Online Appendix Licensing Process Innovations when Losers Messages Determine Royalty Rates

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1 Online Appendi Licensing Pocess Innovations when Loses Messages 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 Byoung Heon Jun Koea Univesity, Seoul Depatment of Economics Apil 3, 213 Intoduction This online appendi has two pats: Pat A A collection of technical poofs that ae not spelled out in the pape Pat B An etension of ou analysis to thee fims, to show how ou analysis can be genealized to moe than two fims. Refeences without pefi, such as a efeence to equation (1) o lemma 1, efe to the pape; efeences with a pefi, such as a efeence to equation (A1) o lemma A1, efe to this appendi. A Poofs of Lemmas and Popositions A.1 Poof of Lemma 1 Poof. Fist, we show that π W () > and π L (y) <. Using the envelope theoem and the fact that q L 2 () < and qw 2 (y) > one has: π W () = P ( )q L 2 ()q W1 () + q W1 () > π L (y) = P ( )q W 2 (y)q L(y) <. Net, we show that z π L (,z,y) z= = q L (y)γ(y), whee γ(y) > 1 fo all y. Shanghai Univesity of Finance and Economics, School of Economics, Guoding Road 777, 2433 Shanghai, China, cuihongf@mail.shufe.edu.cn Sung-Buk Ku An-am Dong 5-1, Seoul , Koea, bhjun@koea.ac.k Institute of Economic Theoy I, Humboldt Univesity at Belin, Spandaue St. 1, 1178 Belin, Gemany, wolfstette@gmail.com 1

2 By the envelope theoem we have z π L (,z,y) z= = q L1 (,z,y) z= (1 P (q W2 ( ) + q L1 ( )) z q W2 (,z,y) ) z= whee = q L(y)γ(y) ( γ(y) : = 1 P (q W2 ( ) + q L1 ( )) z q W2 (,z,y) ) z= > 1 (because P < and z q W2 (,z,y) z= > ). 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 Section A.9). A.2 Pat 3 of the poof of Poposition 2 Compute β () fom (1) fo. By Lemma 1 and the assumed log-concavity of the eliability function, which implies that f () > f () 2 /(1 F()), one has β () = (π W () πl ()) + d ( 1 c ) z π L (,z,y) d f () z= df(y) > (π W () π L ()) + q L()γ() 1 1 F() q L(y)γ(y)dF(y). Thee, π W () and πl () ae the equilibium pofits in the Counot subgames when the winne s cost eduction is : π W () = ma(p(q + q q L()) c + )q, πl() = ma(p(q q W () + q) c)q, and qw () and q L () ae the coesponding equilibium outputs.1 The fist ode conditions of the above maimization poblem ae: P (q W () + q L())q W () + P(q W () + q L()) c + = P (q W () + q L())q L() + P(q W () + q L()) c =. (A.1) Diffeentiating (A.1) w..t., one obtains ( P ( )qw () + P ( ) ) (qw () + q L ()) + P ( )qw () = 1 ( P ( )q L() + P ( ) ) (qw () + q L ()) + P ( )q L () =, fom which one can deive 2P ( ) + P ( )q qw L () = () P ( )(3P ( ) + P ( )(qw () + q L ())) P ( ) + P ( )q q L L () = () P ( )(3P ( ) + P ( )(qw () + q L ())). (A.2) 1 Note that q W () = q W 1 () = q W2 (,z,y) z==y,q L () = q L 2 () = q L1 (,z,y) z==y. 2

3 By the envelope theoem, one obtains π W () = ( ) P ( )q L () + 1 qw () π L () = P ( )q W ()q L(). (A.3) To compute γ(), conside the Counot subgame in the event when both fims epots met the theshold level and fim 1 has lost. The fist ode conditions of that subgame ae: P ( )q W2 ( ) + P( ) c + y = P ( )q L1 ( ) + P( ) c + z =. (A.4) Diffeentiating (A.4) 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))). (A.5) Setting z = y = and using definition (8), we get γ() = 1 + P ( ) + P ( )q W () 3P ( ) + P ( )(q W () + q L ()) (A.6) Combining (A.2), (A.3) and (A.6), we have (π W () π L ())+q L()γ() = 2q L() + q W () + (P ( ) + P ( )q L ())q W () 3P ( ) + P ( )(q W () + q L ()) > 2q L() + q W () (because P,P ). (A.7) Evaluation (A.5) at z = and using (8), one obtains γ(y), which has the same fom as (A.6) with eplaced by y. Hence, we have q L(y)γ(y) = 2q L(y) + (2P ( ) + P ( )q L (y))q L (y) 3P ( ) + P ( )(q W (y) + q L (y)) > 2q L(y). (A.8) Fom (A.7) and (A.8), it follows that β () > 2q L() + qw 1 () 1 F() > 2q L() + qw 1 () 1 F() = q W () >, the second inequality holds because q L ( ) is a deceasing function. 2q L(y)dF(y) 2q L()dF(y) 3

4 A.3 Pat 4 of the poof of Poposition 2 Poof. Conside a fim with cost eduction. If that fim paticipates, and the othe fim tells the tuth, that fim s payoff is Π p (), wheeas if it does not paticipate, its payoff is Π np (): Let Π p () = F()(π W () R) + = F()R + F()π W () Π np () = F()π A + πl(y)df(y). ψ() : = Π p () Π np () (π W () β(y))df(y) + = F()(π A + R) + F()π W () β(y)df(y) + π L(y)dF(y) π L(y)dF(y) (π L(y) + β(y))df(y). Diffeentiate ψ with espect to, and one obtains, using (1), fo : ψ () = f ()π W () + F()π W () (π L() + β()) f () = F()π W () z π L (,,y)df(y) >. By definition of, one has ψ() = ; hence, is implicitly defined as the solution of (1), and we conclude that fims paticipate if and only if. A.4 Poof of Poposition 3 Poof. Π(,z) is pseudoconcave if the coss deivative is positive: z Π(,z) = ( π W () π L (,z,z)) f (z) + z z π L (,z,y)df(y) >. (A.9) We show that this is tue if and z ae sufficiently lage. Because z, this holds if and only if is sufficiently lage. Let c. Then the integal on the RHS of (A.9) vanishes; howeve, the fist tem does not vanish and is positive. Indeed, as we show below, lim,z c ( π W () π L (,z,z)) >. Hence, z Π(,z) > fo = z = c. By continuity, pseudoconcavity holds tue fo,z if is sufficiently lage. Finally, we pove the inequality: lim ( π W () π L (,z,z)) = (4P (Q) + P (Q)Q)(q W1 (c) q L1 (c,c,c)),z c 3P >. (A.1) (Q) + P (Q)Q By the envelope theoem one has π W () = q W1 () ( 1 + P (Q) q L2 () ) π L (,z,z) = q L1 (,z,z) ( 1 + P (Q) q W2 (,z,z) ). (A.11) (A.12) The pofile of unit costs that undelies π W () is (c 1,c 2 ) = (c,c), and the pofile that undelies π L (,z,z) is (c 1,c 2 ) = (c + z,c z). In each case, the sum of unit costs is equal to 2c. By a 4

5 well-known fact, the aggegate equilibium output, Q, is only a function of the sum of unit costs (see Begstom and Vaian, 1985). Theefoe, Q is the same in both equations. The fist ode conditions concening the subgame that undelies the fist equation ae: P(Q) + P (Q)q W1 () (c ) = P(Q) + P (Q)q L2 () c =. (A.13) (A.14) Diffeentiating (A.13) and (A.14) with espect to and solving fo q L2 () one has q L2 () = P (Q) + P (Q)q L2 () P (Q)(3P (Q) + P (Q)Q). (A.15) Similaly, one finds q W2 (,z,z) = P (Q) + P (Q)q W2 (,z,z) P (Q)(3P (Q) + P (Q)Q). (A.) Combining (A.11)-(A.) and using the facts that q L1 (c,c,c) = q L2 (c),q W1 (c) = q W2 (c,c,c) poves inequality (A.1). A.5 Poof of Poposition 4 Poof. Substituting π W (), and π L (,z,y) fom Section A.9 into (A.9), one can easily confim that,z : z Π(,z) = 4 3 z f (z) 8 9 (1 F(z)),z min. (A.17) Eistence and uniqueness of min follow fom the fact that the LHS of (12) in the pape is negative at =, positive at = c, and stictly inceasing in by the assumed hazad ate monotonicity. A.6 Poof of Lemma 3 Poof. The poof of pat 1) is the same as the poof of Lemma 1 (spelled out in Section A.1) ecept that in model II one has z q W2 ( ) = ; theefoe, γ(y) = 1, see the definition of γ in (8) in the pape. 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) (because q L2 (z) = ) ) z= > (because P ( ) < and z q L2 (z) < ) π L (y) = P ( ) y q W2 (y)q L1 (,z,y) < (because y q W2 (y) > ). A.7 Poof of Poposition 8 Poof. The coss deivative of the payoff function is z Π(,z) = ( π W (,z) π L (,z,z)) f (z) + z π W (,z)f(z) + z z π L (,z,y)df(y). (A.18) 5

6 1) Sufficiency: We show that this coss-deivative is positive if,z ae sufficiently lage. Theefoe, Π is pseudoconcave if is sufficiently lage. Let c. Then the integal on the RHS of (A.18) vanishes; howeve, the sum of the fist and second tems on the RHS of (A.18) is positive, as we show net. By the envelope theoem, π W (,z) = q W1 (,z), π L (,z,z) = q L1 (,z,z). (A.19) Hence, lim,z c ( π W (,z) π L (,z,z)) = q W1 (c,c) q L1 (c,c,c) >. By z π W (,z) =z=c >, it follows that fo = z = c, z Π(,z) >. By continuity, pseudoconcavity holds tue fo all, z if is sufficiently lage. 2) Necessity: We show that pseudoconcavity is violated if is not sufficiently lage. Fo this pupose, suppose =. Conside a fim with cost eduction > that epots z =. We show that z Π(,z) z= <, which contadicts pseudoconcavity, because pseudoconcavity equies that Π(,z) is monotone inceasing fo all z <. Diffeentiating Π(, z) with espect to z, and using the candidate tansfe ule β stated in Poposition 7, one has fo all z Π(,z) z= = (π W (,) π L (,,)) f () + (πl() π W (,)) f () ) + ( z π L (,z,y) z= z π L (,z,y) =,z= df(y) = = τz π L (τ,z,y) z= dτdf(y) z q L1 (τ,z,y) z= dτdf(y) <. Thee, the second equality follows fom π W (,) π L (,,), π L () π W (,), the definition of the definite integal and a change in the ode of diffeentiation of π L ; the thid equality follows fom (A.19), and the final inequality follows fom the fact that z q L1 (τ,z,y) z= <. A.8 Poof of Poposition 9 Poof. Denote the tansfe ules in models I and II by βn,β I I and βn II,β II, and define β I := βn() I β I () and β II := βn II () β II (). Using Lemmas 1 and 3, fo all fom the intesection of the domains of the functions β n,β in models I and II: β I = 1 f () > 1 f () z π I L(,z,y) z= df(y) q L(y)dF(y) (by Lemma 1) = β II > (by Lemma 3). 6

7 A.9 Model I with linea demand Suppose P(Q) = 1 Q, Q := q 1 + q 2. In the game with oyalty scheme the equilibium output stategies of fim 1 ae: q W1 () = (1 c+2)/3, q L1 (,z,y) = (1 c+2 2z y)/3. The associated equilibium pofits ae π W () = q W1 () 2 and π L (,z,y) = q L1 (,z,y) 2. The equilibium pofit when no fim has access to the innovation is π A = (1 c) 2 /9. In the game without oyalty scheme q L1 (,z,y) should be eplaced by q L (y) = (1 c y)/3 and π L (,z,y) by π L (y) = q L (y) 2. The tansfe ules, β, β n, ae equal to β() = β n () = R = 4(1 c+)/9 fo <, and fo all : (2 2c + ) β n () =, β() = β n () 4 (1 c y)df(y). 3 9 f () A.1 Model II with linea demand In the game with oyalty scheme, the equilibium output stategies of fim 1, ae q W1 (,z) = (2 2c+3+z)/6, q L1 (,z,y) = (2 2c+3 3z 2y)/6. The associated equilibium pofits ae π W (,z) = q W1 (,z) 2 and π L (,z,y) = q L1 (,z,y) 2. The equilibium pofit when no fim has access to the innovation, π A, is the same as in model I. In the game without oyalties, q L1 (,z,y) should be eplaced by q L (y) = (1 c y)/3 and π L (,z,y) by π L (y) = (q L (y)) 2. The tansfe ules β, β n ae equal to β() = β n () = R = 4(1 c+)/9 fo <, and fo : (2 2c + ) β n () = + 3 β() = β n () 1 3 f () (1 c + 2)F() 9 f () (1 c y)df(y). B Etension to moe than two fims In this section we show how ou analysis can be etended to moe than two fims. Specifically, we conside the case of thee fims, allowing the innovato to awad eithe one o two unesticted licenses, whicheve is moe pofitable. All othe assumptions ae maintained. We poceed as follows: fist, we conside the case when one unesticted license is offeed and show that adding the oyalty scheme is pofitable; second, we conside the case when the innovato offes two unesticted licenses and again show that adding the oyalty scheme is pofitable; finally, we show that offeing one unesticted license togethe with the oyalty scheme is optimal. We conside only the moe plausible model II. B.1 One unesticted license One unesticted license is offeed to thee fims. Fims simultaneously epot thei cost eductions. Denote the highest, the second highest, and the thid highest epoted cost eductions by ˆ (1), ˆ (2) 7

8 and ˆ (3), espectively. The fim that has epoted the highest cost eduction not smalle than is awaded the unesticted license and pays β(ˆ (2) ). The fim(s) that epoted a cost eduction smalle than the highest cost eduction and not smalle than obtain a oyalty license with a oyalty ate equal to thei epoted cost eduction. Fims that epoted a cost eduction below do not obtain a license and pay nothing. Fo convenience we efe to the fim that obtains an unesticted license as winne and to oyalty licensee(s) as lose(s). A fim that epoted a cost eduction smalle than the theshold level qualifies neithe as winne no lose. Afte licensing, all epoted cost eductions ae publicly evealed to all fims. B.1.1 Counot subgames Conside one fim, say fim 1, that has dawn the cost eduction but epots z, while the two othe fims with cost eductions y and v tell the tuth. As a convention let y > v. Case 1): All thee epots met the theshold level and fim 1 won (z y > v ) The innovato allocates the unesticted license to fim 1 and the oyalty licenses to the othe fims and chages them a oyalty ate equal to thei epoted cost eduction.because fims 2 and 3 tell the tuth, they believe to play a Counot game with the pofile of unit costs (c 1,c 2,c 3 ) = (c z,c,c). Denote the associated equilibium stategies of the game the loses believe to play by (q W (z),q L2 (z),q L3 (z)). Howeve, fim 1 pivately knows that its cost eduction is athe than z. Theefoe, fim 1 plays the best eply to (q L2 (z),q L3 (z)): q W1 (,z) = ag ma(p(q + q L2 (z) + q L3 (z)) c + )q. q (B.1) The educed fom pofit function of fim 1 is π W (,z) := (P(q W1 (,z) + q L2 (z) + q L3 (z)) c + )q W1 (,z). (B.2) Case 2): All thee epots met the theshold level and fim 1 lost (y > z and z,y,v ) In that case fims 2 and 3 believe to play a Counot game with the pofile of unit costs (c 1,c 2,c 3 ) = (c,c y,c). Denote the associated equilibium stategies of the game fims 2 and 3 believe to play by (q L1 (y),q W2 (y),q L3 (y)). If no oyalty scheme is used, the equilibium play of fim 1 depends only on the winne s cost eduction y. The associated educed fom pofit function of fim 1 is π Ln (y) := (P(q L1 (y) + q W2 (y) + q L3 (y)) c)q L1 (y). (B.3) Wheeas if the oyalty scheme is adopted, fim 1 pivately knows that its cost eduction is and that it pays a oyalty ate equal to z. Theefoe, fim 1 plays the best eply to (q W2 (y),q L3 (y)): q L1 (,z,y) = agma(p(q + q W2 (y) + q L3 (y)) c + z)q. q (B.4) The associated educed fom pofit function of fim 1 is π L (,z,y) : = (P(q L1 (,z,y) + q W2 (y) + q L3 (y)) c + z)q L1 (,z,y). (B.5) 8

9 On the equilibium path, i.e., fo z =, the payoff of fim 1 in the event when it loses is only a function of the highest cost eduction of othe fims, y; theefoe, we denote the on-the-equilibium path output and pofit of fim 1 when it loses by q L(y) = q L (,z,y) z=, π L(y) = π L (,z,y) z= = π Ln (y). (B.6) Case 3): At least one epot did not meet the theshold level If no one s epot met the theshold level, the equilibium pofit of fim 1 is equal to π A. If fim 1 was the only one whose epot met the theshold level, its equilibium pofit is π W (,z), and if the ival fim with a cost eduction y was the only one whose epot met the theshold level, the equilibium pofit of fim 1 is the same as in the game without oyalty scheme, i.e. π Ln (y). Eample Suppose invese demand is linea, P(Q) = 1 Q,Q = q 1 + q 2 + q 3. Then, the equilibium payoffs of the elevant Counot equilibium subgames ae: π W (,z) = (1 c z)2 (1 c + 2 2z y)2, π L (,z,y) = q L(y) = 1 c y, π Ln (y) = π 4 L(y) = (1 c y)2, π A = (B.7) (1 c)2. (B.8) B.1.2 Licensing mechanism We now constuct the tansfe ule β that induces tuth-telling as a Bayesian Nash equilibium, assues voluntay paticipation, and etacts the highest possible suplus fo given. Simila to the case of two fims, we fist constuct β() fo < to assue voluntay paticipation fo these types. Lemma B.1. Let R = π W (,) π A, (B.9) If one sets β() = R fo all <, all fims with cost eductions ae indiffeent between paticipation and non-paticipation, assuming the othe fims epot tuthfully. Hence, voluntay paticipation is assued fo these types. Poof. Conside the maginal fim, with cost eduction =. If that fim paticipates and all othe fims tell the tuth, that fim s payoff is Π p (), wheeas if it does not paticipate, its payoff is Π np (): Π p () = F() 2 (π W (,) R) + Π np () = F() 2 π A + π L(y) f (12:2) (y,v)dvdy + π Ln (y) f (12:2) (y,v)dvdy + y y π L(y) f (12:2) (y,v)dvdy π Ln (y) f (12:2) (y,v)dvdy, whee f (12:2) = 2 f (y) f (v) is the joint p.d.f. of the highest and lowest ode statistics in a sample of two i.i.d. andom vaiables. The maginal fim with = is indiffeent between paticipation and non-paticipation, Π p () = Π np (), which implies R = π W (,) π A. Like in the case of two fims, lowe types with < will also paticipate since they obtain no license and pay nothing and thus ae not hut by paticipating. Also, tuth-telling is assued fo all types <. 9

10 To deive β fo, conside fim 1 with cost eduction, but epots a cost eduction z, wheeas fims 2 and 3 tell the tuth. Then, using the equilibia of the duopoly subgames, the payoff function of fim 1 in the game with oyalty scheme fo z is z Π(,z) = F() 2 (π W (,z) R) + (π W (,z) β(y)) f (1:2) (y)dy + π L (,z,y) f (1:2) (y)dy, z whee f (1:2) (y) = 2F(y) f (y) is the p.d.f. of the lagest ode statistic in a sample of two i.i.d. andom vaiables. In the licensing game without oyalty scheme, the payoff function is the same, ecept that in the last tem π L (,z,y) must be eplaced by π Ln (y). The tansfe ule β must be such that = agma z Π(,z), fo all [,c]. Poposition B.1. In the mechanism with and without oyalty scheme, set β() = β n () = R fo <, and fo : β() = β n () + 1 c z π L (,z,y) f (1:2) () z= f (1:2) (y)dy (B.1) β n () = π W (,) π Ln () + F() 2 f () zπ W (,z) z=. (B.11) Then β,β n induce tuthtelling povided is sufficiently high, and β,β n ae stictly inceasing fo all. Poof. It is staightfowad to deive the tansfe ules, β and β n, and to show that they ae stictly monotone inceasing. Howeve, one also needs to assue that second ode conditions fo best eplies ae satisfied. Like in the case of two fims, this equies that the theshold level is sufficiently lage. The eact condition will be stated when we compute an eample, below. The paticipation constaints ae satisfied fo all, the poof is simila to that in online appendi A.3. B.1.3 The innovato s epected evenue The epected evenue of the innovato in the licensing games with and without oyalty scheme, G() and G n (), ae: G n () = 3(1 F())F() 2 (π W (,) π A ) + β n (y) f (2:3) (y)dy G() = G n () + (β(y) β n (y)) f (2:3) (y)dy + 6 y (v + y)q L() f () f (y) f (v)dvdyd, whee f (2:3) (y) = 6(1 F(y))F(y) f (y) is the p.d.f. of the second highest ode statistic in a sample of thee i.i.d. andom vaiables. In Figue B.1 we plot the functions G(),G n (), assuming linea invese demand, as in (B.7)- (B.8),c =.49, and F() := c (unifom distibution). As one can easily confim, in the game with oyalty scheme, second-ode conditions (pseudoconcavity of π(,z) in z) ae satisfied only if min =.289. Theefoe, the dashed potion of the G function must be ignoed. Compaing G and G n, this eample shows that adding the oyalty scheme is pofitable also in the case of thee fims and one unesticted license. 1

11 G n G.119 n min Figue B.1: Pofitability of the oyalty scheme with 3 fims and 1 unesticted license B.2 Two unesticted licenses When thee ae thee fims, the innovato may also awad two unesticted licenses. Theefoe, in ode to detemine whethe the oyalty scheme is pofitable when thee ae thee fims, we also need to choose the optimal numbe of unesticted licenses. Fo this pupose we now solve the game fo the case when two unesticted licenses ae offeed. As we will see late, in ou eample it is actually optimal fo the innovato to offe only one unesticted license togethe with the oyalty scheme. When two unesticted licenses ae offeed, the innovato employs the following ules: If (2), two unesticted licenses ae awaded to the fims that epoted the two highest cost eductions, and the winnes pay β(ˆ (3) ). If (1) > (2), one unesticted license is awaded to the fim that epoted the highest cost eduction and the winne pays β( (2) ). The fims that epoted a cost eduction geate o equal to but failed to win (called loses) ae awaded a oyalty license and pay a oyalty ate equal to thei epoted cost eduction. Fims that epot a cost eduction below do not obtain a license and pay nothing. B.2.1 Counot subgames Suppose fim 1 has dawn cost eduction but epots z, 2 wheeas fims 2 and 3 tell the tuth (again, downwad deviations yield the same diffeential equation). Again, by convention y > v. The following Counot subgames occu. Case a) Only fim 1 s epot met the theshold level (z > y v) In this case, the equilibium pofit of fim 1 is the same as that in case 1) of the game with 3 fims and only one unesticted license is offeed, i.e., π Wa (,z) = π W (,z). Case b) At least two epots met the theshold level and fim 1 won (eithe z > v and z,y,v o z,y and v < ) In that case, fims 2 and 3 believe to play a Counot game with the pofile of unit costs (c 1,c 2,c 3 ) = (c z,c y,c). Denote the associated equilibium stategies of the game fims 2 and 3 believe to play by (q W1 (z,y),q W2 (z,y),q L3 (z,y)). 2 Like in the case of one unesticted license, downwad deviations yield the same diffeential equation. 11

12 Howeve, fim 1 pivately knows that its cost eduction is athe than z, and plays the best eply to (q W2 (z,y),q L3 (z,y)): q W (,z,y) = agma(p(q + q W2 (z,y) + q L3 (z,y)) c + )q. q (B.12) The associated educed fom pofit function of fim 1 is then π Wb (,z,y) := (P(q W (,z,y) + q W2 (z,y) + q L3 (z,y)) c + )q W (,z,y). (B.13) Case c): All thee epots met the theshold level and fim 1 lost (v > z and z,y,v ) In that case, both fim 2 and fim 3 won an unesticted license and believe to play a Counot game with the pofile of unit costs (c 1,c 2,c 3 ) = (c,c y,c v). Denote the associated equilibium stategies of the game the winnes believe to play by (q L1 (y,v),q W2 (y,v),q W3 (y,v)). If no oyalty scheme is used, the equilibium stategy of fim 1 depends only on the winnes cost eductions, y and v. The educed fom pofit function of fim 1 is π L2n (y,v) := (P(q L1 (y,v) + q W2 (y,v) + q W3 (y,v)) c)q L1 (y,v). (B.14) Wheeas if the oyalty scheme is adopted, fim 1 pivately knows its cost eduction is, yet pays a oyalty ate z. Theefoe, fim 1 plays the best eply to (q W2 (y,v),q W3 (y,v)): q L (,z,y,v) = agma(p(q + q W2 (y,v) + q W3 (y,v)) c + z)q. q (B.15) The associated educed fom pofit function of fim 1 is π L (,z,y,v) := (P(q L (,z,y,v) + q W2 (y,v) + q W3 (y,v)) c + z)q L (,z,y,v). (B.) On the equilibium path, i.e., fo z =, that payoff of fim 1 when it loses is only a function of winnes cost eductions y and v; and we wite q L(y,v) = q L (,z,y,v) z=, π L(y,v) = π L (,z,y,v) z= = π L2n (y,v). (B.17) Case d) Fim 1 s epot did not meet the theshold level (z < ) If fim 1 s epot is below while both ival fims epots ae not smalle than, the equilibium pofit of fim 1 depends only on the ivals cost eductions and is equal to π L2n (y,v). If fim 1 s epot is below and only the ival fim with highe cost eduction (y) epoted a cost eduction not smalle than, the equilibium pofit of fim 1 depends only on y and is equal to π Ln (y). If no one s epot met the theshold level, the equilibium pofit of fim 1 is equal to π A. Eample Suppose invese demand is linea, as in the eample in section B.1. Then, the equilibium payoffs of the elevant Counot subgames ae: (1 c z)2 (1 c z y)2 π Wa (,z) =, π Wb (,z,y) =, (1 c + 2 2z y v)2 π L (,z,y,v) =, π L(y,v) = π L2n (y,v) = q L(y,v) = 1 c y v, π Ln (y) = 4 (1 c y)2. 12 (1 c y v)2,

13 B.2.2 Licensing mechanism We fist constuct β fo < to assue voluntay paticipation fo these types. Lemma B.2. If one sets β() = R fo all <, whee R 1 ( c = (πwa (,) π A )F() + 2 (π Wb (,,y) π Ln (y)) f (y)dy ), (B.18) 2 F() then all fims with cost eductions ae indiffeent between paticipation and non-paticipation, assuming ival fims epot tuthfully. Hence, voluntay paticipation is assued fo these types. Poof. Conside the maginal fim with cost eduction =. If that fim paticipates, and all othe fims tell the tuth, that fim s payoff is Π p (), wheeas if it does not paticipate, its payoff is Π np (): Π p () = F() 2 (π Wa (,) R ) + + y Π np () = F() 2 π A + π L(y,v) f (12:2) (y,v)dvdy whee f (12:2) (y,v) = 2 f (y) f (v). π Ln (y) f (12:2) (y,v)dvdy + (π Wb (,,y) R ) f (12:2) (y,v)dvdy y π L2n (y,v) f (12:2) (y,v)dvdy, The maginal bidde with = must be indiffeent between paticipating and not paticipating, Π p () = Π np (), which implies (B.18). The types < will also paticipate because they obtain no license and pay nothing and thus ae not hut by paticipation. Simila to the case of one unesticted license, tuth-telling is assued fo all <. Now we deive β fo. Conside fim 1 with cost eduction, but epots a cost eduction z, wheeas fims 2 and 3 tell the tuth. Then, using the equilibia of the duopoly subgames, the payoff function of fim 1 in the game with oyalty scheme fo z is Π(,z) = F() 2 (π Wa (,z) R ) z y y z whee f (12:2) (y,v) = 2 f (y) f (v). z (π Wb (,z,y) β(v)) f (12:2) (y,v)dvdy + π L (,z,y,v) f (12:2) (y,v)dvdy (π Wb (,z,y) R ) f (12:2) (y,v)dvdy z z (π Wb (,z,y) β(v)) f (12:2) (y,v)dvdy In the licensing game without oyalty scheme, the payoff function is the same, ecept that in the last tem π L (,z,y,v) must be eplaced by π L2n (y,v). The tansfe ule β must be such that = agma z Π(,z), fo all [,c]. Poposition B.2. Set β() = β n () = R fo all <, and fo all : β() = β n () + β n () = y 1 2 f ()(1 F()) z π L (,z,y,v) z= f (12:2) (y,v)dvdy ( ( F() 2 z π Wa (,z) z= + 2F() 13 ) z π Wb (,z,y) z= df(y)

14 ) + z π Wb (,z,y) z= f (1:2) (y)dy + 2 f () (π Wb (,,y) π Ln (y,)) f (y)dy. Then β,β n induce tuth-telling and voluntay paticipation, povided is sufficiently high, and β,β n ae stictly inceasing fo. Poof. The deivation of β,β n and the poof of thei monotonicity ae simila to that in the case of two fims. Also simila to othe cases with oyalty scheme, the second ode conditions ae satisfied only when the theshold level is sufficiently lage. B.2.3 The innovato s epected evenue The epected evenue functions of the innovato without and with oyalty scheme ae, G n () = 3(1 F())F() 2 R + 3(1 F()) 2 F()2R + 2 G() = G n () + 2(β(v) β n (v)) f (3:3) (v)dv + 6 y β n (v) f (3:3) (v)dv vq L(,y) f () f (y) f (v)dvdyd whee R is defined in (B.18)) and f (3:3) (v) = 3 ( 1 F(v) 2) f (v) is the p.d.f. of the lowest ode statistic in a sample of thee i.i.d. andom vaiables G 2 license case G 1 license case G n 2 license case n 3.2 min Figue B.2: Pofitability of 1 vs 2 unesticted licenses when thee ae 3 fims In Figue B.2 we plot the innovato s epected evenue with one unesticted license, G() (one license case with oyalty scheme), with two unesticted licenses, G() (two license case with oyalty scheme), and G n () (two license case without oyalty scheme). Like befoe, we assume linea invese demand, c =.49, and F() = /c (unifom distibution). Compaing the plots in Figues B.1-B.2 we conclude that awading one unesticted license, combined with the oyalty scheme, is stictly bette than all othe mechanisms. 3 Refeences Begstom, T. C. and H. Vaian (1985). Two Remaks on Counot Equilibia. Economics Lettes 19, pp Note, the diffeent shapes of G() (one license case with oyalty scheme) in Figues B.1 and B.2 ae due to the diffeent scales we use in these two figues. 14