewinningbidder,undertheassumptionthatbidderssubmitbidsaccordingtothe

same strictly increasing bidding curve. This implies that the outside observer’s best

guessequals 100/√2 79.W.r.t. theguesses for the losingbidders,

/2 /10 . The outside observer’s optimal guess is approximately equal to 35. The

differencebetween the estimates equals about44,which translates to the inflationof

bidsby22.

27

AppendixB:Equilibriumbiddingandrevenuecomparisonforrisk‐aversebidders

In this appendix,weexplore riskaversionas apotentialdriverof the rejectionof the

revenuerankingslaidoutinhypotheses2and3basedonriskneutralbidding.Inorder

todoso,wederivetheequilibriumbiddingcurvesforrisk‐aversebiddersandcompare

themwiththeequilibriumbiddingcurvesforrisk‐neutralbidders.Wekeepthenotation

introducedinAppendixA.Allbiddersevaluatetheirpayoffsaccordingtoutilityfunction

: → suchthat 0 0, 0and 0. Wedenote thestrictly increasing

risk‐averse bidding function and, as previously, the strictly increasing neutral risk

biddingfunction,with , : 0, ̅ → .

The structure of this appendix is as follows: in section B.1 we determine that risk

aversionshiftsequilibriumbiddingcurvesupwardsinafirst‐pricesealed‐bidauctionin

which the winner identity and her payment are revealed to the outside observer

(FPWP). In section B.2 we derive equilibrium bidding curves with risk aversion in a

second‐price sealed‐bid auction in which the winner identity and her payment are

revealed to the outside observer (SPWP) and show that risk aversion depresses

equilibrium bidding under CARA preferences. In section B.3, we compare expected

revenues assuming CARA preferences and show that risk aversion offers a potential

explanationforwhythedatadonotsupporthypotheses2and3.

B.1First‐pricesealed‐bidauctionwinnerpayment

We start by deriving a symmetric equilibrium for FPWP. A bidder with a type ,

pretendingtobeatype ,facesthefollowingproblem:

max , .

TheequilibriumFOCisgivenby

,′ ′

0

28

⟹1

′ 1

1 1

Toestablish 1 ,noticethat forall andthatattheequilibrium

the winning payoff is always higher than the losing payoff. Then, using the same

technicalargumentsthanKrishna(2009,Chapter4,page39)weget

forall 0.WeconcludethatinFPWP,risk‐aversebiddersbidmoreaggressivelythan

risk‐neutralbidders.

B.2Second‐pricesealed‐bidauctionwinnerpayment

TheanalysisforSPWPisanalogoustoFPWP.Forthesakeofcomputationsimplicity,we

useanotherwaythaninAppendixA.2toestablishthebidders’expectedpayoffs.Welet

and representtheoutsideobserver’sestimateofthewinner’svalueandthe

loser’svalue respectively, conditional on the second‐highest valuebeing .Recall that

theidentityofthesecondhighestbidderisunknowntotheoutsideobserver.Atype

loser is the second highest bidder with probability 1 1

and another bidder with probability 1 2 . We

denote and thedensityfunctionsassociatedwith and respectively.Abidderwith

atype ,pretendingtobeatype ,facesthefollowingproblem:

max ,

,

with and .TheFOC,attheequilibrium ,

isgivenby

29

′ ′

0.

Let us now consider CARA preferences such that exp for 0.

Moreover,usingtheparametersfromtheexperiment( 1/2, 3, 0,1 ),we

have , , , 2 1 and 2 .The

FOCbecomes

exp5 14

2 2 1 2 exp38

2 1 exp38

38

2 1 exp38

0.

Therefore,

78

14

1ln 1

38

1 2 .

Weprove /2 5/8forall 0bycontradiction:

∀ 0 ⇔

ln 138

138

1 ∀ 0 ⇔

1 38

1 exp38

1 ∀ 0,

whichisincontradictionwithexp 1forall 0.Finally,notethat

0 0 forall 0.

B.3Revenuecomparison

Inthissection,wecompareexpectedequilibriumrevenueassumingCARApreferences

suchthat exp for 0.Let denotetherevenueintreatment inthe

casebidders’degreeofriskaversionequals 0.Let thebiddingstrategyforSPWP

inthecaseofinfinitelyriskaversebidders,i.e., lim → forall ∈ 0, ̅ .

Equation(B2)implies

30

78

14.

This follows from the observations that 0 ln 1 1 ln 1 , as

1 1forall ∈ 0,1 ,and lim→

ln 1 0.

It follows that the associated revenue, , is equal to . Compared to the risk

neutralcase,riskaversionaffectsequilibriumbiddingstrategiespositively inthe first‐

price sealed‐bid auction and negatively in the second‐price sealed‐bid auction, and

thereforecanleadtoalowerrevenueinthelater:

1116

34

where denotesexpectedrevenueintherisk‐neutralequilibriumofFPWP.In

otherwords,asufficientlyhighlevelofriskaversioncanexplaintherevenueranking

swapobservedinthedatarelativetotheriskneutralcaseonwhichhypothesis3is

based.

Forthesettingwhereonlytheidentityofthewinningbidderisrevealed,let and

denotetheoutsideobserver’sequilibriumvalueestimatesforthewinnerandthelosing

biddersrespectively.Becausethesequantitiesdonotdependonthebidssubmittedon

the equilibrium path, the effect of risk aversion on equilibrium bidding follows from

standardreasoning. Inparticular, theequilibriumbiddingstrategy in thesecond‐price

sealed‐bidauctionisnotaffectedbyriskaversionbecausebiddingvalueplus( )

isaweaklydominantstrategy,regardlessofriskattitude.

Asa consequence,SPWPwith infinitely risk‐aversebiddersyields lower revenue than

SPW:

1116

0.687512

0.72

where denotesexpectedrevenueintherisk‐neutralequilibriumofSPW.We

concludethatforsufficientlyriskaversebidders,therevenuerankingofSPWPandSPW

swapscomparedtotheriskneutralcase,inconsistentwithhypothesis2and

qualitativelyconsistentwiththedata.

31

AppendixC:AdditionalfiguresFigureC.1:Bidssubmitted

Notes:Thesolidlinesrefertotheequilibriumbiddingcurves

32

Figure C.2: Average outside observer’s estimates, and average bidder’s values,conditionalonwinner’spaymentfortheFPauction

Notes:Thedotsrepresenttheoutsideobservers’valueestimates;solidlinesrefertotheactualvaluesofthe bidders; the dashed lines are the outcomes of linear regressions of the outsider observers’ valueestimates on the winner payment; the two panels at the bottom plot the difference in the outsideobservers’valueestimatesbetweenwinningandlosingbidders.

33

FigureC.3:Averageoutsideobserver’sestimatesandaveragebidder’svaluesforeachwinner’spaymentfortheSPauction

Notes:Thedotsrepresenttheoutsideobservers’valueestimates;solidlinesrefertotheactualvaluesofthe bidders; the dashed lines are the outcomes of linear regressions of the outsider observers’ valueestimates on the winner payment; the two panels at the bottom plot the difference in the outsideobservers’valueestimatesbetweenwinningandlosingbidders.

34

OnlineAppendixD:InstructionsfortreatmentFPWP

WELCOME

You are about to participate in an economic experiment. The instructions are simple. If youfollowthemcarefully,youmaymakeasubstantialamountofmoney.Yourearningswillbepaidtoyouineurosattheendoftheexperiment.Thiswillbedoneconfidentially,oneparticipantatatime.

Earningsintheexperimentwillbedenotedby‘francs’.Attheendoftheexperiment,francswillbeexchanged foreuros.Theexchangerate is1euro forevery70 francs.Yourstartingcapitalequals490francs(or7euros).

Theseinstructionsconsistofsevenpageslikethis.Youmaypagebackandforthbyusingyourmousetoclickon ‘previouspage’or ‘nextpage’atthebottomofyourscreen.Atthebottomofyour screen, you will see the button ‘ready’. You can click this when you have completelyfinishedwithallpagesoftheinstructions.

AUCTION

In today’s experiment, youwillparticipate in auctions. In theseauctions, three biddersbid toobtaina fictitiousgood.Thebiddersareobservedbyanoutsideobserver. Intheremainderofthese instructionswewillexplainthewayinwhichtheauctionisorganizedandtherulesyoumustfollow.

ROUNDS

Today’sexperimentconsistsof30rounds.Ineachround,afictitiousgoodisauctioned.

Intheexperiment,youwillbememberofagroup.Thisgroupconsistsofyouandthreeotherparticipants.Itisunknowntoyouandtotheotherparticipantswhoisinwhichgroup.Thefourgroupmembersremaininthesamegroupthroughouttheexperiment.Thus,youwillmeetthesamethreeparticipantsineachofthe30rounds.

In every round, one groupmember is randomly chosen by the computer to play the role ofoutsideobserver.Theremainingthreegroupmembersarethebiddersintheauction.

THEVALUEOFTHEAUCTIONEDGOOD

The value of the fictitious goodwill typically differ from one bidder to the next. To bemoreprecise,ineveryround,thecomputerwilldrawanewvalueforeverybidder.Valuesaredrawnfromtheset{1,2,3,…,100}.

Notethefollowingaboutthevaluefortheobjects:

1. The value for a bidder is determined independently of the values for the other twobidders;

2. Anyvalueintheset{1,2,3,…,100}isequallylikely;3. Eachbidderonlylearnsherownvalue,notthevalueoftheotherbidders;4. Theoutsideobserverisnotinformedaboutthevaluesofanyofthethreebidders.

35

THEAUCTION

Intheauction,eachofthethreebiddersindependentlysubmitabidforthefictitiousgood.Bidsmustbechosen fromtheset {0,1,2,…,200}.Thebidderwith thehighestbidgets thegoodandpayshisbid.Iftwoorthreebidderssubmitthesame(highest)bid,thecomputerwillrandomlydeterminewhichoneobtainsthegood.

THEOUTSIDEOBSERVER

Aftertheauction,theparticipantplayingtheroleofoutsideobserverisaskedtoguessthevaluesofeachofthethreebidders.Beforeshedoesso,sheobtainsinformationabouttheoutcomeoftheauction.Inparticular,sheisinformedaboutwhichbidderwontheauctionandhowmuchthewinnerpaid.

The payoffs of the outside observer depend on the precision of her estimates. Once she hasenteredvalueestimatesforallbidders,thecomputerdrawsoneofthethreebiddersatrandom.If theoutsideobserver’sestimate for thisbidder isexactlycorrect, sheobtains40points.Thefurtherher estimate is away from theactual value, the lower isherpayoff. Specifically, if herestimate deviates points for the actual value, her payoff is equal to 40– . In words: theoutside observer loses one point for every unit her estimate is further away from the actualvalue.

EARNINGSFORTHEBIDDERS

Thepayoffsforthebiddersaredependentonboththeoutcomeoftheauctionandtheestimateoftheoutsideobserver.Ifabidderdoesnotwintheobject,hisearningsinaroundonlydependonthevaluetheoutsideobserverestimatedthisbiddertohave:

(Earnings)=(Theoutsideobserver’svalueestimate)/2So,abidderearnshalfafrancforeveryfrancintheoutsideobserver’svalueestimate.Abidder’searningsdonotdependontheoutsideobserver’sestimatedvaluesfortheothertwobidders.Ifabidderwinstheobject,hisearningsinaroundwilldependonbothhisprofitsintheauctionandtheoutsideobserver’svalueestimate:(Earnings)=(Valueforthegood)–(Winningbid)+(Theoutsideobserver’svalueestimate)/2Notethatabiddergetsthesamepayoffsfromtheoutsideobserver’sestimate,winorlose.

Documents

Signalling in Auctions: Experimental Evidence