Sequential Auctions of Oligopoly Licenses: Bankruptcy and Signaling

Transcription

1 Sequential Autions of Oligopoly Lienses: Bankrupty an Signaling Georgios Katsenos Institut für Mikroökonomik, Leibniz Universität Hannover Deember 2010 Abstrat This paper ompares two proeures for alloating multiple oligopoly lienses: a simultaneous pay-your-bi aution an a sequential first-prie aution, in whih the winning bis are announe at the en of eah roun. When the winners marginal osts are truthfully reveale after the en of the aution, so that the information onveye by their bis annot affet the ensuing market ompetition, the two shemes are alloation an revenue equivalent. In the sequential aution, however, the firms bi in a more informe manner. As a result, the weaker (ex post) of the two oligopolists an win his liense at a lower prie than the one he woul pay in the simultaneous aution. Conversely, the stronger oligopolist pays a higher prie. Hene, the sequential aution results in a more equal istribution of the wealth generate by the oligopolisti market. In aition, the sequential aution suees in selling the two lienses at pries that o not exee their eventual value. Thus, it eliminates one of the reasons for post-aution bankrupty or prie renegotiation, a problem that ourre in several reent liene autions. Finally, when the winners marginal osts have to be inferre from their aution bis, it is possible for the firms to enhane their market profits by signaling a ifferent type. In this ase, the above results exten only in the ase of the Cournot oligopoly, in whih the firms are better-off overstating their strength. In the Bertran oligopoly, in whih the firms signaling inentives are negative, a separating equilibrium exists only in the simultaneous aution. JEL Classifiation: D43, D44, D62. Keywors: First-prie autions, sequential autions, oligopoly, externalities, revenue equivalene, bankrupty, signaling This paper is base on hapter 3 of my issertation, written at the University of Pittsburgh. I am grateful to Anreas Blume for his supervision of my work. I also wish to thank Oliver Boar, Heirun C. Hoppe, Esther Gal-Or, Paul J. Healy, Jak Ohs, Utku Ünver, as well as seminar partiipants at the University of Bielefel, the University of Hannover, the University of Pittsburgh, the University of St Anrews, the Fifth CEPR Shool on Applie Inustrial Organisation, the Jahrestagung 2008 es Vereins für Soialpolitik an the 2009 Meeting of the European Eonomi Assoiation, for helpful omments an suggestions. Any remaining errors are my own. aress: [email protected].

2

3 1 Introution Many important autions have aime at reating new markets or at expaning markets that alreay exist. For example, the raio spetrum autions onute by the Feeral Communiations Commission in the Unite States (an by similar government agenies elsewhere in the worl) have allowe the wireless ommuniation ompanies to expan their servies. In the same manner, the aution of other state-owne resoures, suh as oil fiels or timber trats, have enable the relate ompanies to expan their operations. In aition, in several regulate markets, ompanies nee to ompete perioially for the aquisition or renewal of the liense to supply their prout. Finally, following the evelopment of new tehnologies, ompanies that an benefit from them often bi for the right to use them. 1 Suh autions are typially haraterize by the presene of alloative an informational externalities. In partiular, the value of the resoures or of the lienses that are sol, that is, the market profit that these assets an generate, epens on the entire outome of the aution. First, it epens on the harateristis of the biers against whih a firm will ompete in the market. In aition, it may epen on the ientities of the winning an the losing biers. Finally, when the biers harateristis are privately known, their bis onvey information about them, in a manner that allows for the possibility of signaling. For example, an oligopoly liense is more valuable for its reipient when the firms that gain the other lienses are weaker. Similarly, an oligopolist s overall benefit from aquiring the rights to use a new tehnology inreases when his market ompetitors, whih are prevente from aessing this tehnology, an erive a greater benefit from it. Finally, in both examples, a firm an profit from exaggerating or from unerstating its private information uring the aution. Hene, beause of these externalities, the value of the assets sol is etermine enogenously, by the types of the biers, the final alloation an the information that the aution reveals. 2 In this paper, we stuy a partiular environment in whih suh externalities are present: an aution of two lienses to supply in a Cournot oligopoly or in a Bertran oligopoly for ifferentiate prouts. 3 For eah of the ompeting firms, the value of the autione 1 For a survey of the FCC autions, see Cramton [6, 7]; for a survey of similar autions in Europe, see Jehiel an Molovanu [23] an Klemperer [33]. For information regaring the autions of oil soures or of timber trats, see, respetively, Cramton [8] an Henriks an Porter [16] as well as the referenes therein. For the relation between an aution an the market reate by it, see Dana an Spier [9]. Regaring patent liensing shemes, see Kamien [28], Kamien et al. [29] an Kamien an Tauman [30]. Finally, for a general survey of autions of publi assets an the issues that typially arise in them, see Janssen [19]. 2 This environment iffers from that of an aution with symmetrinterepenent valuations, suh as the one stuie in Milgrom an Weber [36]. There, the value of the assets sol epens on the biers private information only. It is inepenent of the outome of the aution; in partiular, it oes not epen on who wins the aution or on the information that the aution may reveal. 3 For simpliity, we have assume that the sale of the lienses results in the reation of a new market. Our results, however, easily exten to an aution of two lienses to enter an alreay existing oligopoly or monopoly. In suh an aution, the struture of the firms payoffs an inentives is iential to that in our moel. Similarly, one an moify our moel so as to esribe the aution of two lienses to use a proess innovation. Again, the struture of the firms inentives will not hange. 1

4 lienses epens, first, on its own proution osts. In aition, it epens on the proution osts of the winner of the other liense, that is, of the firm against whih it will ompete in the market. Finally, when the firms proution osts annot be reveale otherwise, there is the possibility of signaling through biing. In a Cournot oligopoly, a firm an inrease its market profit by signaling, uring the aution, a stronger type, so as to beguile its opponent into supplying a smaller quantity. Similarly, in a Bertran oligopoly, a firm an inrease its profit by signaling a weaker type, so as to lure its opponent into setting a higher prie. We ompare two proeures for alloating the oligopoly lienses, a simultaneous payyour-bi aution 4 an a sequential first-prie aution. We assume that in eah aution sheme, the seller reveals the same information, namely, the two winning bis. The two shemes iffer, however, in the timing of revelation of that information. In the simultaneous format, all information is reveale at the en of the entire proeure, prior only to market ompetition. On the other han, in the sequential format, some information, the bi that won the first liense, is reveale uring the biing proess, in between the two aution rouns. As a result, in the sequential aution, the firms an bi in a better-informe manner, as some of the unertainty present in this environment is eliminate. When the winners proution osts are truthfully reveale after the aution 5, so that the information onveye by their bis oes not affet the ensuing market ompetition, then for eah aution sheme there exists a symmetri equilibrium in stritly monotone biing strategies. The two lienses are therefore alloate to the two strongest firms. The ifferene in the information struture, however, affets the egree by whih the firms shae their bis an, therefore, the pries that they eventually pay. In the first roun of the sequential aution, the firms know that they will win the liense only if they have the strongest type. Therefore, sine they o not take into aount the possibility of having to ompete against a stronger market opponent, they shae their bis less than they woul o in the simultaneous aution. Consequently, the stronger of the two market ompetitors, as reveale by the outome of the aution, pays a higher prie in the sequential aution than in the simultaneous one. Similarly, the firms partiipating in the seon roun know that, by winning the seon liense, they will neessarily have to fae a stronger market ompetitor. Therefore, on average, they shae their bis more than they woul o in the simultaneous aution. As a result, the weaker of the two market ompetitors pays a higher prie in the simultaneous aution. Despite the ifferenes in the biers behavior, the two aution shemes turn out to be 4 In Krishna [34] an elsewhere, this aution is referre to as a isriminatory aution. 5 The revelation of the two oligopolists proution osts an be the onsequene of the ations that they nee to take in the time perio between the en of the aution an the beginning of the market. Truthful revelation an also be assume when the effets of false signaling are negligible, for example, when the oligopolists an quikly ajust their market strategies. This assumption is present in muh of the literature on autions with externalities, for example, in Jehiel an Molovanu [21]. 2

5 revenue equivalent. 6 The exess of aggression that the biers show in the first roun of the sequential aution, whih stems from the goo news that a vitory in this roun will onvey, is balane, on average, by the exess of restraint that the biers show in the seon roun, in reation to the ba news they have reeive. Therefore, without inurring any ost to the seller, the sequential aution results in a more even istribution of the wealth generate by the introution of the new market. 7 In aition, the sequential aution suees in selling the two lienes at pries that o not exee their (ex post) values. Therefore, it eliminates one of the reasons for post-aution bankrupty or prie renegotiation. Suh problems have ourre in several liense autions, most famously, in the 1996 C-Blok raio spetrum aution in USA. 8 The experiene of these autions iniates that post-aution bankrupty an prie renegotiation an result in severe revenue or welfare losses 9 ; their prevention, therefore, an be an important objetive in the esign of a liense aution. When the oligopolists proution osts are not automatially reveale after the aution, but have to be inferre from their bis, the possibility of signaling is introue. In this ase, the firms ajust their valuations by inorporating the informational rents that they an extrat. In eah aution sheme, there are two inentive trae-offs, the signaling an the non-signaling one. Sine the firms market profit funtions are separable in their real an signalle proution osts, the two trae-offs an be separate an treate inepenently. Therefore, the firms non-signaling inentives an always be balane, as in the ase in whih their osts are truthfully reveale. On the other han, the possibility of balaning the signaling inentives epens on the aution sheme as well as on the type of the market. In the simultaneous aution, the firms an ajust their strategies so that any gains from false signaling an be offset by an inrease in the expete payment, in the ase of the Cournot market, or by a erease in the probability of winning a liense, in the ase of the Bertran market. In the sequential aution, a similar trae-off is possible only in the 6 Sine we assume that the firms proution osts are inepenent, the intuition of the Linkage Priniple for autions with interepenent valuations (f. Milgrom an Weber [36]) oes not apply to our setting. For this priniple to impose a revenue ranking in favor of the aution sheme that reveals more information uring the biing proeure, in this ase, in favor of the sequential aution, the firms proution osts must be affiliate. 7 The seller s onern for a more even istribution of the oligopolists overall profit may stem from a esire to maintain balane, over time, among the ompeting firms. In this ase, therefore, the hoie of a sequential aution an be thought off as an iniret subsiy to the weaker market partiipant. For a isussion of this issue, see Maaslan et al. [35]. 8 For a esription of the bankrupties following the 1996 C-Blok aution, see Boar [3], Zheng [39] an the referenes therein. Other examples inlue the ollapse of the ITV Digital in 2002, following its aquisition of the rights to broaast the English Premier League football mathes, an the eman for prie renegotiation by some of the ompanies that aquire parts of Brazil s teleommuniations ompany Telebras, in For example, in the ase of the C-Blok aution, bankrupty ourts initially reue the obligation of one of the winners, NextWave, from $4.74 billion to $1.02 billion. It was only after eight years of legal battles, uring whih part of the spetrum ha to remain ile, that FCC oul reah a settlement with NextWave, whih allowe FCC to re-aquire some of the lienses an re-aution them in

6 Cournot oligopoly. In the Bertran oligopoly, a firm an always profit from mimiking a weaker type in the first roun, as this woul inrease its expete market profit an erease its expete payment, without hanging its overall probability of winning one of the two lienses. Therefore, in this ase, no symmetri monotone equilibrium exists. Hene, in the Cournot oligopoly, the non-signaling results remain vali uner signaling. The two aution shemes are revenue equivalent, even though the sequential aution favors the weaker (ex-post) of the two oligopolists. On the other han, in the Bertran oligopoly, a omparison between the two aution shemes is not possible, sine a separating equilibrium exists only in the simultaneous aution. 10 Overall, our results regaring the omparison of the two aution shemes are base on the ability of the sequential aution to generate impliitly, by the haraterization of the equilibrium, information about the two winners relative strengths. This information enables the firms to moify their interim valuations for eah of the autione lienses. The first liense beomes more valuable, relative to a liense won in a simultaneous aution, sine its aquisition implies a stronger presene in the market. On the other han, the seon liense beomes less valuable, sine its reipient will have a weaker market presene. As a result, the lienses (an the information that aompanies their aquisition) are sol at ifferent pries than the ones pai in the simultaneous aution. The early stuy of autions with externalities 11 assume that the externalities epen only on the number of alloate objets (an not on the biers types an ientities). In their stuy of the persistene of a monopoly, Gilbert an Newbery [14] show that a monopolist who faes a potential entrant may bi for a tehnologial innovation for whih he has no use, in a preemptive manner. Within the ontext of patent liensing an vertial ontrating, Katz an Shapiro [31], Kamien an Tauman [30] an Kamien et al. [29] ompare some typial liensing mehanisms, suh as autions, fixe fees an royalties, an show the superiority of autions. Optimal selling mehanisms in the presene of type-epenent externalities were first stuie by Jehiel et al. [26, 27]. If the agents private information is multi-imensional, then it is optimal for the seller to employ ientity-epenent threats, whih exploit the negative alloative externalities in orer to extrat surplus from biers that o not aquire the liense. For very strong negative externalities, the biers may even pay the seller not to alloate the liense at all. Clearly, this mehanism is not effiient. In fat, Jehiel an Molovanu [22, 24] show that, with multi-imensional signals, effiieny annot be implemente. If the agents private information is single-imensional, however, effiieny is feasible. Figueroa an Skreta [11] show that sometimes the optimal mehanism is effiient; at other times, though, it alloates the autione objets in a manner involving ranomization. 10 In light of the results regaring information sharing in oligopoly (f. Gal-Or [12, 13]), our signaling results are harly surprising. In partiular, when signaling is possible, a separating equilibrium exists only in the oligopoly in whih the firms are willing to share information about their proution osts. 11 For an extensive survey of the literature on this subjet, see Jehiel an Molovanu [25]. 4

7 Sine the appliation of the optimal mehanism, in partiular, the ifferential treatment of the biers, may not be possible, Jehiel an Molovanu [20, 21] examine relatively simpler selling shemes, suh as autions with fixe reserve pries or entry fees. They fin that some biers may prefer to abstain from the aution, if their partiipation an have an averse effet on biing or on the winner s ientity. Conversely, to enourage partiipation, the seller may set a reserve prie below his own reservation value. In a multi-unit setting, in partiular, in the sale of the rights to use a ost-reuing innovation in an oligopoly, Shmitz [38] an Baghi [2] examine simultaneous autions of a preetermine number of lienses. As the biers information is single-imensional, the lienses are alloate effiiently. In aition, the seller an be better off autioning multiple lienses rather than the exlusive rights to use the tehnology. Finally, in an aution of multiple lienses to enter an alreay existing oligopoly, Hoppe et al. [17] show that the resulting market an be less ompetitive if more lienses are mae available. The problem of bankrupty by an aution winner has been stuie by Zheng [39] an Boar [3]. In their settings, the biers have fixe buget onstraints, refleting the limits of their liability. If the value of the autione asset turns out to be suffiiently high, relative to the winner s bi an buget onstraint, then the winner of the aution an reeive further finaning for his bi; otherwise, he elares bankrupty, at the ost of his buget. The asset s value is fully etermine after the en of the aution, epening on the realization of an exogenous ranom variable. On the other han, in our setting, the value of a liense is etermine within the aution, by the biers types an the inue alloation. Signaling in autions with externalities was introue in Goeree [15], who examine the aution of a single liense to ompete against a monopolist with known marginal ost. Das Varma [10], in a problem of biing for the aquisition of a ost-reuing patent, ientifie onitions for the existene of equilibrium in the presene of negative informational externalities. Katzman an Rhoes-Kropf [32] extene the stuy of signaling to more general shemes of information revelation, showing the revenue equivalene of the aution shemes that result to the same alloation an reveal the same information. Finally, Molnár an Virág [37] etermine the revenue maximizing alloation an information mehanism in environments with post-aution interation. The present work ontributes to the existing literature on autions with externalities by extening the analysis of the multi-unit ase to sequential autions. When there is no signaling, we show that both the simultaneous an the sequential sheme lea to an effiient alloation of the lienses 12 while they raise the same revenue for the seller. In aition, we emonstrate the effets of the better-informe biing that is allowe by the sequential format, showing that it favors the weaker, ex-post, of the two oligopolists. Hene, in the absene of informational externalities, we onlue that the sequential aution an be reommene as a poliy evie to an autioneer who wishes to ahieve a 12 Sine the biers private information is single-imensional, the effiieny result in our setting oes not ontrait Jehiel an Molovanu [22, 24]. 5

8 more even istribution of the wealth generate by the reation of the oligopoly. Finally, we explore the impliations of signaling, showing that the inentive to unerstate one s strength, whih is present in the ase of the Bertran oligopoly, eliminates the possibility of effiient alloation. In the next two setions, we present the moel esribing our problem an we analyze the firms behavior in the oligopoly reate by the aution of the two lienses. In setion 4, we examine the two aution proeures when there is no signaling, eriving symmetri equilibria in stritly monotone biing strategies an omparing them. In setion 5, we analyze the ase of positive signaling, whih is present in the Cournot oligopoly, showing that the non-signaling results fully exten. In setion 6, we stuy the ase of negative signaling, present in the Bertran oligopoly, showing that a symmetri equilibrium in stritly monotone strategies exists only for the simultaneous aution. Finally, we onlue in setion 7. Long proofs have been plae at the appenix. 2 General Moel We stuy the aution of 2 lienses for partiipating in a newly forme oligopoly, whih will take the form of either Cournot ompetition or Bertran ompetition for ifferentiate prouts. The market profits of the two oligopolists epen, respetively, on the quantities they supply to the market or on the pries they set for their prout. These eisions epen, in turn, on the oligopolists proution osts. There are N > 2 firms ompeting for the aquisition of the oligopoly lienses. The firms have linear proution tehnologies without fixe osts. Therefore, for eah firm i, its tehnology is haraterize by the privately known marginal ost, whih is rawn inepenently, at the beginning of the game, from a istribution F : [, ] [0, 1], where 0 < < < We assume that F is twie ifferentiable, with a ensity funtion f : [, ] R that has full support. Finally, we assume that the inverse hazar rate [1 F ()]/f() is ereasing. 14 For any firm i, we enote by 1 i an 2 i the ranom variables esribing respetively the lowest an the seon-lowest marginal osts of firm i s ompetitors (an by 1 13 In partiular, this last assumption implies that < This assumption is satisfie by many well-known istributions, suh as the uniform, exponential, normal, power (for α 1), Weibull (for α 1) an gamma (for α 1) istributions. It is onsistent with the assumption of logonave istribution of the firms strength, mae elsewhere in the literature, in partiular, in Das Varma [10] an Goeree [15]. If the firms strength θ [, ], efine by θ() = ( ), is istribute aoring to a logonave ensity f, then the rate F (θ)/ f(θ) must be inreasing. This, in turn, implies that the hazar rate [1 F ()]/f() must be ereasing. For the efinition an properties of logonave probability ensity funtions, onsult An [1] an Caplin an Nalebuff [5]; for more on the assumption of inreasing inverse hazar rate, onsult Hoppe et al. [18]. 6

9 an 2 the values that these ranom variables may take). In aition, we enote by G( 1 ) = 1 [1 F ( 1 )] N 1 the umulative istribution funtion of 1 i an by g( 1 ) = (N 1) [1 F ( 1 )] N 2 f( 1 ) the orresponing ensity funtion. Finally, we enote by G( 1, 2 ), for 1 2, the joint umulative istribution funtion of 1 i an 2 i an by g( 1, 2 ) = (N 1)(N 2) [1 F ( 2 )] N 3 f( 2 ) f( 1 ), for 1 2, the joint ensity funtion. We onsier two aution formats: a. A simultaneous pay-your-bi aution, with the two winning bis announe at the en of the aution. b. A sequene of two first-prie autions, with the winning bi announe at the en of eah aution. In both formats, the winners bis are publily known by the en of the aution proess, so that the information they reveal affets the ensuing market ompetition. Furthermore, in the sequential aution, the winning bi in the first roun beomes known prior to the beginning of the seon roun, so that the information it onveys also affets the biing for the seon liense. Sine there are no reserve pries in any of the autions, we an assume that the two lienses are always sol, even at a zero prie. We will restrit attention to equilibria in symmetri strategies, stritly monotone in the firm s own marginal ost. Therefore, in the simultaneous aution, eah firm i bis b i = β( ), aoring to its marginal ost an the strategy β : [, ] R. Similarly, in the sequential aution, eah firm i bis b 1 i aoring to its marginal ost an the strategy = β 1 ( ) in the first roun, β 1 : [, ] R. If it fails to win the first roun, then firm i bis b 2 i = β 2 ( b 1 i, b 1 ) in the seon roun, aoring to its marginal ost, the first-roun history h 1 i = (b 1 i, b 1 ) H 1 sq = R R, onsisting of the privately known bi b 1 i an the publily known prie b 1, an the strategy 7

10 β 2 : [, ] H 1 sq R. Following the ompletion of the aution, eah of the winning firms enters the oligopoly. The information that firm i has at the en of the simultaneous aution, h i = (b i, b 1, b 2 ) H sim = R R R, onsists of its privately known bi b i an the publily known pries b 1, b the information that firm i has at the en of the sequential aution is Similarly, h 2 i = (b 1 i, b 2 i, b 1, b 2 ) H 2 sq = R ( R { } ) R R, allowing for the absene of a seon-roun bi, in the ase of a first-roun win. Given this information an its marginal ost, firm i will supply q i = q( b i, b 1, b 2 ) or q i = q( b 1 i, b 2 i, b 1, b 2 ) in the Cournot oligopoly, aoring to the strategy q : [, ] H R, for H {H sim, H 2 sq}, following respetively a simultaneous or a sequential aution. Similarly, in the Bertran oligopoly, firm i will set a prie p i p i = p( b 1 i, b 2 i, b 1, b 2 ), aoring to the strategy = p( b i, b 1, b 2 ) or p : [, ] H R, for H {H sim, H 2 sq}, following respetively a simultaneous or a sequential aution. Overall, we will impose a stronger symmetry requirement, one that rules out the possibility of using past histories as a labeling evie for asymmetri ontinuation strategies. This assumption will rule out, in partiular, asymmetri supply or prie-setting strategies for the two oligopolists. 16 In the sequel, we will use the strit monotoniity of the biing strategies, with respet to the firm s own marginal ost, to simplify the notation in the following manner: 15 Without loss of generality, the two pries are in esening orer. 16 For example, in the sequential setting, this assumption oes not allow the possibility of presribing ifferent oligopoly strategies to the winners of the first an the seon sequential autions. 8

11 Notation 1 In the sequential aution, we will enote 17 firm i s seon-perio bi b 2 i, following a first-perio bi b 1 i an a prie b 1 = β 1 ( 1 ), by β 2 ( b 1 i, β 1 ( 1 )) β 2 ( 1 ). In the Cournot oligopoly, following either a simultaneous or a sequential aution, we will enote firm i s supplie quantity, q i = q( b i, b 1, b 2 ) or q i = q( b 1 i, b 2 i, b 1, b 2 ), by q i q(, j ), where an j are the marginal osts orresponing to the bis, uner the equilibrium biing strategies, with whih firms i an j won their oligopoly lienses. Similarly, in the Bertran oligopoly, following either of the two aution formats, we will enote firm i s requeste prie, p i = p( b i, b 1, b 2 ) or p i = p( b 1 i, b 2 i, b 1, b 2 ), by p i p(, j ). The notational simplifiation of the oligopoly supply or prie setting strategies is also base on the inepenene of these strategies of the firm s privately known bis. Inee, as it will turn out, eah oligopolist s behavior epens only on its own marginal ost,, its opponent s inferre marginal ost, j, an its own marginal ost as pereive by its opponent,. Sine the osts an j are inferre by the publily known pries, the privately known bis provie no information. The game payoff of firm i, in ase it wins a liense, equals its profit from the oligopoly minus the prie that it pai for the liense. Otherwise, if it oes not win any liense, it equals zero. The solution onept is that of perfet Bayesian equilibrium. The players must therefore behave optimally at eah eision point, given their knowlege of the other players strategies an their beliefs. On the equilibrium path, these beliefs are forme by applying Bayes rule while, off the equilibrium path, they are arbitrary. 17 This simplifiation is ustomary in sequential autions; for example, see Krishna [34], hapter 15. It is base on the strit monotoniity of the equilibrium biing strategy β 1 as well as on the inepenene of the equilibrium biing strategy β 2 of the first-roun bi b 1 i. 9

12 3 Market Competition When the two winners marginal osts, an j, are truthfully reveale at the en of the aution, the firms annot use their bis to manipulate their market opportunities, that is, no signaling is possible. In this ase, eah oligopolist supplies a quantity q i = q NS (, j ) or sets a prie p i = p NS (, j ), whih is inepenent of the bis submitte to or the pries reporte in the aution. When the oligopolists must infer their opponent s marginal ost by the reporte pries, the opportunity of signaling arises. In this ase, the two winning bis b t i an b t j perfetly reveal, through the inversion of the orresponing strategies β t (.) an β t (.), the marginal osts an j that the winners mimike in the aution. 18 Therefore, eah firm supplies q i = q(, j ) or sets a prie p i = p(, j ). Sine we onsier only unilateral eviations, in examining the inentives of player i we will assume that j = j, so that q i = q(, j ) an p i = p(, j ). Clearly, in the equilibrium path, the two firms reveal their marginal osts truthfully. As a result, for all, j [, ], we have q(, j ) = q NS (, j ) an p(, j ) = p NS (, j ). In analyzing the firms market behavior, therefore, we will onsier only the ase of signaling, treating the absene of signaling as one of its partiular outomes. 3.1 Cournot Oligopoly We onsier a Cournot oligopoly, in whih the inverse eman funtion is given by p = 1 q, where p is the market prie an q = q i q j is the aggregate supply of oligopolists i an j. If firm i reveals its marginal ost truthfully, then, by supplying q i [0, 1 q j ] in response to q j [0, 1], it will make a market profit π(q i, q j ) = q i (1 q i q j ). Therefore, in equilibrium, firm i will supply 19 for a profit of q(, j ) = 1 3 (1 j 2 ), π(, j ) = ( 1 3 )2 (1 j 2 ) In the sequential aution, it is possible for a bier to eviate into mimiking two ifferent types, if he oes not win in the first roun. In this ase, however, only the type mimike in the seon aution will be reveale an, therefore, be relevant in the analysis of the post-aution ompetition. 19 The assumption of < 1 2 ensures that the market oes not beome a monopoly. In equilibrium, it is optimal for a firm to supply a positive quantity to the market, regarless of its own marginal ost an its beliefs about the marginal ost of the other firm. 10

13 Off the equilibrium path, if firm i mimis a type in the aution that it wins, firm j will supply q( j j, ) = 1 3 (1 2 j ). Therefore, firm i will maximize its profit by supplying In this ase, its market profit will be q(, j ) = 1 3 (1 j ). π(, j ) = ( 1 3 )2 (1 j ) 2. Clearly, we have π 2 = π < 0, so, prior to market ompetition, uring the aution proess, eah firm has an inentive to overstate its power by mimiking a lower marginal ost. 3.2 Bertran Oligopoly We onsier a Bertran oligopoly, in whih eah firm i faes a linear eman funtion q i = 1 p i γ p j, where p i an p j are the pries set respetively by firm i an its rival, firm j, while γ [0, 1) is a parameter refleting the egree of prout ifferentiation. 20 Sine γ < 1, the eman fae by eah firm is more responsive to hanges in the prie harge by this firm than to hanges in the prie harge by its rival. If firm i reveals its marginal ost truthfully, then, by setting a prie p i [0, 1 γ p j ] in response to a prie p j [0, 1], it will make a market profit π(p i, p j ) = (1 p i γ p j ) (p i ). Therefore, in equilibrium, firm i will set a prie for a profit of p(, j ) = 2 γ γ j 2 4 γ 2, π(, j ) = [2 γ γ j (2 γ 2 ) ] 2 (4 γ 2 ) 2. Off equilibrium, if firm i mimis a type, firm j will set a prie p j = p( j j, ). Therefore, firm i will be best-off by setting a prie p(, j ) = 2(2 γ) 2γ j (4 γ 2 ) γ 2 2(4 γ 2 ) 20 Our results will not hange, if we onsier a more general linear eman funtion q i = α β q i γ p j, for β γ, as the inue equilibrium market profit funtion will emonstrate the same properties. 11,

14 for a market profit of In this market, we have π(, j ) = [ 2(2 γ) 2γ j (4 γ 2 ) γ 2 ] 2 4 (4 γ 2 ) 2. π 2 = π > 0, so, uring the aution proess, eah firm has an inentive to unerstate its power by mimiking a higher marginal ost. 3.3 General Remarks For both oligopolies, the market profit funtion of eah firm i is ereasing in its own marginal ost an inreasing in its rival s marginal ost j. That is, an π 1 = π < 0 Furthermore, for all [, ], we have π 3 = π j > 0. [π(, )] = π 1(, ) π 2 (, ) π 3 (, ) < 0, so that, with truthful revelation of the firms marginal osts, a firm s market profit will be affete more by a hange in its own marginal ost than by the same hange in its rival s marginal ost. The two oligopolies iffer in the sign of the erivative π 2 = π, that is, in the signaling inentives of the firms. In the Cournot oligopoly, there is positive signaling, that is, eah firm has an inentive to overstate its power. On the other han, in the Bertran oligopoly, there is negative signaling, that is, eah firm is better off unerstating its power. Other than that, in partiular, when there is no signaling, the profit funtions in the two oligopolies inue the same, qualitatively, inentives The opposite signaling inentives orrespon to the istintion between strategi substitutes an strategi omplements, introue by Bulow et al. [4]. In partiular, in the Cournot oligopoly that we have esribe, quantities are strategi substitutes, sine an inrease in q i auses a erease in the q j (as well as a erease in the profit of firm j). On the other han, in the Bertran oligopoly, pries are strategi omplements, sine an inrease in p i auses an inrease in p j (an in the profit of firm j). 12

15 4 No Signaling When signaling is not possible, the value of eah oligopoly liense, π NS (, j ) = π(, j ), is fully etermine by the atual marginal osts of the two firms that ompete in the market. Mimiking a ifferent type uring the aution proess annot affet a firm s potential market profits. It only affets the firm s probability of winning the aution an its expete payment in it. 4.1 Simultaneous Aution Suppose that all firms follow a stritly ereasing biing strategy b = β() an onsier firm i with marginal ost. Then, by mimiking a type [, ] uring the aution, firm i will win a liense if an only if 2 i. In this ase, the atual value of this liense will be equal to firm i s market profit, π NS (, 1 i), whih epens on the marginal ost 1 i of the winner of the other liense. Therefore, for a bi b i = β( ), the expete total payoff of firm i is Π( ) = P[ 2 i ] [ E 1 i [π NS (, 1 i) 2 i ] β( ) ] or, by expaning the term for the firm s expete market profit, Π( ) = i π NS (, 1 ) (N 1)[1 F ( )] N 2 f( 1 ) 1 π NS (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 P[ 2 i ] β( ). The first-orer onition with respet to results in the equation i {P[ 2 i ] β( )} = π NS (, 1 ) (N 1)(N 2) [1 F ( )] N 3 f( )f( 1 ) 1, whih requires, in a manner that is stanar for pay-your-bi aution shemes, that any inrease in the firm s expete market profit from a eviation to must be offset by an inrease in the firm s expete payment in the aution. 13

16 The ifferential equation erive from the first-orer onition, along with the bounary onition expressing the equilibrium behavior of the weakest type =, β( ) = π NS (, ) f(), whih guarantees the uniqueness of the solution to that ifferential equation, provies the equilibrium biing strategy for this setting. Proposition 1 In the simultaneous pay-your-bi aution of two oligopoly lienses, in whih the winners marginal osts are reveale truthfully after the aution, the following strategy onstitutes a symmetri separating equilibrium: β( ) = 2 π NS ( 2, 1 ) (N 1)(N 2) [1 F (2 )] N 3 f( 2 )f( 1 ) P[ 2 i ] 1 2. The strategy β( ) an also be expresse as where β( ) = v NS ( 2 ) (N 1)(N 2) [1 F (2 )] N 3 F ( 2 )f( 2 ) P[ 2 i ] v NS () = π NS (, 1 ) f(1 ) F () 1, for [, ], is the expete market profit of a firm with marginal ost, assuming that its market opponent is stronger. Therefore, in equilibrium, eah firm submits a bi equal to the expete market profit of the strongest non-winning firm. The value of the lienes that the two winners of the aution gain is etermine enogenously, as a funtion of the marginal osts of the winning bis. Sine these osts are unknown prior to the en of the aution proess, it is possible for a firm, when its market opponent turns out to be stronger than expete, to aquire a liene at a prie above its ex-post value. 22 2, 22 We emphasize the ifferene between this phenomenon an the winner s urse for an aution with interepenent valuations (as well as for our environment). The winner s urse refers to the ba news that a vitory in suh an aution onveys, namely, that the winner s estimate of the value of the autione objet has been the most optimisti one. In the equilibrium path, the winner s urse is eliminate by means of an ajustment of the biers estimates. Still, it is possible that the losing biers private information will be very negative, so as to efy the winner s reasonable expetation an to result in a value that is below the prie the winner must pay. It is this phenomenon to whih we refer as the winner s regret. 14

17 Corollary 2 In the simultaneous aution, the firm with the lowest marginal ost gains a liense at a prie below its ex-post value. The firm with the seon-lowest marginal ost, however, may gain a liense at a prie above its ex-post value. Hene, in equilibrium, the stronger of the two oligopolists will always make a positive profit. On the other han, the weaker oligopolist may regret his partiipation to the market, beause of the prie of the liense. 4.2 Sequential Aution When the two lienses are alloate by means of a sequene of first-prie autions, then, assuming that the firms follow stritly monotone biing strategies, the winning bi in the first aution reveals the marginal ost 1 of the strongest oligopolist. This information affets the biing for the seon liense in two istint manners. First, it allows the remaining firms to learn, prior to the seon aution, the atual value of the liense for whih they ompete. In aition, the reveale marginal ost 1 forms a lower boun for the marginal osts of the remaining firms. Therefore, after the en of the first aution, the firms upate their beliefs, so that for [ 1, ]. F () = F () F (1 ), 1 F ( 1 ) Sine the privately known first-perio bis o not affet the firms behavior in the seon perio, the seon aution takes the form of a stanar first-prie aution with inepenent private values. Lemma 3 Suppose that N 1 firms, whose marginal osts are i.i.. aoring to the istribution funtion F ( ) on [, ], ompete in a first-prie aution for a liense to partiipate in an oligopoly against a firm with known marginal ost 1 [, ]. In aition, suppose that the firms believe that the unknown marginal osts are boune below by the value 1. Then, assuming that the winner s marginal ost is reveale truthfully at the en of the aution, the following strategy onstitutes a symmetri equilibrium: For a marginal ost 1, firm i bis β 2 ( 1 ) = π NS (, 1 ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2, while for a marginal ost < 1, firm i bis b 2 = β 2 ( 1 1 ). 15

18 A marginal ost < 1 orrespons to an event off the equilibrium path, namely, to the ase in whih firm i shoul have won the first liense but i not bi aoring to the strategy β 1 that was presribe in the first aution. 23 Suh a firm enters the seon aution knowing that it has the highest valuation an it woul be best-off biing as if it ha marginal ost 1. For the analysis of the firms behavior in the first aution, we will nee the strategy β 2 ( ) to be ereasing with respet to the marginal ost. Without this onition, the strategy β 1 ( ) that we erive may fail to be stritly ereasing, thus invaliating the argument leaing to it. Notie, therefore, that the erivative of β 2 ( ) equals to [ β 2 ( ) ] = π NS (, ) (N 2)f() 1 F ( ) π NS 2 (, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 π NS (, ) (N 2)[1 F ()]N 3 f() (N 2)f(), [1 F ( )] N 2 1 F ( ) or, after integrating the last term by parts, to [ β 2 ( ) ] = π NS 2 (, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 π NS 1 (, ) [1 F ()]N 2 [1 F ( )] (N 2)f(). N 2 1 F ( ) Sine the erivative π NS 1 ( 1 ) is ereasing in while the erivative π NS 2 ( 1 ) is inreasing in, we have [ β 2 ( ) ] π NS 2 (, ) π1 NS (, ) [ E[ 1 i,j 1 i,j ] ] (N 2)f(), 1 F ( ) 23 Restriting attention to the game esribe in Lemma 3, notie that the possibility of < 1 oes not ontrait the firms beliefs an, therefore, oes not violate the onsisteny requirement in the efinition of Nash equilibrium. We an simply assume that prior to the raw of the privately known marginal osts, eah firm attahes zero probability to the event < 1, for all i. 16

19 where 1 i,j enotes the lowest value among N-2 realizations of the marginal ost. Hene, sine π1 NS < 0 < π2 NS, if the hazar ratio f( )/[1 F ( )] is too small, then the erivative [ β 2 ( ) ] an be positive. We avoi this possibility by imposing the following onition: Assumption 1 The istribution of the firms marginal osts satisfies the inequality { [1 F ()] N 2 π NS sup [1 F ( )] N 2 2 (, ) π NS 1 (, ) } 1 F (i ) (N 2) f( ), for all [, ]. In the ase of uniformly istribute marginal osts, this assumption is satisfie for the Cournot oligopoly that we have esribe. Inee, for U[, ], it requires that N 1 2(N 2), whih is true for all N 3. On the other han, for a Bertran oligopoly with parameter γ [0, 1), the assumption reues to requiring that N 2 N 1 γ 2 γ 2, whih is satisfie only if the number of firms, N, is suffiiently large, relative to γ. Assumption 1 requires that the inverse hazar rate oes not erease too rapily. More preisely, as it is shown by the proof of the next Lemma, the inequality 1 F () f() > πns π NS 2 (, ) 1 (, ) 1 F ( ) f( ) remains vali for a suffiiently large interval of values, so that the negative term in the equation of [β 2 ( )] ominates the positive one. Lemma 4 Uner Assumption 1, the funtion β 2 ( ) is ereasing in [, ]. 17

20 In the first aution, suppose that all firms follow a stritly monotone biing strategy b 1 = β 1 () an onsier firm i with marginal ost. Then, by mimiking a type [, ] in this aution, firm i will win the first liense if an only if 1 i. In this ase, the atual value of this liense will be equal to firm i s market profit, π NS (, 1 i), whih epens on the marginal ost 1 i of the winner of the seon liense. Therefore, by biing b i = β( ) for, firm i expets a total payoff Π( ) = [ π NS (, 1 ) β 1 ( ) ] (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 i [ π NS (, 1 ) β 2 ( 1 ) ] (N 1)[1 F ( )] N 2 f( 1 ) 1, while by biing b i = β( ) for, firm i expets a total payoff Π( ) = [ π NS (, 1 ) β 1 ( ) ] (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 i i [ π NS (, 1 ) β 2 ( 1 ) ] (N 1)[1 F ( )] N 2 f( 1 ) 1 [ π NS ( 1, 1 ) β 2 ( 1 1 ) ] (N 1)[1 F ( 1 )] N 2 f( 1 ) 1. In the seon ase, the extra term results from the possibility of selling the first liense to a firm with marginal ost 1 [, ]. In this ase, firm i bis b 2 = β 2 ( 1 1 ) in the seon aution, knowing that it has the lowest marginal ost among the remaining firms. In both ases, the neessary first-orer onition at the enpoint = results in the ifferential equation { [1 F ( )] N 1 β 1 ( )} = β 2 ( ) (N 1) [1 F ( )] N 2 f( ). By solving this ifferential equation, along with the bounary onition β 1 ( ) = π NS (, ) that expresses the biing behavior of the weakest possible type, we get the strategy β 1 ( ) = β 2 ( 1 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 1, whih is part of the equilibrium in our sequential aution. 18

21 Proposition 5 In a sequential first-prie aution of two oligopoly lienses, in whih the winners bis are announe at the en of eah roun an their marginal osts are truthfully reveale at the en of the entire aution, the following strategy profile onstitutes a symmetri separating equilibrium: In the first aution, eah firm i bis β 1 ( ) = 1 π NS ( 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) [1 F ( )] N In the seon aution, if firm i has a marginal ost 1 = (β 1 ) 1 (b 1 ), where b 1 is the prie at whih the first liense was sol, then it bis β 2 ( 1 ) = π NS ( 2, 1 ) (N 2)[1 F (2 )] N 3 f( 2 ) [1 F ( )] N 2 2 while with a marginal ost < 1, it bis b 2 = β 2 ( 1 1 ). The equation efining the biing strategy β 1 ( ) is a non-arbitrage onition for the winner of the first aution, whih is, as it has turne out, the firm with the lowest marginal ost. If this firm oes not partiipate in the first aution, then it an win the seon aution with a bi equal to β 2 ( 1 1 ), where 1 is the reveale lowest ompeting marginal ost. Therefore, to be inifferent, this firm must bi in the first aution an amount equal to its expete bi in the seon roun. In the sequential first-prie aution, the marginal ost of the strongest firm is known at the time of the biing for the seon liense, sine it is reveale after the en of the first roun. Hene, in the sequential aution, unlike the ase of the simultaneous aution, it is possible for both winning firms to avoi gaining a liense at a prie above its ex-post value. Corollary 6 In the sequential first-prie aution, both lienses are sol at pries below their ex-post values. Proof: The proof for the liense sol in the first aution, to the firm with the lowest marginal 19

22 ost, follows from a iret argument, iential to the one for the simultaneous aution. In aition, in the seon aution, the remaining biers know the marginal ost 1 of the first firm an, therefore, their value for the liense. As a result, they never bi an amount above that value. The revelation of the marginal ost of the strongest firm after the en of the first aution makes the seon liense less profitable for the remaining firms. As a result, these firms biing for the seon liense beomes less aggressive. Corollary 7 In the sequential first-prie aution, the pries at whih the two lienses are sol form a super-martingale: E 1 i [β 2 ( 1 i ) ] β 1 ( ). Proof: Suppose that the first liense is sol at a prie b 1 = β 1 ( ), orresponing to a marginal ost. Then, onitional on this information, the expete prie for the seon liense will be E 1 i [β 2 ( 1 i ) b 1 = β 1 ( )] = E 1 i [β 2 ( 1 i ) 1 i ] an, sine β 2 ( 1 1 ) > β 2 ( 1 ), for all 1 [, ], E 1 i [β 2 ( 1 i ) b 1 = β 1 ( )] < E 1 i [β 2 ( 1 i 1 i) 1 i ] = β 1 ( ), as require for the result. The super-martingale property implies that the (ex-ante) expete prie of the seon liense is lower than the expete prie of the first liense: E i, 1 i [ β2 ( 1 i ) ] = E i [ E 1 i [β 2 ( 1 i ) b 1 = β 1 ( )] ] < E i [ β 1 ( ) ]. 20

23 Hene, the information reveale in the proess of the sequential aution makes the expete pries erease. 4.3 Comparison of Aution Shemes The two aution shemes that we have examine, the simultaneous pay-your-bi aution an the sequential first-prie aution, have turne out to be alloation equivalent. The lienses are alloate to the two strongest firms, that is, to the firms with the lowest marginal osts. However, the manner in whih the firms bi in eah sheme is ifferent. In the simultaneous aution, the firms submit their bis without knowing the atual value of the lienses that they try to aquire. This value is etermine enogenously, by the marginal osts of the firms that will ompete in the market, an is reveale only at the en of the aution. In aition, the firms annot know whether, in ase they win one of the two lienses, they will fae a stronger or a weaker market ompetitor. Therefore, while biing, they nee to take both possibilities into aount. On the other han, in the sequential aution, the firms biing for the seon liense know its atual value, sine the marginal ost of the first oligopolist has been reveale by the winning bi in the first aution. Furthermore, in the aution for the first liense, the firms know that if they win, then they will fae a weaker market ompetitor. Therefore, they an bi more aggressively, sine they are protete from the more negative of the two possibilities. The following result shows that these informational ifferenes o not affet the revenue generate by the autioneer in the two shemes. Proposition 8 The simultaneous pay-your-bi aution an the sequential first-prie aution of two oligopoly lienses result in the same expete revenue for the autioneer. Proof: It is easy to show, by hanging the orer of integration in the efinition of β( ), that P[ 2 i ] β( ) = [1 F ( )] N 1 β 1 ( ) (N 1)[1 F ( )] N 2 F ( ) i β 2 ( 1 ) f(1 ) F ( ) 1. This means that the expete payments of a firm with marginal ost in the simultaneous aution, R D NS (), an in the sequential aution, R S NS (), are equal. 21

24 Therefore, sine this is true for any [, ], it follows that N R D NS( )f( ) = N R S NS( )f( ), so that the two aution shemes raise the same expete revenue. Hene, the autioneer is inifferent, with respet to the revenue he expets to raise, between the two aution shemes. Similarly, the biers are inifferent, with respet to the payments they expet to make, between the simultaneous an the sequential aution. The two aution formats, however, alloate eah of the two lienses at a ifferent prie. Proposition 9 The stronger of the two oligopolists pays a higher prie for his liense in the sequential first-prie aution than in the simultaneous pay-your-bi aution; the weaker oligopolist pays a lower prie for his liense in the sequential aution than in the simultaneous aution: β 1 ( ) > β( ) > E 1 i [ β 2 ( 1 i) 1 i < ]. Therefore, in the first aution of the sequential format, the firms bi more aggressively than in the simultaneous aution, knowing that if they win, they will neessarily fae a weaker market ompetitor. On the other han, the firms partiipating in the seon aution bi less aggressively, on average, sine they know that they will have to fae a stronger market ompetitor. Corollary 10 The stronger of the two oligopolists makes a higher total profit in the simultaneous payyour-bi aution. The weaker oligopolist makes a higher total profit in the sequential first-prie aution. Proof: Sine both aution formats result to the same market supply an profits, any hange in the firms total profits will be the onsequene of a hange in the pries that the firms pay for their lienses. Therefore, the result follows iretly from Proposition 9. Hene, an autioneer aiming at a more equal istribution of the wealth generate in the oligopolisti market will prefer the sequential first-prie aution to the simultaneous pay-your-bi aution. 22

25 5 Positive Signaling: Cournot Competition When signaling is possible, we nee to revise the firms valuations for the oligopoly lienses so as to inorporate to them the informational rents that the firms an extrat. In the ase of Cournot ompetition, in whih the signaling inentives are positive, the firms valuations shall be ajuste upwars. To emonstrate the nee for this ajustment, onsier an aution of a single liense to ompete against a monopolist with known marginal ost 1. When signaling is not possible, a firm i with marginal ost [, ] woul be willing to bi for the liense an amount up to π NS (, 1 ) = π(, 1 ). If signaling beomes possible, then, by mimiking a marginally stronger type <, firm i an inrease its market profit, in ase it wins the liense, by approximately π 2 (, 1 ) > 0. Therefore, the maximal amount that the firm woul be willing to bi exees π NS (, 1 ). Sine the effets of mimiking a ifferent type epen on the aution format an on the equilibrium strategies that the biers use, the manner in whih the firms ajust their valuations will also epen on these elements. Therefore, the firms valuations will be ifferent in eah aution environment that we onsier. Overall, uner positive signaling, a firm s eviation to signaling a stronger type will have two effets. First, assuming that the biing strategies are monotone, it will inrease the probability of aquiring a liense. Seon, it will inrease the profitability of the liense that the firm may win. Hene, to offset both these effets, the firms must bi more aggressively than they woul o if signaling were not possible. 5.1 Simultaneous Aution Suppose that all firms follow a stritly ereasing biing strategy b = β() an onsier firm i with marginal ost. If firm i mimis a type [, ] uring the aution, by biing b i = β( ), then its expete total payoff will be Π( ) = i π(, 1 ) (N 1)[1 F ( )] N 2 f( 1 ) 1 π(, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 P[ 2 i ] β( ). 23

26 The first-orer onition with respet to results in the equation {P[ 2 i ] β( )} = i π 2 (, 1 ) (N 1) [1 F ( )] N 2 f( 1 ) 1 π 2 (, 1 ) (N 1) [1 F ( 1 )] N 2 f( 1 ) 1 i π(, 1 ) (N 1)(N 2) [1 F ( )] N 3 f( )f( 1 ) 1, whih requires that any inrease in the firm s expete market profit from a eviation to, as this may be augmente by the expete gains from false signaling 24, must be offset by an inrease in the firm s expete payment in the aution. The ifferential equation erive from the first-orer onition, along with the bounary onition expressing the behavior of the weakest type, =, β( ) = provies the equilibrium strategy for this setting. π(, ) f(), Proposition 11 In the simultaneous pay-your-bi aution of two Cournot oligopoly lienses, in whih the winners bis are reveale at the en of the aution, there is a symmetri separating equilibrium given by the strategy β( ) = 2 π( 2 2, 1 ) (N 1)(N 2) [1 F (2 )] N 3 f( 2 ) f( 1 ) P[ 2 i ] π 2 ( 2 2, 1 ) (N 1) [1 F (2 )] N 2 f( 1 ) P[ 2 i ] π 2 ( 2 2, 1 ) (N 1) [1 F (1 )] N 2 f( 1 ) P[ 2 2 i ] In this equilibrium, the firm with the lowest marginal ost gains its liense at a prie below its ex-post value. The firm with the seon-lowest marginal ost, however, may gain its liense at a prie above its ex-post value. 24 In partiular, when < 1 i < 2 i, the hange in the firm s expete market profit is entirely the onsequene of false signaling. 24

27 Proof: Notie that the strategy β( ) an be expresse as β( ) = u( 2 ) (N 1)(N 2) [1 F (2 )] N 3 F ( 2 ) f( 2 ) P[ 2 i ] 2, where u() = π(, 1 ) f(1 ) F () 1 [ π 2 (, 1 ) 1 F () (N 2)f() ] f(1 ) F () 1 [ π 2 (, 1 ) 1 F () (N 2)f() ] [1 F (1 )] N 2 f( 1 ) [1 F ()] N 2 F () 1 for [, ], is the valuation of a firm with marginal ost, assuming that its market opponent is stronger an taking into aount the informational rents from false signaling. Therefore, the proof of this result parallels the one of Proposition 1, with u() in plae of v NS (). Its etails, in partiular, the argument establishing that u() is ereasing, an be foun in the Appenix. The first term in the biing strategy, β NS ( ) = 2 π( 2 2, 1 ) (N 1)(N 2) [1 F (2 )] N 3 f( 1 )f( 2 ) P[ 2 i ] 1 2, orrespons to the amount that a firm with marginal ost woul bi, if signaling were not possible. The seon an thir terms, 25 β S ( ) = 2 π 2 ( 2 2, 1 ) (N 1)[1 F (2 )] N 2 f( 1 ) P[ 2 i ] 1 2 π 2 ( 2 2, 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) P[ 2 2 i ] 1 2, 25 The two ouble integrals o not allow, of ourse, for the possibility of 1 i > 2 i. Rather, in eah ase, the outer integral etermines a value 2 [, ] suh that 2 2 i, while 1 i 2 i. This reates two possibilities, namely, either 1 i 2 2 i, orresponing to the first signaling term, or 2 1 i 2 i, orresponing to the seon signaling term. 25

28 orrespon to the amount by whih the firms shoul augment their bis so as to offset possible gains from false signaling by the other firms. Even though there an be no false signaling in equilibrium, without this amount, it woul be possible for a firm to eviate into mimiking a stronger type an, therefore, to inrease both the probability of winning a liense an, through false signaling, the value of that liense. 5.2 Sequential Aution In the sequential aution, the winning bi in the first roun reveals the marginal ost 1 of the strongest oligopolist. Therefore, similarly to the ase in whih signaling is not possible, in the seon roun, the firms know preisely the value of the liense for whih they bi. In aition, they know that the other firms marginal osts are boune below by 1 ; thus, they upate their beliefs, so that F () = F () F (1 ). 1 F ( 1 ) Sine the privately known first-perio bis o not affet the firms inentives in the seon aution, our seon-perio biing environment belongs to the lass of autions stuie by Das Varma [10], Goeree [15] an Katzman an Rhoes-Kropf [32]. In the following Lemma, we apply their analysis to our setting: Lemma 12 Suppose that N 1 firms, whose marginal osts are i.i.. aoring to the istribution funtion F (.) on [, ], ompete in a first-prie aution for a liense to partiipate in a Cournot oligopoly against a firm with known marginal ost 1. In aition, suppose that the firms believe that the unknown marginal osts are boune below by the value 1 [, ]. Then, assuming that the winner s bi is reveale at the en of the aution, the following strategy onstitutes a symmetri equilibrium: For a marginal ost 1, firm i bis β 2 ( 1 ) = [ ] π(, 1 ) π 2 (, 1 1 F () (N 2)[1 F ()] N 3 f() ), (N 2)f() [1 F ( )] N 2 while for a marginal ost < 1, firm i bis b 2 = β 2 ( 1 1 ). In aition, for any 1 [, ], the strategy β 2 ( 1 ) is stritly ereasing in ( 1, ], so that, along the equilibrium path, an aution prie b 2 < β 2 ( 1 1 ) fully reveals the marginal ost of the winning firm. 26

29 Every firm i submits a bi that is equal to the value that its strongest ompetitor is expete to have for the liense, assuming that this ompetitor has marginal ost, β 2 NS( 1 ) = π(, 1 ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2, augmente by the amount neee to offset possible gains from false signaling by its ompetitors, namely, β 2 S( 1 ) = π 2 (, 1 ) [1 F ()]N 2. [1 F ( )] N 2 Without this amount, it woul be possible for a firm i to eviate into mimiking < an to inrease both the probability of winning a liense an, through false signaling, the value of that liense. For the seon gain to be offset, eah firm nees to bi above β 2 NS ( 1 ), by an amount at least as large as β 2 S ( 1 ). To ensure that the strategy β 2 ( ) is ereasing with respet to the marginal ost, we will nee to moify Assumption 1 in the following manner: Assumption 2 The istribution of the firms marginal osts satisfies the inequality { } [1 F ()] N 2 v2 (, ) 1 F (i ) sup [1 F ( )] N 2 ṽ 1 (, ) (N 2) f( ), for all [, ], where ṽ 1 (, 1 ) = [π(, 1 )] [π 2(, 1 )] 1 F () (N 2) f(). For the Cournot uopoly that we have esribe, { } v2 (, ) sup ṽ 1 (, ) { } π1 (, ) = sup, π 2 (, ) so that Assumption 2 reues to Assumption 1. U[, ], the assumption is always satisfie. In partiular, for marginal osts Corollary 13 Uner Assumption 2, the funtion β 2 ( ) is ereasing in [, ]. 27

30 In the first aution, arguing in the same manner as in the non-signaling ase, suppose that all firms follow a stritly monotone biing strategy β 1 () an onsier firm i with marginal ost. By biing b i = β( ) for, firm i expets a total payoff Π( ) = [ π(, 1 ) β 1 ( ) ] (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 i [ π(, 1 ) β 2 ( 1 ) ] (N 1)[1 F ( )] N 2 f( 1 ) 1, while by biing b i = β( ) for, firm i expets a total payoff Π( ) = [ π(, 1 ) β 1 ( ) ] (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 i i [ π(, 1 ) β 2 ( 1 ) ] (N 1)[1 F ( )] N 2 f( 1 ) 1 [ π( 1, 1 ) β 2 ( 1 1 ) ] (N 1)[1 F ( 1 )] N 2 f( 1 ) 1. In both ases, the neessary first-orer onition at the enpoint = results in the ifferential equation { [1 F ( )] N 1 β 1 ( )} = π 2 (, 1 ) (N 1) [1 F ( 1 )] N 2 f( 1 ) 1 β 2 ( ) (N 1) [1 F ( )] N 2 f( ). By solving this ifferential equation, along with the bounary onition we get the strategy β 1 ( ) = β 1 ( ) = π(, ), β 2 ( 1 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 1 2, for the equilibrium of the sequential aution. 28

31 Proposition 14 In a sequential first-prie aution of two Cournot oligopoly lienses, in whih the winners bis are reveale at the en of eah aution, the following strategy profile onstitutes a symmetri separating equilibrium: In the first aution, eah firm i bis β 1 ( ) = 1 π( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (2 )] N 2 f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N In the seon aution, if firm i has a marginal ost 1 = (β 1 ) 1 (b 1 ), where b 1 is the prie at whih the first liense was sol, then it bis β 2 ( 1 ) = π( 2 2, 1 ) (N 2)[1 F (2 )] N 3 f( 2 ) [1 F ( )] N 2 2 π 2 ( 2 2, 1 ) [1 F (2 )] N 2 [1 F ( )] N 2 2, while with a marginal ost < 1, it bis b 2 = β 2 ( 1 1 ). Proof: Notie that the strategy β 1 ( ) an be expresse as β( ) = v( 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 1, where v() = π(, 1 ) (N 2) [1 F (1 )] N 3 f( 1 ) [1 F ()] N 2 1 [ π 2 (, 1 ) [ π 2 (, 1 ) 1 F () (N 2)f() ] (N 2) [1 F (1 )] N 3 f( 1 ) 1 [1 F ()] N 2 1 F () (N 2)f() ] (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ()] N 1 1, 29

32 for [, ], is the valuation of a firm with marginal ost, assuming that its market opponent is weaker. Therefore, the proof of this result parallels the one of Proposition 11, with v() in plae of u(). Its etails an be foun in the Appenix. Similarly to the ase of non-signaling, the equation efining the biing strategy β 1 ( ) is a non-arbitrage onition for the firm with the lowest marginal ost,. For this firm to be inifferent between winning the first or the seon aution, its bi in the first roun must exee its expete bi in the seon roun by preisely its expete gain from signaling a stronger type. The first term of the biing strategy for the first aution, β 1 NS( ) = 1 π( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 1 )f( 2 ) [1 F ( )] N 1 2 1, orrespons, again, to the amount that firm i woul bi if signaling were not possible. The remaining two terms, β 1 S( ) = 1 π 2 ( 2 2, 1 ) (N 1)[1 F (2 )] N 2 f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 2 1, orrespon to the amounts that the firms must a to their bis in orer to offset possible gains from false signaling by their ompetitors. Beause of the information reveale in the first roun of the sequential aution, the weaker of the two oligopolists is able to avoi the possibility of winning his liense at a prie above its ex-post value. Thus, along the equilibrium path, both oligopolists make a positive profit. In aition, the revelation of the marginal ost of the strongest firm makes the seon liense less profitable an, therefore, the firms biing for it less aggressive. As a result, the pries of the two lienses form a super-martingale, E 1 i [β 2 ( 1 i ) ] β 1 ( ), so that the expete prie of the seon liense is lower than that of the first liense. 30

33 5.3 Comparison of Aution Shemes Our analysis of the simultaneous an the sequential autions uner positive signaling parallels the analysis of the same autions without signaling. Non-surprisingly, so o the results regaring the omparison of the equilibria that we erive. Proposition 15 The simultaneous pay-your-bi aution an the sequential first-prie aution of two Cournot oligopoly lienses result in the same expete revenue for the autioneer. Therefore, the informational ifferenes between the two aution shemes o not affet the expete revenue of the autioneer or the expete payment of the biers. Proposition 16 The stronger of the two oligopolists pays a higher prie for his liense in the sequential first-prie aution than in the simultaneous pay-your-bi aution; the weaker oligopolist pays a lower prie for his liense in the sequential aution than in the simultaneous aution: β 1 ( ) > β( ) > E 1 i [ β 2 ( 1 i) 1 i < ]. Therefore, the stronger of the two oligopolists makes a higher total profit in the simultaneous pay-your-bi aution while the weaker oligopolist makes a higher total profit in the sequential first-prie aution. Hene, an autioneer aiming at a more equal istribution of the wealth generate in the Cournot uopoly will still prefer the sequential aution to the simultaneous one, even when signaling is possible. 31

34 6 Negative Signaling: Bertran Competition In the ase of Bertran ompetition, the firms have an inentive to signal a weaker type. Therefore, opposite to the ase of Cournot ompetition, the firms valuations shall be ajuste ownwars. Beause of this ajustment, if the firms signaling inentive is too strong, it is possible that a positive measure of bier types will have valuations below zero. To avoi this problem, we nee to assume the presene of a large number of firms ompeting in the aution. Uner this assumption, we an onstrut an equilibrium in stritly monotone biing strategies for the simultaneous aution. On the other han, in the sequential aution, sine it is not possible to balane the biers signaling profits from eviating into waiting for the seon roun, suh an equilibrium turns out not to exist. 6.1 Simultaneous Aution By repeating the argument that we use for the Cournot oligopoly, that is, by assuming the use of a stritly ereasing biing strategy b = β() an onsiering the neessary first-orer onition for the expete payoff funtion Π( ) of some firm i at =, we an erive the equilibrium for this setting. Proposition 17 In the simultaneous pay-your-bi aution of two Bertran oligopoly lienses, with the winners bis reveale at the en of the aution, if there are suffiiently many biers, then there is a symmetri separating equilibrium given by the strategy β( ) = 2 π( 2 2, 1 ) (N 1)(N 2) [1 F (2 )] N 3 f( 2 ) f( 1 ) P[ 2 i ] π 2 ( 2 2, 1 ) (N 1) [1 F (2 )] N 2 f( 1 ) P[ 2 i ] π 2 ( 2 2, 1 ) (N 1) [1 F (1 )] N 2 f( 1 ) P[ 2 2 i ] In this equilibrium, the firm with the lowest marginal ost gains its liense at a prie below its ex-post value. The firm with the seon-lowest marginal ost, however, may gain its liense at a prie above its ex-post value. Notie that by unerstating its strength, a firm gains in terms of its expete market profit an of a lower payment in the aution, assuming that it wins an oligopoly liense. On the other han, it suffers the ost of a lower probability of winning the aution. This 32

35 ost inreases as the number of the biers in the aution, N, beomes larger. Therefore, if N is suffiiently large, the ost is so severe that it an always ounter-balane possible gains from false signaling. 6.2 Sequential Aution In the sequential aution, the firms inentive to signal a weaker type turns out to be too strong. Contrary to the ase of the simultaneous aution, it is not possible to onstrut a symmetri separating equilibrium. Proposition 18 In a sequential first-prie aution of two Bertran oligopoly lienses, in whih the winners bis are reveale at the en of eah aution, there is no symmetri equilibrium in monotone strategies. In the presene of a suffiiently large number of biers, as shown in Das Varma [10], the strategy β 2 ( 1 ), given in Lemma 12, forms the unique symmetri equilibrium for the ontinuation game that follows the alloation of the first liense to a firm with marginal ost 1 [, ]. In aition, by aapting Assumption 2 to the Bertran oligopoly setting, one an show that β 2 ( ) is ereasing in. Finally, by repliating the argument leaing to Proposition 14, one an erive the biing strategy β 1 ( ), iential to the one use in the Cournot oligopoly, as the unique solution to the neessary first-orer onition. This strategy, however, annot be part of an equilibrium. Although, the non-signaling omponent of β 2 ( 1 ) is suffiiently more aggressive than the non-signaling omponent of β 1 ( ), so that to just eliminate the inentive to wait for the seon roun (if signaling were not possible), the signaling omponent of β 2 ( 1 ) annot ounter-balane the orresponing omponent of β 1 ( ). As a result, eah firm has a profitable eviation from β 1 into waiting for the seon roun. In partiular, trying to iminish the potential gains from signaling by inreasing the number of firms, as in the ase of the simultaneous aution, annot proue any result. The eviation into waiting for the seon roun oes not ost any firm in terms of the probability of aquiring a liense, so, hanging the number of firms is ineffetive. 33

36 7 Conlusion We have examine two multi-unit aution shemes with alloative an, possibly, informational externalities, in partiular, two autions of oligopoly lienses. When there is no signaling, we have provie a rationale for the use of a sequential proeure. The information release uring this proeure leas to more informative biing. Even though this oes not affet the seller s expete revenue, or the biers expete payments, the two winners are protete from the possibility of regret, that is, from buying a liense at a prie that exees its ex-post value. In aition, the strongest oligopolist has to pay a higher prie for his liense than he woul pay in a simultaneous aution, whereas the weaker oligopolist pays a lower prie. Therefore, the sequential aution results in a more even istribution of the wealth generate in the oligopoly. When signaling is possible, these results remain vali only in the ase of positive signaling inentives, as in the Cournot oligopoly. On the other han, with negative signaling inentives, as in the Bertran oligopoly, there is no symmetri monotone equilibrium for the sequential aution. Hene, in this environment, an effiient alloation is ahieve only by means of a simultaneous aution. The two aution formats will ease to be revenue equivalent, if we onsier affiliate marginal osts. In this ase, aoring to the intuition of the linkage priniple, the sequential format will ominate, in terms of revenue, the simultaneous aution. The two autions will also generate ifferent expete seller revenues, if the firms fae partiipation osts. In partiular, if the winning bi in the first roun of the sequential aution is suffiiently low, then some bier types that woul otherwise not partiipate may eie to bi in the seon roun. In this ase, however, the aution sheme that is preferable for the seller may epen on the istribution of the firms marginal osts. A seller may also inrease his expete revenue by aopting ifferent information revelation rules an, therefore, allowing for ifferent signaling possibilities. Aoring to the intuition erive from the stuy of the aution of a single liense, shemes that reveal more information about the winners will be revenue ominant in the ase of positive signaling inentives, while shemes that isable signaling will be ominant in the ase of negative signaling inentives. Finally, it woul be interesting to investigate experimentally the biing behavior in a sequential aution with negative informational externalities. An experimental stuy may reveal patterns of behavior that an be of interest to sellers that woul like to onsier the use of a sequential aution sheme. These extensions are the subjet of future researh. 34

37 Appenix: Proof of Results Proof of Proposition 1: It is straightforwar to verify that the funtion β( ) is a solution to the ifferential equation that resulte from the neessary first-orer onition. In aition, by using L Hospital s rule, it is easy to hek that lim β() = π NS (, ) f(), as require by the bounary onition. Sine the equation that proue the strategy β( ) was only a neessary onition, we still nee to establish that it is optimal for any bier i with marginal ost to bi b i = β( ), if all other biers follow this biing strategy. Suppose that firm i bis b i = β( ), for [, ] while having a marginal ost. Then, by hanging its bi marginally, that is, by mimiking a marginally ifferent type, it an hange its expete payoff by Π ( ) = { P[ 2 i ] β( ) } i π NS (, 1 ) (N 1)(N 2) [1 F ( )] N 3 f( )f( 1 ) 1 Substituting the expression for β( ) results in Π ( ) = i i π NS (, 1 ) (N 1)(N 2)[1 F ( )] N 3 f( )f( 1 ) 1 π NS (, 1 ) (N 1)(N 2)[1 F ( )] N 3 f( )f( 1 ) 1 Sine the funtion π NS (, 1 ) is ereasing in the marginal ost, the hange in the firm s expete payoff is Π ( ) > 0, for < ; = 0, for = ; < 0, for >, showing that the firm s expete profit Π( ) attains its maximum at =. 35

38 To show that the strategy β( ) is ereasing, we an alulate its erivative to be β ( ) = P[ 2 i ] P[ 2 i ] [ v( ) v( 2 ) ] P[ 2 2 i 2 ] P[ 2 i 2, i] where v NS () = π NS (, 1 ) f(1 ) F () 1 for [, ], is the expete market profit of the strongest non-winning firm. Therefore, if the funtion v() is ereasing, we an onlue that β ( ) < P[ 2 i ] P[ 2 i ] [ v( ) v( ) ] = 0, as require for the strategy β( ) to be ereasing. By ifferentiating the funtion v NS (), we get v NS () = π NS (, ) f() F () π NS 1 (, 1 ) f(1 ) F () 1 π NS (, 1 ) f(1 ) F () 1 f() F (), an, after integrating the last term by parts, v NS () = π NS 1 (, 1 ) f(1 ) F () 1 π NS 2 (, 1 ) F (1 ) F () 1 f() F (). Sine the firms marginal osts are istribute in a logonave manner, the expression f()/f () is ereasing. Therefore, v NS () [ π NS 1 (, 1 ) π NS 2 (, 1 ) ] f(1 ) F () 1, 36

39 an, sine π NS 1 π NS 2 < 0, we onlue that ompleting the argument. v NS () < 0, Proof of Corollary 2: Suppose that firm i has the lowest marginal ost among all firms, namely, [, ]. Then, in equilibrium, it will win one of the two lienses, at a prie β( ), for a market profit π NS (, 1 ), where 1 is the lowest ompeting marginal ost. Its overall payoff, therefore, will be π NS (, 1 ) β( ). Sine the funtion π NS (, 1 ) is inreasing in 1, so, it suffies to show that π NS (, 1 ) β( ) π NS (, ) β( ), P[ 2 i ] {π NS (, ) β( )} 0. Notie that this last inequality is true for the bounary value =. Furthermore, { P[ 2 i ] [π NS (, ) β( )] } = (N 1)(N 2)[1 F ( )] N 3 F ( )f( ) π NS (, ) { [1 F ( )] N 1 (N 1)[1 F ( )] N 2 F ( ) } [π NS (, )] i π NS (, 1 ) (N 1)(N 2) [1 F ( )] N 3 f( )f( 1 ) 1 By integrating the last term by parts, this erivative beomes { P[ 2 i ] [π NS (, ) β( )] } = { [1 F ( )] N 1 (N 1)[1 F ( )] N 2 F ( ) } [π NS (, )] i π NS 2 (, 1 ) (N 1)(N 2) [1 F ( )] N 3 F ( ) f( 1 ) 1. Sine both terms are negative, it follows that 37

40 { P[ 2 i ] [π NS (, ) β( )] } 0, proving firm i s realize payoff to be always positive. To show that the firm with the seon-lowest marginal ost may win a liense at a prie above its ex-post value, onsier a firm with marginal ost =. In equilibrium, suh a firm will bi β( ) = π NS (, ) f(), the expete value of the liense, given that N 2 firms have marginal ost equal to. If the firm wins the liense, then, in the market, it may fae an opponent with marginal ost. In this ase, it will make a market profit π NS (, ) π NS (, ), for all [, ]. Therefore, π NS (, ) < for a negative overall profit. π NS (, ) f(), Proof of Lemma 4: The erivative of the funtion β 2 ( ) with respet to the variable is equal to [ β 2 ( ) ] = π NS 1 (, ) w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2, where w(, ) = πns 2 (, ) π1 NS (, ) f()/[1 F ( )] f()/[1 F ()]. The expression w(, ) is negative for =, positive for =, ontinuous an inreasing with respet to [, ]. Hene, there exists a value = ( ) (, ) suh that w(, ) < 0, for [, ); = 0, for = ; > 0, for (, ]. It follows that 38

41 [ β 2 ( ) ] < π NS 1 (, ) w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 an, sine Assumption 1 implies that w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 0, we an onlue that β 2 ( ) is ereasing. Proof of Proposition 5: Suppose that all firms follow the biing strategy (β 1, β 2 ) an onsier firm i with marginal ost [, ]. The optimality of biing β 2 ( 1 ) in the seon aution, following the sale of the first liense at a prie b 1 orresponing to a marginal ost 1 = (β 1 ) 1 (b 1 ), has been establishe in Lemma Therefore, we only nee to examine the optimality of biing β 1 ( ) in the first aution. Obviously, firm i annot gain from submitting a bi above β 1 () or below β 1 ( ). So, suppose that it mimis a type [, ], that is, it bis β 1 ( ). If, then, by hanging its bi marginally, firm i will hange its expete payoff by Π ( ) = π NS (, ) (N 1)[1 F ( )] N 2 f( ) π NS (, ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ) { [1 F ( )] N 1 β 1 ( ) }. After substituting the appropriate expression for the last term, the hange in the expete payoff of firm i beomes 26 In ase the first liense is sol at a prie b 1 > β 1 (), an event outsie the equilibrium path, we an assume that the remaining firms attribute a marginal ost 1 = to the winner of the liense. Similarly, for b 1 < β 1 ( ), we an assume that 1 =. 39

42 Π ( ) = π NS (, ) (N 1)[1 F ( )] N 2 f( ) π NS (, ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ). The ifferene between the last two terms equals to β 2 ( ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ) = i an sine π NS (, 1 ) < π NS ( 2, 1 ) for all 2 [, ], π NS ( 2, ) (N 2)[1 F ( 2 )] N 3 f( 2 ) 2 (N 1)f( ) β 2 (, ) (N 1)[1 F ( )] N 2 f( ) β 2 (, ) (N 1)[1 F ( )] N 2 f( ) > π NS (, ) (N 1)[1 F ( )] N 2 f( ) π NS (, ) (N 1)[1 F ( )] N 2 f( ). Hene, we an onlue that with equality only when =. Π ( ) 0, Similarly, if firm i mimis a marginal ost, then, by hanging its bi marginally, it will hange its expete payoff by Π ( ) = β 2 (, ) (N 1)[1 F ( )] N 2 f( ) { [1 F ( )] N 1 β 1 ( ) }. 40

43 After substituting the seon term, we get with equality only when =. Π ( ) 0, We have therefore shown that the erivative of the firm s expete profit is Π ( ) > 0, for < ; = 0, for = ; < 0, for >, as require for the firm s expete profit Π( ) to attain its maximum at =. To show that the biing strategy β 1 ( ) is stritly ereasing, notie that we an write its erivative as β 1 ( ) = (N 1)f( [ i) β 2 ( ) β 2 ( 2 2 ) (N 1)[1 F ] (2 )] N 2 f( 2 ) 2 1 F ( ) [1 F ( )] N 1 an, sine Assumption 1 implies that β 2 ( ) is ereasing, we an onlue that β ( ) < (N 1)f() 1 F ( ) [ β 2 ( ) β 2 ( ) ] = 0, as require for the biing strategy β 1 to be stritly ereasing. Proof of Proposition 9: First, notie that by rearranging the terms of the equation relating the biing strategies β( ), β 1 ( ) an β 2 ( 1 ), given in the proof of Proposition 8, we get [1 F ( )] N 1 [ β 1 ( ) β( ) ] = (N 1)[1 F ( )] N 2 F ( ) [ i ] β( ) β 2 ( 1 ) f(1 ) F ( ) 1. Therefore, for the entire result, it suffies to show that β 1 ( ) > β( ). 41

44 Using the efinitions of the strategies β( ) an β 1 ( ), we an show, by means of a iret alulation, that β 1 ( ) > β( ) if an only if i 1 π NS ( 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) [1 F ( )] N > π NS ( 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) (N 1)[1 F ( )] N 2 F ( ) 2 1. That is, we shall show that E 1 i, 2 i [ πns ( 2, 1 ) 2 i 1 i ] > E 1 i, 2 i [ πns ( 2, 1 ) 2 i 1 i ]. Sine the market profit funtion π NS ( 2, 1 ) is inreasing in 1, we have E 1 i, 2 [ πns ( 2 i i, 1 i) 2 i 1 i ] = E 2 i [ E 1 i [π NS ( 2 i, 1 i) 1 i [, 2 i] ] 2 i ] > E 2 i [ E 1 i [π NS ( 2 i, ) 1 i [, 2 i] ] 2 i ] = E 2 i [ π NS ( 2 i, ) 2 i ] = E 2 i [ E 1 i [π NS ( 2 i, ) 1 i [, ] ] 2 i ] > E 2 i [ E 1 i [π NS ( 2 i, 1 i) 1 i [, ] ] 2 i ] = E 1 i, 2 [ πns ( 2 i i, 1 i) 2 i 1 i ], as require for the result. Proof of Proposition 11: It is straightforwar to verify that the funtion β( ) is a solution to the ifferential equation that resulte from the neessary first-orer onition. In aition, by using L Hospital s rule, it is easy to hek that as require by the bounary onition. lim β() = π(, ) f(), 42

45 Sine the equation that proue the strategy β( ) was only a neessary onition, we still nee to establish that it is optimal for any bier i with marginal ost to bi b i = β( ), if all other biers follow this biing strategy. Suppose that firm i bis b i = β( ), for [, ] while having a marginal ost. 27 Then, by hanging its bi marginally, that is, by mimiking a marginally ifferent type, it an hange its expete payoff by Π ( ) = { P[ 2 i ] β( ) } i i π(, 1 ) (N 1)(N 2) [1 F ( )] N 3 f( )f( 1 ) 1 π 2 (, 1 ) (N 1) [1 F ( )] N 2 f( 1 ) 1 π 2 (, 1 ) (N 1) [1 F ( 1 )] N 2 f( 1 ) 1. Substituting β( ) an gathering the orresponing terms together result to Π ( ) = i i [π(, 1 ) π(, 1 )] (N 1)(N 2)[1 F ( )] N 3 f( )f( 1 ) 1 [π 2 (, 1 ) π 2 (, 1 )] (N 1) [1 F ( )] N 2 f( 1 ) 1 [π 2 (, 1 ) π 2 (, 1 )] (N 1) [1 F ( 1 )] N 2 f( 1 ) 1. Sine the funtions π(, 1 ) an π 2 (, 1 ) are ereasing in, the hange in the firm s expete profit is Π ( ) > 0, for < ; = 0, for = ; < 0, for >, showing that the firm s expete profit Π( ) attains its maximum at =. To show that the strategy β( ) is ereasing, we an alulate its erivative to be 27 If a liense is sol at a prie b > β(), an event outsie the equilibrium path, we an assume that the remaining firms attribute a marginal ost = to the winner of that liense. Similarly, for b < β( ), we an assume that =. In either ase, no firm an profit from mimiking a type [, ]. 43

46 β ( ) = P[ 2 i ] P[ 2 i ] [ v( ) v( 2 ) ] P[ 2 2 i 2 ] P[ 2 i 2, i] where v( ) = i i π(, 1 ) f(1 ) F ( 1 ) 1 [ ] π 2 (, 1 1 F ( ) ) (N 2)f( ) [ ] π 2 (, 1 1 F ( ) ) (N 2)f( ) f(1 ) F ( 1 ) 1 [1 F (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1. Therefore, if the funtion v( ) is ereasing, we an onlue that β ( ) < P[ 2 i ] P[ 2 i ] [ v( ) v( ) ] = 0 For the monotoniity of the funtion v( ), it suffies to show that eah of the three terms in its efinition is ereasing. We emonstrate the result for the thir term only, sine the argument for the first two integrals is similar. By rewriting this term as [ ] π 2 (, 1 [1 F ( )] 2 ) (N 2)F ( )f( ) we an alulate its erivative to be [1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 1, { [ ] π 2 (, 1 1 F ( ) ) (N 2)f( ) 1 F ( ) π 2 (, ) (N 2)f( ) [ π 2 (, 1 ) [ π 2 (, 1 ) [1 F } (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1 ] [1 F ( )] 2 (N 2)F ( )f( ) ] 1 F ( ) (N 2)F ( ) = [1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 1 (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N

47 By integrating the last term by parts, we get { [ π 2 (, 1 ) 1 ] 1 F ( ) (N 2)f( ) [ π 2 (, 1 ) [ π 2 (, 1 ) [1 F } (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1 ] [1 F ( )] 2 (N 2)F ( )f( ) 1 F ( ) (N 2)F ( ) = [1 F (1 )] N 2 f( 1 ) 1 [1 F ( )] N 1 ] [1 F (1 )] N 1 [1 F ( )] N 1 1, an, sine the assumption of the ereasing inverse hazar ratio 1 F () f( ) term implies that the [1 F ( )] 2 (N 1)(N 2)F ( )f( ) = [1 F ()] (N 1)F ( ) [1 F ( )] (N 2)f( ) is also ereasing, we an rop a negative term from the first integral, so as to get { [ ] π 2 (, 1 1 F ( ) ) (N 2)f( ) [1 F } (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1 [ π 2 (, 1 [1 F ( )] 2 ) ] (N 2)F ( )f( ) [1 F (1 )] N 2 f( 1 ) 1 [1 F ( )] N 1 [ π 2( 1 i, 1 1 F ( ) ) ] (N 2)F ( ) [1 F (1 )] N 1 [1 F ( )] N 1 1. Using again the assumption of the ereasing inverse hazar rate, { [ ] π 2 (, 1 1 F ( ) ) (N 2)f( ) [ π 2 (, 1 ) ] [1 F } (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1 1 F ( ) (N 2)F ( ) [1 F (1 )] N 1 [1 F ( )] N 1 1 [ π 2( 1 i, 1 1 F ( ) ) ] (N 2)F ( ) [1 F (1 )] N 1 [1 F ( )] N 1 1. Finally, sine [ π 2 (, 1 ) ] 1 [ π 2(, 1 ) ] = [ π 21 (, 1 ) π 22 (, 1 ) π 23 (, 1 ) ] < 0, 45

48 for all marginal osts [, ] an all 1 [, ], we an onlue that the erivative is negative, as esire. The argument establishing that the stronger of the two oligopolists always makes a positive profit, unlike the weaker oligopolist who may regret his partiipation to the market, is iential to that in the proof of Corollary 2, with π(, 1 ) in plae of π NS (, 1 ). It is therefore omitte. Proof of Lemma 12: For arbitrary 1 [, ], suppose that all firms follow the biing strategy β 2 (. 1 ) an onsier firm i with marginal ost [ 1, ]. Obviously, firm i annot profit by biing above β 2 ( 1 1 ) or below β 2 ( 1 ). 28 So, suppose that firm i mimis a type [ 1, ], that is, it bis β 2 ( 1 ). Then its expete payoff will be Π( ) = [1 F ( )] N 2 [ π(, 1 ) β 2 ( 1 ) ]. By hanging its bi marginally, firm i will hange its expete payoff by Π ( ) = π(, 1 ) (N 2)[1 F ( )] N 3 f( i ) π 2 (, 1 ) [1 F ( )] N 2 { [1 F ( )] N 2 β 2 ( 1 ) }. i By alulating the erivative in the last term, we get Π ( ) = [ π(, 1 ) π(, 1 ) ] (N 2)[1 F ( )] N 3 f( i ) [ π 2 (, 1 ) π 2 (, 1 ) ] [1 F ( )] N 2. Sine both the profit funtion π(, 1 ) an the erivative π 2 (, 1 ) are ereasing in, it follows that Π ( ) > 0, for < ; = 0, for = ; < 0, for >, 28 If the liense is sol at a prie b 2 > β 2 ( 1 1 ), an event outsie the equilibrium path, we an assume that the remaining firms, in partiular, the ompeting oligopolist, will attribute a marginal ost 2 = 1 to the winner of the liense. Similarly, for b 2 < β 2 ( 1 ), we an assume that 2 =. 46

49 as require for the optimality of biing β 2 ( 1 ). In aition, when firm i has marginal ost [, 1 ], the previous analysis shows that Π 1 ( ) < 0, for all [ 1, ]. Hene, firm i is best-off biing β 2 ( 1 1 ). Finally, for the monotoniity of the strategy β 2 ( 1 ), notie that, sine the inverse hazar rate [1 F ()]/f() is ereasing, the expression v(, 1 ) = π(, 1 ) π 2 (, 1 ) is ereasing in. Therefore, the erivative β 2 ( 1 ) = v(, 1 ) (N 2)f() 1 F ( ) 1 F () (N 2)f() v(, 1 ) (N 2)[1 F ()]N 3 f() (N 2)f() [1 F ( )] N 2 1 F ( ) is negative, showing that the strategy β 2 ( 1 1 ) is stritly ereasing in ( 1, ]. Proof of Lemma 13: The erivative of β 2 ( ) equals to [ β 2 ( ) ] = v(, ) (N 2)f(), 1 F ( ) v 2 (, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 v(, ) (N 2)[1 F ()]N 3 f() (N 2)f(), [1 F ( )] N 2 1 F ( ) or, after integrating the last term by parts, to [ β 2 ( ) ] = v 2 (, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 [1 F ()]N 2 v 1 (, ) [1 F ( )] (N 2)f(). N 2 1 F ( ) For all [, ], sine the inverse hazar ratio [1 F ()]/f() is ereasing, we have 47

50 [ β 2 ( ) ] < v 2 ( ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 [1 F ()]N 2 ṽ 1 ( ) [1 F ( )] (N 2)f(), N 2 1 F ( ) where Sine the expression ṽ 1 ( 1 ) = π (, 1 ) π 2 (, 1 1 F ( ) ) (N 2)f( ). w(, ) = v 2( ) ṽ 1 ( ) f()/[1 F ( )] f()/[1 F ()] = 1 2 f()/[1 F ( )] f()/[1 F ()] is negative for =, positive for =, ontinuous an inreasing with respet to [, ], there exists a value = ( ) (, ) suh that w(, ) < 0, for [, ); = 0, for = ; > 0, for (, ]. Therefore, sine the funtion ṽ 1 (, 1 ) is ereasing in [, ] an positive, we have [ β 2 ( ) ] < < ṽ 1 ( ) ṽ 1 ( ) w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 ṽ 1 ( ) w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 Assumption 2 implies that w(, ) (N 2)[1 F ()]N 3 f() [1 F ( )] N 2 0, whih suffies for β 2 ( ) to be ereasing. 48

51 Proof of Proposition 14: Suppose that all firms follow the biing strategy (β 1, β 2 ) an onsier firm i with marginal ost [, ]. The optimality of biing β 2 ( 1 ) in the seon aution, following the sale of the first liense at a prie b 1 orresponing to a marginal ost 1 = (β 1 ) 1 (b 1 ), has been establishe in Lemma Therefore, we only nee to examine the optimality of biing β 1 ( ) in the first aution. Obviously, firm i annot gain from submitting a bi above β 1 () or below β 1 ( ). So, suppose that it mimis a type [, ], that is, it bis β 1 ( ). If, then, by hanging its bi marginally, firm i will hange its expete payoff by Π ( ) = π(, ) (N 1)[1 F ( )] N 2 f( ) π(, ) (N 1)[1 F ( )] N 2 f( ) π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1, β 2 ( ) (N 1)[1 F ( )] N 2 f( ) { [1 F ( )] N 1 β 1 ( ) }. After substituting the appropriate expression for the last term, the hange in the expete payoff of firm i beomes Π ( ) = π(, ) (N 1)[1 F ( )] N 2 f( ) π(, ) (N 1)[1 F ( )] N 2 f( ) π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 β 2 ( ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ). The ifferene between the last two terms equals to 29 In ase the first liense is sol at a prie b 1 > β 1 (), an event outsie the equilibrium path, we an assume that the remaining firms attribute a marginal ost 1 = to the winner of the liense. Similarly, for b 1 < β 1 ( ), we an assume that 1 =. 49

52 β 2 ( ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ) = i π( 2 2, ) (N 2)[1 F ( 2 )] N 3 f( 2 ) 2 (N 1)f( ) i π 2 ( 2 2, ) [1 F ( 2 )] N 2 2 (N 1)f( ). Sine π(, 1 ) is ereasing in, we have β 2 ( ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ) > i π( 2, ) (N 2)[1 F ( 2 )] N 3 f( 2 ) 2 (N 1)f( ) i π 2 ( 2 2, ) [1 F ( 2 )] N 2 2 (N 1)f( ) an, by integrating the first term by parts, β 2 ( ) (N 1)[1 F ( )] N 2 f( ) β 2 ( ) (N 1)[1 F ( )] N 2 f( ) > π(, ) (N 1)[1 F ( )] N 2 f( ) π(, ) (N 1)[1 F ( )] N 2 f( ) i π 2 ( 2, ) [1 F ( 2 )] N 2 2 (N 1)f( ) i π 2 ( 2 2, ) [1 F ( 2 )] N 2 2 (N 1)f( ). Hene, we get 50

53 Π ( ) π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 i π 2 ( 2, ) [1 F ( 2 )] N 2 2 (N 1)f( ) i π 2 ( 2 2, ) [1 F ( 2 )] N 2 2 (N 1)f( ) an, sine the erivative π 2 (,, 1 ) is ereasing in, we an onlue that with equality only when =. Π ( ) 0, Similarly, if firm i mimis a marginal ost, then, by hanging its bi marginally, it will hange its expete payoff by Π ( ) = π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1, β 2 ( ) (N 1)[1 F ( )] N 2 f( ) { [1 F ( )] N 1 β 1 ( ) }. By substituting the appropriate expression for the last term, we get Π ( ) = π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 whih implies that π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1, with equality only when =. Π ( ) 0, 51

54 We have therefore shown that the erivative of the firm s expete profit is Π ( ) > 0, for < ; = 0, for = ; < 0, for >, as require for the firm s expete profit Π( ) to attain its maximum at =. Finally, to show that the biing strategy β 1 ( ) is stritly ereasing, notie that we an write its erivative as β 1 ( ) = (N 1)f() 1 F ( ) [ v 1 ( ) v 1 ( 2 ) (N 1)[1 F ] (2 )] N 2 f( 2 ) 2, [1 F ( )] N 1 where v 1 ( ) = β 2 ( ) [ ] π 2 (, 1 1 F ( ) ) (N 1)f( ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 1. Beause of Assumption 2, the term β 2 ( ) is ereasing in. In aition, by an argument similar to the one use for the orresponing term in β( ), we an show that the seon term is also ereasing in. Therefore, the funtion v 1 is ereasing. It follows that β ( ) < (N 1)f() 1 F ( ) [ v( ) v( ) ] = 0, as require for the biing strategy β 1 to be stritly ereasing. Proof of Proposition 16: First, notie that by rearranging the terms of the equation relating the biing strategies β( ), β 1 ( ) an β 2 ( 1 ), given in the proof of Proposition 8, we get [1 F ( )] N 1 [ β 1 ( ) β( ) ] = (N 1)[1 F ( )] N 2 F ( ) [ i ] β( ) β 2 ( 1 ) f(1 ) F ( ) 1. 52

55 Therefore, for the entire result, it suffies to show that β 1 ( ) > β( ). Using the efinitions of the strategies β( ) an β 1 ( ), we an show, by means of a iret alulation, that β 1 ( ) > β( ) if an only if 1 1 v( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N > i v( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) (N 1)[1 F ( )] N 2 F ( ) 2 1, where v( 2 2, 1 ) = π( 2 2, 1 ) π 2 ( 2 2, 1 ) 1 F (2 ) (n 2)f( 2 ). Sine the seon ouble integral is positive, it suffies to show that i 1 v( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) [1 F ( )] N > v( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) (N 1)[1 F ( )] N 2 F ( ) 2 1, that is, to show that E 1 i, 2 i [ v(2 i 2 i, 1 i) 2 i 1 i ] > E 1 i, 2 i [ v(2 i 2 i, 1 i) 2 i 1 i ]. Sine the funtion v( 2 2, 1 ) is inreasing in 1, we have 53

56 E 1 i, 2 [ v(2 i i 2 i, 1 i) 2 i 1 i ] = E 2 i [ E 1 i [v( 2 i 2 i, 1 i) 1 i [, 2 i] ] 2 i ] > E 2 i [ E 1 i [v( 2 i 2 i, ) 1 i [, 2 i] ] 2 i ] = E 2 i [ v( 2 i 2 i, ) 2 i ] = E 2 i [ E 1 i [v( 2 i 2 i, ) 1 i [, ] ] 2 i ] > E 2 i [ E 1 i [v( 2 i 2 i, 1 i) 1 i [, ] ] 2 i ] = E 1 i, 2 [ v(2 i i 2 i, 1 i) 2 i 1 i ], as require for the result. Proof of Proposition 17: Similarly to the proof of Proposition 11, notie that the strategy β( ) an be expresse as β( ) = u( 2 2 ) (N 1)(N 2) [1 F (2 )] N 3 F ( 2 ) f( 2 ) P[ 2 i ] 2, where u( ) = π(, 1 ) f(1 ) F ( ) 1 [ π 2 (, 1 ) [ π 2 (, 1 ) 1 F ( ) (N 2)f( ) 1 F ( ) (N 2)f( ) ] ] f(1 ) F ( ) 1 [1 F (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1, for, [, ]. In partiular, u() = u( ) is the valuation of a firm with marginal ost, assuming that its market opponent is stronger. Therefore, we have Π ( ) = (N 1)(N 2) [1 F ( )] N 3 F ( ) f( ) [ u( ) u( ) ] 54

57 an 2 Π ( ) = (N 1)(N 2) [1 F ( )] N 3 F ( ) f( ) u 2 ( ). Hene, if we an show that u 2 ( ) > 0, then, sine Π ( ) = 0, we an onlue that > 0, for Π i < ; ( ) = 0, for = ; < 0, for >, as it suffies for firm i s expete profit funtion Π( ) to attain its maximum at =. Notie that u 2 ( ) = i i π 1 (, 1 ) f(1 ) F ( ) 1 [ ] π 21 (, 1 1 F ( ) ) (N 2)f( ) [ ] π 21 (, 1 ) 1 F ( ) (N 2)f( ) f(1 ) F ( ) 1 [1 F (1 )] N 2 f( 1 ) [1 F ( )] N 2 F ( ) 1. Therefore, if N is suffiiently large, then the positive term ominates the negative ones, so that u 2 ( ) > 0. Moreover, it is possible to fin N suh that for N > N we have u 2 ( ) > 0, for all, [, ], uniformly. The rest of the proof is iential to that of Proposition 11, so, it is omitte. Proof of Proposition 18: Suppose, ontrary to our assertion, that there exists a symmetri equilibrium in monotone biing strategies (β 1, β 2 ). Sine β 1 is stritly ereasing, the announement of the first-roun winning bi reveals the marginal ost 1 of the strongest oligopolist. Therefore, in the seon roun, the firms upate their beliefs, so that F () = F () F (1 ), 1 F ( 1 ) 55

58 for all [ 1, ]. If the number of firms partiipating in the seon aution, N 1, is suffiiently large, so as to satisfy the inequality π 2 (, ) π 12 (, ) > [1 F ()] (N 2) f(), for all [, ], then, as shown in Das Varma [10], the strategy β 2 ( 1 ) = π( 2 2, 1 ) (N 2)[1 F (2 )] N 3 f( 2 ) [1 F ( )] N 2 2 π 2 ( 2 2, 1 ) [1 F (2 )] N 2 [1 F ( )] N 2 2, for 1, forms the unique equilibrium in the aution of the seon liense. In partiular, for a marginal ost < 1, firm i bis b 2 = β 2 ( 1 1 ). By assuming, as we i in the ase of the Cournot oligopoly, that { } [1 F ()] N 2 v2 (, ) 1 F (i ) sup [1 F ( )] N 2 ṽ 1 (, ) (N 2) f( ), for all [, ], where ṽ 1 (, 1 ) = [π(, 1 )] [π 2(, 1 )] 1 F () (N 2) f(), we an ensure that the strategy β 2 ( ) is ereasing in the marginal ost. In the first aution, the optimization of the profit funtion Π( ) of a firm with marginal ost results in the strategy β 1 ( ) = 1 π( 2 2, 1 ) (N 1)(N 2)[1 F (2 )] N 3 f( 2 )f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (2 )] N 2 f( 1 ) [1 F ( )] N π 2 ( 2 2, 1 ) (N 1)[1 F (1 )] N 2 f( 1 ) [1 F ( )] N 1 2 1, 56

59 as the unique solution of the ifferential equation erive by the neessary first-orer onition Π 1 ( ) = 0. To hek suffiieny, we an alulate, for, Π ( ) = π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1 π 2 (, 1 ) (N 1)[1 F ( 1 )] N 2 f( 1 ) 1. Sine π 2 (, 1 ) > π 2 (, 1 ), for, we onlue that Π ( ) > 0, showing that the firm s eviation from β 1 ( ) to β 1 ( ), for >, is profitable. Hene, the strategy β 1 ( ) fails to support an equilibrium. Sine the strategy β 1 was the unique solution to the neessary onition, we onlue that the sequential aution has no equilibrium in stritly monotone strategies. 57

60 Referenes [1] An, M. Y. (1998), Logonavity versus Logonvexity: A Complete Charaterization, Journal of Eonomi Theory, 80(2), [2] Baghi, A. (2005), How to Commerialize Tehnology Using Autions, Mimeo. [3] Boar, S. (2007), Biing into the Re: A Moel of Post-Aution Bankrupty, Journal of Finane, 62(6), [4] Bulow, J., Geanakoplos, J. an Klemperer, P. (1985), Multimarket Oligopoly: Strategi Substitutes an Complements, Journal of Politial Eonomy, 93(3), [5] Caplin, A. an Nalebuff, B. (1991), Aggregation an Imperfet Competition: On the Existene of Equilibrium, Eonometria, 59(1), [6] Cramton, P. (1997), The FCC Spetrum Autions: An Early Assessment, Journal of Eonomis an Management Strategy, 6(3), Reprinte in D.L. Alexaner (e.), Teleommuniations Poliy, Praeger Publishers, [7] Cramton, P. (2002), Spetrum Autions, in M. Cave, S. Majumar, an I. Vogelsang (es.), Hanbook of Teleommuniations Eonomis, Amsteram: Elsevier Siene. [8] Cramton, P. (2006), How Best to Aution Oil Rights, in M. Humphreys, J. Sahs, an J. Stiglitz (es.), Esaping the Resoure Curse, forthoming. [9] Dana, J. an Spier, K. (1994), Designing an Inustry: Government Autions with Enogenous Market Struture, Journal of Publi Eonomis, 53(1), [10] Das Varma, G. (2003), Biing for a Proess Innovation uner Alternative Moes of Competition, International Journal of Inustrial Organization, 21(1), [11] Figueroa, A. an Skreta, V. (2009), The Role of Outsie Options in Mehanism Design, Mimeo. [12] Gal-Or, E. (1985), Information Sharing in Oligopoly, Eonometria, 53(2), , [13] Gal-Or, E. (1986), Information Transmission - Cournot vs. Bertran, Review of Eonomi Stuies, 53(1) Reprinte in A.F. Daughety (e.) Cournot Oligopoly, Cambrige University Press, [14] Gilbert, R. an Newbery, D. (1982), Preemptive Patenting an the Persistene of Monopoly, Amerian Eonomi Review, 72, [15] Goeree, J. (2003), Biing for the Future: Signaling in Autions with an Aftermarket, Journal of Eonomi Theory, 108(2),

61 [16] Henriks, K. an Porter, R.H. (2003), The Timing an Iniene of Exploratory Drilling on Offshore Wilat Trats, Amerian Eonomi Review, 86, [17] Hoppe, H., Jehiel, P. an Molovanu, B. (2006), Liene Autions an Market Struture, Journal of Eonomis an Management Strategy, 15, [18] Hoppe, H., Molovanu, B. an Ozenoren, E. (2007), Coarse Mathing an Prie Disrimination, Mimeo. [19] Janssen, M. (2004), Autioning Publi Assets, Cambrige: Cambrige University Press. [20] Jehiel, P. an Molovanu, B. (1996), Strategi Non-Partiipation, RAND Journal of Eonomis, 27(1), [21] Jehiel, P. an Molovanu, B. (2001), Autions with Downstream Competition among Buyers, RAND Journal of Eonomis, 31(4), [22] Jehiel, P. an Molovanu, B. (2001), Effiient Design with Interepenent Valuations, Eonometria, 69, [23] Jehiel, P. an Molovanu, B. (2003), An Eonomi Perspetive on Autions, Eonomi Poliy, 36, [24] Jehiel, P. an Molovanu, B. (2004), The Design of an Effiient Private Inustry, Journal of the European Eonomi Assoiation, 2, [25] Jehiel, P. an Molovanu, B. (2005), Alloative an Informational Externalities in Autions an Relate Mehanisms, in R. Blunell, W. Newey, an T. Persson (es.), Proeeings of the 9th Worl Congress of the Eonometri Soiety, Cambrige: Cambrige University Press. [26] Jehiel, P., Molovanu, B. an Stahetti, E. (1996), How (not) to Sell Nulear Weapons, Amerian Eonomi Review, 86(4), [27] Jehiel, P., Molovanu, B. an Stahetti, E. (1999), Multiimensional Mehanism Design for Autions with Externalities, Journal of Eonomi Theory, 85(2), [28] Kamien, M. (1992), Patent Liensing, in R.J. Aumann an S. Hart (es.), Hanbook of Game Theory, Amsteram: North-Hollan. [29] Kamien, M., Oren, S. an Tauman, Y. (1992), Optimal Liensing of Cost- Reuing Innovations, Journal of Mathematial Eonomis, 21, [30] Kamien, M. an Tauman, Y. (1986), Fees versus Royalties an the Private Value of a Patent, Quarterly Journal of Eonomis 101, [31] Katz, M. an Shapiro, C. (1986), How to Liense Intangible Property, Quarterly Journal of Eonomis 101,

62 [32] Katzman, B. an Rhoes-Kropf, M. (2002), The Consequenes of Information Reveale in Aution, Mimeo. [33] Klemperer, P. (2002), How (Not) to Run Autions: the European 3G Teleom Autions, European Eonomi Review, 46, Also reprinte in G. Illing (e.), Spetrum Autions an Competition in Teleommuniations, [34] Krishna, V. (2002), Aution Theory, San Diego; Lonon an Syney: Elsevier Siene, Aaemi Press. [35] Maaslan, E. Montangie, Y. an van er Bergh, R. (2004), Levelling the Playing Fiel in Autions an the Prohibition of State Ai, in M. Janssen (e.), Autioning Publi Assets, Cambrige: Cambrige University Press. [36] Milgrom, R.P. an Weber, R. (1983), A Theory of Autions an Competitive Biing, Eonometria, 50(5), [37] Molnár, J. an Virág, G. (2006), Revenue Maximizing Autions with Market Interation an Signaling, Mimeo. [38] Shmitz, P.W. (2002), On Monopolisti Liensing Strategies uner Asymmetri Information, Journal of Eonomi Theory 106, [39] Zheng C.Z. (2001), High Bis an Broke Winners, Journal of Eonomi Theory, 100,