Common Value Auctions with Costly Entry

Transcription

1 Common Value Autions with Costly Entry Pauli Murto Juuso Välimäki June, 205 preliminary and inomplete Abstrat We onsider a model where potential bidders onsider paying an entry ost to partiipate in an aution. The value of the objet sold depends on an unknown state of the world, and the bidders have onditionally i.i.d. signals on the state. We onsider mostly the ase where entry deisions are taken after observing the signal. We ompare rst- and seond-prie aution formats and show that for many symmetri equilibria of the game, rst-prie aution results in higher expeted revenue to the seller. Introdution Bidding in an aution is often ostly. At the very least, eah bidder loses the opportunity ost of time spent in preparing the bid and paying attention to the eventual outome. When the objet for sale is valuable and information is dispersed among potential bidders, these osts an be substantial. We onsider a setting where a large number of potential bidders have observed a signal on its true value. The value of the objet is ommon to all bidders, but the bidders are di erentially informed about the true value. For the most part in this paper, we onsider the ase where the true value of the objet depends on a binary random variable! 2 f0; g (state of the world), and we also assume that the signals take binary values i 2 f H ; L g.

2 At the beginning of the game, eah potential bidder deides whether to enter the aution at a positive ost > 0: We onsider the ase where a single objet is for sale (and disuss extensions to the ase of a xed number of objets). Furthermore, we assume anonymity on the part of the bidders so that entry involves a oordination problem. Entry an be pro table only when a limited number of other bidders enter. We analyze the symmetri equilibria of the entry game under rst-prie and seond-prie aution rules. Not surprisingly, equilibrium entry deisions are in mixed strategies for both types of autions. Conditional on entry, optimal bidding strategies are qualitatively quite di erent. In the rst-prie aution, equilibrium bids are mixed strategies both for bidders with signal H and those with signal L. In the seond-prie aution, bidding onditional on the more pessimisti signal L is in pure strategies, but bidding onditional on H is in mixed strategies when there are many potential bidders. Only in the ase with just two potential bidders, the bids of both types of entrants are in pure strategies in the seond prie aution. Our main result is that in ontrast to most ndings in ommon value autions, the rst-prie aution often dominates the seond-prie aution in terms of the expeted revenue to the seller. To understand this result, it is useful to onsider the entry deisions of a soial planner under the onstraint of symmetri strategies (i.e. onditional on the signal, bidders use idential entry strategies). Sine the model is one of ommon values, alloative e ieny is not an issue, and as a result, the planner maximizes the probability of alloating the objet weighted by its value in the two states subjet to paying the entry osts. The planner gains from adding a bidder only when no other bidders are present. In a seond-prie aution, bidders with high signals make a positive pro t in the aution if there are no other bidders of high type. If she is the only bidder in the aution, the high type bidder reeives her entire marginal ontribution to the soial welfare. If bidders with low signals are present, she pays a positive prie but still makes a positive expeted pro t due to a more optimisti signal. Hene the inentives to enter are stronger than soially optimal for the bidders with high signals. In a rst-prie aution, eah bidder pays her own bid. In any symmetri 2

3 equilibrium, an entering bidder must believe that there is a positive probability that no other bidders enter. This leads to mixed strategy equilibria in the bidding stage, and hene also to potential ompliations in evaluating the expeted payo s of the two types of bidders. While it is lear that the bidders with low signals must have zero bids in the support of their bid distribution, it is more surprising that for some parameter values this is also the ase for the bidders with high signals. Sine there are no mass points in the bid distributions (by standard arguments), this implies that both types of bidders are willing to plae bids that win at zero prie if and only if no other bidder has entered. Hene the expeted payo to both types of bidders oinides with their marginal ontribution to soial welfare, and as a result equilibrium entry is at soially optimal level. Sine entry deisions are in mixed strategies, entering bidders must make a zero expeted payo. This implies that the seller gets the entire expeted soial surplus in revenue. Sine the entry deisions in the rst-prie aution are expeted surplus maximizing, the seller must gain relative to the seondprie aution. It is also possible that zero bids are not in the support of the bidding strategies for bidders with high signals. For the ase of two potential bidders, we show that the rst-prie aution still dominates the seond prie aution in terms of expeted revenue. For a large number of potential bidders, we show that for small entry osts this ranking an be reversed. For high entry osts, the rst-prie aution yields a higher revenue in this ase as well. Two additional features of the symmetri seond-prie equilibria with many potential bidders deserve speial mention. The equilibrium bid onditional on a low signal is not uniquely determined and the bids onditional on high signal are mixed. In order to understand these results, it is useful to reall winners urse for ommon value autions. As usual, the equilibrium bid in a seond prie aution is given by the expeted value of the objet onditional on having the highest bid, onditional on tying the seond highest bid and onditional on winning the objet. Suppose that all the high type bidders submit the same bid. With uniform rationing, the probability of winning is the highest when the number of tying bids is the smallest. Sine the signals are a liated with the true value of the objet, this is bad news on 3

4 the expeted value. By deviating to a slightly higher prie, a bidder with a high signal wins in all ases inluding those in whih there are large numbers of other bidders. Hene a pure strategy equilibrium annot exist. Consider next the inentives of the low type bidders. In a seond prie aution, they an win only in the ases where no bidders with high signal have entered. If all low type bidders pool at the same bid, winning is more likely when there are fewer other bidders with low signals. Again by a liated values, this is positive news on the value of the objet. By deviating to a slightly higher prie, a low type bidder wins the objet in all ases with no bidders with high signals present. In a sense, low bidders experiene a winner s blessing at the pooled bid and as a onsequene, a ontinuum of pooled equilibrium pries exists. In the last setion of this paper, we outline some generalizations of our ndings. First of all, the revenue ranking favoring the rst-prie aution remains true for some parameter values of the model even if the entry deision is taken at the ex ante stage, i.e. before knowing own signal realization. At this moment, we are not sure how often this happens (relative to the interim ase that we mostly study). Randomized entry deisions give a strong inentive for plaing a zero bid in the rst prie aution if it is likely that no ompetition is present. If bidders with both types of signals plae suh a bid, then the timing of information revelation is immaterial. We also disuss the extension to settings with multiple signals and also the ase where many idential objets are for sale. We should stress that we are not laiming a general result showing the uniform superiority of rst-prie aution when partiipation in the aution is random. As the entry ost! 0; the usual reasons based on linkage prinipal beome relatively more important, and seond-prie autions are likely to dominate. Our goal is to demonstrate that in a non-trivial set of models, limited and random entry may reverse the usual revenue omparisons.. Related Literature Autions with endogenous entry have been modeled in two separate frameworks. In the rst, entry deisions are taken at an ex ante stage where all 4

5 bidders are idential. Potential bidders learn their private information only upon paying the entry ost. Hene these models an be though of as games with endogenous information aquisition. Frenh & MCormik (984) gives the rst analysis of an aution with an entry fee in the IPV ase. Harstad (990) and Levin & Smith (994) analyze the a liated values ase. These papers show that due to business stealing, entry is exessive relative to soial optimum. They also show that seond-prie autions results in higher expeted revenues than the rst-prie aution. In the other strand, bidders deide on entry only after knowing their own signals. Samuleson (985) and Stegeman (996) are early papers in the IPV setting where this question has been taken up. Due to revenue equivalene in the IPV ase, omparisons aross aution formats are not very interesting. To the best of our knowledge, ommon values autions have not been analyzed in this setting. Hene our paper is the rst to ask how the aution format a ets information aggregation through entry. Finally some reent papers have analyzed ommon values autions with some similarities to our paper. Lauermann & Wolinsky (203) analyze rstprie autions where an informed seller hooses the number of bidders to invite to an aution. In their setting, it is also important to aount for the winner e et when omputing the expeted value of the objet. Atakan & Ekmeki (204) onsider a ommon value aution where the winner in the aution has to take an additional ation after winning the aution. This leads to a non-monotoniity in the value of winning the aution that has some resemblane to the fores in our model that lead to non-monotoni entry (i.e. bidders with both types of signals enter with positive probability). 2 Binary model We start by laying out the basi model. The state of the world is a binary random variable! 2 f0; g with a prior probability q = Prf! = g: The ommon value of the objet in state! is v (!) for all the bidders, and we assume that v () > v (0) > 0: 5

6 At the outset, eah potential bidder i observes a binary signal i 2 f h ; l g: Let Pr = l j! = 0 = Pr = h j! = := > =2 and denote by q h := Pr! = = h and q l := Pr! = = l the posterior based on a high and low signal, respetively. For example, if prior is q = =2, then q l = and q h =. The signals are assumed to be i.i.d. onditional on the state of the world. After observing i ; eah player i deides whether to pay an entry ost > 0 with < v (0) and submit a bid b i in an aution for a single objet or whether to stay out and reeive a ertain payo of 0. At the moment of bidding, i does not know how many other bidders have hosen to partiipate in the aution. Furthermore, we distinguish between two alternative aution formats: the rst-prie aution (FPA) and the seond-prie aution (SPA). The entry strategy of potential bidder i spei es the probability of entry: i : f h ; l g! [0; ]: We use h i for i h and l i for i l. Sine we onentrate on symmetri equilibria, we often omit subsripts. Similarly a bid strategy is a funtion b i : f h ; l g! (R + ) ; where we have allowed for randomized bids. The bidders are risk neutral and bid to maximize their expeted pro t from the aution. 3 A two-player version We start the analysis with the simple ase where there are only two potential bidders. There are now just two players, and we look for a symmetri mixed strategy equilibrium. We are interested in seeing how the equilibrium entry deisions depend on the format of the aution hosen. We onsider the symmetri ase with Prf = h j! = g = Prf = l j! = 0g for notational simpliity. The results in this paper go through with asymmetri spei ations too. 6

7 We start by haraterizing the e ient solution, i.e. the symmetri entry probabilities that a utilitarian planner would hoose. The planner s problem is to max V h ; l := h ; h l q h ( ) l 2 v () 2 h + ( ) l i h + ( q) ( ) h l 2 v (0) 2 l + ( ) h i : From the rst order onditions to this onave maximization problem, we get: b h = v () ( ) v (0) ; 2 v () v (0) b l = v (0) ( ) v () : 2 v (0) v () This is a valid solution if b l 0 and b h : These restritions are satis ed if v (0) ( ) v () v (0) v () and v () v (0) : If the rst inequality is violated, then only high signal players enter in the e ient solution. We shall see that in this ase ompetitive entry followed by the SPA and the FPA also result in e ient entry levels and idential expeted revenues to the seller. In the ase of an interior solution, we shall see that FPA dominates SPA in terms of expeted revenue. Notie that the seond inequality restrition is needed only beause here the number of potential entrants is limited to two. In the model with a large number of potential bidders a orner solution b h = would be very ostly to the planner. Lemma Fix h 2 (0; ) in the planner s problem and let b l h be the onditionally optimal entry rate by the low type bidders. Then b l h is dereasing in h and V h; ; b l h is stritly onave in h : 7

8 Proof. By straightforward omputation. 3. Entry game with seond-prie aution With only two bidders, the realized prie is the lower of the two bids if both players enter, and zero if only one player enters. Equilibrium inferene about the value of the good is straightforward. As usual in ommon value autions, the optimal bid is obtained by assuming that both bidders have submitted the same bid. It is lear that this an our only if the bidders have observed the same signals. Hene, if both types enter with positive probability in equilibrium, we an write p h = E! [v (!) = 2 = h ]; p l = E! [v (!) = 2 = l ]: The bidding strategies of both types are thus pure. If only high type bidders enter in equilibrium, then their bids remain as above and low type bidders bid any amount below b h in any sequentially rational ontinuation following a deviation sine they want to win only onditional on having entered alone. Sine the expeted pro t to the deviating low type bidder is exatly her ontribution to the expeted soial surplus, she will hoose to stay out of the market if high types enter e iently whenever the planner s solution is not an interior solution. Notie that high signal bidders also ollet exatly their expeted marginal ontribution at the e ient pro le. We onlude that e ient entry is an equilibrium in the entry game followed by seond-prie aution as long as b l = 0 in the e ient solution. Suppose next that b l > 0: In this ase, the soial planner gains from a high type entry if and only if there are no other bidders present in the market. In the SPA, high type bidders gain their soial ontribution if there are no other bidders present, but in addition, they also make a positive pro t whenever another bidder with low signal has entered. Hene, with b l > 0; the expeted private pro t of a high signal type exeeds the soial bene t due to a business stealing e et, and equilibrium level of h exeeds the onditional soial optimum (i.e. optimum given l ). Sine the low type bidders make a positive pro t only when there are no other bidders present, 8

9 their private pro t oinides with their marginal soial ontribution. Sine the onditionally e ient level of l h is dereasing in h ; we onlude that: Proposition Suppose b l > 0: Then the entry equilibrium h S ; S l followed by a seond-prie aution is haraterized by: i) h S > bh and ii) l S < bl : 3.2 Entry game with rst-prie aution Consider next the ase of a rst-prie aution. Using standard arguments, we an show that symmetri equilibria in this ase must be in atomless mixed strategies. Denote by s > 0 the equilibrium entry probability and by b s (p) the equilibrium bid distribution for s 2 fh; lg. We an then let s (p) = s b (p) denote the probability that i has entered the aution and plaed a bid above level p, onditional on having observed signal s. Let supp s () denote the support of the bid distribution b s. A rst simple observation is that if both types of potential bidders enter with a positive probability and a bid of 0 is in the support of their bid distributions, then entry must be at e ient level. Sine there are no atoms, a bid of zero wins only if there are no other ative bidders. If this bid is in the support of both types, we onlude that potential bidders of both types earn an expeted pro t exatly equal to their marginal ontribution and hene entry must be at the onstrained e ient level. We onsider next the ase where 0 is not neessarily in the support of the bid distributions. Suppose that both types of bidders enter with positive probability and that a bid p is in the support of the bid strategies of both types of potential bidders. Sine both must be indi erent between entering and not, and sine the two types of bidders have di erent assessments of the relative probabilities of the two states, we must have indi erene between entering with bid p and staying out aross the two states. In other words, the following must hold: h (p) ( ) l (p) (v () p) = l (p) ( ) h (p) (v (0) p) = ; 9

10 whih leads to h (p) = l (p) = v () p 2 v () p v (0) p 2 v (0) p ( ) v (0) p v (0) p ( ) v () p v () p : ; () This observation leads to the following lemma that is key to all our revenue omparisons. Lemma 2 Suppose that 0 < l < h < 0: Then either i) 0 2 supp s () for s 2 fh; lg or ii) supp l () = 0; p l and supp h () = p l ; p h for some 0 < p l < p h < v () : Proof. First of all, it is obvious that 0 2 supp l () [ supp h () sine otherwise the bidder submitting the lowest bid would have a pro table deviation. We prove the laim by showing that whenever 0 =2 supp h () ; the supports are non-overlapping intervals with 0 2 supp l () : To show this, we notie that if the supports overlap on a non-empty interval, then equations () hold over this interval, and we an di erentiate the equations to get: d h (p) ( ) = 2 + dp 2 (v () p) (v (0) p) 2 ; d l (p) ( ) = 2 + dp 2 (v (0) p) (v () p) 2 : Di erentiating for a seond time yields: d 2 h (p) 2 ( ) = dp (v () p) (v (0) p) 3 ; (2) d 2 l (p) dp 2 = 2 2 We see here that d l (p) < 0 for all p, dp and that the following impliation holds: d h (p) dp ( ) 3 + (v (0) p) (v () p) 3 < 0: 0 =) d2 l (p) dp 2 > 0: 0

11 The rst property implies immediately that if bid p is in the support of l () ; then so is p 0 < p demonstrating 0 2 supp l () : Sine the onlusion of the seond impliation is false by the seond line in equation (2), we know that its premise must be false as well and hene d h (p) dp < 0: By the same logi as above, 0 2 supp h () : Hene we onlude that either 0 2 supp h () or the supports do not overlap. Gaps in supp l () [ supp h () and mass points in either distribution are ruled out by usual arguments. Notie also that by the seond line in equation () for p 0 < p fp 0 ; pg fsupp l () \ supp h ()g ) l (p 0 ) < l (p) : This rules out the possibility that supp l () is a union of disjoint intervals. Finally, we need to show that in the ase of non-overlapping interval supports 0 2supp l () : Let p max be the maximal bid in the union of the two supports. Sine we have assumed that 0 < l < h < 0; we have E v (!) p max j h > E v (!) p max j l : Hene p max 2supp h (p) ;and hene also 0 2supp l () : Lemma 3 If the equilibrium entry rates in the rst-prie equilibrium h ; l 6= b h ; b l, then h > b h and l < b l : Proof. Against any xed rate l of entry for the low types, the equilibrium rate of entry is at least b h l. For a lower rate, entry followed by p = 0 would yield a pro table deviation. The onlusion of the lemma follows immediately. With the help of these lemmas, we are ready to prove the main result of this setion. Theorem In the two-bidder model, the rst-prie aution yields always a higher expeted revenue than the seond prie aution.

12 Proof. i) If 0 2 supp h () \supp l () ; then entry rates b h ; b l 2 are onsistent with the First-Prie Aution. Sine p h ; p l = (0; 0) is in the support of the bidders, their expeted payo oinides with the expeted payo of the planner. Sine the bidders expeted payo s in this equilibrium are zero, the revenue in the aution oinides with the maximal expeted soial surplus. Proposition shows that in the seond-prie aution, entry rates di er from the planner s solution. Sine the bidders expeted payo s are non-negative in any equilibrium, the laim follows from the strit onavity of the planner s optimization problem. ii) If 0 =2 supp h () in the equilibrium of the game with rst-prie aution. Then, by the previous lemma, supp l () = 0; p l and supp h () = p l ; p h in the rst-prie aution. Let h ; l denote the equilibrium entry rates for the game with rstprie aution bidding. Keep entry rates xed at h ; l, but swith the aution format to seond-prie aution. Bidder with l is still indi erent between entering and not beause in rst-prie aution by bidding 0 she gets the same alloation and prie as in the seond prie aution. On the other hand, p l is also in supp l (p). At that prie, she gets the good with probability if and only if there is no opponent of type h : Denote this event by A: Thus, onsider the alloation rule where a bidder gets the good if and only if event A happens. Compare two di erent priing rules for event A as follows: i) pay p l for sure, and ii) pay p = 0 if no low type presents, and otherwise pay p = E vj = l ; a low type present. A bidder of type l is indi erent between these two situations sine i) is her outome in rst prie aution for bid b = 0, whih is within her bidding support, and ii) is her outome in seond prie aution (if she outbids equilibrium bid by an in nitesimal "). But then a bidder type h prefers ase ii) to i), beause she nds it more likely that p = 0 than bidder type l. Sine p l 2supp h () in the rst-prie aution, her expeted pro t in that aution must be : Hene, the high type makes a stritly higher pro t in the seond prie aution. Hene, there must be more entry by bidders of type h in the seond-prie aution if h < ; and if h = ; then the rent to h is higher in the seond-prie aution. By Lemma and Lemma 3, we onlude that the seond-prie aution yields a 2

13 lower soial surplus and hene also lower expeted revenue to the seller. 4 Many potential bidders We onsider the limiting model where the number of potential bidders N! : By usual arguments, given a sequene of symmetri entry strategies for nite games f s N g, s = l; h, suh that N s N! s, the realized number of entering agents onverges to a Poisson random variable with parameter h + ( ) l in state! = and with parameter ( ) h + l in state! = 0. Hene, we will be looking for equilibrium "entry intensities" l and h. As in the previous setion, we start by onsidering the soial planner s problem: She hooses entry intensities l and h to maximize soial surplus net of entry ost. The objetive funtion of the soial planner is given by: W l ; h h = q e h ( ) l v () h + ( ) l i h + ( q) e ( )h l v (0) ( ) h + l i : The rst term in square brakets omputes the bene t and ost of entry if the state is high and the seond orresponds to the low state. To simplify the formulas slightly, write () = h + ( ) l ; (0) = ( ) h + l ; for the Poisson parameter onditional on the state of the world. This is a onave problem with rst-order onditions for interior solutions given by: q e () v () + ( q) ( ) e (0) v (0) = 0; q ( ) e () v () + ( q) e (0) v (0) = 0: This is satis ed when e () v () = ; e (0) v (0) = ; 3

14 or b h = b l = v () v (0) log ( ) log ; 2 v () v (0) ( ) log + log : 2 For this to yield a valid solution, we must have b l > 0, so our assumption in terms of model parameters is: v (0) v () log > ( ) log or > log (v ()) log () log (v (0)) log (). Notie that as! 0, the right hand side onverges to. Therefore, we always have l > 0 for low enough. On the other hand, by inreasing towards v (0), at some point b l redues to zero and we get a orner solution where only high types enter. Notie also that we have immediately the result that b h > b l sine b h b l v () = 2 log. v (0) 4. Seond-prie aution with unobserved entry With more than two potential bidders, the e et of onditioning upon winning the aution is more ompliated than in the standard ase of a xed number of bidders with a ontinuum of signal. In the standard ase, the onditioning event is that the winner s bid ties with the highest bid amongst other bidders. Under inreasing strategies this again translates into having a tie between the two highest signals. In our ase, winning the aution gives also information about the number of other entrants to the aution. In aordane with our interest in symmetri equilibria, we assume symmetri rationing in ase of tied bids. This implies that a bidder is more likely to win the objet if there are fewer bidders. This 4

15 in turn gives information on the value of the objet for sale if entry takes plae at di erent rates for di erent signals. It is easy to show that in any symmetri equilibrium of the SPA, bidders with low signals must bid below the bidders with high signals. If high signal bidders bid aording to a pure strategy, the probability of winning is ; where n h is the (random) number of entrants with high signals. Hene n h winning is evidene of low n h and by a liated signals this is also evidene in favor of f! = 0g: By the usual logi of SPA, the equilibrium bid must equal the onditional expeted value of the objet upon winning. By a small upward deviation, any bidder with a high signal wins the objet for sure. Under this onditioning event, the value of the objet is stritly larger than when submitting the assumed ommon equilibrium bid. Hene bidders with high signals annot use the same pure bid in equilibrium, and we must onsider a mixed strategy equilibrium for those bidders. Bidders with low signals fae a di erent updating. Again, with symmetri rationing winning the objet gives evidene of a small number of bidders in the aution. In ontrast to the bidders with high signals, this is now good news about the value of the objet. This translates into a multipliity of symmetri pure equilibrium bids for bidders with low signals. By deviating to a higher prie (still below the lowest bid of the high signal biders), any low signal bidder wins regardless of the number of other low signal bidders (as long as there are no high type bidders). But the expeted value of the objet is smaller under the new onditioning event. This winner s blessing e et makes it possible to sustain di erent pure symmetri equilibrium bids for the bidders with low signals. It is lear that equilibrium entry with a seond-prie aution is always distorted if bidders with both signals enter with positive intensity. To see this, assume e ient entry. Then by the same arguments as in the previous setion, at least the high type gets more than her ontribution to soial welfare. It is also possible that low type gets more (in a partiular equilibrium). So it is possible that too many bidders of both types enter in equilibrium (similarly to the traditional business stealing e et). Let us derive next the equilibrium, where for given entry rates, the low signal type bids the highest bid onsistent with equilibrium. This is the 5

16 equilibrium where a low type entrant gains exatly her ontribution to soial welfare. The bid of a player with = h when equilibrium entry rates are h and is denoted b l l ; h. The entry rate of l onditional on state! is given by: ( ) low l for! = (!) = l for! = 0 Denote the event that no bidder with h enter by A: The probability of getting the objet (for bidder of type l ), onditional on state! and event A is (for simpliity denote = low (!)): Pr (win j! and A) = e + 2 e ! e + 4 = = e. 3 3! e + ::: e + e + e + 2 2! e + 3 3! e + 4 4! e + For the right onditioning event for the equilibrium bid, let B denote the event that at least one (other) bidder of type l entered. We ompute the probability of winning the aution in event B onditional on the state! and event A as follows: Pr (win and B j! and A) = e e = e e. Then we an ompute the likelihood ratio on the states for a low type that gets the objet at prie b l. This deomposes information into ) the prior belief, 2) own signal, 3) the event that no high types enter, 4) other low types enter and the player in question wins q l q l = q q e h e ( )h q e h e = q e ( )h : = L: e ( )l ( ) l e ( )l ( ) l ( )l l e l l e l ( ) l ( e )l e l l e l 6

17 Similarly, we an ompute the belief ratio for high type who gets the objet at prie b l : q h q = q h q e h ( )l e > L. e ( )h e l We ompute the bid of l in the equilibrium where she is indi erent between winning the aution and losing it at b l : This is obviously the highest possible pure strategy equilibrium bid by l and we all it the highest equilibrium. b l l ; h = L + L v () + v (0). (3) + L Given b l l ; h, we an ompute the expeted payo s V l and V h for l and h respetively. Sine we onentrate on the highest equilibrium, the the low type gets positive payo only if no other bidders present, so V l is easy to ompute: V l = q l e h ( ) l v () + q l e ( )h l v (0) : The expeted payo of h is: V h = q h e h v () + q h e ( )h v (0) e ( )l b l e l b l ; where b l depends on l and h through equation (3). In equilibrium, l and h must be suh that V l = V h =. We do not have losed form solutions for equilibrium l and h, but it an be shown that h > h and l < l, where the starred rates denote the soially e ient entry rates. Hene we onlude that similar to the two-bidder ase, the entry rate of h is distorted upwards and the entry of l is distorted downwards. 4.2 First prie aution with unobserved entry We onstrut the symmetri equilibrium by onsidering di erent prie regions. We us use notation h (p) and l (p) to denote Poisson entry intensity of high and low type, who bid above p. 7

18 4.2. Overlapping prie intervals If p is within the equilibrium prie support for both players, then it must be the ase that both types are indi erent between state realization. Therefore, posteriors q h and q l play no role, and we have: e h (p) ( ) l (p) (v () p) = e ( )h (p) l (p) (v (0) p) = or v () h (p) + ( ) l (p) = log v (0) ( ) h (p) + l (p) = log p p and we get h (p) = l (p) = v () p log 2 v () ( ) log 2 v (0) ( ) log p p ; (4) v (0) p + log : (5) If the supports of h () and l () overlap on an interval of positive length, then within this interval, the above equalities hold for all p, and we an di erentiate the entry intensities to get our rst results: h p (p) = ; whenever l p (p) = 2 v (0) 2 v () p p v () v (0) v (0) v () log > ( ) log : By de nition (in the interior of the relevant supports), these derivatives must be negative. 8 p p,

19 In any equilibrium, we have always h (p) > 0, and l p (p) < 0. To have a simple equilibrium, where p 2 supp h \ supp l ) [0; p] supp h \ supp l ; (6) we should have h p (p) < 0 for p < p 0, where p 0 = p : l (p) = 0. This is the ase for high enough. For low, equilibrium must have a di erent struture. We ompute the seond derivatives of the entry intensities as follows: h pp (p) = l pp (p) = 2 2 (v (0) p) 2 (v () p) 2 (v () p) 2 (v (0) p) 2 We see here that the following impliation holds: h p (p) 0 =) h pp (p) > 0; whih implies that the only possible equilibrium that does not satisfy ondition (6) is the following: supp l () = 0; p l and supp l () = p l ; p h for some p h > p l. Hene the entry distributions in the rst-prie aution with many bidders look qualitatively similar to the two-bidder ase. We summarize our ndings in the following lemma. Lemma 4 In the entry game with rst-prie aution, we have either i) 0 2 supp s () for s 2 fh; lg or ii) supp l () = 0; p l and supp h () = p l ; p h for some 0 < p l < p h < v () : Highest possible prie The highest possible prie, p, an be easily solved by onsidering the highest bidding high type bidder. He is ertain to get the good, so upon getting it his belief is unhanged. On the other hand, free entry means that his value-prie margin must be. So, we have q h (v () p) + q h (v (0) p) = ;. 9

20 or where v := v () v (0). p = q h v () + q h v (0) = q h v + v (0) ; Prie range above low type Take p 2supp h () \ supp l () C, where h enter but l do not enter. Then by Lemma 4, we have l (p) = 0, and h (p) > 0. Any bidder of type l that submits bid p makes an expeted loss: q l e h (p) (v () p) + q l e ( )h (p) (v (0) p) < and a bidder of type h breaks even in expetation: q h e h (p) (v () p) + q h e ( )h (p) (v (0) p) = : For this to hold, we must have e h (p) (v () p) > > e ( )h (p) (v (0) p). It is lear that for p < p, the indi erene equation for high type is solved by some dereasing funtion h (p). If h () is onstruted to maintain indi erene for the bidders with h ; then at some p, low type wants to enter. This an always be guaranteed to happen if the optimal entry rate b l in the planners problem is stritly positive. There is some p 0 < p suh that q l e h (p) (v () p) + q l e ( )h (p) (v (0) p) > (<) for p < (>) p 0. We nd p 0 by requiring q l e h (p) (v () p) + q l e ( )h (p) (v (0) p) = q h e h (p) (v () p) + q h e ( )h (p) (v (0) p) and this is of ourse satis ed when e h (p) (v () p) = e ( )h (p) (v (0) p) = : 20

21 Clearly there is a unique p 0 that solves this, and p 0 < p. So, the range of pries where only high type enters, is [p 0 ; p]. If is high enough, we have a simple equilibrium where l submit bids within [0; p 0 ] and h submit bids within [0; p]. Notie that sine h (0) and l (0) given in (4) and (5) are equal to the e ient entry rates, this equilibrium is e ient, and hene maximizes expeted revenue to the seller aross all possible mehanisms Prie range where only low type is ative Suppose that only l submits bids in some [p ; p + ] for some 0 p < p +. Let h := h (p + ) denote the onstant entry intensity above p 2 [p ; p + ] for h. Sine type h does not want to enter and low type is indi erent, so q h e h ( ) l (p) (v () p) + q h e ( )h l (p) (v (0) p) < ; q l e h ( ) l (p) (v () p) + q l e ( )h l (p) (v (0) p) =. For this to hold, we must have e ( )h l (p) (v (0) p) > > e h ( ) l (p) (v () p). We an show that for some parameters, there is an equilibrium where l bids within [0; p 00 ] and h within [0; p 0 ] [ [p 00 ; p], where p 0 < p 00. Notie that this equilibrium is also e ient. For other parameters, we an have an equilibrium, where l bids within 0; p l and h within p l ; p. Sine the prie support of high type does not extend to zero, this equilibrium is not e ient. 4.3 Revenue omparisons We have shown that for some ases where both types of bidders enter with positive intensity, zero bids are in the support of the bid distribution for both bidder types. Hene the rst-prie aution results in soially optimal entry in this ase. Sine the bidders expeted surplus is zero by onstrution, expeted revenue must equal expeted soial surplus. This implies that FPA gives the highest possible revenue to the seller subjet to individual rationality by the bidders. 2

22 We also showed that all equilibria of the SPA involve distorted entry pro les. Sine the bidders still make a zero expeted pro t, this implies that the expeted revenue in any symmetri equilibrium of the SPA falls below the expeted revenue in FPA. For small > 0; we have found a numerial example where a symmetri equilibrium of the SPA dominates the FPA in terms of expeted revenue. Hene the lean revenue ranking of the two-bidder game no longer holds. We an show, however, that FPA dominates for high enough entry osts. 5 Further remarks 5. Entry ost at ex ante stage In this subsetion, we onsider equilibria in a version of the model where entry deision is made before learning the signals. This orresponds to the ase where inspeting the good for sale is ostly. We maintain the assumption that the number of atual bidders that have paid the ost is not dislosed prior to the bidding stage. This is the only signi ant modeling di erene in omparison to Levin & Smith (994). The example demonstrate that the randomness in the number of partiipants allows the bidders in a seond-prie aution to get higher pro t than in the rst prie aution. We assume here that there are only two potential entrants. Consider the following speial ase of the binary model: and In this ase, the posteriors are h = Pr i = h j! = ; l = Pr i = l j! = 0 = : q h = ; q l = q h q ( h ) : The haraterization of soially optimal entry using symmetri strategies is even easier than before. Simply ompute the symmetri entry probability 22

23 to get W () : = max q ( ) 2 v () 2 + ( q) ( ) 2 v (0) 2 : Hene the optimal entry rate is given by: = qv () + ( q) v (0) qv () + ( q) v (0) := v : v Consider a bidding equilibrium in a rst prie aution that is held under the assumption that entry has taken plae at rate : Sine entry is independent of the true state, both types of bidders believe to be bidding alone with probability ( ). With probability ; there is a ompeting bidder. A bidder of type h believes that her opponent is of type h with probability h. A bidder of type l believes that her opponent is of type h with probability q l h : Hene for h ; the three possible events {no ompetitor, ompetitor of type h, ompetitor of type l } have probabilities ; h ; h ; and for l ; the orresponding probabilities are ; h q l ; h q l : Finally, let bq l denote the posterior probability that a bidder of type l assigns on f! = g if she is told that there is no ompetitor of type h ; i.e. bq l := ql h q l h q l : Again, it is easy to see that all possible equilibria involve mixed strategies for both types of players. Assume then rst that the supports in the distributions are intervals without overlap suh that l bids on [0; p l ] and h bids on [p l ; p h ]: Furthermore let and v l := q l v () + q l v (0) ; bv l := bq l v () + bq l v (0) : Then we have from the optimality onditions of the two types of bidders: ( ) v l = h q l bv l p l ; 23

24 ( ) v () h v () p l = v () p h : If we have h! ; we see from the inequality that p l must onverge to zero. Also, q l! 0 and bq l! 0 and therefore the rst equation beomes ( ) v (0) = v (0) : Sine does not depend on h ; we get a ontradition. Hene we onlude that for h! ; ordered supports are not possible. By arguments similar to the previous setions, we also onlude that when h is large enough, then 0 is in the support of both bid distributions. In this ase, the bidding strategy b h ; b l = (0; 0) is an optimal ex ante strategy for a player onsidering entry. The expeted payo from this strategy is the value of the objet in eah state of the world if the other bidder does not enter, i.e. ( ) (qv () + ( q) v (0)) ; and the ost is : Hene by de nition of, the player is indi erent between entering and not, and we have an overall equilibrium in the game with symmetri entry rate and zero payo s to the bidders. A full analysis of this model (inluding the many bidders ase) is left for future researh. 5.2 More signals A trivial extension to a pure strategy equilibrium in a model with a ontinuum of signals is possible immediately for our model. Consider a ontinuum of signals with only two possible likelihood ratios for the states of the world. Lumping signals with the same likelihood ratios into two aggregate signals yields a model that is equivalent to the urrent disrete-signal ase. The mixed strategy equilibria of the urrent paper an then be translated into a pure strategy equilibrium of the ontinuous model. As long as the signals satisfy monotone likelihood ratio property, there is good reason to hope that the results would hold for nearby signal strutures (in the sense that onditional signal distributions are lose to the ontinuous version of the urrent model in the weak topology). 24

25 More generally, the ase with more signals (e.g. a ontinuum) seems relatively easily handled in the ase of a large number of bidders. The planner an onentrate on the two signals with the highest and the lowest likelihood ratios for the states. By mixing these appropriately, any feasible entry pro le an be generated. Equilibria in rst-prie and seond-prie autions an then be onstruted where only bidders with these signals enter. 5.3 More objets We are urrently working on an extension to the ase with k objets for sale. A omparison of disriminatory autions and k + st prie autions is then possible. For payo alulations in the disriminatory aution, eah bidder trades o the probability of having one of the k highest bids to the ost of the bid. Hene the equations for bidder indi erene are similar to the ones in the rst-prie aution of Setion 4. For the k + st prie aution, the same issues on non-unique equilibrium (pure) bids for l and randomized bids for h persist. This extension also raises the interesting issue of information aggregation for large k: It seems lear that full information aggregation in the sense of Pesendorfer & Swinkels (997) or Kremer (2002) will not be possible. As long as > 0; numbers of entrants remain random, and pries in k + st prie autions will not onverge to the true value of the objet. Determining the limit distribution of the realized prie when! 0 is also an interesting question for future researh. Referenes Atakan, A. & M. Ekmeki Autions, Ations, and the Failure of Information Aggregation. Amerian Eonomi Review (forthoming). Frenh, K. & R. MCormik Sealed Bids, Sunk Costs, and the Proess of Competition. Journal of Business 57: Harstad, R Alternative Common Value Proedures: Revenue Comparisons with Free Entry. Journal of Politial Eonomy 98:

26 Kremer, I Information Aggregation in Common Value Autions. Eonometria 70: Lauermann, S. & A. Wolinsky A Common Value Aution with Bidder Soliitation.. Levin, Dan & James Smith Equilibrium in Autions with Entry. Amerian Eonomi Review 84: Pesendorfer, W. & J. Swinkels The Loser s Curse and Information Aggregation in Common Value Autions. Eonometria 65(6): Samuleson, W Competitive Bidding with Entry Costs. Eonomis Letters pp Stegeman, M Partiipation Costs and E ient Autions. Journal of Eonomi Theory 7: