Auctioning Process Innovations when Losers Bids Determine Royalty Rates

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1 Auctioning Pocess Innovations when Loses Bids Detemine Royalty Rates Cuihong Fan Shanghai Univesity of Finance and Economics School of Economics Elma G. Wolfstette Humboldt-Univesity at Belin and Koea Univesity, Seoul Decembe 14, 2009 Byoung Heon Jun Koea Univesity, Seoul Depatment of Economics We conside a licensing mechanism fo pocess innovations that combines a license auction with oyalty contacts to those who lose the auction. Fims bids ae dual signals of thei cost eductions: the winning bid signals the own cost eduction to ival oligopolists, wheeas the losing bid influences the beliefs of the innovato who uses that infomation to set the oyalty ate. We deive conditions fo eistence of a sepaating equilibium, eplain why a sufficiently high eseve pice is essential fo such an equilibium, and show that the innovato geneally benefits fom the poposed mechanism. KEYWORDS: Patents, licensing, auctions, oyalty, innovation, R&D, mechanism design. JEL CLASSIFICATIONS: D21, D43, D44, D45. Financial suppot was eceived fom Koea Univesity, the BK21 Pogam, the Deutsche Foschungsgemeinschaft DFG), SFB Tansegio 15, Govenance and Efficiency of Economic Systems, the Humanities and Social Sciences Reseach Foundation of Ministy of Education of China Gant No. 09YJA790133), and the Leading Academic Discipline Pogam, 211 Poject fo Shanghai Univesity of Finance and Economics 3d phase). Shanghai Univesity of Finance and Economics, School of Economics, Guoding Road 777, Shanghai, China, cuihongf@mail.shufe.edu.cn Sung-Buk Ku An-am Dong 5-1, Seoul , Koea, bhjun@koea.ac.k Coesponding Autho: Institute of Economic Theoy I, Humboldt Univesity at Belin, Spandaue St. 1, Belin, Gemany, wolfstette@gmail.com 1

2 1 Intoduction This pape evisits the analysis of license auctions of a non-dastic pocess innovation by an outside innovato to a Counot oligopoly. The cost eductions induced by that innovation ae of the pivate values type and ae fims pivate infomation. Afte the auction the innovato and fims update thei pio beliefs based on obseved bids. The main novel featue is that we eplace the standad license auction by a supeio mechanism that combines a standad license auction with a mandatoy oyalty contact fo those who lose the auction. Thee, the innovato has two souces of evenue: the equilibium pice paid by the winne of the license auction, and the oyalty income paid by those who lose the auction. This licensing mechanism gives ise to a dual signaling poblem: If a bidde wins the auction, his bid signals the own cost eduction to ival fims, wheeas if he loses, his bid signals the own cost eduction to the innovato who sets the oyalty ate equal to the epected cost eduction. Biddes take into account that they can influence othes beliefs with thei bid. Specifically, a bidde gains a stategic advantage in the oligopoly game with an inflated bid that signals a highe than tue cost eduction, povided this bid happens to win the auction. In tun, a bidde can fool the innovato to set the oyalty ate below the tue cost eduction with a deflated bid, povided this bid happens to lose the auction. Of couse, no such misleading signaling occus on the equilibium path of a sepaating equilibium. If a sepaating equilibium eists, the maginal benefit of signaling a cost eduction that deviates fom the tue cost eduction must be matched by a coesponding maginal cost in such a way that both kinds of signaling ae deteed in all states of the wold. If bids can only influence the beliefs of ival fims, misleading signals ae easily deteed by choosing an appopiate steepness of the bid function. As a esult, the possibility to influence the beliefs of ival fims with a winning bid, simply eets an upwad pessue on equilibium bids. Similaly, the possibility to influence the beliefs of the innovato with the losing bid eets a downwad pessue on equilibium bids. Howeve, one cannot dete biddes fom signaling to the innovato a lowe than tue cost eduction by adjusting the steepness of the bid function alone. In addition, the innovato must set a sufficiently high eseve pice. Without it, biddes with a low cost eduction would always signal a zeo cost eduction, then they lose the auction, yet obtain the innovation fo fee. In the face of the dual signaling poblem, whee bids can influence the beliefs of ival fims and of the innovato, a sepaating equilibium bid function eists only in combination with a sufficiently high eseve pice. Of couse, a selle can typically incease his evenue by adding a eseve pice equiement. Howeve, in the pesent famewok, the ole of the eseve pice is moe fundamental, because without it, no sepaating equilibium eists. We analyze two specifications of the model that diffe eclusively in the infomation available to fims in the downsteam oligopoly game. In the fist highly stylized specification, efeed to as model I, fims cost eductions become common knowledge among fims afte the auction and befoe the oligopoly game is played. Howeve, the innovato emains uninfomed about cost 2

3 eductions, and he can only update his pio beliefs afte obseving bids. The innovato uses this infomation to set oyalty ates fo loses, and fims use thei knowledge about the infomation available to the innovato to pedict the oyalty ate to be paid by those who lose the auction. In the second specification, efeed to as model II, fims emain uninfomed about thei ivals cost eductions afte the auction. Like the innovato they can, howeve, update thei pio beliefs fom the obseved bids. Fims use this updated infomation to assess the epected costs of thei ivals, to pedict the oyalty ates the loses have to pay and to pedict thei ivals beliefs about the oyalty ate they themselves have to pay if they lose the auction. Altogethe, model II is moe plausible. Nevetheless, model I is useful as a benchmak fo compaison with the liteatue, and it pepaes nicely the moe comple analysis of model II. Thee is lage liteatue on patent licensing in oligopoly by an outside innovato, among which the following contibutions ae closely elated to the pesent pape. In thei seminal contibutions, Kamien 1992), Kamien and Tauman 1984, 1986), Katz and Shapio 1985, 1986) show that auctioning a esticted numbe of licenses is stictly moe pofitable fo the innovato than othe mechanisms, such as pue oyalty contacts, fied-fee licensing, and two-pat taiffs. 1 The limitation of the classical liteatue is that it assumes complete infomation both in the auction and in the subsequent oligopoly game. Late Jehiel and Moldovanu 2000) intoduce incomplete infomation at the bidding stage, combined with complete infomation in the oligopoly game, which is the infomation stuctue to which we aleady efeed as model I). They show that patent licensing unde incomplete infomation is a pime eample of an auction with negative etenalities, whee biddes payoffs depend not only on thei own types, but also on the type of the playe who wins the auction. In an auction with negative etenalities, the theoy of optimal mechanism design does not apply, which suggests a piecemeal appoach to seaching fo good mechanisms. And, unlike in standad pivate value auctions, the eseve pice plays a much less pominent ole, since the selle in an auction that is subject to negative etenalities has a stonge incentive to induce paticipation. As Jehiel and Moldovanu 2000, p. 777) put it succinctly: when the selle sells moe often, the buyes ae moe afaid that the good will fall into the hands of the competito, and they bid moe aggessively. Moe ecently, Das Vama 2003) and Goeee 2003) econside that model unde the moe plausible assumption that fims do not know each othe s cost eductions afte the auction and befoe the oligopoly game is played, which coesponds to the infomation stuctue in ou model II. This intoduces the possibility to signal the own cost eduction to ival fims. Das Vama 2003) consides both Counot and Betand competition with poduct diffeentiation). He shows that a sepaating equilibium eists unde Counot competition with linea demand, but geneally fails to eist unde Betand competition when goods ae substitutes. 1 Recently, Sen 2005) shows that if one takes into account that the numbe of licenses must be an intege, pue oyalty contacts can be supeio to license auctions. Howeve, this esult is again evesed if one genealizes the fomat of license auctions see Giebe and Wolfstette, 2008). See also Sen and Tauman 2007) who analyze an auction of oyalty contacts. 3

4 Goeee 2003) assumes educed fom payoff functions of the oligopoly subgames that ae geneally satisfied fo Counot but not fo Betand maket games with substitutes. He compaes all thee standad auction fomats fist-, second pice, and English auctions. Inteestingly, he finds that the second pice auction is the most pofitable auction fom fo the innovato, since the upwad pessue on equilibium bids due to the possibility of signaling to ival fims is the most damatic thee. We follow this insight, and assume in ou analysis that the innovato adopts a second-pice auction. With this fomat the potential of signaling with the winning bid gives ise to the most damatic upwad pessue on equilibium bids, and thus maimizes the auction evenue, while the potential to signal with the losing bid is not affected by the choice of auction fomat. In a peceding pape, Giebe and Wolfstette 2008) intoduce an optional oyalty scheme to the license auction unde complete infomation, and show that such a mechanism is stictly moe pofitable than the standad license auction. 2 Howeve, unde incomplete infomation, adding oyalty contacts fo the loses of the auction woks in a diffeent way. Fist of all, one cannot gant oyalty contacts as an option, but one must make them mandatoy, second, adding oyalty contacts fo the loses leads to a eduction in auction evenue, and thid, adding oyalty contacts leads to a comple dual signaling poblem whee both the winning and the losing bids signal infomation to ival fims esp. to the innovato. Ou main findings can be summaized as follows: 1) The equilibium bid function is stictly monotone inceasing unde faily standad conditions concening the pobability distibution of fims cost eductions, povided the innovato sets a sufficiently high eseve pice. 2) The eseve pice plays a cucial ole in assuing eistence of a sepaating equilibium; without it, no sepaating equilibium eists, unlike in the standad license auction whee the eseve pice plays a mino ole. 3) Adding the oyalty contact fo loses advesely affects bidding, which lowes the innovato s evenue fom the auction. 4) Howeve, the additional oyalty income weighs moe than that loss in auction evenue, unless the pobability distibution of cost eductions ehibits an eteme concentation on low values. Theefoe, the poposed mechanism is geneally moe pofitable. 5) Adding the oyalty scheme has a stonge effect on equilibium bids fo low than fo high cost eductions. Theefoe, adding the oyalty scheme is paticulaly pofitable when the pobability distibution ehibits a concentation on high cost eductions, since it entails a elatively small loss in auction evenue togethe with a high oyalty income. The pape is oganized as follows: In Section 2 we pesent the model and intoduce basic assumptions. In Section 3 we analyze model I whee we also show in detail why the eseve pice plays a cucial ole in assuing that the fist ode conditions of the equilibium bidding poblem yield global maima. In Section 4 we analyze the moe plausible model II which nicely complements and etends the analysis of model I. We find stonge esults fo model II, and show that the oyalty scheme is even moe pofitable in model II. In Section 5 we discuss ou esults and sketch some etensions. Some of the poofs ae contained in the Appendi. 2 In a companion pape Fan, Jun, and Wolfstette 2009) conside the licensing poblem assuming fims daw impefect signals of an unknown common value cost eduction. 4

5 2 The Model An outside innovato auctions the ight to use a pocess innovation potected by a patent to a Counot duopoly. Pio to the innovation, fims have the same constant unit cost c. The innovation educes fims unit cost by an amount that depends on who uses it. At the time when auction game is played, fims cost eductions ae thei pivate infomation pivate values case) and unknown both to thei competito and to the innovato. The innovato employs the following licensing mechanism: one license is auctioned to the highest bidde in a second-pice auction with the povision that the lose must accept a oyalty contact. The winne of the auction can use the innovation at no futhe cost. The lose of the auction can also use the innovation, and has to pay a fied oyalty ate pe output unit. The oyalty ate is set to be equal to the lose s epected cost eduction, conditional on the infomation evealed to the innovato by his bid. Afte the auction, the innovato publishes all bids. Having obseved the bids the duopolists compete in a Counot maket game. We conside two models that diffe in the infomation available in the duopoly game: In model I, fims know each othe s cost eductions afte the auction and befoe the duopoly game is played, and in model II cost eductions emain pivate infomation, and fims can only update thei beliefs about each othe s cost eductions afte obseving thei ival s bid. In both models, the innovato geneally does not know fims cost eductions, and can only infe cost eductions fom obseved bids. Fims and the innovato ae isk neutal and invese maket demand PQ) is a deceasing and concave function of aggegate output Q := q 1 + q 2. 3 The fim specific cost eductions induced by the innovation, denoted by, y ae iid andom vaiables, dawn fom the c.d.f. F : [0, c] [0, 1], with positive density f eveywhee. Both F and the eliability function 1 F ae assumed to be log-concave which ules out that F has pats that ae highly conve and highly concave. The log-concavity of 1 F is equivalent to the well-known hazad ate monotonicity. 4 We conside only non-dastic innovations whose eclusive use does not popel a monopoly. If PQ) = 1 Q, innovations ae non-dastic if and only if c [0, 1 /2). We denote the two highest ode statistics of the sample of cost eductions by X 1), X 2), and the associated p.d.f. of X 2) by g 2 2 ) = 21 F 2 )) f 2 ) and the joint p.d.f. of X 1), X 2) by g 12 1, 2 ) = 2 f 1 ) f 2 ). And we denote the equilibium duopoly pofits of the winne and the lose of the auction by π W and π L, espectively; and the equilibium pofit when both fims abstain fom bidding by π nn. 3 This assues that the duopoly game has a unique equilibium see Szidaovszky and Yakowitz, 1977). 4 A sufficient condition fo the log-concavity of both F and 1 F is the log-concavity of f. See Lemma A2 in Goeee and Offeman 2003) togethe with Theoem 1 in Bagnoli and Begstom 2005). 5

6 3 Model I complete infomation in the duopoly game) Following Jehiel and Moldovanu 2000) we fist conside a highly stylized model in which fims lean about each othe s cost eductions befoe they play the duopoly game. Jehiel and Moldovanu efeed to the auctioning of a patent license as a pime eample of auctions with negative etenalities. Thei main finding was that in the pesence of a negative etenality, using a eseve pice becomes less attactive fo the auctionee. Among othe esults we will show that the eseve pice plays a cucial ole if one eplaces the simple license auction consideed by Jehiel and Moldovanu 2000) by the geneally supeio mechanism poposed hee. 3.1 Benchmak: The game without oyalty contact fo the lose The auctioning of one license without oyalty contact fo the lose has aleady been analyzed in Jehiel and Moldovanu 2000, Sect. 4). We biefly eview thei esults fo the case of a duopoly, which seve as a benchmak. Since only the winne of the auction has access to the innovation, the equilibium pofits in the duopoly subgames depend only on the cost eduction of the winne of the license auction, denoted by. And the equilibium pofit of the winne, π W ), is inceasing and that of the lose, π L ), is deceasing in. In a second-pice auction with eseve pice, R, the winne pays the eseve pice if he is the only bidde and othewise pays the second highest bid. The game has an equilibium in weakly dominant stategies. Thee, biddes play cutoff stategies, and bid tuthfully if and only if thei cost eduction is at least as high as the cutoff value, and do not bid othewise. The equilibium cutoff value is such that the maginal bidde, with a cost eduction equal to, is indiffeent between bidding and not bidding, i.e., π W ) R = π nn. This yields a unique elationship between and R, that allows us to eliminate R, and compute the innovato s epected pofit as a function of. The equilibium bid function, β n, and the innovato s epected evenue, G n ) ae see Sect. 4, Jehiel and Moldovanu, 2000), whee R = π W ) π nn, β n ) = π W ) π L ), fo 1) G n ) = 2F)1 F)) R + And the optimal cutoff value n solves the following condition: 1 F) f ) = π W ) π nn π W ) π W 2 ) π L 2 )) g 2 2 )d 2. 2) + 1 F) π nn π L )) F) π W ). 3) It is useful to compae this with a hypothetical wold in which thee is no negative etenality in the sense that the lose s equilibium duopoly pofit is not affected by the winne s cost eduction. 6

7 In that hypothetical wold, the equilibium bidding stategy and the innovato s epected evenue would be equal to G 0 ) = 2π W ) π nn ) F)1 F)) + β 0 ) = π W ) π nn 4) yielding the optimal cutoff value 0 that solves the equation, 1 F) f ) π W 2 ) π nn ) g 2 2 )d 2, 5) = π W ) π nn π W ). 6) Compaing 3) and 6) and using the assumed hazad ate monotonicity it follows immediately: Poposition 1 Jehiel and Moldovanu 2000)). The optimal eseve pice with negative etenality is lowe than that in a standad auction without negative etenality: n < 0. Essentially, in the pesence of the negative etenality, the innovato has an incentive to lowe the eseve pice, because a lowe eseve pice makes it moe likely that both fims bid, and the winne pays π W ) π L ), which is moe than the eseve pice, R = π W ) π nn that the winne pays if only one fim bids. 3.2 The game with oyalty contact fo the lose Fo the moment, suppose cost eductions become common knowledge afte the auction to fims as well as to the innovato. Then neithe the duopoly no the bidding game is affected by adding the oyalty contact. And it follows immediately: Poposition 2. When costs eductions ae common knowledge afte the auction, adding the oyalty scheme to the auction is pofitable fo the innovato and inceases welfae. Adding the oyalty scheme leaves the lose s cost unchanged, since the oyalty ate is equal to his cost eduction. Theefoe, neithe the duopoly no the bidding game is affected. The innovato then eans the same income in the auction, yet the oyalty contact adds a positive oyalty income. The equilibium payoffs of the duopolists emain unchanged, but the innovato s equilibium epected evenue inceases; hence, welfae inceases. Howeve, if cost eductions become common knowledge only among fims, while the innovato emains uninfomed, the innovato can only update his pio beliefs about cost eductions fom obseved bids, and then sets the oyalty ate equal to the infeed cost eduction of the lose. This, in tun, induces biddes to use thei bids to influence the innovato s beliefs about thei cost eduction and thus the oyalty ate they have to pay in the event when they lose the auction. We employ the following pocedue to solve the game. As a woking hypothesis we assume that the bidding game has a symmetic and monotone inceasing equilibium, β : [,c] R which we will confim), which allows the innovato to daw a pefect infeence fom obseved bids to the undelying cost eductions and allows the winne of the auction to infe the oyalty ate paid 7

8 by the lose. We conside one bidde, say bidde 1 with cost eduction, who assumes that his ival plays the stictly inceasing equilibium stategy β. Without loss of geneality we estict deviating bids to b [β), βc)], because bidding outside that ange is obviously dominated. Theefoe, bidding accoding to the stategy β as if the cost eduction wee equal to z,c) captues all elevant deviating bids. We fist solve all duopoly subgames that may occu if bidde 1 unilateally deviates fom the equilibium bid while eveyone ival and innovato) believes that fims play the equilibium bidding stategy β Downsteam duopoly subgames Suppose fim 1 has dawn the cost eduction but bids as if it had dawn cost eduction z, while fim 2 has played the stictly inceasing equilibium bidding stategy. 5 In the continuation duopoly game, the following subgames occu, depending upon the tue and petended cost eductions of fim 1,, z, and the cost eduction of fim 2, y. When both fims bid and fim 1 has won the auction Let z > y and, y. The innovato infes that fim 2 has cost eduction y and chages a oyalty ate equal to y. It is then common knowledge among fims that the pofile of unit costs is c 1,c 2 ) = c,c). Denote the Counot equilibium stategies fo this cost pofile by q W1 ),q L2 )). Theefoe, the educed fom pofit function of fim 1, conditional on winning the auction, is π W ) := PqW1 ) + q L2 )) c + ) q W1 ). When both fims bid and fim 1 has lost the auction Let y > z and, y. The innovato infes that the cost eduction of fim 1 is equal to z and then chages a oyalty ate equal to z. It is then common knowledge among fims that the pofile of unit costs is c 1,c 2 ) = c + z,c y). Denote the equilibium stategies of that duopoly subgame by q L1, z, y)),q W2, z, y). Theefoe, the educed fom pofit function of fim 1, conditional on losing the auction, is π L, z, y) := Pq W2, z, y) + q L1, z, y)) c + z ) q L1, z, y). On the equilibium path, i.e., fo z =, the equilibium stategy of fim 1 when it lost the auction and the associated educed fom payoff ae only a function of fim 2 s cost eduction, y; theefoe, we wite: ql y) = q L1, z, y) z= and πl y) = π L, z, y) z=. Similaly, we wite qw y) = q W 2, z, y) z=. When at least one fim did not bid If no one has made a bid, the game is just the default game without innovation; in this case the equilibium pofit of fim 1 is equal to π nn. If fim 1 was the only bidde, its equilibium pofit is the same as in the event when both fims bid and fim 1 has won the auction, and if fim 2 was the only bidde, the equilibium pofit of fim 1 is the same as in the game without oyalty scheme, and is eclusively a function of the winne s cost eduction, π L y), as eplained in section 3.1. Lemma 1. In the elevant duopoly subgames one has z π L, z, y) z= = ql y)γ y), whee γ y) : = 1 P q W2 ) + q L1 )) z q W2, z, y) ) z= > 1. 7) 5 The case of z is slightly diffeent, yet yields the same payoff function,, z) and diffeential equation, and hence is omitted. 8

9 and π W ) > 0, πl y) < 0. If demand is linea, γ y) = 4 /3 fo a summay account of the linea model see Appendi A.5, A.6). The poof is in Appendi A Auction subgame Using the equilibia of the continuation duopoly subgames, fim 1 s payoff function is, z) = F)π W ) R) + + z z π W ) βy))d Fy) π L, z, y)d Fy), whee R = π W ) π nn. 8) In equilibium, the bid function β is such that it is the best eply of fim 1 to bid β), athe than βz), z. Theefoe, β is an equilibium if and only if = agma z, z), fo all [,c]. Poposition 3. In equilibium fims with bid accoding to the stictly inceasing stategy, β), povided the eseve pice is sufficiently high, and abstain fom bidding if <, β) = π W ) π L ) + 1 f ) z π L,, y)d Fy) < β n ) 9) Poof. 1) Suppose. Diffeentiating the epected payoff function of fim 1,, z), 8), with espect to z, and then setting z =, one obtains, π W ) β)) f ) + z π L, z, y) z= d Fy) πl ) f ) = 0. 10) This implies the asseted bid function 9) and, by Lemma 1, β) < β n ) fo all. 2) The poof of the asseted monotonicity of β is in Appendi A.2, 3) The above assumes that biddes play a cutoff stategy, and bid if and only if. In Appendi A.3 we pove fomally that the equilibium stategy is indeed such a cutoff stategy. 4) Having established sufficient conditions fo the monotonicity of the bid function one must also confim that the undelying fist-ode conditions 10) yield a global maimum fo each. This is assued if the function, z) is pseudoconcave in z, which is assued if and only if z, z) 0. 6 As we eplain below, this second-ode condition is satisfied only if the innovato adopts a sufficiently high eseve pice. 6 A function of one vaiable is pseudoconcave if it is inceasing to the left of the stationay point and deceasing to the ight. Biddes payoff function, z) is pseudoconcave in z if z 0, fo all, z, since the sign of that coss deivative implies z < z, z) > z z, z) = 0, z > z, z) < z, ) = 0. Pseudoconcavity obviously implies that evey stationay point is a global maimum. 9

10 Fom the point of view of the innovato, adding the oyalty scheme has an advese effect on equilibium bids. This signalling effect has the following intepetation. If the intoduction of the oyalty scheme did not affect the equilibium stategy, each bidde would benefit fom bidding below equilibium in ode to signal a cost eduction that is lowe than tue cost eduction. This incentive to signal can only be eliminated by pointwise loweing the equilibium bid function and by intoducing a sufficiently high eseve pice. This indicates that the innovato faces a tade-off between income eaned in the auction and oyalty income eaned fom the duopoly game. To see why a sufficiently high eseve pice is needed, suppose no eseve pice is used and hence = 0, and assume pe absudum that β is the equilibium bid function. Then, a bidde with a cost eduction > 0 who bids β) eans the payoff, ), wheeas if he deviates and bids β0) he eans,0), with,0), ) = 0 π L,0, y) π L,, y))d Fy) + 0 π W ) βy))d Fy). 0 π L,, y)d Fy) Evidently, π L,0, y) > π L,, y) and fo all y <, π W ) > β) > βy). Theefoe, if is sufficiently small, it follows that,0) >, ), which contadicts the assumption that β is an equilibium stategy. Specifically, Poposition 4. If demand is linea, the smallest eseve pice and the associated cutoff value min that assue that the second-ode conditions ae satisfied is implicitly defined as the unique solution of 1 F) 2 = 0. 11) f ) 3 Poof. Substituting π W ), R, βy) and π L, z, y) fom Appendi A.5 into the payoff function of fim 1 8), one can easily confim that, z : z, z) = 4 3 z f z) Fz)) 0 min., z) is pseudoconcave and hence has a global maimum at z = fo all if and only if z, z) 0 see footnote 6), which is equivalent to the condition that min. Finally, the eistence and uniqueness of min follows fom the fact that the LHS of 11) is negative at = 0, positive at = c, and stictly inceasing in by the hazad ate monotonicity. To illustate the ole of the eseve pice, conside the following eample which indicates why 9) is an equilibium only if min. 10

11 Eample 1. Let F) := /c unifom distibution), in which case min = 2c /5, and suppose c = 0.49, = If = 0, the stationay point z, z) z= = 0 is a local but not a global maimum see the left side of Figue 1). Theefoe, the best eply is to bid β0), thus lose the auction yet obtain the innovation fo fee. Wheeas if = min, that stationay point is a global maimum see the ight side of Figue 1) z z z z Figue 1: Left: Stationay point at z = is not a global maimum fo = 0; Right: Stationay point at z = is a global maimum fo min The pupose of a sufficiently high eseve pice is to dete fims with low cost eductions fom bidding as if they had dawn no cost eduction, by making a bid equal to β0). Of couse, the innovato could also live without a eseve pice and an equilibium that involves pooling at low levels of. Howeve, in that case he would ean neithe oyalty income no income in the auction wheneve at least one bidde has a low level of. Wheeas, if he adds the eseve pice, he eans at least the eseve pice fom the fim who has a cost eduction Is it pofitable to add oyalty contacts fo the lose? Fo the innovato, adding the oyalty scheme has a benefit and a cost. The benefit is that he eans oyalty income fom the lose wheneve both fims bid in the auction, without diectly affecting the payoff of the winne in the downsteam duopoly game, since, in equilibium, the lose pays a oyalty ate equal to his cost eduction. The cost is that the innovato needs to set a elatively high eseve pice, which involves the isk that only one o even no fim bids, and that bidding is pointwise lowe than without oyalty scheme. Nevetheless, adding the oyalty scheme is pofitable fo all standad pobability distibutions, even though one can constuct eamples in which it is not pofitable. The latte occus if the pobability distibution ehibits a high concentation on low cost eductions. The innovato s epected evenue in the game including oyalties, G), has thee components: the collected eseve pice, the epected payment fom the winning bidde when both fims paticipate, and the oyalty income fom the lose: G) = 2F)1 F)) R + β 2 )g 2 2 )d q L 1)g 12 1, 2 )d 2 d 1. 11

12 Define ) := G) G n ). Substituting G n ) see 2)), β, and R = π W ) π nn, one finds afte changing the ode of integation and a bit of eaanging: ) = 2 = 2 2 ql ) 1) f 2 ) + 1 F 2 )) z π L 2, z, 1 ) z=2 d F1 )d 2 2 ql 1) 2 1 F ) 2) γ 1 ) d F 1 )d F 2 ), by Lemma 1). 12) 2 f 2 ) To obtain futhe esults, we now assume that the demand function is linea. We denote the maimizes of G n, G, and by n,,, espectively, and the oot of ) by ˆ. Lemma 2. Suppose demand is linea. Then, > min and ) > 0 fo all ˆ,c). The poof is in Appendi A.4. Poposition 5. Suppose demand is linea. Adding the oyalty scheme inceases the innovato s epected evenue: G ) > G n n ) if n > ˆ. Poof. The assumption that n > ˆ implies n ) > 0 by Lemma 2. Theefoe, G ) G n ) G n n ) + n ) > G n n ). Whethe n is eithe smalle o geate than ˆ depends on the pobability distibution F. We now state two eamples in which n > ˆ, and then show how one must change the pobability distibution to make the oyalty scheme unpofitable fo the innovato. In Figue 2 we plot the innovato s epected evenue with and without oyalty scheme, G), G n ), fo the case of the unifom distibution, F : [0,c] [0,1], F) = /c, assuming c = In this case, if the oyalty scheme is adopted, the innovato must set a high eseve pice esp. cutoff value min = Although this eposes the innovato to a high isk of not selling his innovation, altogethe, adding the oyalty scheme is pofitable, since G ) > G n ) G Gn min Figue 2: Unifom distibution Adding the oyalty scheme is pofitable fo many standad pobability distibutions. Stating fom the unifom distibution, one finds that if pobability mass is shifted to high cost eductions, the oyalty scheme becomes even moe pofitable. Howeve, if the pobability distibution 12

13 ehibits a sufficiently high concentation on low cost eductions, the oyalty schemes becomes less pofitable, and the evenue anking can be evesed. We illustate this with tuncated eponential distibutions in Figues 3 and 4. 7 The pobability distibution in Figue 3 ehibits a concentation on high cost eductions; wheeas the pobability distibution in Figue 4 ehibits a high concentation on low cost eductions. Figues 3 and 4 plot the innovato s epected evenue fo the coesponding pobability distibutions. As one skews the distibution towads high cost eductions, the oyalty scheme becomes moe pofitable; howeve, as one concentates pobability mass on low cost eductions, as in Figue 4, the evenue anking is evesed and it no longe pays to employ the oyalty scheme c Λ c Λ F, Λ 5 c Λ G Gn min Figue 3: Tuncated eponential distibution with a concentation on high cost eductions: cdf left) and associated epected evenue of the innovato ight) The intuition fo these findings is as follows. We know fom the analysis of the bid functions β and β n that adding the oyalty scheme has a stonge effect on equilibium bids fo low than fo high cost eductions see pat 2 of the poof of Poposition 3). Also, equilibium oyalty ates ae smalle fo low than fo high cost eductions. Theefoe, if the pobability distibution ehibits a high concentation on low cost eductions, adding the oyalty scheme is not vey appealing since it gives ise to a high loss in auction evenue togethe with a low epected oyalty income. Wheeas if the pobability distibution has a high concentation on high cost eductions, adding the oyalty scheme is appealing since it entails a elatively small loss in auction evenue combined with a high gain in oyalty income. 4 Model II incomplete infomation in the duopoly game) We now tun to the moe plausible model in which fims do not lean each othe s cost eduction afte the auction and befoe they play the duopoly game. In this case, bids not only signal fims cost eductions to the innovato but also to thei ival. Like in model I, each fim would like to signal weakness to the innovato, and make the innovato believe that thei cost eduction is low, since a low cost eduction tanslates into a low oyalty ate in the event when they lose the auction. Howeve, unlike in model I, each fim would also like to signal stength to its ival and make him believe that one s cost eduction is high, since this will make the ival play less 7 Both eamples assume c = 0.3, and both distibutions ae consistent with ou assumption of log-concave eliability functions. 13

14 c Λ c Λ F, Λ 40 c Λ Gn G min Figue 4: Tuncated eponential distibution with a concentation on low cost eductions: cdf left) and associated epected evenue of the innovato ight) aggessively in the duopoly game. Both signaling consideations will affect equilibium bids in the auction. As we will show, model II yields moe geneal esults and altogethe adding the oyalty scheme is moe pofitable than in model I. We solve the pefect equilibium of the auction game followed by the incomplete infomation duopoly game. Theeby we employ the same methodology as in model I. Note, howeve, that unlike in model I, fims do not lean each othe s cost eductions afte the auction. Theefoe, both the innovato and fims use the infomation evealed by obseved bids to update thei pio beliefs. The innovato uses this infomation to set the oyalty ate fo the lose, and fims use it to pedict thei ival s cost eduction and the oyalty ate paid by the lose. Since the game without oyalty scheme coesponds to a game analyzed by Goeee 2003) and Das Vama 2003), we focus on the game with oyalty scheme and only mention casually what changes if no oyalty scheme is adopted. 4.1 Downsteam duopoly subgames Suppose fim 1 has dawn cost eduction but bids as if it had dawn cost eduction z, while fim 2 has played the stictly monotone inceasing equilibium bidding stategy β. 8 In the continuation duopoly game, the following subgames occu, depending upon,z and y When both fims bid and fim 1 has won the auction Let z > y and, y. Fim 1 pivately knows its cost eduction is ; wheeas fim 2 the lose) believes that the winne s cost eduction is equal to z. Theefoe, fim 2 believes to play a duopoly subgame with the pofile of unit costs c 1,c 2 ) = c z,c). Denote the associated equilibium stategies of that game the lose believes to play by q W z),q L2 z)). 8 Like in model I, the case of z is slightly diffeent, yet yields the same payoff function,, z), and diffeential equation, and hence is omitted. 14

15 Fim 1 anticipates that the lose plays q L2 z). But since fim 1 pivately knows that its cost eduction is equal to athe than z it plays the best eply: q W1, z) = ag ma Pq + ql2 z)) c + ) q. 13) q The educed fom pofit function of fim 1, conditional on winning the auction, is π W, z) := PqW1, z) + q L2 z)) c + ) q W1, z) When both fims bid and fim 1 has lost the auction Let y > z and, y. Fim 2 believes to play a Counot duopoly subgame with unit costs c 1,c 2 ) = c,c y). Denote the associated equilibium stategies of the game that fim 2 the winne) believes to play by q L y),q W2 y)). If the oyalty scheme is adopted, fim 1 pivately knows that its cost eduction is yet pays a oyalty ate z that eceeds its cost eduction. Theefoe, fim 1 plays the following best eply to q W2 y): q L1, z, y) = agma Pq + qw2 y)) c + z ) q. 14) q The associated educed fom pofit function of fim 1 conditional on losing the auction is then π L, z, y) := Pq W2 y) + q L1, z, y)) c + z ) q L1, z, y). Also note that on the equilibium path, fo z =, that payoff is only a function of the cost eduction of fim 2, y; theefoe, we wite q L y) = q L, z, y) z= and π L y) = π L, z, y) z=. Wheeas if no oyalty scheme is used, the equilibium play of fim 1 depends only on its ival s cost eduction and theefoe is independent of and z. Hence, in this case, the educed fom pofit function of fim 1 conditional on losing the auction is π L y) := Pq W2 y) + q L y)) c ) q L y). We stess that in model II the equilibium stategy that fim 2 plays in the event when fim 1 lost the auction, q W2, is only a function of its own cost eduction, y, wheeas in model I it is a function of, z, and y. This is due to the fact that in model II both fim 2 and the innovato believe that the cost eduction of fim 1 is equal to z, and theefoe fim 2 believes that the effective unit costs ae equal to c 1 = c z + z = c, and c 2 = c y. Wheeas in model I, the innovato believes that the cost eduction of fim 1 is equal to z but fim 2 knows that it is equal to ; theefoe, fim 2 believes that the effective unit costs ae equal to c 1 = c + z, and c 2 = c y When at least one fim did not bid If no one has made a bid, the game is just the default game without innovation; in this case the equilibium pofit of fim 1 is equal to π nn. If fim 1 was the only bidde, its equilibium pofit is the same as in subgame 4.1.1, and if fim 2 was the only bidde, the equilibium pofit of fim 1 is the same as in the game without oyalty scheme, and is eclusively a function of the winne s cost eduction, π L y), as eplained in section Lemma 3. In the elevant duopoly subgames one has: 1) z π L, z, y) z= = ql y), and 2) d d π W, ) > 0, πl y) < 0. 15

16 Poof. The poof of pat 1) is the same as the poof of Lemma 1 spelled out in Appendi A.1) if one takes into account that in model II one has z q W2 ) = 0 and theefoe γ y) = 1, see 7). Pat 2) follows fom the envelope theoem: d d π W, ) = = ) P ) q L2 z)q W1, z) + q W1, z) + P ) z q L2 z)q W1, z) z= q W1, z) + P ) z q L2 z)q W1, z) > 0 since P ) < 0 and z q L2 z) < 0) ) z= π L y) = P ) y q W2 y)q L1, z, y) < 0 since y q W2 y) > 0). since q L2 z) = 0) 4.2 Auction subgame Suppose fim 1 has dawn the cost eduction but bids as if its cost eduction wee equal to z. Assume that fim 2 plays the stictly monotone inceasing equilibium stategy βy). Then, the payoff function of fim 1 in the mechanism with oyalty scheme is, z) = F)π W, z) R) + z π W, z) βy))d Fy) + z π L, z, y)d Fy), whee simila to model I, the following elationship applies to the eseve pice R and the cutoff value : R = π W,) π nn. In the mechanism without oyalty scheme, the payoff function is the same, ecept that in the last tem π L, z, y) is eplaced by π L y), as eplained in the above analysis of the elevant duopoly subgames. Poposition 6. The equilibium bidding stategies with esp. without oyalty scheme, β),β n ), ae, fo all : β) = β n ) + 1 z π L, z, y) f ) z= d Fy) 15) β n ) = π W, ) πl F) ) + f ) zπ W, z) z= 16) Wheeas β is stictly monotone inceasing only if is sufficiently lage, β n is stictly monotone inceasing fo all. Poof. Fo the deivation of β n see Goeee 2003, Poposition 2). To poof the monotonicity of β n, compute the deivative of β n. This deivative has two pats. The fist pat is positive, πw, ) πl )) ) F) > 0, by Lemma 3. The second pat, which is equal to d f ) zπ W,z) z= /d, is also positive by the assumption that F is log-concave. 16

17 The deivation of β is simila to that in model I. To pove monotonicity of β, it is sufficient to show that β) β n ) is stictly monotone inceasing, i.e. β) β n )), is also positive, as we confim below: d 1 c ) z π L,, y)d Fy) d f ) = 1 z π L,, y) + zz π L,, y))d Fy) f ) z π L,, ) f ) c f ) 2 z π L,, y)d Fy) = z π L,, ) f ) f ) 2 1 > z π L,, ) + 1 F) 1 > z π L,, ) + 1 F) z π L,, y)d Fy) step a) = z π L,, ) + z π L,, ) = 0. z π L,, y)d Fy) step b) z π L,, )d Fy) step c) The diffeent steps in this assessment ae eplained as follows: step a) follows fom the facts that z π L,, y) = ql y) = 0 and zzπ L,, y) = z ql y) = 0 see Lemma 3); step b) follows fom the assumed log-concavity of the eliability function, which implies that f ) > f )2 /1 F)), togethe with the fact that z π L,, y) < 0; step c) follows fom the fact that z π L,, y) = ql y) by Lemma 3), which is monotone inceasing in y. Theefoe, the poof of monotonicity of β applies to all concave invese demand functions. Also note that β) < β n ) by the fact that z π L,, y) = ql y) < 0. The ole of the eseve pice, esp., to assue that the second-ode conditions ae satisfied, is simila to model I. Poposition 7. The intoduction of the oyalty scheme educes equilibium bids pointwise by a smalle amount than in model I. Poof. Distinguish the equilibium bid functions in models I esp. II by witing βn I,βI, esp. βn II,βII and define β I := βn I ) β I ), esp. β I I := βn I I ) β I I ). Recall that due to γ y) > 1: z πl I, z, y) z= = ql y)γ y) < q L y) model I) 17) z πl I I, z, y) z= = ql y) model II) 18) Theefoe, fo all fom the intesection of the domains of these bid functions, β I = 1 f ) = 1 f ) z π I L, z, y) z= d Fy) γ y)ql y)d Fy) by Lemma 1) 17

18 > 1 f ) = β I I > 0. ql y)d Fy) since γ y) > 1) The intuition fo this esult is as follows. Conside the equilibium bid function in model II without oyalty scheme, which by definition of an equilibium ehibits equality of the maginal benefit and the maginal cost of an incemental change in the bid fom βn I I I I ) to βn + ε).9 Now intoduce the oyalty scheme, while maintaining the candidate equilibium bid function. Then, the maginal benefit of bidding highe does not change while it gives ise to a positive maginal cost, due to the fact that when the bidde loses the auction, he pays a oyalty ate that eceed his cost eduction. This is also tue in model I. Howeve, wheeas in model II, the ival fim believes that the effective cost is equal to c, in model I the ival knows that the oyalty ate eceeds the cost eduction so that the effective cost is highe than c. To eestablish an equilibium, the bid function has to be loweed pointwise, to bing the unchanged maginal benefit in balance with the highe maginal cost. But since that maginal cost is highe in model I than in model II, the bid function has to be loweed moe in model I than in model II. 4.3 The innovato s epected evenue The above esult suggests that adding the oyalty scheme is moe pofitable in model II than in model I, since it has a smalle advese effect on the equilibium bid and thus on the innovato s auction evenue. The epected evenue of the innovato in the mechanism with esp. without oyalty scheme, G), G n ), is G n ) = 21 F)) F)R + G) = 21 F))F)R + = G n ) + β n 2 )g 2 2 )d 2 β 2 )g 2 2 )d 2 + β 2 ) β n 2 ))g 2 2 )d q L 1)g 12 1, 2 )d 2 )d 1 2 q L 1)g 12 1, 2 )d 2 )d 1. Define ) := G) G n ). Afte substituting β and R and using the fact that z π L 2, 2, 1 ) = ql 1), by Lemma 3, one obtains, ) = 2 ql 1) 2 1 F ) 2) d F 1 )d F 2 ), 2 f 2 ) Poposition 8. The intoduction of the oyalty scheme inceases the innovato epected evenue moe than in model I, fo all fom the intesection of the domains of these functions. 9 The benefit is the epected evenue fom winning the auction, net afte deducting the epected pice, and the cost is the loss in the event of losing the auction and paying a oyalty ate that eceed the own cost eduction. 18

19 Poof. Distinguish in model I esp. II by witing I ), II ). Using 17), 18), and the fact that γ y) > 1 in model I, one finds fo all fom the intesection of the domains of these functions, I ) = F ) 2) γ 1 ) d F 1 )d F 2 ) < 2 2 q L 1) 2 q L 1) f 2 ) 2 1 F 2) f 2 ) ) d F 1 )d F 2 ) = II ). As in model I, define ˆ as the oot of ), and, n, as the maimizes of G, G n,, espectively. The following esults hold fo all concave invese demand functions unlike in model I whee the coesponding esults hold only fo a class of concave demand functions). Lemma 4. ) > 0 fo all ˆ,c). Poof. Let ϕ) := 1 F)) / f ). It is staightfowad to see that is the unique solution of ϕ) = 0. The emainde of the poof is the same as the poof of Lemma 2. Note, howeve, that the functions ϕ diffe in models I and II.) Poposition 9. Adding the oyalty scheme inceases the innovato s epected evenue, G ) > G n n ) if n > ˆ. Poof. The assumption that n > ˆ implies n ) > 0 by Lemma 4. Theefoe, G ) G n ) G n n ) + n ) > G n n ). Fo moe specific esults tun to the case of linea demand, which was assumed in model I. In that case we find that adding the oyalty scheme is pofitable fo each of the pobability distibutions consideed in model I, even fo the distibution that ehibits a high concentation on low cost eductions. This fact is illustated in Figues 5. Thee, the figue on the left coesponds to the pobability distibution plotted in Figue 3, and the figue on the ight coesponds to the distibution plotted in Figue 4. Evidently, and unlike in model I, adding the oyalty scheme is pofitable even fo the pobability distibution that ehibits a high concentation on low cost eductions. These eamples illustate that adding the oyalty scheme is moe pofitable in model II than in model I, and is paticulaly appealing if the pobability distibution ehibits a high concentation on high cost eductions, as aleady eplained intuitively at the end of section 3. 19

20 Gn G Gn G min min Figue 5: Epected evenue fo tuncated eponential distibutions with a concentation on high cost eductions left) and on low cost eductions ight) 5 Discussion In the pesent pape we econside the licensing of a pocess innovation to a Counot duopoly unde incomplete infomation assuming the pivate values paadigm. Unlike the pevious liteatue, we assume that the innovato combines a estictive license auction with a mandatoy oyalty contacts fo the fim who loses the auction. We conside two specifications of the model: one in which the cost eductions become common knowledge among fims afte the auction and bids only seve as signals of the undelying cost eduction to the innovato and one in which cost eductions emain pivate infomation and both the innovato and fims can only update thei beliefs using the infomation evealed by bids. In both models, the innovato uses the infomation evealed by bids to set the oyalty ate to be paid by the lose of the auction. Ou main finding is that adding the oyalty scheme to the license auction advesely affects equilibium bidding in the auction, yet is geneally pofitable unless cost eductions ae highly concentated on low values, and is moe pofitable in model II than in model I. The limitation of the pesent pape is that we only conside the case of two fims. If the oligopoly consists of moe than two fims it emains attactive to add the oyalty scheme to the license auction. Howeve, it becomes also an issue how many licenses should be auctioned. This issue has been at cente stage in the classical liteatue on patent licensing unde complete infomation see Kamien, 1992, Giebe and Wolfstette, 2008), but has not been addessed as yet in the famewok of incomplete infomation. Anothe concen is whethe the pivate value paadigm is appopiate to analyze the cost eductions fo fims that seve the same maket and employ simila technologies. If instead one assumes a common value famewok, each fim s epected cost eduction is a function of the signals obseved by all fims, and so is the oyalty ate set by the innovato, unless the lagest signal is a sufficient statistic of the unknown cost eduction, in which case only the lagest signal mattes see Fan, Jun, and Wolfstette, 2009). A Appendi A.1 Poof of Lemma 1 Poof. We show that z π L, z, y) z= = ql y)γ y), whee γ y) > 1 fo all y. 20

21 By the envelope theoem we have z π L, z, y) z= = q L1, z, y) 1 P q W2 ) + q L1 )) z q W2, z, y) ) z= whee = ql y)γ y) γ y) : = 1 P q W2 ) + q L1 )) z q W2, z, y) ) z= > 1 since P < 0 and z q W2, z, y) z= > 0). Note, both P ) and z q W2, z, y) z= ae only functions of y. If demand is linea, P ) = 1, z q W2, z, y) z= = 1 /3; hence, γ y) = 4 /3 see Appendi A.5). Net we show that π W ) > 0 and πl y) < 0. Again, using the envelope theoem and the fact that q L 2 ) < 0 and qw 2 y) > 0 one has: π W ) = P )q L 2 )q W1 ) + q W1 ) > 0 πl y) = P )qw 2 y)ql y) < 0. A.2 Pat 2 of the poof of Poposition 3 Compute β ) fom 9). By Lemma 1 and the assumed log-concavity of the eliability function, which implies that f ) > f )2 /1 F)), this deivative can be witten as β ) = π W ) πl )) + d 1 c ) z π L,, y)d Fy) d f ) > π W ) πl )) + q L )γ ) 1 1 F) ql y)γ y)d Fy). Thee, π W ) and πl ) ae the equilibium pofits in the Counot subgames when the winning fim s signal is : π W ) = ma Pq + q q L )) c + ) q, π L ) = ma q Pq W ) + q) c ) q, and qw ) and q L ) ae the coesponding equilibium outputs.10 The fist ode conditions of the above maimization poblem ae: P q W ) + q L ))q W ) + Pq W ) + q L )) c + = 0 P q W ) + q L ))q L ) + Pq W ) + q L )) c = 0. 19) 10 Note that q W ) = q W 1 ) = q W2, z, y) z==y and q L ) = q L 2 ) = q L1, z, y) z==y. 21

22 Diffeentiating 19) w..t., one obtains fom which one can deive P )qw ) + P ) ) qw ) + q L )) + P )qw ) = 1 P )ql ) + P ) ) qw ) + q L )) + P )ql ) = 0, W ) = 2P ) + P )ql ) P )[3P ) + P ) qw ) + q L )) ] q q L ) = P ) + P )q L ) P )[3P ) + P ) q W ) + q L )) ]. 20) By the envelope theoem, one obtains π W ) = P )ql )q ) + 1 W ) 21) πl ) = P )qw )q L ). To compute γ ), which is defined as γ y) := 1 P q W2 ) + q L1 )) z q W2, z, y) ) z= see equation 7)), conside the Counot subgame in the event when both fims bid and fim 1 has lost the auction. The fist ode conditions of that subgame ae: P )q W2 ) + P ) c + y = 0 P )q L1 ) + P ) c + z = 0. 22) Diffeentiating 22) w..t. z, one obtains P ) + P )q W2, z, y) z q W2, z, y) = P )[3P ) + P ) q W2, z, y) + q L1, z, y) ) ]. 23) Setting z = y =, we get Combining 20), 21) and 24), we have P ) + P )qw γ ) = 1 + ) 3P ) + P ) qw ) + 24) q L )) π W ) πl )) + q L )γ ) = 2q L ) + q W ) + P ) + P )q L )) q W ) 3P ) + P ) q W ) + q L )) > 2q L ) + q W ) since P 0, P 0). 25) Fom 23) evaluating at z = and 7), one obtains γ y), which has the same fom as 24) with eplaced by y. Hence, we have 2P ql y)γ y) = 2q L y) + ) + P )ql y)) ql y) 3P ) + P ) qw y) + q L y)) > 2q L y). 26) 22