Licensing process innovations when losersʼ messages determine royalty rates

Games and Economic Behavior 82 (2013) 388–402

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Licensing process innovations when losers’ messages determine royalty rates ✩ Cuihong Fan a,∗ , Byoung Heon Jun b , Elmar G. Wolfstetter c,d a

Shanghai University of Finance and Economics, School of Economics, Guoding Road 777, 200433 Shanghai, China Korea University, Department of Economics, Sung-Buk Ku An-am Dong 5-1, Seoul 136-701, Republic of Korea Humboldt University at Berlin, Department of Economics, Spandauer Str. 1, 10178 Berlin, Germany d Korea University, Seoul, Republic of Korea b c

a r t i c l e

i n f o

Article history: Received 20 April 2010 Available online 23 August 2013 JEL classification: D21 D43 D44 D45

a b s t r a c t We consider a licensing mechanism for process innovations that awards a limited number of unrestricted licenses to those firms that report the highest cost reductions, combined with royalty licenses to others. Firms’ messages are dual signals of their cost reductions: the message of those who win an unrestricted license signals their cost reduction to rival firms, while losers’ messages influence the royalty rate set by the innovator. We explain why a sufficiently high threshold level for awarding the unrestricted license is essential to induce truth-telling, show that the innovator generally benefits from the proposed mechanism, and derive conditions for implementability by a modified second-price auction. © 2013 Elsevier Inc. All rights reserved.

Keywords: Patents Licensing Auctions Royalty Innovation R&D Mechanism design

1. Introduction This paper revisits the licensing of a non-drastic process innovation by an outside innovator to a Cournot oligopoly. The cost reductions induced by the innovation are of the private values type and are firms’ private information. The main feature is that we replace the standard license auction by a superior mechanism that awards a limited number of unrestricted licenses to those firms that report the highest cost reductions and royalty licenses to the remaining firms, provided that their reported cost reductions meet a threshold level set by the innovator. Thus, the innovator has two sources of revenue: the transfers paid by those who win the unrestricted licenses and the royalty income paid by those who obtain the royalty licenses. ✩ Financial support was received from the Humanities and Social Sciences Research Foundation of the Ministry of Education of China (Grant 09YJA790133) and the “Innovation Program of Shanghai Municipal Education Commission” (Grant 12ZS076), the National Research Foundation of Korea funded by the Korean Government (NRF-2010-330-B00085), and the Deutsche Forschungsgemeinschaft (DFG), SFB Transregio 15, “Governance and Efficiency of Economic Systems.” Detailed comments by the Advisory Editor and the anonymous referees and discussions with Michael Peters and David Salant are gratefully acknowledged. Corresponding author. E-mail addresses: [email protected] (C. Fan), [email protected] (B.H. Jun), [email protected] (E.G. Wolfstetter).

*

0899-8256/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.geb.2013.08.003

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This licensing mechanism gives rise to a dual signaling problem: If a firm wins an unrestricted license, its message signals the own cost reduction to rival firms, whereas if it loses, its message signals the own cost reduction also to the innovator who sets the royalty rate equal to the reported cost reduction. Firms take into account that they can influence others’ beliefs with their messages. Specifically, a firm gains a strategic advantage in the oligopoly game with an “inflated message” that signals a higher than true cost reduction, provided it happens to win. In turn, a firm can fool the innovator to set the royalty rate below the true cost reduction with a “deflated message”, provided it happens to lose. Of course, no such “misleading” signaling occurs on the truth-telling equilibrium path. In equilibrium, the marginal benefit of reporting a cost reduction that deviates from the true cost reduction must be matched by a corresponding marginal cost in such a way that both kinds of signaling are deterred in all states of the world. If messages can only influence the beliefs of rival firms, misleading signals are easily deterred by choosing an appropriate steepness of the winner’s transfer rule. As a result, the possibility to influence the beliefs of rival firms with a winning message, simply exerts an upward pressure on winners’ transfers. Similarly, the possibility to influence the beliefs of the innovator with the losing message exerts a downward pressure on winners’ transfers. However, one cannot deter firms from reporting to the innovator a lower than true cost reduction by adjusting the steepness of the winners’ transfer rule alone. In addition, the innovator must set a sufficiently high threshold level or the support of the distribution of cost reductions must be sufficiently bounded away from zero. Otherwise, firms with a low cost reduction would always report a zero cost reduction in order to lose and obtain the innovation for free. We analyze two specifications of the model that differ in the information available to firms in the downstream oligopoly game. Whereas in model I firms’ cost reductions become common knowledge among firms after licenses have been awarded and before the oligopoly game is played, in model II firms remain uninformed about their rivals’ cost reductions after licensing. In both models the innovator does not know firms’ cost reductions, and the winners of the unrestricted licenses remain uninformed about the royalty rates paid by the losers. Altogether, model II is more plausible. Nevertheless, model I is useful as a benchmark and prepares nicely for the more complex analysis of model II. There is an extensive literature on patent licensing in oligopoly by an outside innovator, among which the following contributions are closely related to the present paper. In their seminal contributions, Kamien (1992), Kamien and Tauman (1984, 1986), and Katz and Shapiro (1985, 1986) show that auctioning a limited number of licenses is more profitable for the innovator than other mechanisms, such as pure royalty contracts, fixed-fee licensing, and two-part tariffs.1 The limitation of the classical literature is that it assumes complete information both in the auction and in the downstream oligopoly game. Jehiel and Moldovanu (2000) introduce incomplete information at the bidding stage combined with complete information in the oligopoly game. They show that, because the loser of the auction is adversely affected by the winner’s cost reduction (negative externality), the seller has a stronger incentive to induce participation, and therefore the reserve price plays a less prominent role than in the standard auction framework. Later, Das Varma (2003) and Goeree (2003) reconsider that model assuming that firms do not know each other’s cost reductions after the auction and before the oligopoly game is played. This introduces the possibility to signal the own cost reduction to rival firms. In a preceding paper, Giebe and Wolfstetter (2008) introduce royalties into license auctions assuming complete information. There, the innovator charges the loser of the auction a royalty rate equal to the cost reduction induced by the innovation. As a result, the royalty scheme has no effect on equilibrium bids because the losers’ payoffs in the oligopoly game are not affected. The innovator’s expected revenue is increased if no loser is crowded out of the market. The challenge of that paper was to show that the optimal number of auctioned licenses is such that no losers are crowded out and royalties are paid. Unlike Giebe and Wolfstetter (2008), the present paper assumes that firms’ cost reductions are their private information, unknown both to their rivals and to the innovator. Under incomplete information, adding the royalty scheme works in a different way: First, it gives rise to a complex dual signaling problem where the winning and the losing messages signal information both to rival firms and to the innovator; second, it leads to lower transfers to be paid by the winners. Therefore, the innovator faces a trade-off between revenue earned from winners of the unrestricted licenses and the royalty income from losers. Our main findings are summarized as follows: 1) A transfer rule that induces truth-telling exists provided the innovator sets a sufficiently high threshold level. 2) Adding royalty licenses for losers exerts a downward pressure on winners’ transfers. 3) However, the additional royalty income weighs more than the loss in revenue from the winners. 4) Adding royalty licenses is particularly profitable when the probability distribution exhibits a concentration on high cost reductions. The paper is organized as follows: In Section 2 we present the model. In Section 3 we analyze model I and show that the threshold level plays a crucial role in assuring the existence of a transfer rule that induces truth-telling. In Section 4 we analyze model II. In Section 5 we show that the equilibrium outcome of the licensing game can also be implemented by replacing the direct mechanism with a modified license auction. In Section 6 we show that our analysis can be extended

1 Sen (2005) shows that if one takes into account that the number of licenses must be an integer, pure royalty contracts can be superior to license auctions. However, this result is again reversed if one generalizes the format of license auctions (see Giebe and Wolfstetter, 2008). See also Sen and Tauman (2007) who analyze an auction of royalty contracts.

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to more than two firms. In Section 7 we discuss our results and explain why the assumed restrictions on royalty rates are compelling due to antitrust concerns. Some of the proofs and supplementary results are relegated to an Online Appendix. 2. The model We consider a dynamic licensing game designed by an outside innovator of a non-drastic process innovation and played between two firms that interact in a duopoly market. In the first stage the innovator sets the licensing rule and firms compete for licenses constrained by this rule. In the second stage firms play a Cournot duopoly game. Prior to the innovation, firms have the same unit cost c. Using the innovation reduces unit costs by an amount xi ∈ X := [0, c ] that depends on who uses it. Cost reductions are firms’ private information, unknown to their rival and to the innovator. They are i.i.d. random variables, drawn from the c.d.f. F : X → [0, 1] with positive p.d.f. f everywhere. The innovator employs the following direct (revelation) mechanism that prescribes the allocation of licenses and transfers as a function of reported cost reductions xˆ = (ˆx1 , xˆ 2 ) ∈ X 2 . If firm i has reported the highest cost reduction not smaller than a threshold level r it is awarded an unrestricted license and pays β(ˆx j ) (r and β are to be determined in our analysis). If it has reported the lowest cost reduction not smaller than r it is awarded a royalty license with a royalty rate equal to its reported cost reduction xˆ i and pays xˆ i qi , where, q i denotes its output that is determined in the downstream Cournot market game. Firms that report a cost reduction below r do not obtain a license and pay nothing.2 Timing of the game: 1) The innovator announces the licensing mechanism. 2) Firms simultaneously report their cost reductions. 3) The innovator awards licenses and either the true cost reductions become known (model I) or the reported cost reductions are revealed (model II). 4) Firms play a Cournot market game. 5) The innovator observes the output of the royalty licensee (if a royalty license has been awarded) and collects payments. Firms and the innovator are risk neutral and inverse market demand P ( Q ) is a decreasing and concave function of aggregate output Q := q1 + q2 .3 Both F and the reliability function 1 − F are log-concave, which rules out that F has parts that are highly convex or highly concave.4 The log-concavity of 1 − F is equivalent to the usual hazard rate monotonicity.5 We treat the threshold level r as a parameter, but will show that it must be set sufficiently high to assure existence of a transfer rule that induces truth-telling and extract surplus. For convenience we refer to the recipient of the unrestricted license as “winner” and to the royalty licensee as “loser”. A firm that reported a cost reduction lower than the threshold level r qualifies neither for winning nor for losing. By design, the above mechanism is reminiscent of a second-price auction. We prove existence of a strictly increasing transfer rule β that induces voluntary participation and truth-telling as a Bayesian Nash equilibrium (BNE), provided the threshold level is sufficiently large.6 If this condition is met, the equilibrium outcome can also be implemented by a modified second-price auction subject to a reserve price in which the firm that makes the lowest bid must accept a royalty license with the royalty rate set by a proclaimed rule as a function of the lowest bid (see Section 5). 3. Model I Following Jehiel and Moldovanu (2000) we first consider a model in which firms learn about each other’s cost reductions before they play the duopoly game. Jehiel and Moldovanu (2000) consider a license auction with a reserve price and refer to the auctioning of a patent license as a prime example of auctions with negative externalities. Their main finding is that in the presence of a negative externality, using a high reserve price that induces a high threshold level becomes less profitable for the auctioneer. The purpose of this section is to show that if one supplements the licensing mechanism with the proposed royalty scheme, the threshold level plays a crucial role and the expected revenue of the innovator is generally increased. 3.1. Benchmark: The game without royalty license for the loser We briefly review the results of Jehiel and Moldovanu (2000) in light of our direct mechanism, which serve as a benchmark. 2 Stated formally, the direct mechanism ( X 2 , L (ˆx)) consists of firms’ strategy space, X 2 (which is their type space) and the outcome function L (ˆx) := (k(ˆx), κ (ˆx), t (ˆx)), where ki (ˆx) and κi (ˆx) denote the probability that firm i is awarded the unrestricted and the royalty license respectively, and t (ˆx) the expected transfer from firm i to the innovator, t i (ˆx) = ki (ˆx)β(ˆx j ) + κi (ˆx)ˆxi qi , where qi is the output of firm i if it is awarded the royalty license, and ki (ˆx), κi (ˆx) ∈ {0, 1} with k1 (ˆx) = 1 if xˆ 1  max{ˆx2 , r }, κ1 (ˆx) = 1 if r  xˆ 1 < xˆ 2 , k2 (ˆx) = 1 if xˆ 2  r ∧ xˆ 2 > xˆ 1 , κ2 (ˆx) = 1 if r  xˆ 2  xˆ 1 . 3

This assures that the duopoly game has a unique equilibrium (see Szidarovszky and Yakowitz, 1977). F and 1 − F are log-concave if f is log-concave. See Lemma A2 in Goeree and Offerman (2003) together with Theorem 1 in Bagnoli and Bergstrom (2005). 5 At some point we assume that the support of the distribution is bounded away from zero, by considering the family of truncations of F from below, denoted by H : [d, c ] → [0, 1], d  0, H (x) := ( F (x) − F (d))/(1 − F (d)). Truncations preserve the log-concavity of the c.d.f. and the reliability function. 6 Bayesian Nash equilibrium refers to the reduced message game induced by the direct mechanism, where the downstream Cournot game with the posterior belief induced by messages is replaced by its equilibrium payoff. Altogether, we are using the equilibrium concept of a perfect Bayesian equilibrium to solve the dynamic (message cum Cournot) game played between firms. 4

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Because in a licensing game without royalty scheme only the winner has access to the innovation, the equilibrium profits in the duopoly subgames depend only on the winner’s cost reduction, denoted by x. The equilibrium profit of the winner, πW (x), is increasing and that of the loser, πL (x), is decreasing in x. The status quo equilibrium profit, π A , that applies if no firm has access to the innovation is, of course, independent of x. Consider the following variation of our direct mechanism: firms report their cost reductions (ˆx1 , xˆ 2 ); the firm i that reported the highest cost reduction not smaller than a given threshold level r is awarded an unrestricted license and pays βn (ˆx j ) while the other firm j gets no license and pays nothing. If both firms’ reported cost reductions are lower than r, then no firm gets a license. Let

R := π W (r ) − π A .

(1)

Then the following transfer rule βn induces truth-telling as a Bayesian Nash equilibrium and voluntary participation, and the innovator’s expected revenue is equal to G n (r ) (see Section 4, Jehiel and Moldovanu, 2000):



βn (x) =

πW (x) − πL (x), if x  r , R,

(2)

otherwise,





G n (r ) = 2F (r ) 1 − F (r )



c

πW (r ) − π A +





πW ( y ) − πL ( y ) g2 ( y ) dy ,

(3)

r

where g 2 ( y ) = 2(1 − F ( y )) f ( y ) is the p.d.f. of the second highest order statistic of the sample of two i.i.d. cost reductions. Note that βn has a discontinuity at x = r, because π W (r ) − π L (r ) > R, which is due to the negative externality. We mention that Jehiel and Moldovanu (2000) analyze a second-price license auction with a reserve price R. There the equilibrium bid function is equal to the strictly increasing function b(x) := π W (x) − π L (x) and the reserve price induces a threshold level r that solves the equation R = π W (r ) − π A so that bidders submit a bid if and only if x  r. By the revelation principle it follows immediately that the above direct mechanism with transfer rule (2) is truthfully implementable. The optimal threshold level rn∗ solves the following condition:

1 − F (r ) f (r )

=

πW (r ) − π A 1 − F (r ) (π A − πL (r )) + .  (r )  (r ) πW F (r ) πW

(4)

It is useful to compare this with a hypothetical world in which there is no negative externality, i.e., the loser’s equilibrium duopoly profit is not affected by the winner’s cost reduction. In that case, the transfer rule that induces truth-telling and voluntary participation and the associated innovator’s expected revenue would be equal to

 β0 (x) =

πW (x) − π A , if x  r , R,

(5)

otherwise,

   G 0 (r ) = 2F (r ) 1 − F (r ) π W (r ) − π A +

c





πW ( y ) − π A g2 ( y ) dy ,

(6)

r

yielding the optimal threshold level r0 that solves the equation,

1 − F (r ) f (r )

=

πW (r ) − π A .  (r ) πW

(7)

Comparing (4) and (7) and using the assumed hazard rate monotonicity one obtains rn∗ < r0 , and because R is strictly increasing in r, it follows immediately: Proposition 1. (See Jehiel and Moldovanu (2000).) The optimal threshold level, r (and hence R), is lower than that in a standard auction without negative externalities. Essentially, in the presence of negative externalities, the innovator has an incentive to lower the reserve price (and associated threshold level) because a lower reserve price makes it more likely that both firms submit a bid and the winner pays π W (r ) − π L (r ), which is more than the reserve price, R = π W (r ) − π A that the winner pays if only one firm bids. 3.2. The game with royalty license for the loser An innovation is a public good. This suggests that the innovator leaves money on the table if he excludes the loser from using the innovation.

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Under complete information the innovator can easily extract that surplus by letting the loser of the auction use the innovation in exchange for a royalty rate equal to the cost reduction induced by the innovation, without losing auction revenue (Giebe and Wolfstetter, 2008). However, if the innovator does not know cost reductions and uses the direct mechanism, he must rely on firms’ reported cost reductions, set the royalty rate equal to the loser’s reported cost reduction provided it meets the threshold level, and set the transfer rule β in such a way that truth-telling is a Bayesian Nash equilibrium and voluntary participation is assured. We employ the following procedure to construct that transfer rule β . Consider one firm that unilaterally deviates from truth-telling. We then state conditions concerning the β function that make such deviations unprofitable. These conditions yield a unique β function for x  r. Unilateral deviations from truth-telling lead into duopoly subgames that are off the equilibrium path. Therefore, in order to compute the payoff of the firm that deviates from truth-telling, we must first solve all relevant duopoly subgames. 3.2.1. Downstream duopoly “subgames” Suppose one firm, say firm 1, has drawn cost reduction x but reports cost reduction z  x, whereas firm 2 tells the truth.7 In the continuation duopoly game, the following “subgames” occur, depending upon the true and reported cost reductions of firm 1, denoted by x and z, and the true cost reduction of firm 2, denoted by y. When both reports met the threshold level and firm 1 won. Let z  y  r.8 The innovator allocates the unrestricted license to firm 1 and the royalty license to firm 2 and charges firm 2 a royalty rate equal to y. It is then common knowledge among firms that the profile of unit costs is (c 1 , c 2 ) = (c − x, c ). Denote the Cournot equilibrium strategies by (q W 1 (x), q L 2 (x)), and the reduced form profit function of firm 1, conditional on winning, is π W (x) := ( P (q W 1 (x) + q L 2 (x)) − c + x)q W 1 (x). When both reports met the threshold level and firm 1 lost. Let y > z  r. The innovator charges firm 1 a royalty rate equal to z. It is then common knowledge among firms that the profile of unit costs is (c 1 , c 2 ) = (c − x + z, c − y ). Denote the equilibrium strategies of that subgame by (q L 1 (x, z, y ), q W 2 (x, z, y )), and the reduced form profit function of firm 1, conditional on losing, is

 





πL (x, z, y ) := P q W 2 (x, z, y ) + q L 1 (x, z, y ) − c + x − z q L 1 (x, z, y ). On the equilibrium path, i.e., for z = x, the equilibrium strategy of firm 1 when it lost and the associated reduced form payoff are only a function of firm 2’s cost reduction, y; therefore, we write: q∗L ( y ) := q L 1 (x, z, y )|z=x and π L∗ ( y ) := .π L (x, z, y )|z=x . Similarly, we write q∗W ( y ) := .q W 2 (x, z, y )|z=x . When at least one report did not meet the threshold level. If no one’s report met the threshold level, the game is just the default game without innovation, and the equilibrium profit of firm 1 is equal to π A . If firm 1 was the only one whose report met the threshold level, its equilibrium profit is the same as in the event when both firms’ reports met the threshold level and firm 1 won. If firm 2 was the only one whose report met the threshold level, the equilibrium profit of firm 1 is the same as in the game without royalty scheme, and is exclusively a function of the winner’s cost reduction, π L ( y ), as explained in Section 3.1. 

 (x) > 0, π ∗ ( y ) < 0, and ∂ π (x, z, y )| ∗ Lemma 1. For all x, y ∈ [r , c ]: π W z L z=x = −q L ( y )γ ( y ), with L

 





γ ( y ) := 1 − P  q W 2 (·) + q L 1 (·) ∂z q W 2 (x, z, y ) z=x > 1.

(8)

If demand is linear, γ ( y ) = 4/3 (for a summary account of the linear model see Online Appendix A.9). The proof is in Online Appendix A.1. Lemma 1 is pivotal in the proof of monotonicity of the transfer rules and in assessing the profitability of the royalty scheme. 3.2.2. Licensing mechanism We now derive the strictly increasing transfer rule β that induces truth-telling as a Bayesian Nash equilibrium, assures voluntary participation, and extracts surplus for given r.9 The derivation of β is motivated by the second-price auction with reserve price. We set β to be flat for all x < r so that, if the mechanism allocates an unrestricted license but no royalty license, the transfer to be paid by the winner is The case of z  x is slightly different, yet yields the same payoff function, Π(x, z) and differential equation, and hence is omitted. We assume without loss of generality that firm 1 is favored in the event of a tie (see footnote 2). Note, the tie rule does not affect payoffs because cost reductions are continuous random variables. 9 Of course, truth-telling can also be induced without requiring strict monotonicity. For example, one may award licenses by a lottery, which may be appealing if the distribution exhibits a concentration on cost reductions below rmin . However, with more mass on low values, rmin also becomes smaller, which makes these alternative mechanisms less attractive. 7 8

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independent of the lowest reported cost reduction.10 All firms with cost reductions at or below the threshold level r should be indifferent between participation and non-participation. Lemma 2. If one sets β(x) = R for all x < r, all firms with cost reductions x  r are indifferent between participation and nonparticipation, assuming the other firm reports truthfully. Hence, voluntary participation is assured for these types. Proof. Consider the marginal firm with cost reduction x = r. If that firm participates its payoff is Π p (r ), whereas if it does not participate, its payoff is Πnp (r ):

  Π p (r ) = F (r ) πW (r ) − R +

c

πL∗ ( y ) dF ( y ),

r

c Πnp (r ) = F (r )π A +

πL∗ ( y ) dF ( y ).

r

By (1) these are the same. Obviously, low types with x < r will also participate since they are not hurt by participating (they get no license and pay nothing). 2 We now proceed to characterize β for x  r. For this purpose, consider a firm, say firm 1, that has drawn cost reduction x, but unilaterally deviates from truth-telling and reports a cost reduction z  x, whereas firm 2 tells the truth. Then, using the equilibria of the duopoly subgames, the payoff function of firm 1 is

  Π(x, z) = F (r ) πW (x) − R +

z



 πW (x) − β( y ) dF ( y ) +

r

c

πL (x, z, y ) dF ( y ).

(9)

z

The transfer rule β must be such that x = arg maxz Π(x, z), for all x ∈ [r , c ]. Proposition 2. Set

 β(x) =

πW (x) − πL∗ (x) + R,

1 f (x)

c x

∂z π L (x, z, y )|z=x dF ( y ), if x  r , otherwise.

(10)

Then, truth-telling is an equilibrium, provided r is sufficiently high, β is strictly increasing on [r , c ], β(x) < βn (x), ∀x ∈ [r , c ), and β(c ) = βn (c ). In addition, βn (x) − β(x) is strictly decreasing if demand is linear (the latter result is used in the interpretation of Proposition 6). Proof. 1) Suppose firm 1 with cost reduction x  r reports a cost reduction z  x. Then, it obtains the payoff Π(x, z), stated in (9). Differentiating Π(x, z) with respect to z and setting z = x one obtains,



 πW (x) − β(x) f (x) +

c

∂z π L (x, z, y )|z=x dF ( y ) − π L∗ (x) f (x) = 0.

(11)

x

This implies (10) for x  r and, by Lemma 1, β(x) < βn (x) for all x  r. 2) Having derived the functional form of β one must also confirm that the underlying first-order conditions (11) yield a global maximum for each x. This is assured if the function Π(x, z) is pseudoconcave in z, which requires ∂zx Π(x, z)  0.11 As we explain in Proposition 3, this “second-order condition” is satisfied if the innovator adopts a sufficiently high threshold level r. 3) The proof of monotonicity of β is in Online Appendix A.2. The proof of monotonicity of βn (x) − β(x) is the same as the proof of Proposition 7, except that in steps a) and c) of the proof one has ∂z π L (x, z, y )|z=x = −q∗L ( y )γ , where γ = 4/3 by Lemma 1. 4) We show in Online Appendix A.3 that participation constraints are satisfied for all x  r. 5) Truth-telling is obviously assured for all types x < r, because if they report z = x they either win and pay more than winning is worth (if z  r) or lose and pay a royalty rate that exceeds their cost reduction or this has no effect (if z < r). 2 10

There is some degree of freedom in choosing that function for x < r that affects neither firms’ incentives nor the innovator’s equilibrium payoff. A function of one variable is pseudoconcave if it is increasing to the left of the stationary point and decreasing to the right. Firms’ payoff function Π(x, z) is pseudoconcave in z if ∂zx Π  0, for all x, z, because the sign of that cross derivative implies z < x ⇒ ∂z Π(x, z) > ∂z Π( z, z) = 0, and z > x ⇒ ∂z Π(x, z) < ∂z Π(x, x) = 0. Pseudoconcavity obviously implies that every stationary point is a global maximum. 11

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Fig. 1. Left: Stationary point at z = x is not a global maximum for r = 0; right: Stationary point at z = x is a global maximum for r = rmin .

Comparing β and βn one can see that adding the royalty scheme has an adverse effect on the revenue earned from the winner, which has the following interpretation. If we introduce the royalty scheme without changing the transfer rule βn , each firm would benefit from “low balling”, i.e., from reporting a cost reduction that is lower than its true cost reduction. This incentive to low-ball can only be eliminated by pointwise lowering the βn function, and by introducing a sufficiently high threshold level r. This indicates that the innovator faces a trade-off between revenue earned from selling the unrestricted license and royalty income. To see why a sufficiently high threshold level r is needed to induce truth-telling, we show that: Proposition 3. Π(x, z) is pseudoconcave if r is sufficiently large. The proof is in Online Appendix A.4. To obtain some insight into how large is a “sufficiently large” r, consider the model with linear demand. In that case we can strengthen Proposition 3 to: Proposition 4. Suppose demand is linear. Then, Π(x, z) is pseudoconcave if and only if r  rmin , where rmin is implicitly defined as the unique solution of

r−

1 − F (r ) 2 f (r )

3

= 0.

(12)

Specifically, if F (x) = x/c (uniform distribution), rmin = 2c /5. The proof is in Online Appendix A.5. Using the example of linear demand, a uniform distribution, c = 0.49, and x = 0.296, the role of the threshold level is illustrated in Fig. 1.12 There we plot Π(x, z) for r = 0 (figure on the left) and for r = rmin (figure on the right). If r = 0, the stationary point ∂z Π(x, z)|z=x = 0 is a local but not a global maximum for the assumed x = 0.296. Therefore, the best reply of a firm with x = 0.296 is to report a cost reduction equal to zero; the firm thus becomes a loser, yet obtains the innovation for free. Whereas if r = rmin , that stationary point is a global maximum (see the right side of Fig. 1). The purpose of a sufficiently high threshold level is to deter firms from “low balling” by reporting a zero cost reduction. Of course, the second-order conditions can also be assured by assuming that the support of the distribution is sufficiently bounded away from zero, employing the family of truncated distributions from below, H , with support [d, c ], 0  d < c, defined in Section 2.13 3.2.3. Is it profitable to add the royalty scheme? For the innovator, adding the royalty scheme has a benefit and a cost. The benefit is that he earns royalty income when both firms’ reported cost reductions meet the threshold level r, without directly affecting the payoff of the winner in the duopoly game, because, in equilibrium, the loser pays a royalty rate equal to his cost reduction. The cost is that the innovator needs to set a relatively high threshold level, which involves the risk that only the unrestricted license or no license is awarded, and that β is pointwise lower than βn . Nevertheless, adding the royalty scheme is profitable for all standard probability distributions, even though one can construct examples in which it is not profitable. The latter occurs if the probability distribution exhibits a high concentration on low cost reductions.

12 Because we assume a non-drastic innovation whose exclusive use does not propel a monopoly, the cost reduction is bounded from above. If demand is linear, P ( Q ) = 1 − Q , which we assume in all examples, innovations are non-drastic if and only if c ∈ [0, 1/2). 13 To prove this, go to Eq. (A17) in Online Appendix A.5 and replace F by H . Then, the condition ∂zx Π(x, z)  0 in (A17) becomes a condition concerning d, which is equivalent to r  rmin .

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The innovator’s expected revenue in the game with royalty scheme, G (r ), has three components: the expected transfer of the winner when only one firm’s report meets r, the expected transfer of the winner when both reports meet r, and the expected royalty payment:

  G (r ) = 2F (r ) 1 − F (r ) R +

c

c  x β( y ) g 2 ( y ) dy +

r

r

yq∗L (x) g 12 (x, y ) dy dx,

r

where g 12 (x, y ) = 2 f (x) f ( y ) is the joint p.d.f. of the highest and second highest cost reductions. The maximizer of G (r ) is denoted by r ∗ . Define (r ) := G (r ) − G n (r ). Substituting G n (r ) (see (3)), β , and R = π W (r ) − π A , one finds after changing the order of integration and a bit of rearranging:

c c (r ) = 2 r







yq∗L (x) f ( y ) + 1 − F ( y ) ∂z π L ( y , z, x)|z= y dF (x) dy

y

c c =2 r





q∗L (x) y −

y

1 − F ( y) f ( y)



γ (x) dF (x) dF ( y ).

(13)

To obtain further results, we now assume that the demand function is linear. In that case, 4/3 (see Lemma 1).

γ is constant and equal to

Proposition 5 (Sufficient condition). Consider truncations of F from below, H : [d, c ] → [0, 1], d  0. Then, there exists d∗ ∈ (0, c ) such that (r ) > 0, ∀r and ∀d  d∗ . Hence, G (r ∗ ) > G n (rn∗ ). H (x)) Proof. Define φ(x) := x − (1− γ = x − 43 (1−f (Fx()x)) . By the assumed hazard rate monotonicity, φ is strictly increasing. H  (x) Therefore, φ(x)  0, ∀x ∈ [d, c ] if and only if φ(d)  0. Because φ(0) < 0, φ(c ) > 0, and φ is strictly increasing, it follows that there exists a unique d∗ such that φ(d∗ ) = 0, and we conclude that (r ) > 0, ∀r ∈ [d, c ) if d > d∗ . For example, if F is the uniform distribution, then d∗ = 4c /7. 2

In order to understand intuitively why truncations from below guarantee the profitability of the royalty scheme, note that (assuming linear demand): Proposition 6. Truncations from below do not affect the gap between βn (x) and β(x). Proof.

βn (x) − β(x) =

1

4

H  (x) 3

c x

q∗L ( y ) H  ( y ) dy =

1 − F (d) 4 f (x)

c

3 x

q∗L ( y )

f ( y) 1 − F (d)

dy =

1

4

f (x) 3

c

q∗L ( y ) f ( y ) dy .

2

x

Because truncations from below imply a first-order stochastic dominance shift of the distribution function, in view of Proposition 6 truncations from below shift probability mass from cost reductions at which the royalty scheme involves a large loss in revenue earned from the winner, because of a large gap between βn and β , to those where that loss is small. At the same time, royalty rates are smaller for low than for high cost reductions. Therefore, truncations from below also increase royalty income. Both effects increase the profitability of the royalty scheme. Altogether, adding the royalty scheme is profitable for many standard probability distributions, including the uniform distribution with support [0, c ]. Starting from the uniform distribution, one finds that if probability mass is shifted to high cost reductions, the royalty scheme becomes even more profitable. Whereas, if the probability distribution exhibits a sufficiently high concentration on low cost reductions, the royalty scheme becomes less profitable, and the revenue ranking can be reversed. We close with two examples, one in which the royalty scheme is profitable, and one where it is not. Both examples assume truncated exponential probability distributions and c = 0.3. These distributions are consistent with the assumed log-concavity of reliability functions. The probability distribution plotted in Fig. 2 exhibits a concentration on high cost reductions; whereas that in Fig. 3 exhibits a concentration on low cost reductions. Evidently, as one skews the distribution towards high cost reductions, the royalty scheme becomes more profitable; whereas, if one concentrates probability mass on low cost reductions, the revenue ranking is reversed and it no longer pays to adopt the royalty scheme.

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Fig. 2. Truncated exponential distribution with a concentration on high cost reductions: c.d.f. (left) and the associated expected revenue of the innovator (right).

Fig. 3. Truncated exponential distribution with a concentration on low cost reductions: c.d.f. (left) and the associated expected revenue of the innovator (right).

4. Model II We now turn to the more plausible model where firms do not learn each other’s cost reduction after licenses have been awarded and before they play the duopoly game. Like in model I, firms would like to signal weakness to the innovator, and thus reduce the royalty rate in the event when they are awarded the royalty license. However, unlike in model I, firms would also like to signal strength to their rival and make him believe that their own cost reduction is “high” in order to induce the rival to play less aggressively in the duopoly game. Both signaling incentives must be taken into account in designing the transfer rule that induces truth-telling and voluntary participation. In the following we employ the same procedure to construct the transfer rule β as in model I. We show that model II yields more general results, because none of our results assume linear demand, and adding the royalty scheme is more profitable in model II than in model I. Because the game without royalty scheme corresponds to the game analyzed by Goeree (2003) and Das Varma (2003), we focus on the game with royalty scheme and only mention casually what changes if no royalty scheme is adopted. 4.1. Downstream duopoly “subgames” Suppose firm 1 has drawn cost reduction x but reports cost reduction z  x, whereas firm 2 tells the truth.14 The following duopoly “subgames” occur, depending upon x,z and the true cost reduction of firm 2, y. 4.1.1. When both reports met the threshold level and firm 1 won Let z  y  r. The innovator allocates the unrestricted license to firm 1 and the royalty license to firm 2 and charges firm 2 a royalty rate equal to y. Firm 1 privately knows its cost reduction is x, whereas firm 2 (the loser) believes that firm 1’s cost reduction is z. Therefore, firm 2 believes to play a duopoly subgame with unit costs (c 1 , c 2 ) = (c − z, c ). Denote the associated equilibrium strategies of the game that the loser believes to play by (q W ( z), q L 2 ( z)). Firm 1 anticipates that the loser plays q L 2 ( z). But because firm 1 privately knows that its cost reduction is x rather than z it plays the best reply:

14

Like in model I, the case of z  x is slightly different, yet yields the same payoff function, Π(x, z), and differential equation, and hence is omitted.

C. Fan et al. / Games and Economic Behavior 82 (2013) 388–402

 





q W 1 (x, z) = arg max P q + q L 2 ( z) − c + x q.

397

(14)

q

The reduced form profit function of firm 1, conditional on winning, is

 





πW (x, z) := P q W 1 (x, z) + q L 2 (z) − c + x q W 1 (x, z). 4.1.2. When both reports met the threshold level and firm 1 lost Let y > z  r. The innovator allocates the royalty license to firm 1 and charges firm 1 a royalty rate equal to z. Firm 2 believes to play a Cournot duopoly subgame with unit costs (c 1 , c 2 ) = (c , c − y ). Denote the associated equilibrium strategies of the game that firm 2 (the winner) believes to play by (q L ( y ), q W 2 ( y )). If the royalty scheme is adopted, firm 1 privately knows that its cost reduction is x yet pays a royalty rate z. Therefore, firm 1 plays the following best reply to q W 2 ( y ):

 





q L 1 (x, z, y ) = arg max P q + q W 2 ( y ) − c + x − z q.

(15)

q

The associated reduced form profit function of firm 1 conditional on losing is then π L (x, z, y ) := ( P (q W 2 ( y ) + q L 1 (x, z, y )) − c + x − z)q L 1 (x, z, y ). Also note that on the equilibrium path, for z = x, that payoff is only a function of the cost reduction of firm 2, y; therefore, we write q∗L ( y ) := q L 1 (x, z, y )|z=x and π L∗ ( y ) := π L (x, z, y )|z=x . If no royalty scheme is used, the equilibrium play of firm 1 depends only on its rival’s cost reduction y. Hence, the reduced form profit function of firm 1 conditional on losing is π L ( y ) := ( P (q W 2 ( y ) + q L ( y )) − c )q L ( y ). We stress that in model II the equilibrium output of firm 2 in the event when firm 1 has lost, q W 2 , is only a function of its own cost reduction, y, whereas in model I it is a function of x, z, and y. This is due to the fact that in model II firm 2 believes that firm 1’s cost reduction is z, which is equal to the royalty rate set by the innovator. Therefore firm 2 believes that the effective unit costs are (c 1 , c 2 ) = (c − z + z = c , c − y ). Whereas in model I, firm 2 knows that firm 1’s cost reduction is x, but the innovator charges firm 1 a royalty rate equal to z; therefore, firm 2 believes that effective unit costs are (c 1 , c 2 ) = (c − x + z, c − y ). 4.1.3. When at least one report did not meet the threshold level If no one’s report met the threshold level, the game is just the default game without innovation, and the equilibrium profit of firm 1 is π A . If firm 1 was the only one whose report met the threshold level, its equilibrium profit is the same as in subgame 4.1.1; if firm 2 was the only one whose report met the threshold level, the equilibrium profit of firm 1 is the same as in the game without royalty scheme, and is exclusively a function of the winner’s cost reduction, π L ( y ), as explained in Section 4.1.2. Lemma 3. For all x, y ∈ [r , c ]: 1) ∂z π L (x, z, y )|z=x = −q∗L ( y ) and 2)

d dx



πW (x, x) > 0, πL∗ ( y ) < 0.

The proof is in Online Appendix A.6. Lemma 3 plays a similar role as Lemma 1 in model I. 4.2. Licensing mechanism We construct the transfer rule β that induces truth-telling as a Bayesian Nash equilibrium and assures voluntary participation and extract surplus for given r, using the same procedure as in model I. Similar to model I, we set β(x) = R = π W (r , r )− π A for all x < r so that all firms with cost reductions x  r are indifferent between participation and non-participation, given that the other firm reports truthfully. Hence voluntary participation is assured for these types. Also, truth-telling is obviously assured for all types x < r. We proceed to derive β for x  r. Consider firm 1 with cost reduction x, but reports a cost reduction z  x, whereas firm 2 tells the truth. Then, using the equilibria of the duopoly subgames, the payoff function of firm 1 in the game with royalty scheme for z  x  r is

  Π(x, z) = F (r ) πW (x, z) − R +

z r



 πW (x, z) − β( y ) dF ( y ) +

c

πL (x, z, y ) dF ( y ). z

In the game without royalty scheme, the payoff function is the same, except that in the last term by

π L ( y ).

The transfer rule β must be such that x = arg maxz Π(x, z), for all x ∈ [r , c ].

πL (x, z, y ) is replaced

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Proposition 7. In the mechanism with and without royalty scheme, set β(x) = βn (x) = R = π W (r , r ) − π A for x < r, and for x  r

β(x) = βn (x) +

1 f (x)

c ∂z π L (x, z, y )|z=x dF ( y ),

(16)

x

βn (x) = πW (x, x) − π L∗ (x) +

F (x) f (x)

.∂z πW (x, z)|z=x .

(17)

Then β, βn induce truth-telling and voluntary participation, provided r is sufficiently high, and are strictly increasing on [r , c ]. Moreover, βn (x) − β(x) is strictly decreasing, and β(x) < βn (x), ∀x ∈ [r , c ) with β(c ) = βn (c ). Proof. For the solution and proof of monotonicity of βn see Goeree (2003, Proposition 2). The derivation of β is similar to that in model I. To prove monotonicity of β , it is sufficient to show that β(x) − βn (x) is strictly increasing, i.e. (β(x) − βn (x)) > 0 for all x  r, which we confirm below:



d dx

=

1 f (x) 1 f (x)



c

∂z π L (x, z, y )|z=x dF ( y ) x

c



 ∂zx π L (x, z, y )|z=x + ∂zz π L (x, z, y )|z=x dF ( y )

x

−∂z π L (x, z, x)|z=x −

= −∂z π L (x, z, x)|z=x −

> −∂z π L (x, z, x)|z=x +

> −∂z π L (x, z, x)|z=x +

f  (x) f (x)2 f  (x) f (x)2

c ∂z π L (x, z, y )|z=x dF ( y ) x

c ∂z π L (x, z, y )|z=x dF ( y ) (step a) x

1 1 − F (x) 1 1 − F (x)

c ∂z π L (x, z, y )|z=x dF ( y ) (step b) x

c ∂z π L (x, z, x)|z=x dF ( y ) (step c) x

= −∂z π L (x, z, x)|z=x + .∂z π L (x, z, x)|z=x = 0. The different steps in this assessment are explained as follows: step a) follows from the facts that ∂zx π L (x, z, y )|z=x = −∂x q∗L ( y ) = 0 and ∂zz πL (x, z, y )|z=x = −∂z q∗L ( y ) = 0 (see Lemma 3); step b) follows from the assumed log-concavity of the reliability function, which implies that f  (x) > − f (x)2 /(1 − F (x)), together with the fact that ∂z π L (x, z, y )|z=x < 0; step c) follows from the fact that ∂z π L (x, z, y )|z=x = −q∗L ( y ) (by Lemma 3), which is strictly increasing in y. Note that β(x) < βn (x) for all x  r by Lemma 3. The participation constraints are satisfied for all x  r, the proof is similar to that in Online Appendix A.3. The only difference in the proof is that π W (x) should be replaced by π W (x, x) and R = π W (r ) − π A by R = π W (r , r ) − π A . The role of the threshold level in assuring the second-order conditions is explained below. 2 Similar to model I, we find that a sufficiently high threshold level is required for pseudoconcavity of the payoff function

Π(x, z). Unlike in model I, this condition is not only sufficient but also necessary.15 Proposition 8. Π(x, z) is pseudoconcave if and only if r is sufficiently large. The proof is in Online Appendix A.7. Proposition 9. Adding the royalty scheme reduces the transfers β(x) pointwise for all x  r by a smaller amount than in model I. The proof is in Online Appendix A.8.

15

In model I this condition was also necessary only if demand is linear.

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The intuition for this result is as follows. Consider the transfer rule βn without royalty scheme in model II, which, in order to induce truth-telling, must exhibit equality of the marginal benefit and the marginal cost of an incremental change in the report from x to x + ε .16 Now add the royalty scheme, while maintaining the candidate βn function. Then, the marginal benefit of reporting a higher cost reduction does not change whereas it gives rise to a positive marginal cost, due to the fact that when the firm loses, it pays a royalty rate that exceeds its cost reduction. To reestablish equality, the transfer βn has to be lowered pointwise, to bring the unchanged marginal benefit in balance with the higher marginal cost. This is also true in model I. However, in model II the rival firm believes that the effective cost is equal to c, whereas in model I the rival knows that the royalty rate exceeds the cost reduction so that the effective cost is higher than c. Because the marginal cost is higher in model I than in model II, βn has to be lowered more in model I than in model II. 4.3. The innovator’s expected revenue The above result suggests that adding the royalty scheme is more profitable in model II than in model I, because it has a smaller adverse effect on β and thus on the innovator’s revenue earned from the winner. The expected revenues of the innovator in the game with and without royalty scheme, G (r ) and G n (r ), are

c



 G n (r ) = 2 1 − F (r ) F (r ) R +

βn ( y ) g 2 ( y ) dy , r



 G (r ) = 2 1 − F (r ) F (r ) R +

 c  x

c β( y ) g 2 ( y ) dy + r

c = G n (r ) +

yq L (x) g 12 (x, y ) dy dx, r



 β( y ) − βn ( y ) g 2 ( y ) dy +

r

 ∗

r

 c  x

 ∗

yq L (x) g 12 (x, y ) dy dx. r

r

Define (r ) := G (r ) − G n (r ). After substituting β and βn and using the fact that ∂z π L ( y , z, x)|z= y = −q∗L (x) by Lemma 3, one obtains,

c c (r ) = 2 r



q∗L (x) y −

1 − F ( y)

y



f ( y)

dF (x) dF ( y ).

Proposition 10. In model II adding the royalty scheme increases the innovator’ expected revenue more than in model I, for all r from the intersection of the domains of these functions. Proof. Distinguish  in models I and II by writing  I (r ) and II (r ). Using the fact that in model I ∂z π L (x, z, y )|z=x = −q∗L ( y )γ ( y ) and γ ( y ) > 1 (by Lemma 1) and that in model II ∂z πL (x, z, y )|z=x = −q∗L ( y ) (by Lemma 3), one finds for all r from the intersection of the domains of the functions G and G n ,

c c I

 (r ) = 2 r

y

c c <2 r



q∗L (x) y −

y

1 − F ( y) f ( y)



γ (x) dF (x) dF ( y )

 1 − F ( y) q L (x) y − dF (x) dF ( y ) = II (r ). f ( y) ∗

2

The following results hold for all concave inverse demand functions (unlike in model I where similar results hold only for a class of linear demand functions). Proposition 11 (Sufficient condition). Consider truncations of F from below, H : [d, c ] → [0, 1], d  0. Then, there exists d∗ ∈ (0, c ) such that (r ) > 0, ∀r and ∀d  d∗ . Hence, G (r ∗ ) > G n (rn∗ ). Proof. The proof is the same as that of Proposition 5, except that now φ(x) := x − (1 − H (x))/ H  (x), independent of the form of demand function. 2 16 The benefit is the expected revenue from increased probability of winning and from signaling strength, and the cost is the increased expected transfer paid by the winner.

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Fig. 4. Expected revenue for truncated exponential distributions with a concentration on high cost reductions (left) and on low cost reductions (right).

Similar to model I: Proposition 12. Truncations from below do not affect the gap between βn (x) and β(x). Therefore, truncations from below unambiguously increase the innovator’s revenues earned from both the winner and the loser. The proof is the same as that of Proposition 6, except that 4/3 is replaced by 1. Similar to what we observed in model I, we see that truncations from below shift probability mass to high values in the sense of first-order stochastic dominance, without changing the gap between the functions β and βn . Therefore, truncations from below unambiguously raise the revenues earned from both firms. Moreover, unlike in model I (where we had to invoke linear demand), these results hold for all concave inverse demand functions. Finally, we illustrate the performance of the royalty scheme with two examples, which employ the same demand function and probability distributions as in model I (see Online Appendix A.10). In Fig. 4, the figure on the left corresponds to the probability distribution plotted in Fig. 2, and the figure on the right corresponds to the distribution plotted in Fig. 3. Unlike in model I, adding the royalty scheme is profitable even for the probability distribution that exhibits a concentration on low cost reductions. Altogether, these examples indicate that adding the royalty scheme is more profitable in model II than in model I, and is particularly appealing if the probability distribution exhibits a concentration on high cost reductions, as already explained intuitively at the end of Section 3. The enhanced profitability in model II is due to the fact that the potential to signal strength to one’s rival exerts an upward pressure on the winner’s transfer, which counters the downward pressure due to the potential to signal to the innovator. 5. Implementation If the transfer rule β is strictly increasing on [r , c ], the equilibrium outcome of the above licensing game can also be achieved if one replaces the direct mechanism by a modified second-price license auction with reserve price R and a royalty rate rule φ that prescribes the royalty rate as a function of the losing bid b.17 As an illustration consider model I with linear demand and uniformly distributed cost reductions. In that case, implementation is achieved by the modified second-price license auction with the following reserve price and royalty rate rule:

R=

4r (1 − c + r ) 9

,

φ(b) =



25 + 9b − 46c + 19c 2 − 5(1 − c ).

The constructed indirect mechanism is useful for practical applications. 6. Robustness The current analysis focuses on two firms. One may wonder whether it can be extended to more than two firms. If one increases the number of firms, two issues come up: the innovator must optimize the number of unrestricted licenses, and he must choose among various possible transfer rules that apply if more than one unrestricted license is awarded. The first of these issues has been at center stage in the classical literature on patent licensing under complete information (see Kamien, 1992; Giebe and Wolfstetter, 2008), but is more challenging in the present framework of incomplete information. 17 For this purpose set φ(b) := β −1 (b), for all b ∈ [β(r ), β(c )], and R = π W (r , r ) − π A . Then, one finds that the equilibrium bid function is equal to β for all x ∈ [r , c ], and in equilibrium the royalty rate is equal to the loser’s cost reduction. The required monotonicity of β is assured by Proposition 2 in model I and Proposition 7 in model II.

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Fig. 5. Left: one unrestricted license with/without royalty; right: one vs. two unrestricted licenses.

In order to confirm robustness, we carried out a full scale analysis of the case of three firms. With three firms, the innovator may award either one or two unrestricted licenses. Denote the highest, second highest, and third highest reported cost reductions by xˆ (1) , xˆ (2) , and xˆ (3) . We assume that the innovator employs the following transfer rule: if only one unrestricted license is offered, the firm that reported the highest cost reduction not smaller than r is awarded the unrestricted license and pays β(ˆx(2) ); if two unrestricted licenses are offered, the firms that reported the two highest cost reductions not smaller than r are awarded the unrestricted licenses and both have to pay β(ˆx(3) ).18 The analysis is a bit more involved because it covers a larger number of “subgames”. Due to space limitations we only review the results of a simple example; the detailed analysis is relegated to part B of the Online Appendix. Fig. 5 illustrates our findings with an example that assumes linear demand, unit cost c = 0.49, and uniformly distributed cost reductions, F (x) = x/c. In the left panel of Fig. 5 we plot the innovator’s expected revenues with and without royalty scheme, G (r ), G n (r ) for the case of one unrestricted license.19 Evidently, adding the royalty scheme is profitable in this case. In the right panel we plot the innovator’s expected revenues for the case of two unrestricted licenses, with and without royalty scheme, and the case of one unrestricted license with royalty scheme. Altogether, these figures indicate that the innovator’s expected revenue is maximized if he awards one unrestricted license, combined with the proposed royalty scheme.20 Another concern is whether the private value paradigm is appropriate to analyze the cost reductions of firms that serve the same market and employ similar technologies. If instead one assumes a common value framework, each firm’s expected cost reduction is a function of the signals observed by all firms, and so is the royalty rate set by the innovator, unless the largest signal is a sufficient statistic of the unknown cost reduction, in which case only the largest signal matters. In a companion paper we show that the main results of the present paper also extend to this common value framework (see Fan et al., forthcoming). 7. Discussion In the present paper we reconsider the licensing of a process innovation to a Cournot oligopoly. Unlike the previous literature, we assume incomplete information of the private values type, and analyze a licensing mechanism that awards a limited number of unrestricted licenses to those firms that report the highest cost reductions, combined with royalty licenses to others. One crucial feature of our paper is that we neither allow arbitrary royalty payments nor allow the mechanism to prescribe outputs. These restrictions are justified on the following grounds: If one would permit royalty rates that exceed the loser’s cost reduction, the innovator would charge the loser an excessively high royalty rate in order to extract more surplus from the winner. In the extreme, the innovator would set the equilibrium royalty rate so high that it induces losers to exit from the market. The same outcome could also be achieved more directly if the mechanism could prescribe outputs. Of course, antitrust authorities do not permit such anti-competitive conduct. Antitrust authorities have repeatedly invoked “abuse of dominance” laws to deal with excessively high royalty rates. For example, in 2007 the European Commission accused Rambus, Inc., a dominant producer of DRAMs, of having infringed Article 82 of the EC Treaty. After 18 months of legal disputes, the case was settled when Rambus agreed to put a five-year cap on its royalty rates (see Rey and Salant, 2012).

18 If only one firm’s report met the threshold level, then only one unrestricted license is awarded to that firm who then pays β(ˆx(2) ), like in the case when only one unrestricted license is offered. 19 As one can easily confirm, second-order conditions (pseudoconcavity of π (x, z) in z) are satisfied only if r  rmin = 0.2089. 20 Note, the different shapes of G (r ) (one license case with royalty scheme) in the left and the right panel are due to the different scales we use in these two figures.

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These observations also suggest that an economically meaningful theory of optimal licensing must incorporate restrictions that reflect antitrust concerns. Since antitrust rules are somewhat vague and subject to changing legal interpretation, solving the optimal mechanism design problem raises fundamental issues that are beyond the scope of the present paper. Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.geb.2013.08.003. References Bagnoli, M., Bergstrom, T., 2005. Log-concave probability and its applications. Econ. Theory 26, 445–469. Das Varma, G., 2003. Bidding for a process innovation under alternative modes of competition. Int. J. Ind. Organ. 21, 15–37. Fan, C., Jun, B., Wolfstetter, E., forthcoming. Licensing a common value innovation in oligopoly when signaling strength may backfire. Int. J. Game Theory. http://dx.doi.org/10.1007/s00182-013-0391-9. Giebe, T., Wolfstetter, E., 2008. License auctions with royalty contracts for (winners and) losers. Games Econ. Behav. 63, 91–106. Goeree, J., 2003. Bidding for the future: Signaling in auctions with an aftermarket. J. Econ. Theory 108, 345–364. Goeree, J., Offerman, T., 2003. Competitive bidding in auctions with private and common values. Econ. J. 113, 598–613. Jehiel, P., Moldovanu, B., 2000. Auctions with downstream interaction among buyers. RAND J. Econ. 31, 768–791. Kamien, M.I., 1992. Patent licensing. In: Aumann, R., Hart, S. (Eds.), Handbook of Game Theory, vol. I. Elsevier Science, pp. 331–354. Kamien, M., Tauman, Y., 1984. The private value of a patent: a game theoretic analysis. J. Econ. 4, 93–118. Kamien, M.I., Tauman, Y., 1986. Fee versus royalties and the private value of a patent. Quart. J. Econ. 101, 471–491. Katz, M.L., Shapiro, C., 1985. On the licensing of innovations. RAND J. Econ. 16 (4), 504–520. Katz, M.L., Shapiro, C., 1986. How to license intangible property. Quart. J. Econ. 101, 567–589. Rey, P., Salant, D., 2012. Abuse of dominance and licensing of intellectual property. Int. J. Ind. Organ. 30, 518–527. Sen, D., 2005. Fee versus royalty reconsidered. Games Econ. Behav. 53, 141–147. Sen, D., Tauman, Y., 2007. General licensing schemes for a cost-reducing innovation. Games Econ. Behav. 59, 163–186. Szidarovszky, F., Yakowitz, S., 1977. A new proof of the existence and uniqueness of the Cournot equilibrium. Int. Econ. Rev. 18, 787–789.