Common Value Auctions with Costly Entry

Transcription

1 Common Vaue Auctions wit Costy Entry Paui Murto Juuso Väimäki Marc 25, 2019 Abstract We anayze an affiiated common vaues auction wit costy participation wit an unknown number of competing bidders. We ca suc auctions informa auctions. We contrast te symmetric equiibria of informa first-price auctions wit te we-understood symmetric equiibria of forma auctions were te number of entrants is common knowedge at te bidding stage. Wit endogenous entry, te informa first-price auction often yieds a iger expected payoff tan any of te standard forma auctions. JEL Cassification: D44 1 Introduction Entering an auction often entais significant costs. In addition to te opportunity cost of time and effort spent on being pysicay present at an auction site, acquiring information about one s own vauation for te good is often costy. On top of tis, te preparation of bids may be costy as a resut of concerns for due diigence. If entry costs are sunk at a stage were te eventua number of participants in te auction is unknown, it is natura to consider te performance of various auction formats wit a random number of bidders. We ca suc auctions informa auctions. In tis paper, we sow tat accounting for tis uncertainty eads to new types of bidding equiibria and to new revenue resuts in singe object auctions wit affiiated common vaues. In particuar, we demonstrate te superior performance of first-price auctions in common vaue auctions wit an undiscosed number of bidders wen entry costs are significant. Te previous iterature on auctions wit costy participation as focused on sequentia decisions by te bidders. Entry decisions are taken in te first stage. In te second stage, a entering bidders see te reaized number of participants and bid optimay in te ensuing auction. In tis paper, Aato University Scoo of Business, paui.murto@aato.fi Aato University Scoo of Business, juuso.vaimaki@aato.fi 1

2 we reax te assumption tat te set of bidders is common knowedge at te bidding stage. In an informa auction, eac individua bidder knows ony te equiibrium entry strategy, but not te reaized number of participating bidders at te time of coosing er bid. Tis auction format is meant to capture open seing procedures were interested bidders are invited to submit bids and te igest bidder wins and pays er own bid. Wit correated vaues, different types of bidders perceive te vaue of te good and teir competitive environment differenty. By affiiation, bidders wit iger signas are intrinsicay more optimistic about te vaue of te good. Simiary by affiiation, tey beieve tat oter bidders are more ikey to be optimistic. If bids are increasing in signa, tis means tat bidders wit ig signas face more aggressive bidding from oter bidders. Of course, tese two different effects are present in auctions witout entry costs. Te second effect gains in importance once we add te entry costs to te mode. If te cost is sunk before te auction takes pace, osing to competing bidders becomes a more important consideration. We sow tat in any symmetric equiibrium of te auction modes tat we consider bidders wit iger signas pay te entry cost wit a iger probabiity tan te ower types but in many cases ower types aso enter. We sow tat uncertainty and more importanty different beiefs about te number of opponents give rise to non-monotonic bidding in te sense tat participating bidders wit ig types sometimes ose to ower types at te auction stage. We concentrate on symmetric equiibria of suc modes. By tis soution concept, we empasize settings were te set of potentia bidders is not we known in advance. Exampes of suc cases incude internet auctions and takeover bid contests. If te set of truy interested potentia bidders is not known in advance, any mecanism based on eiciting a bidders types fais if tere are costs for contacting te potentia bidders. We consider two different types of entry costs. In our main mode, potentia bidders ave aready observed a signa on te vaue of te object wen deciding weter to pay te entry cost. We ca tis te interim entry mode. In te appendix, we aso discuss te ex ante entry case were te cost is paid at an initia stage were a potentia bidders are symmetricay informed. In bot of tese formats, bids are decided before te uncertainty on oter potentia bidders participation decisions is resoved. Te anaysis of suc auctions is ard because in contrast to te case of forma auctions, it is possibe tat te game as no symmetric equiibria in monotone strategies. 1 Witout monotonicity, it is not cear ow one soud proceed in te standard affiiated mode wit a continuum of signas for eac bidder as in Migrom and Weber Landsberger 2007 is an eary paper sowing tat existence of monotone equiibria often fais in auctions wit participation costs. 2

3 In order to make progress, we consider a finite minera rigts mode were te unknown common vaue of te object is a binary random variabe. Te signas are drawn from a finite set and tey are assumed to be independent across te potentia bidders conditiona on te true vaue of te object. In an informa auction, eac bidder is uncertain about te number of competing bidders. Wit correated signas, different types of bidders ave different beiefs on te type profies and reaized numbers of teir competitors. Two types of non-monotonicities emerge. Entry decisions may be non-monotonic in te sense tat mutipe types of bidders may enter wit positive probabiity, and te bidding strategies may be non-monotone in te sense tat a bid from a ower type wins over a bid of a iger type wit positive probabiity. One key feature of our equiibrium construction is tat ony a imited number of bidder types enter te auction in equiibrium. Wit a binary underying state of te word, we can base our anaysis on te bidders expected payoffs conditiona on te state. Wit affiiated types, te beiefs of te potentia bidders on te state of te word are monotone in teir types. Since we assume a arge number of potentia bidders, expected winnings at te auction stage must equa te entry cost in any equiibrium of te game. Suppose tat te expected payoff at a fixed bid in te support of te equiibrium bid functions differs across te two states. Witout oss of generaity, assume tat te bid generates a stricty iger payoff in state 1 tan in state 0. In equiibrium, tis bid can ony be made by tose bidders wose private signa assigns te igest probabiity to state 1. To see tis, notice tat if anoter bidder makes tis bid, ten er payoff must be at east equa to er entry cost. But by affiiation, te most optimistic bidder in er assessment on te probabiity of state 1 makes ten a stricty positive profit contradicting our requirement of zero profits in equiibrium. We sow tat a te bid distributions tat can emerge in a symmetric equiibrium of our mode can be generated in a mode were we consider ony te most extreme types in te set of potentia bidders, i.e. te two bidders wit te most extreme beiefs on te two states. For tis reason, te main anaysis in tis paper is concentrated on te two-type case. We caracterize te bidding equiibria of te informa first-price auction in tis case. In te appendix, we sow tat our main resuts remain vaid for te ex ante entry case as we. In tat case, we must restrict our attention to a mode wit binary types since a bidder types are now present at te bidding stage of te game. Since we do not ave anaytica resuts on te bidding stage wit mutipe bidder types, tis restriction on types is not witout oss of generaity, in contrast to te case wit interim entry. We start our anaysis by soving te mode wit ony two potentia bidders. In tis case, we sow tat te expected revenue to te seer from an informa first-price auction exceeds at east weaky te expected revenue from standard forma auction formats in te affiiated common vaues auction. Tis impies tat in contrast to te inkage principe, te seer may be better off by witoding 3

4 information from te bidders. 2 Migrom and Weber 1982 demonstrate te revenue superiority of te forma second-price auction over forma first-price auction for affiiated common vaue auctions. Tis resut togeter wit a set of resuts demonstrating te good revenue properties of pubic information discosure are aso known as te inkage principe. Te key idea is tat for a fixed own bid, any auction format tat increases te inkage between own information and te perception of oter payers bids increases te expected payment. To see ow tis principe fais in our informa auctions, consider an equiibrium in te second-price auction were ow type bidders bid beow te bid of te ig types. By pacing a bid between tese two bids, a deviating bidder wins if and ony if no ig bidders participate in te auction. In tis case, te payment is eiter te ow bid if tere is competition, or zero if te oter bidder did not participate. By affiiation, it is more ikey tat no bidders wit a ow signa participate if te vaue of te object is ig. But tis means tat te expected payment of te ig type bidder is ower tan te expected payment of te ow bidder. Wit more tan two bidders, informa auctions may potentiay possess mutipe equiibria. If te winner of an auction is tied wit oter bidders wit positive probabiity, information conditiona on winning te object refects te rationing amongst te igest bidders. Tis is not accounted for in te standard conditioning events of auctions wit atomess bid distributions. 3 We sow tat in te informa first-price auction, symmetric equiibrium bidding strategies cannot ave atoms in te interim entry case. Tis eads to a unique symmetric equiibrium outcome in terms of bid distributions in te overa game determining bot entry and bids. We mode te game wit many bidders as a Poisson game were te type of eac potentia payer is positivey correated wit te vaue of te object. Te number of entrants of eac type is drawn from a Poisson distribution wit a parameter tat depends on te true binary vaue of te object. An equiibrium of tis game is a distribution of bids suc tat a entrants can cover teir cost by teir expected profit and no potentia bidder of eiter type can make a positive profit by entering. Wen comparing informa auctions to forma auctions, a second consideration emerges. Te price paid in te forma auction is determined by te reaized number of participating bidders. Wenever a bidder is te soe participant, se gets te object for free. For a forma auction wit n 1 oter participants, te equiibrium bid by te ow bidders is equa to te vaue of te object conditiona on n ow signas. Due to affiiation, tis payment is decreasing in n. For ow vaues of 2 Wit binary signas, forma first-price and second-price auctions resut in te same expected revenue, te same expected bidder rents and ence te same entry decisions. 3 Pesendorfer and Swinkes 1997 point out tis possibiity in te case of forma auctions and Lauermann and Woinsky 2015 discuss tis issue in a first-price auction wit an unknown number of bidders. 4

5 te entry cost, te probabiity of te event tat no oter bidders are present vanises. Since a ig type bidder assigns a iger probabiity to ower n, we see tat te expected payment of te bidder is positivey inked wit te type and te usua inkage principe appies. Tis expains our finding tat forma second-price auction dominates informa first-price auction if te entry cost is very ow. But wen te entry costs are not too sma, we sow tat te expected revenue ranking from te two-bidder case carries over to te Poisson game wit endogenous entry. We get our strongest resuts wen te entry cost is reativey ig and te expected number of entrants is reativey ow, and te affiiation in te signas is strong. In tis case, zero bid is in te support of te equiibrium bidding strategies of bot types of payers in te informa first price auction. Wit atomess bidding strategies tis means tat te payoff of eac type of bidder coincides wit te vaue of te good conditiona on being te ony entrant. Tis private benefit is aso te maximum socia benefit from inducing additiona entry wen restricted to symmetric strategies. We concude tat te symmetric equiibrium entry rates maximize socia wefare in te cass of symmetric entry strategies. Since we ave arge numbers of potentia bidders, te expected payoff to bidders net of entry costs must be zero and as a resut te seer receives te maxima symmetric surpus as er expected revenue in tis cass of auctions. For tese parameter vaues we see ten tat te informa first price auction is te revenue maximizing mecanism in te cass of symmetric mecanisms. 1.1 Reated Literature Endogenous entry into auctions as been modeed in two separate frameworks. In te first, entry decisions are taken at an ex ante stage were a bidders are identica. Potentia bidders earn teir private information ony upon paying te entry cost. Hence tese modes can be toug of as games wit endogenous information acquisition. 4 Frenc and McCormick 1984 gives te first anaysis of an auction wit an entry fee in te IPV case. Harstad 1990 and Levin and Smit 1994 anayze te affiiated interdependent vaues case. Tese papers sow tat due to business steaing, entry is excessive reative to socia optimum. Tey aso sow tat second-price auctions dominate te first-price auction in terms of expected revenue. A of tese papers proceed under te assumption tat te number of entering bidders is known at te moment wen bids are submitted. Our informa auctions are tus not covered at a in tese papers. In te oter strand, bidders decide on entry ony after knowing teir own signas. Samueson 4 Te equiibrium determination of information accuracy in common vaues auctions started wit Mattews 1977 and Mattews 1984 and Persico 2000 extended tis ine of researc to revenue comparisons for different auction formats. Since te number of bidders and equiibrium information acquisition decisions are deterministic, equiibrium bidding in tese papers is sti as in standard affiiated auctions modes. 5

6 1985 and Stegeman 1996 are eary papers in te independent private vaues setting were tis question as been taken up. Due to revenue equivaence, comparisons across auction formats are not very interesting. To te best of our knowedge, common vaues auctions ave not been anayzed in tis setting. 5 Finay some recent papers ave anayzed common vaues auctions wit some simiarities to our paper. Pekec and Tsetin 2008 provides an exampe were informa first-price auction resuts in a iger expected revenue tan an informa second-price auction. Te distribution of te bidders in tat paper is somewat extreme and not derived from entry decisions. Lauermann and Woinsky 2017 and Lauermann and Woinsky 2019 anayze first-price auctions were an informed seer cooses te number of bidders to invite to an auction. Te bidders do not observe ow many oters were invited and ence te bidding stage anaysis is as in our mode wit an exogenous entry rate. Tese papers do not compare revenues across different auction formats and since te distribution of entering bidders resuts from an optima invitation decision by te seer, te anaysis is quite different from our paper. Atakan and Ekmekci 2014 consider a common vaue auction were te winner in te auction as to take an additiona action after winning te auction. Tis eads to a non-monotonicity in te vaue of winning te auction tat as some resembance to te forces in our mode tat ead to non-monotonic entry i.e. bidders wit bot types of signas enter wit positive probabiity. Since we concentrate on symmetric equiibria of a game wit a arge number of potentia entrants, our mode as some simiarities to te urn-ba modes of matcing. Simiar to tose modes, our insistence on symmetric equiibria can be seen as a way of capturing a friction in te market tat precudes coordinated asymmetric decisions. A recent exampe of suc modes is Kim and Kircer 2015 tat studies matcing wit private vaues uncertainty. Tis approac as aso been used in Jeie and Lamy 2015 a procurement auction setting wit private asymmetric vaues. Tese modes ave not covered te case of common vaues to te best of our knowedge. Te paper is structured as foows. In Section 2, we introduce te main ideas of te paper in te simpest possibe setting wit ony two potentia bidders. Section 3 presents te informa first-price auction and sows tat in equiibrium at most two types enter. Te section aso anayzes equiibrium bidding in a mode wit an exogenous random number of bidders wit binary types. Section 4 anayzes equiibrium entry decisions and sows tat equiibrium outcomes are essentiay unique. Section 5 compares te expected revenues across different auction formats and Section 6 discusses te modeing assumptions in tis paper. A proofs of te resuts are in Appendix B. 5 Battacarya, Roberts, and Sweeting 2014 and Sweeting and Battacarya 2015 anayze IPV modes wit seective entry were entry decisions are conditioned on private information on te true vauation. 6

7 2 Two-Payer Mode We start wit te anaysis of te mode wit two types of bidders and ony two potentia bidders. By doing tis, we can introduce te main ideas in te paper wit minima notation and witout anaytica compications tat are unavoidabe wit many bidders. Two potentia bidders wit a signa θ {, } must decide weter to pay an entry cost c to participate in an auction for a singe indivisibe good. Te vaue of te good v ω is binary ω {0, 1} wit v 1 > v 0 > c and te signas are i.i.d. conditiona on v. By affiiation, et α := Pr{ θ = ω = 1} > Pr{ θ = ω = 0} =: β. Let q = Pr{ ω = 1} be te prior probabiity on te vaue of te object. Te bidders are risk-neutra and tey maximize teir payoff at te auction stage net of te entry cost. 2.1 Symmetric Socia Optimum In tis subsection, we anayze a socia panner s probem to obtain a usefu bencmark for symmetric equiibria of various auction formats. Hence, we restrict te panners feasibe strategies to be aso symmetric. Te panner cooses entry probabiities π, π for payers wit signas and respectivey to max 0 π,,π 1 q[ 1 1 απ 1 α π 2 v 1 2c απ + 1 α π ] q [ 1 1 βπ 1 β π 2 v 0 2c βπ + 1 β π ]. Te two first-order conditions for an interior soution to tis concave quadratic probem can be written as: q 1 απ 1 α π v q 1 βπ 1 β π v 0 = c, 2 q 1 απ 1 α π v q 1 βπ 1 β π v 0 = c, 3 were q θ is te posterior of type θ bidder: q = q = αq αq + β 1 q > 1 α q 1 α q + 1 β 1 q. 7

8 Te two ines state tat te cost of adding a payer of eac type must equa te benefit. Te benefit is reaized ony if te oter bidder did not participate. Notice aso tat an immediate impication of tis first-order condition is tat te benefits be equaized aso across states since q > q. Since v 0 < v 1, we see from tis tat π opt > π opt in te soutions to tese equations. For competeness, et us note tat te first order condition is vaid ony for 0 < π opt Te first inequaity 0 < π opt ods if Te second inequaity π opt < 1 ods if α β > v 1 c/v 1 v 0 c/v 0. α β > v 1 v 0. < π opt < 1. Finay, note tat if π opt = 0, ten π opt > 0 if q v q v 0 > c. We see tat we get interior soutions if entry costs are sma and if te ratio of te vauations is not too arge. Te reason wy te panner wants bot types of payers to participate is tat te types of te payers are correated wit te state of te word. Ideay te panner woud taior te entry probabiities to te state of te word, but tis information is not avaiabe to er at te beginning of te game. By inducing entry from bot types of payers, te panner can baance te benefits of entry across te two states. In te case wit a arge number of potentia bidders, te equivaent of te second inequaity is not binding since expected number of entrants is aways bounded by v 1 c equiibrium of te mode. in any soution optima or 2.2 Informa First-Price Auction We anayze next te symmetric Bayes-Nas equiibria of te game were eac potentia bidder decides simutaneousy weter to enter and wat to bid conditiona on entry. Formay, te strategy of payer eac payer maps er type θ to a probabiity of participating π θ and a bid distribution F θ. We use λ ω,θ to denote te probabiity wit wic a payer of type θ enters in state ω if te payers use te symmetric entry strategies π, π : λ 1, = απ, λ 0, = βπ, λ 1, = 1 α π, λ 0, = 1 β π. Wit tis notation, we can write te expected payoff U θ p at te bidding stage to type θ from 8

9 bidding p wen oter payers use strategy π, π, F, F as: U θ p : = q θ 1 λ 1, 1 F p λ 1, 1 F p v 1 p + 1 q θ 1 λ 0, 1 F p λ 0, 1 F p v 0 p = : q θ R 1 p + 1 q θ R 0 p, were R ω p denote te expected rent in state ω from bid p. In a symmetric Bayes-Nas equiibrium, for a p in te support of F θ, p maximizes U θ p. We denote te igest bid in te union of te supports of te two bid distributions by p max. Our next proposition provides a caracterization of te unique bidding equiibrium for te bidding stage of te informa first-price auctions wit 0 < π π < 1. 6 We stress ere tat tis caracterization does not inge on π, π being an equiibrium entry pair. Tis is important for our discussion of te aternative ex ante entry mode in Appendix A were π = π by construction since payers do not know teir types wen coosing entry. Proposition 1 Assume exogenous entry probabiities 0 < π π < 1.Te informa first-price auction wit two potentia bidders as a unique symmetric equiibrium, were bot types of bidders use atomess mixed strategies. Te supports of te two bid distributions F θ for θ {, } satisfy 1. 0 suppf, p max suppf, suppf suppf = [0, p max ], 2. Eiter 0 suppf or tere exists a p > 0 suc tat suppf = [0, p ] and suppf = [p, p max ]. Te first property to notice is tat te bid distributions contain no atoms. Wie tis is a standard feature of auction modes wit a known number of bidders, te resut is not true in genera in modes wit a random number of participating bidders. Wit ony two bidders, tere is a singe event tat can resut in a tied winning bid: bot bidders submit te same bid. In tis case, winning te object conveys no additiona information and because of tis, te usua argument tat impies atomess distributions ods. In te next section, we sow tat wit many potentia bidders, te interim entry mode as a unique equiibrium and it is in atomess strategies. For te case wit ex ante entry, tis is not true. Te second key feature is tat te supports of te distribution take very specific forms. Low type bidders aways ave te zero bid in teir equiibrium bid support. Tis means tat tey earn a positive payoff ony if te oter bidder does not participate. But te expected payoff from tis bid is exacty te payer s contribution to te socia surpus. If zero bids are aso in te support of te 6 We dea wit te corner soutions separeatey in te subsection on revenue comparisons. 9

10 ig type bidder, ten bot types of payers earn as teir equiibrium payoffs exacty teir margina contribution. In te oter possibe case, we know tat we can compute te expected payoff to te ig type bidder by cacuating er payoff at te igest bid tat te ow type bidders make in equiibrium. Tese observations turn out to be usefu for te revenue comparisons beow. 2.3 Forma Auctions As a bencmark for comparison, consider forma standard auctions, were te bidders know te number of oter bidders at te time of pacing teir bids. We ave sown in Ci, Murto, and Väimäki 2019 tat bot first- and second-price auctions ave a unique symmetric equiibrium in te current environment. In forma auctions, it is naturay optima for any bidder to bid zero if tere are no oter bidders. Wenever tere are at east two bidders present, te unique equiibrium in te first-price auction resuts in te bid b = E v θ 1 = θ 2 = for te ow type.hig type bidders mix on an interva [b, p ], p > b. Te ow type bidder gets a stricty positive payoff equa to er expected vaue of te object ony if se is te ony bidder present. Notice tat tis is aso te socia vaue of entry. Te ig type bidder earns an information rent on top of tis socia vaue since at bid b se wins wit positive probabiity and earns a stricty positive expected payoff equa to te difference between te expected vaue of te object based on er ig signa and tat based on a ow signa. Te forma second price auction as aso a unique symmetric equiibrium were θ {, } submits te bid b θ = E v θ 1 = θ 2 = θ Te ig type gets in tis case exacty te same information rent as in te first-price auction. We can ten concude tat forma first-price auction and forma second-price auction resut in te same payoffs to bot types. Furtermore we see tat te expected payoff woud be te same aso in te informa second-price auction since te tying event is uniquey determined. Hence it is sufficient to compare te expected revenues between te informa first-price auction and te forma second-price auction Revenue Comparisons wit Exogenous Entry Wit two potentia bidders, we get an unambiguous revenue ranking for te auction formats tat we consider: te informa first-price auction is superior in terms of expected revenue to te oter 10

11 formats. Note tat in a mode wit common vaues, aocation is aways efficient so tat an increase in revenue is at te expense of rents to te bidders. Hence, iger expected revenue is equivaent to ower bidder rents. Proposition 2 Wit two potentia bidders and exogenous entry probabiities 0 < π π < 1, te informa first-price auction generates a iger expected revenue tan te forma auctions. Equivaenty, te expected rents to te bidders are ower in te informa first-price auction tan in te forma auctions. Te key step in te proof of te proposition compares te expected payments of te ig type bidders across te informa first-price auction and te forma second-price auction. Even toug te types temseves are affiiated in te usua sense, te fact tat ow types enter wit a ower probabiity in a ig state generates a different type of dependence between te types of participating bidders resuting in different expected payments. If te mode ad more signas, ten affiiation in te types woud add a counteracting effect improving te revenue performance of te forma second-price auction reative to te first-price auction. 2.5 Interim Entry Equiibrium wit Two Potentia Bidders We concude tis section wit te anaysis of entry decisions. Tis specification is particuary reevant for cases were differentia seection of te bidders pays a key roe. Te interesting situation is te one were bot types enter wit positive probabiity. Any suc equiibrium trades off two forces in a way tat makes entry viabe for bot types. Hig types are more optimistic about te vaue of te object. Low types on te oter and, find a ow eve of competition more ikey. Te differences in te equiibrium outcomes of different auction formats resut from te different ways in wic tey baance tis trade-off. Tis feeds directy into different revenue properties of te auctions as we sow in tis section. Our first observation reates equiibrium socia surpus to te expected revenue of te seer. If bot types are indifferent between entering or not, i.e. is entry is wit interior probabiities 0 < π π < 1, ten te bidders earn no expected rent in te game. In oter words, teir expected payoff in equiibrium is fuy dissipated by te entry cost c. In our mode wit quasiinear payoffs, tis impies tat te seer s expected revenue coincides wit te socia surpus generated. Wit tis observation, we can conduct our revenue comparisons in terms of te socia surpus generated in te symmetric equiibrium of te game, as in Levin and Smit We can view te first order conditions 2 and 3 as te panner s reaction curves. For eac fixed vaue of π, 2 gives te sociay optima eve π π and for eac eve of π, 3 gives te sociay optima eve π π. 11

12 Te private costs of a potentia entrant coincide wit te socia cost. Since te private benefit is at east equa to te socia benefit, we see tat our auction formats generate excessive entry reative to socia optimum. 7 By te caracterization resut in te previous subsection, ow types aways ave te zero bid in te support of teir bid distribution. Tis means tat teir entry is conditionay efficient given te entry rate of te ig types. Tis impies, in particuar, tat symmetric entry equiibria in a of te auction formats tat we consider ie on te panner s reaction curve π π. Equiibrium entry π π probabiity of te ow types is decreasing and inear in π. Tis observation togeter wit te concavity of te socia objective function impies tat te auction format tat generates ess entry by ig types generates a iger socia surpus. Wenever we ave interior entry probabiities, a comparison of te entry rates of te ig types ten aso gives us a revenue ranking for te auction formats. Consider next te equiibrium payoff to te ig types at te bidding stage of te informa first-price auction. We can sow tat for eac eve of π, tere is a unique eve of π tat V π π, π := max U λ p = c, p π suc were U λ p is te equiibrium payoff to te ig type in te informa first-price auction parametrized by entry probabiities Furtermore, we can sow tat π Let π, π sove By continuity and π π unique. π opt λ 1, = απ π, λ 0, = βπ π, λ 1, = 1 α π, λ 0, = 1 β π. π is continuous in π and tat for a π, π π π π. π π π = π, π = π. π π π, we know tat at any equiibrium soution, π. Using te properties of te bidding equiibrium, we can aso sow tat π π opt We omit te detais ere since te reasoning is competey anaogoue to te case wit Poisson entry proved in fu in Proposition 10. Reca from Proposition 1 tat it is possibe tat zero is aso in te support of te bids made by te ig type in te bidding equiibrium. In tis case, te bidders earn exacty by teir socia contribution at te auction stage and π, π and is π = π π. As a resut, te entry rates ten sociay 7 By bidding zero in eiter of te auction formats, bot types of bidders can secure at east te socia vaue i.e. te vaue of te object in te event tat tere are no oter bidders. 12

13 optima, π = π opt, π = π opt, and te revenue coincides wit te optima socia surpus in te panner s probem. Tis impies tat tere can be no oter symmetric game forms wit iger revenues were entry decisions are taken based on own types ony. Of course in a correated mode, one coud improve on te performance of te auction if entry coud be conditioned on te vector of reported types. In te spirit of costy participation, any communication prior to deciding entry soud aso ave an associated cost, wic we take ere to be proibitivey ig. We can derive a simiar equiibrium reaction curve for te rate of entry π F π in te forma auctions tat keeps ig types indifferent between entering and not entering. Proposition 2 sows tat wit fixed entry rates, te ig types earn more in forma auctions tan in te informa firstprice auction. Wen entry rates are endogenous, tis must be compensated by a iger entry rate in te forma auction so tat π F π π π π π. We can summarize our discussion in te foowing proposition sowing tat te revenue ranking from te mode wit exogenous entry ods aso wit equiibrium entry. Proposition 3 Wit interim entry decisions, informa first-price auction generates a iger expected revenue tan te forma auctions. We end tis section wit a few words regarding te mode wit ex ante entry. We defer te forma anaysis of tat mode to Appendix A. Wit two potentia entrants, very itte canges reative to te mode wit interim entry. Since bidders do not know teir types at te moment of coosing entry, tere is a singe entry rate π = π = π. Since Propositions 1 and 2 cover tis case as we, we concude tat te informa first-price auction resuts in smaer expected equiibrium payoffs to te bidders tan forma auctions. Tese payoffs aways exceed weaky te payers margina contribution to socia surpus, and as a resut, entry rate is distorted upwards reative to te symmetric panner optimum in te concave panner s probem. Since te informa first-price auction resuts in smaer distortions in te entry rate, te concusion from te interim case remains vaid and te informa first-price auction dominates in terms of expected revenue. 3 Informa First-Price Auction wit Poisson Entry In our main mode, we assume tat te number of entrants is a Poisson random variabe wit an endogenousy given parameter. Since te semina work of Myerson on games wit uncertain numbers of participants, te Poisson game mode as been widey used in modes of information economics. 8 It is a tractabe mode tat aows te random number of participating bidders to be 8 For a discussion of te Poisson entry mode in te context of a procurement auction wit private vaues, see Jeie and Lamy

14 correated wit te state of te word te true vaue of te object. As in te previous section wit two potentia participants, tis correation can be rationaized by entry decisions based on affiiated signas. To see ow one arrives naturay at te Poisson mode, assume for te moment tat tere are N potentia entrants tat observe signas θ {1,..., M} wit a conditiona distribution α ω,m =: Pr{ θ = m ω = ω } for m {1,..., M}, were state is ω {0, 1}. We abe te signas so tat tey satisfy strict monotone ikeiood ratio property: α 1,m α 0,m is increasing in m. If eac entrant wit signa m enters wit probabiity π m,n, ten te number of ig type entrants is a binomia random variabe wit parameters α 1,m π m,n, N and α 0,m π m,n, N in states ω = 1 and ω = 0, respectivey. Keeping te expected number of entrants α ω,m π m,n N α ω,m π m constant, et N increase towards infinity and consider te binomia variabes Binα ω,m π m,n, N. Te imiting random number Nω m of entrants of type m in state ω as ten a Poisson distribution wit parameter λ ω,m := α ω,m π m. Te random variabes Nω m are furtermore independent. As before, we use te notation N m to denote te tota random number of participating bidders of type m. Motivated by tis imiting argument, we mode te entry game directy as a Poisson game were π m are endogenousy determined parameters. Eac potentia bidder perceives te number of oter participants of type m to be given by a Poisson random variabe Nω m wit parameter λ ω,m = α ω,m π m tat depends on π m and te distribution of te signa θ. By a symmetric equiibrium, we mean a pair b, π were b : {1,..., M} R + is te equiibrium bidding strategy for participating bidders and π : {1,..., M} R + is te equiibrium entry rate. Te equiibrium condition for entry rate π = π 1,..., π M is tat given b, π, no potentia bidder as a iger expected payoff tan c at te auction stage and if p suppf m for some m, ten er expected payoff in te auction is c. 3.1 Symmetric Panner s Optimum We start again wit te symmetric surpus maximizing bencmark were a socia panner cooses π = π 1,..., π M to maximize te expected gain from aocating te object net of te expected entry cost. Te panner s objective is to max π 0 W π, were W π is te expected tota surpus wit a given entry profie: W π = q[v 1 1 e Σmα 1,mπ m cσm α 1,m π m ] + 1 q [v 0 1 e Σmα 0,mπ m cσm α 0,m π m ]. 14

15 Te first order conditions for interior soutions to tis concave probem are given by: q m v 1 e Σmα 1,mπ m + 1 q m v 0 e Σmα 0,mπ m = c, for a m. Its unique soution is given by: v 1 e Σmα 1,mπ m = c, 4 v 0 e Σmα 0,mπ m = c. Taking ogaritms, we ave v 1 Σ m α 1,m π m = og, c v 0 Σ m α 0,m π m = og. c Since we ave two inear equations in M variabes, te soutions to tis pair of equations are not unique. By affiiation, we ave α 1,M α 0,M α 1,m α 0,m α 1,1 α 0,1. Hence we ave a positive soution for π = π 1,..., π M ony if v 1 α 1,1 π 1 + α 1,M π M = og, c v 0 α 0,1 π 1 + α 0,M π M = og, c as a positive soution. Again by affiiation, we see tat tis is te case ony if v 1 v 0 α 0,M og < α 1,M og. c c If tis condition is satisfied, a soution π 1, 0,..., 0, π M wit π M > π 1 > 0 to tis system exists. Obviousy oter soutions to tis probem exist, but te aggregate amount of entry is determinate for bot states across a suc soutions. We can compute a tresod c suc tat positive entry by mutipe types takes pace if te entry cost is beow c: α 1,M ogv0 α 0,M ogv1 c = e α 1,M α 0,M > 0. For c 0, c we ave an interior soution wit π opt M Wen c c, we ave a corner soution were π opt m entry rate for te θ M types is soved from > πopt q M v 1 e α 1,M π M + 1 q M v 0 e α 0,M π M = c > 0. = 0 for a m < M. In tat case, te optima

16 Tis gives an interior soution π opt M > 0 as ong as c < c, were c = q v q v 0. To summarize, for ow entry cost c < c te sociay optima entry profie features te two extreme types entering wit positive rate, for intermediate entry cost c [ c, c ony igest type enters wit positive rate, and for ig entry cost c c tere is no entry. Note tat te tresod c is te expected vaue of te object for a payer tat as observed te igest signa. Tis is intuitive: as ong as entry cost c is beow te expected vaue of te object for te most optimistic potentia entrant, it is sociay optima to ave at east some entry. 3.2 No Pooing wit Interim Entry In any interim entry equiibrium, entrants earn an expected profit of c in te bidding stage. Hence for a bids p in te union of te bid supports of a bidders, te equiibrium payoff of any bidder submitting p must be c and te payoff for any oter type cannot exceed c. Oterwise we woud ave a contradiction to interior entry probabiities. Wit a random number of bidders, te anaysis of pooing bids i.e. bids tat at east one of te types cooses wit a stricty positive probabiity is more deicate tan in te case of a fixed number of bidders or wit two potentia participants. Wit uniform tie-breaking for igest bids, a bidder submitting a pooing bid is more ikey to win if te number of tying bids is sma. Te number of tying bids contains information on te reaization of N, N. Since N0 θ and N1 θ ave different distributions, tis information is in turn informative on te state of te word and ence on te vaue of te object. Te additiona information contained in te event of winning te auction must be accounted for wen cacuating te optima bid. We ca tis effect te rationing effect of winning. As observed by Pesendorfer and Swinkes 1997, it is not possibe to ave pooing bids tat are made ony by te igest types. If tey were te ony bidders to poo on a bid p, ten te rationing effect woud be negative: by bidding p, a win is more ikey wen tere are few tied bidders. But if ony ig type bidders bid p, ten winning wit a tied bid decreases te posterior on {ω = 1} and te vaue of te object conditiona on winning is ower tan te vaue conditiona on te event tat te bidder is tied for te igest bid. By bidding p + ε, for a sma enoug ε, te bidder wins in te event of a tied bid witout any rationing and makes a positive gain from te deviation. As a resut, pooing by ig types ony is not possibe. To rue out oter types of pooing bids, we note first tat tere cannot be pooing at any p < v 0. Tis is because for suc a ow bid, winning is profitabe in bot states and ence a sma upward deviation woud be stricty profitabe. On te oter and, wit interim entry ony 16

17 te igest type M can bid above v 0. Winning wit a bid p v 0 is profitabe ony if ω = 1. By affiiation, Pr{ω = 1 m} is increasing in m. Terefore, if m gets in expectation c by bidding p v 0, ten type M gets stricty more tan c by bidding p, wic is not compatibe wit interim entry equiibrium. Since it is not possibe to ave pooing bids tat are made ony by ig types, pooing is not possibe for any p v 0 eiter. We ave ence proved: Proposition 4 Tere are no atoms in te bidding distribution b of a symmetric interim entry equiibrium b, π. 3.3 Entry by Extreme Types We consider next te types of bidders tat can enter in eqiibria wit atomess bidding strategies at te auction stage. As in Section 2, denote by R ω p te expected rent at te auction stage in state ω from bid p: M R ω p = v ω p e λω,m1 Fmp. For eac p, tere are tree possibiities: eiter i R 1 p = R 0 p, ii R 1 p > R 0 p, or iii R 1 p < R 0 p. A types of bidders are indifferent beween entering and submitting a bid p in case i. In case ii, ony type θ M can make bid p in equiibrium wit interior entry probabiities. If m < M makes te bid, se must earn at east c. Since er expected payoff at te bidding stage is R 0 p + q m R 1 p R 0 p, we see tat type is not indifferent contradicting interior entry. In case iii, we ave simiary tat ony type 1 can enter in suc an equiibrium. Te equiibrium construction in Proposition 6 sows tat te bid distribution is fuy pinned down by te parameters of te bidders tat bid in cases ii and iii. Te ony remaining indeterminacy concerns te exact composition of bidders tat make bids were case i ods. But te requirement tat te expected payoffs be equaized across te two states determines te aggregate distributions of bids for eac state in a manner competey anaogous to te payoff equaization in te panner s probem across te two states. As in te panner s case, tese aggregate distributions can be generated wit positive entry by types 1 and M. We formaize tis discussion in te foowing proposition, Proposition 5 Let b, π 1, π M denote an equiibrium of a reduced mode, were ony types {1, M} exist. Ten b, π 1, 0,..., 0, π M remains an equiibrium of te fu mode tat aows entry by a types {1,..., M}. Moreover, if tere is anoter equiibrium b, π in te fu mode, tis equiibrium is equivaent to an equiibrium of te reduced mode in terms of induced probabiity distribution of bids and revenues. m=1 17

18 Based on tese two resuts, we restrict our attention for te rest of tis section to atomess bidding equiibria of a mode wit binary bidder types θ {, }, were corresponds to type 1 and corresponds to type M. 3.4 Bidding Equiibrium wit Exogenous Entry by Extreme Types We start by anayzing te bidding stage in te case were te number of entrants is determined by exogenousy given entry rates π and π. A bidder wit signa θ {, } cooses er optima bid in informa auctions depending on er updated probabiity on te state q θ := Pr {ω = 1 θ } and er conditiona distribution on te reaized number of oter bidders wit signa θ. Te number of bidders wit signa θ in state ω is a Poisson random variabe Nω, θ were te parameter λ ω,θ depends on π θ troug: λ 1, = απ, λ 0, = βπ, λ 1, = 1 α π, λ 0, = 1 β π, were α > β impies tat λ 1, > λ 0, and λ 1, < λ 0,. Wie te entry probabiities π and π are treated ere as exogenous, tey wi be endogenized in te next section. Since we consider ere te bidding beavior of an individua bidder, we use Nω θ to denote te number of bidders excuding te bidder under consideration Symmetric Equiibrium in Atomess Strategies Consider equiibrium bidding in te informa first price auction wit an exogenousy given distribution of bidders. Te bidder submitting te igest bid p 0 wins te object and pays er bid, any ties are broken symmetricay between igest bidders, and bidders tat do not win make no payments. We sow in tis section tat tis mode as a unique symmetric equiibrium in atomess strategies. Based on q θ and te entry rates λ = {λ ω,θ }, eac bidder of type θ computes er posterior beiefs on te numbers of oter participants and forms expectations about teir bids. As in te Section 2, we denote by F θ p te continuous c.d.f. of bids beow eve p by any bidder wit signa θ. Since te nubers of participating bidders are drawn from independent Poisson distributions and since te randomizations over bids are independent across bidders, te probabiity of winning in state ω at 9 Note tat in a Poisson mode, an individua bidder perceives te number of oter bidders distributed according to te same distribution as an outsider sees te number of a bidders in te game conditiona on state. 18

19 bid p is given by: Pr N = n ω F p n Pr N = n ω F p n n 0 n 0 = λ ω, n e λ ω, λ F n p n ω, n e λ ω, F! n p n! n 0 n 0 = e λ ω,1 F p e λ ω,1 F p. Using tis winning probabiity, we can compute te expected payoff from bid p to a bidder of type θ wen te oter payers bid according to te profie F = F, F as foows: U λ θ p = q θ e λ 1,1 F p e λ 1,1 F p v 1 p + 1 q θ e λ 0,1 F p e λ 0,1 F p v 0 p. A symmetric bidding equiibrium for entry rates λ is a bid profie F λ suc tat for a p suppf λ θ, p maximizes Uθ λ p. For te remainder of tis section, we fix λ and omit it in te notation for te bid distributions. Te main resut of tis section caracterizes atomess symmetric equiibrium bid distributions. Tis caracterization is remarkaby simiar to Proposition 1 in te two-bidder case. If anyting, te proof of te resut is simper tan in te two-bidder case and te resut is sarper in te sense tat it gives a necessary and sufficient condition for te two quaitativey different types of equiibria. Proposition 6 Te informa first-price auction as a unique symmetric equiibrium in atomess bidding strategies. Te support of te bid distributiobn of te ow types contains 0. If 1 e λ 0, 1 e λ 1, < v 1 v 0, ten te bid supports are non-overapping intervas wit a singe point in common. If 1 e λ 0, 1 e λ 1, > v 1 v 0, ten te bid supports intersect and 0 is contained in te support of bot types Symmetric bidding equiibria in forma auctions For competeness, we record ere te foowing caracterization of te unique symmetric equiibria in forma auctions from Ci, Murto, and Väimäki

20 Proposition 7 In te forma first-price auction wit two bidder types, tere is a unique symmetric equiibrium. Given tat tere are n 2 participants, ow type bidders poo at bid p n = E [ v θ =, N = n 1, N = 0 ] for n 2 and ig types ave an atomess bidding support [ p n, p n ] wit some p n > p n. Wit n = 1, te ony participant bids zero. In te forma second-price auction, tere is a unique symmetric equiibrium. In tis equiibrium, ow types poo at bid p n = E [ v θ =, N = n 1, N = 0 ] and ig types ave an atomess bidding support [p n, p n], were p n = E [ v θ =, N = n 2, N = 1 ], p n = E [ v θ =, N n 2, N 1 ]. Wit n = 1, te ony participant bids zero. It is tus easy to compute te expected payoff at te entry stage if te bidding stage is in a forma auction. In te next two sections, we compare te overa revenues in te games were we account for bot te bidding stage and te costy entry stage. 4 Endogenous Entry In tis section, we endogenize te entry decisions. By Proposition 5, we can pin down te expected equiibrium rents from a possibe bids by considering modes wit binary types θ {, }.We foow te same strategy as in Section 2 and anayze te properties of te interim entry equiibria troug a comparison wit te panner s soution. 4.1 Private and Socia Incentives for Entry We consider first a ypotetica game were te object is given for free to te entrant if tere is a singe entrant. Wit two or more entrants, te object is witdrawn from te market and te entrants just pay te entry cost. Denote by V 0 θ π, π te expected vaue of a bidder wit signa θ at te auction stage wen te symmetric entry profie is π, π : V 0 π, π = q e απ 1 απ v q e βπ 1 βπ v 0, V 0 π, π = q e απ 1 απ v q e βπ 1 βπ v 0. 20

21 Any π, π suc tat V 0 π, π = V 0 π, π = c in te above equations guarantees tat every potentia entrant is indifferent between entering and staying out. By inspection, we see tat tis is te same condition as in 4, and ence private entry incentives coincide wit socia incentives. To understand wy tis is te case, note tat in tis ypotetica situation eac entrant gets exacty er margina contribution to te socia wefare as er payoff. Since te vaue of te object is common to a te payers, an entrant contributes to te tota surpus if and ony if se is te ony entrant. Since te object is given for free in suc a situation er private benefit equas er margina contribution. Terefore, entry incentives coincide exacty wit te socia vaue of entry. As we sow beow, a te different auction mecanisms give at east weaky too ig entry incentives for te ig types. As in te two-bidder case, quantifying tis excess incentive across different auction formats gives us our revenue comparisons. In Figure 1, we iustrate te socia optimum in te case wit interior entry probabiities by drawing te panner s reaction curves π π : = arg max, π, π 0 π π : = arg max, π, π 0 in te π, π pane. Te socia optimum is at te intersection of tese curves. As sown above, tese curves are aso te indifference contours V 0 π, π = c and V 0 π, π = c for a potentia entrant of type and, respectivey, wo gets er margina contribution as expected payoff. We wi see tat many of te auctions considered ere give an additiona reward to potentia entrants on top of teir socia contribution. Tis distorts te entry rates from sociay optima eves. Since bidding zero is in te support of te ow type bidders in te informa first-price auction, teir equiibrium entry incentives coincide wit te panner s incentives conditiona on te entry rate of te ig types. As a resut, teir equiibrium reaction curve coincides wit te panner s reaction curve. For our revenue comparisons it is usefu to anayze ow distortions to te ig types entry rate cange te tota surpus. We denote by W π te tota surpus as a function of te ig types entry rate, wen ow types adjust entry optimay: W π := max π 0 W π, π. Te foowing Lemma sows tat W π is singe peaked in π wit its peak at π opt. Tis is iustrated in Figure 1 by arrows aong π π tat point towards increasing socia surpus. Lemma 8 Let π opt > 0 denote te sociay optima ig type entry rate. We ave dw π dπ > 0 for π < π opt = 0 for π = π opt < 0 for π > π opt 21. 5

22 4.2 Interim Entry Equiibrium Consider first te case were c c and te panner s soution as π opt = 0. Suppose tat tis is te case aso in equiibrium. Since a te entering bidders ave observed a ig signa, it is easy to see tat in a te auction formats tat we consider, te expected payoff to te entering bidders is given by te probabiity tat no oter bidder entered times te expected vaue of te object conditiona on tat event. Since te updated beief of a ig type bidder on {ω = 1} is given by q = equating te expected benefit from entry to te cost of entry gives: qα qα+1 qβ, q e απ v q e βπ v 0 = c. Tis coincides wit te panner s optimaity condition. Hence te symmetric equiibrium entry profie in a of te auction formats tat we consider coincides wit te panner s optima soution. Te key reason for tis is te ack of eterogeneity in te bidders information. In equiibrium te bidders are indifferent between entering and not entering. Hence teir expected payoff must be at teir outside option of 0. Since te auctions generate maxima socia surpus under te restriction to symmetric entry profies in tis case and since bidders expected payoff is at zero, te seer coects te entire expected socia surpus in expected revenues. Hence a tese auction formats are aso revenue maximizing witin te cass of symmetric mecanisms were we require symmetry at te entry stage as we as at te bidding stage. Proposition 9 If c c < c, ten ony te ig type enters and a te auction formats are efficient and ence revenue equivaent. If c c, ten tere is no entry in te panner s soution nor in any auction format. We move next to te more interesting case were te panner s soution induces entry by bot types. In tis case we see immediatey tat te second-price auction eads to suboptima entry decisions. Tis concusion foows from a very simpe argument sowing tat a bidder wit a ig signa earns a iger private benefit in te auction stage tan teir socia contribution. In a mode wit common vaues, additiona entry is sociay vauabe ony if no oter bidder participates in te auction. In a second price auction, te bidder wit a ig type gets te socia benefit, but se aso receives an extra information rent wen bidding against bidders wit ow signas. Tis is an immediate consequence of te usua ogic in modes wit affiiated vaues. As a resut, entry modes wit a second price auction in te bidding stage feature excessive entry by te ig types reative to te panner s soution. Our main resut is tat te game wit interim entry decisions foowed by an informa first-price auction for te object as a unique symmetric equiibrium. Tis narrows down te set of possibe 22