Rabah Amir
email: rabah-amir@uiowa.edu

David Encaoua
email: encaoua@univ-paris1.fr

Yassine Lefouili
email: yassine.lefouili@tse-fr.eu

Claude D'aspremont

Bertrand Gobillard

Christian Helmers

Patrick Rey

Isabel Ruhmer

Cédric Schneider

Yair Tauman

Jean Tirole

Yaron Yehezkel

Optimal Licensing of Uncertain Patents in the Shadow of Litigation

Keywords: Licensing mechanisms, Uncertain patents, Patent litigation, Cost comparative statics. JEL Classi…cation: D45, L10, O32, O34

This paper investigates the choice of a licensing mechanism by the holder of a patent whose validity is uncertain. We provide su¢cient conditions of a general nature under which the licensor prefers to use a per-unit royalty contract. In particular we show that this is the case for the holders of weak patents if the strategic e¤ect of an increase in a potential licensee's unit cost on the equilibrium industry pro…t is positive. The latter condition is shown to hold in a Cournot (resp. Bertrand) oligopoly with homogeneous (resp. di¤erentiated) products under general assumptions on the demands faced by …rms. As a byproduct of our analysis, we contribute to the literature on the cost paradox in oligopoly by o¤ering some new insights of independent interest regarding the e¤ects of cost variations on Cournot and Bertrand equilibria.

Introduction

Since the seminal contribution by [START_REF] Arrow | Economic welfare and the allocation of resources for invention[END_REF], analyzing the licensing contracts o¤ered by patent holders has become an important topic in the economics of innovation and technology di¤usion.

Arrow compared the revenues that an outside innovator obtains from licensing a cost-reducing innovation to a competitive industry and to a monopolistic industry. He showed that when a per-unit royalty is charged, a perfectly competitive industry generates higher licensing revenues than a monopolistic one. Subsequently, Katz andShapiro (1985, 1986) and [START_REF] Kamien | Patent licensing[END_REF]Tauman (1984, 1986) analyzed di¤erent licensing mechanisms (…xed fee, auction and per-unit royalty) when the potential licensees are members of an oligopoly. A key insight of the theoretical literature that has built on those seminal papers is that the optimal licensing mechanism depends on many factors, including the type of downstream competition, the degree of di¤erentiation between products and whether the patent holder is active or not in the downstream market. These three factors have been shown to be critical in the sense that predictions regarding the optimal mechanism can be completely overturned by varying any of them. 1A common feature of the existing papers on the comparison of di¤erent licensing mechanisms is that patents are viewed as certain or "ironclad" rights, the validity of which is unquestionable. This clearly contradicts what we observe in practice: about half of the patents that are challenged before US courts are invalidated [START_REF] Allison | Empirical evidence on the validity of litigated patents[END_REF]). 2 It is now largely recognized that a patent is not a perfectly enforceable right, as are other forms of property. Patents correspond much more to uncertain or probabilistic rights because they only give a right to try to exclude by asserting the patent in court [START_REF] Ayres | Limiting patentees' market power without reducing innovation incentives: the perverse bene…ts of uncertainty and non-injunctive remedies[END_REF][START_REF] Shapiro | Antitrust limits to patent settlements[END_REF][START_REF] Lemley | Probabilistic patents[END_REF]. Moreover, this uncertainty is strengthened by the fact that many applications are granted patent protection by the patent o¢ce (PO) even though they probably do not meet one or several of the statutory requirements: belonging to the patentable subject matters, utility, novelty and non-obviousness (or inventiveness). As a result, some of the uncertain patents are weak in the sense that they have a high probability of being invalidated by a court if challenged by a third party.

The proliferation of uncertain patents is due to several reasons. First, the major patent o¢ces (USPTO, EPO and JPO) have insu¢cient resources to ensure an e¤ective review process for the huge and growing number of patent applications [START_REF] Friebel | Objectives and Incentives at the European Patent O¢ce[END_REF]. Second, mistakes are unavoidable because the patentability requirements are di¢cult to assess, especially for newly patentable subject matters such as software, business methods and research tools. Third, the incentives provided to the examiners are inadequate for making them fully prosecute and reject applications that do not meet the standards [START_REF] Farrell | Incentives to challenge and defend patents: why litigation won't reliably …x patent o¢ce errors and why administrative patent review might help[END_REF][START_REF] Langinier | Monetary and mplicit incentives of the patent examiners[END_REF][START_REF] Lei | Why weak patents? Rational ignorance or pro-"customer" tilt?[END_REF]. This paper investigates the optimal licensing mechanism from the perspective of a licensor holding an uncertain patent and facing the threat of patent litigation. We consider a model in which a patent holder makes a license o¤er to potential licensees who can challenge the patent's validity if they do not accept the o¤er. The potential users of the patented technology are assumed to compete with each other in the output market but the type of competition between them is not speci…ed. 3We focus throughout the paper on the case when the patent holder prefers to deter litigation rather than accomodate it. This scenario is supported by both theoretical arguments and empirical evidence that we discuss in detail in Section 3. Moreover, when the patent holder …nds it optimal to accomodate litigation then it is maximizing its licensing revenues should the patent be ruled valid.

Therefore, it is essentially acting as the holder of an ironclad patent. In that case, the problem of choosing a licensing mechanism for an uncertain patent is the same as its (extensively studied) counterpart for an ironclad patent.

In our baseline model the licensing contracts can involve the payment of either …xed fees or perunit royalties. We show that two opposite forces drive the patent holder's choice between these two instruments. On the one hand, the use of per-unit royalties allows the patent holder to increase the size of the aggregate pro…ts (the sum of the licensor's and licensees' pro…ts) by relaxing competition between the licensees. Per-unit royalties have this e¢ciency e¤ect whenever a mild condition on the strategic e¤ect of marginal costs on the licensees' pro…ts is satis…ed. On the other hand, …xed fee licensing results in a lower pro…t for a …rm that challenges the patent but fails to invalidate it. This punishment e¤ect makes the expected payo¤ of a challenger lower when …xed fees are used.

Therefore, the rent that the patent holder has to leave to the licensees for litigation to be deterred is lower under a …xed fee contract than under a per-unit royalty contract.

When an unsuccessful challenger is driven out of the market under both types of contracts, there is no punishment e¤ect. In that case, the patent holder chooses the mechanism that maximizes the aggregate pro…ts, i.e. the per-unit royalty mechanism. However, when the punishment e¤ect is at work, per-unit royalties will be preferred by the patent holder only if the magnitude of the latter is lower than that the magnitude of the e¢ciency e¤ect. We show that this holds for su¢ciently weak patents. The key reason for this is that the punishment e¤ect is a second-order e¤ect for weak patents while the e¢ciency e¤ect is a …rst-order one. We also show that the superiority of per-unit royalty contracts for the licensing of weak patents remains true in three extensions of our baseline model that account for (i) litigation costs, (ii) the possibility of using two-part tari¤ contracts, and (iii) the possibility that the patent holder competes against the licensees in the output market.

Our paper contributes to three strands of literature:

1. The literature that compares various licensing mechanisms. 4 Our contribution is to extend that comparison to patents whose validity is uncertain. This issue deserves scrutiny in light of the growing proliferation of uncertain patents and, in particular, weak ones. 5 Our main result can be seen as a novel justi…cation, based on the uncertainty over patent validity, for the use of per-unit royalties instead of …xed fees in licensing contracts. Explaining why patent holders prefer to use royalty contracts is important because, despite the frequent use of per-unit royalties in practice, 6 a number of seminal theoretical models on licensing have concluded that the optimal licensing contract should involve the payment of a …xed fee (see e.g. [START_REF] Kamien | Patent licensing[END_REF]. 7 A key di¤erence with the literature considering ironclad patents is that the superiority of per-unit royalties for the licensing of weak patents is a very robust result: It is independent of the type of downstream competition, the degree of product di¤erentiation and whether the patent holder is active or not in the output market.

2. The literature on the licensing and litigation of uncertain patents [START_REF] Aoki | Licensing vs. Litigation: The e¤ect of the legal system on incentives to innovate[END_REF][START_REF] Farrell | How strong are weak patents?[END_REF][START_REF] Encaoua | Licensing "weak" patents[END_REF][START_REF] Choi | Patent pools and cross-licensing in the shadow of patent litigation[END_REF]. This literature has been mostly concerned with the ine¢ciencies stemming from the low private incentives to litigate a weak patent. The social harm of uncertain patents depends on how they are licensed out and, therefore, we argue that it is crucial to get a better understanding of the licensing mechanisms the holders of those patents use. 8

3. The literature on the e¤ects of cost variations on oligopolists' pro…ts [START_REF] Seade | Pro…table cost increases and the shifting of taxation[END_REF][START_REF] Kimmel | E¤ects of cost changes on oligopolists' pro…ts[END_REF][START_REF] Février | Idiosyncratic shocks in a asymmetric Cournot oligopoly[END_REF]. While the existing literature has focused on the (ambiguous) sign of the overall e¤ect of cost variations on equilibrium pro…ts, we investigate the sign of the strategic e¤ect of a unilateral cost increase on industry pro…ts. We show that this sign is unambiguously positive in the two most usual imperfect competition models under general (and commonly invoked) assumptions on the demand functions. An ancillary result is that we provide a lower bound on the overall e¤ect of a unilateral cost increase on industry pro…ts. 9

The remainder of the paper is organized as follows. Section 2 presents the model. Section 3 5 Bessen and Meurer (2008) argue that the inclusion of software and business methods in the patentable subject matters in the US has resulted in an increase in the share of weak patents among all the patents issued by the USPTO.

6 See e.g. [START_REF] Taylor | The Economic Impact of the Patent System[END_REF] and [START_REF] Rostoker | A survey of corporate licensing[END_REF]. In particular, royalties are widely used for the licensing of standard-essential patents.

7 Various other reasons have been put forward in the literature on ironclad patents to explain the use of royalties, including risk aversion [START_REF] Bousquet | Risk sharing in licensing[END_REF], asymmetry of information [START_REF] Gallini | Technology transfer under asymmetric information[END_REF][START_REF] Macho-Stadler | Contrats de licence et asymétrie d'information[END_REF][START_REF] Beggs | The licensing of patents under asymmetric information[END_REF]Sen 2005a), moral hazard (Macho-Stadler et al., 1996;[START_REF] Choi | Technology transfer with moral hazard[END_REF], product di¤erentiation [START_REF] Muto | On licensing policies in Bertrand competition[END_REF][START_REF] Wang | On licensing under Bertrand competition[END_REF][START_REF] Caballero-Sanz | Optimal licensing in a spatial model[END_REF][START_REF] Poddar | On patent licensing and spatial competition[END_REF][START_REF] Stamatopoulos | Licensing of a quality-improving innovation[END_REF], strategic delegation [START_REF] Saracho | Patent licensing under strategic delegation[END_REF], integer nature of the number of licensees (Sen, 2005b), variation in the quality of innovation [START_REF] Rockett | The quality of licensed technology[END_REF]. 8 In [START_REF] Farrell | How strong are weak patents?[END_REF] and [START_REF] Encaoua | Licensing "weak" patents[END_REF], the licensor is assumed to o¤er two-part tari¤ licensing contracts. However, in both papers a technical ad hoc assumption on the shape of the (endogenous) licensing revenue function is made to simplify the analysis by immediately guaranteeing that pure per-unit royalty contracts are optimal for the licensors of weak patents (in the class of contracts deterring litigation). In sharp contrast to the former papers, our result that the holder of a weak patent …nds it optimal to use a per-unit royalty licensing contract is implied by a mild condition which has a natural economic interpretation and is shown to hold with broad generality in standard oligopoly models with general demand functions (as are all the assumptions made in our model).

9 Besides the three strands of literature discussed above, this paper can also be related to the literature on pre-grant licensing (see e.g. [START_REF] Gans | The impact of uncertain intellectual property rights on the market for ideas: evidence from patent grant delays[END_REF]. The licensing of a technology covered by an uncertain patent and the licensing of a technology that is not patented yet have in common that they happen before some type of uncertainty is removed or reduced. However, they are di¤erent in that the public good nature of patent invalidation has strategic implications that are absent in the case of pre-grant licensing. 4 characterizes the licensing contracts that deter litigation. In Section 4 we provide su¢cient conditions under which the per-unit royalty mechanism is preferred by the patent holder. In Section 5, three extensions of our baseline model are considered. In Section 6, we show that the assumptions made on the equilibrium pro…ts in our model and the (su¢cient) conditions ensuring the superiority of the per-unit royalty mechanism in Section 4 are satis…ed with broad generality for both a Cournot oligopoly with homogenous goods and a Bertrand oligopoly with di¤erentiated goods. Section 7 concludes. All proofs are relegated to the Appendix.

The model

We consider an industry consisting of n 2 symmetric risk-neutral …rms producing at marginal cost c (…xed production costs are assumed to be zero). A …rm P outside the industry holds a patent covering a technology that, if used, allows a …rm to reduce its unit cost from c to c

where 2]0; c[.

We consider the following three-stage game: 10

First stage: The patent holder P o¤ers all …rms a licensing contract 11 whereby a licensee can use the patented technology against payment of a per-unit royalty r 2 [0; ] 12 or a …xed fee F 0: 13

Second stage: The n …rms in the industry simultaneously and independently decide whether to purchase a license. If a …rm does not accept the license o¤er, it can challenge the patent's validity before a court. 14 The outcome of such a trial is uncertain: with probability > 0 the patent is upheld by the court and with probability 1 it is invalidated. Hence, the parameter may be interpreted as the patent's quality or the patent's strength. If the patent is upheld, then a …rm that does not purchase the license uses the old technology, thus producing at marginal cost c whereas a …rm that accepted the license o¤er uses the new technology and pays the per-unit royalty r or the …xed fee F to the patent holder. 15 If the patent is invalidated, all the …rms, including those that accepted the license o¤er can use for free the new technology and their common marginal cost is c .

Third stage:

The n …rms produce under the cost structure inherited from the second stage. We do not specify the type of competition that occurs. We only assume that there exists a unique equilibrium of the competition game for any cost structure and we set some general assumptions on the equilibrium pro…t functions. For this purpose, denote e (k; c) (respectively i (k; c)) 16 the equilibrium pro…t function, gross of any potential …xed cost (e.g. a …xed license fee) of a …rm producing with marginal cost c c (respectively with marginal cost c) when k n …rms produce at marginal cost c and the remaining n k …rms produce at the marginal cost c. 17 We assume that the pre-licensing equilibrium pro…ts of the n …rms is strictly positive, i.e. e (n; c) > 0.

We now make the following general assumptions for any given n and k = 1; :::; n:

A1. The equilibrium pro…ts of an e¢cient …rm and an ine¢cient …rm, i.e. e (k; c) and i (k; c) respectively, are both continuously di¤erentiable in c over the subset of [0; c] in which i (k; c) > 0: Furthermore, the equilibrium output q e (n; c) when all n …rms produce at marginal cost c is continuously di¤erentiable with respect to c and strictly positive over [0; c].

A2. If the …rms are symmetric (in terms of e¢ciency), an identical increase in all …rms' marginal costs leads to a decrease in each …rm's equilibrium pro…t: @ e @c (n; c) < 0. A3. An ine¢cient …rm's equilibrium pro…t is increasing in the e¢cient …rms' marginal cost: If i (k; c) > 0 then @ i @c (k; c) > 0 and if i (k; c) = 0 then i (k; c 0 ) = 0 for any c 0 < c: A4. A …rm's pro…t is decreasing in the number of e¢cient …rms in the industry: for any c < c and any k < n it holds that e (k; c) > e (k + 1; c) and i (k; c) i (k + 1; c):

A5. A …rm's pro…t increases as it moves from the subgroup of ine¢cient …rms to the subgroup of e¢cient …rms: for any c < c and any k < n it holds that i (k; c) < e (k + 1; c):

As we shall argue in precise detail in Section 6, all these assumptions are satis…ed with broad generality in the two most widely used oligopoly models with general demand functions.

Licenses deterring litigation

If litigation occurs then, with probability , the patent is upheld by the court (thus becoming an ironclad right) and, with probability 1 ; it is invalidated and the technology can then be used for free by all …rms. Thus, if the patent holder expects its license o¤er to trigger litigation, it should make an o¤er that maximizes the licensing revenues it will collect if the patent is ruled valid by the court. The patent holder would then essentially act as if the patent were ironclad. In other words, the uncertainty over the patent's validity does not a¤ect the licensing terms in that scenario.

We therefore consider in what follows only the class of license o¤ers deterring litigation, for which the uncertainty over patent validity does matter. In doing so, we follow [START_REF] Farrell | How strong are weak patents?[END_REF] who also focus on license o¤ers such that litigation is avoided because they aim to investigate the social costs of the uncertainty over patent validity (which is resolved if litigation occurs). Farrell and Shapiro actually show that focusing on litigation-deterring contracts is much less restrictive than one might think: they show that the patent hoder …nds it optimal to deter litigation (even in the absence of litigation costs) in (i) a symmetric Cournot oligopoly with linear inverse demand and costs as long as the number of licensees is three or more, and (ii) a symmetric Bertrand oligopoly with linear costs and a linear demand system (or, more generally, linear pass-through) as long as the number of licensees is two or more. This theoretical …nding is supported by empirical evidence: the vast majority of patent disputes are settled using licensing agreements before a court decides whether the patent is valid or not [START_REF] Allison | Empirical evidence on the validity of litigated patents[END_REF][START_REF] Allison | Empirical evidence on the validity of litigated patents[END_REF]Lemley and[START_REF] Lemley | Probabilistic patents[END_REF]. Also, [START_REF] Zuniga | Who licenses out patents and why?[END_REF] report that deterring litigation is one of the main motives put forward by …rms when they are asked why they license out their patents.

We start our analysis by determining the license o¤ers that deter litigation under each mechanism.

Per-unit royalty mechanism

Let us …rst examine a …rm's incentives to challenge the patent's validity when the patent holder makes a license o¤er involving the payment of a per-unit royalty r 2 [0; ]. A …rm that decides not to purchase a license is always (weakly) better o¤ challenging the patent's validity: if no other …rm challenges the patent's validity it gets a payo¤ i (n 1; c + r) + ( 1) e (n; c ) which is strictly greater than the pro…t i (n 1; c + r) it would get by not challenging the patent, and if some other …rm challenges the patent's validity then it is indi¤erent between challenging and not.

Thus, a situation where one or more …rms do not buy a license and no …rm challenges the patent's validity can never be an equilibrium of the second stage subgame. It follows that a license o¤er deters litigation if and only if it is accepted by all …rms.

Assume that the patent holder makes a license o¤er (in the …rst stage) involving the payment of a per-unit royalty r < : Let us show that in this case, any outcome with k n 2 licensees cannot be an equilibrium. We have already shown that a situation where not all …rms buy a license and no …rm challenges the patent cannot be an equilibrium so we can focus on situations with k n 2 licensees and at least one non-licensee challenging the patent. Any of the other n k 1 1 nonlicensees has an incentive to unilaterally deviate by buying a license since it would get an expected pro…t of e (k + 1; c + r) + ( 1) e (n; c ) instead of the strictly lower expected pro…t i (k; c + r) + ( 1) e (n; c ) if it remains a non-licensee. Therefore, any equilibrium of the second stage subgame involves at least n 1 …rms if r < . The latter result extends to the case r = if it is assumed, as will be the case from now on, that a …rm which is indi¤erent between getting a license and not buying a license purchases one.

Let us now write the condition under which all …rms accepting the license o¤er r is an equilibrium of the second stage subgame. A …rm anticipating that all other …rms will purchase a license gets a pro…t equal to e (n; c + r) if it accepts the license o¤er. If it does not and challenges the patent's validity then with probability ; the patent is upheld by the court and the challenger gets a pro…t equal to i (n 1; c + r) and, with probability 1 , the challenger gets a pro…t of e (n; c )

(and so do all other …rms). Thus, a …rm challenging the patent's validity when all other …rms accept the license o¤er, gets an expected pro…t of i (n 1; c + r) + ( 1) e (n; c ): Therefore, all …rms accepting the license o¤er is an equilibrium if and only if:

e (n; c + r) i (n 1; c + r) + (1 ) e (n; c ): (1)
The next lemma characterizes the values of the per-unit royalty set by the licensor that induce all …rms to buy a license (thus deterring litigation).

Lemma 1 De…ne r( ) as the unique solution in r to the following equation:

e (n; c + r) = i (n 1; c + r) + (1 ) e (n; c ):
Then all …rms accepting to pay a per-unit royalty r is an equilibrium if and and only if r r( ):

Fixed fee mechanism

The previous observation that a license o¤er deters litigation if and only if it is accepted by all …rms remains true when the licensor uses a …xed fee scheme. For a license o¤er involving the payment of a …xed fee F to be accepted by all …rms, the following condition must hold:

e (n; c ) F i (n 1; c ) + (1 ) e (n; c )
which can be rewritten as:

F e (n; c ) i (n 1; c ) : (2)
The next lemma is the counterpart of Lemma 1 for the …xed fee mechanism.

Lemma 2 All …rms accepting to pay the …xed fee F to use the patented technology is an equilibrium

if and only if F F ( ) [ e (n; c ) i (n 1; c )]:
4 Optimal licensing mechanism

The optimal per-unit royalty contract deterring litigation solves the following maximization program:

max r 0 nrq e (n; c ) s.t. r r( )
while the optimal …xed fee contract deterring litigation is given by

max F 0 nF s.t. F F ( )
It is obvious that the constraint of the latter maximization program is binding at the optimum. For the sake of exposition we will also also assume that the constraint of the program de…ning the optimal per-unit royalty contract is binding at the optimum. 18 Note however that relaxing this assumption can only strengthen the subsequent results about the superiority of the per-unit royalty mechanism.

E¢ciency e¤ect versus punishment e¤ect

Denoting Pr ( ) (resp. PF ( )) the licensing revenues generated by the optimal per-unit royalty (resp.

…xed fee) contract deterring litigation, we have

Pr ( ) = nr( )q e (n; c + r( )) and PF ( ) = n F ( ) = n e (n; c ) i (n 1; c ) This, combined with e (n; c + r( )) = i (n 1; c + r( )) + (1 ) e (n; c ), yields Pr ( ) PF ( ) = A( ) + B( ):
where

A( ) n [ e (n; c + r( )) + r( )q e (n; c + r( )) e (n; c )] and B( ) n i (n 1; c + r( )) i (n 1; c )
The term A( ) captures an e¢ciency e¤ect : it is the di¤erence between the aggregate pro…ts generated under the optimal per-unit royalty contract and those under the optimal …xed fee contract.

This term is always positive if the following condition holds:

e (n; c 0 ) e (n; c) > (c 0 c)q e (n; c 0 ) for any c; c 0 2 [ c ; c] such that c < c 0 . ( 3
)
The decomposition e (n; c 0 ) e (n; c) = (c 0 c)q e (n; c 0 ) + p e (n; c 0 ) c q e (n; c 0 ) (p e (n; c) c) q e (n; c)

shows that the overall e¤ect of an increase in marginal costs on equilibrium pro…ts can be split into a cost e¤ect, captured by the term (c 0 c)q e (n; c 0 ), and a strategic e¤ect, captured by the term (p e (n; c 0 ) c) q e (n; c 0 ) (p e (n; c) c) q e (n; c). The cost e¤ect is obviously negative. Condition (3) means that the strategic e¤ect of a marginal cost increase on equilibrium pro…ts is positive.

Let us now explain why Condition (3) is su¢cient to ensure that the per-unit royalty mechanism generates higher aggregate pro…ts than the …xed fee mechanism. Per-unit royalties do not a¤ect the marginal cost of production from an aggregate perspective: the additional marginal cost incurred by the licensees is fully captured by the licensor. However, they distort the …rms' price/output decisions since the e¤ective marginal cost of the licensees increases. The e¤ect of this distorsion on the aggregate pro…ts is the same as the (previously de…ned) strategic e¤ect of a marginal cost increase on the licensees' pro…ts. Therefore whenever the latter is positive, per-unit royalties increase aggregate pro…ts with respect to …xed fees.

To the best of our knowledge, Condition (3) has not been studied in the literature on the e¤ects of cost variations on oligopolists' pro…ts which has mainly focused on the overall e¤ect of cost changes on pro…ts (e.g. [START_REF] Seade | Pro…table cost increases and the shifting of taxation[END_REF][START_REF] Kimmel | E¤ects of cost changes on oligopolists' pro…ts[END_REF][START_REF] Février | Idiosyncratic shocks in a asymmetric Cournot oligopoly[END_REF]. In Section 6, we will establish that this condition is satis…ed with broad generality in the widely used settings of Cournot competition with homogeneous products and Bertrand competition with di¤erentiated products.

Let us now turn to B( ). Since a challenger gets the same pro…t under both mechanisms if the patent is invalidated, B( ) is the di¤erence between the expected pro…ts a challenger would get under the optimal per-unit royalty contract deterring litigation and its …xed fee counterpart. It captures a punishment e¤ect: an unsuccessful challenger gets a (weakly) lower pro…t under the …xed fee mechanism than under the per-unit royalty mechanism because its licensed competitors are more e¢cient under the former. Therefore, the rent that the patent holder needs to leave to each licensee in order to deter litigation is (weakly) lower under the …xed fee mechanism.

We can therefore conclude that, under Condition (3), the choice of a licensing mechanism is driven by two con ‡icting considerations. On the one hand, the e¢ciency e¤ect gives the patent holder an incentive to use the mechanism that maximizes aggregate pro…ts, i.e. the per-unit royalty mechanism. On the other hand, the punishment e¤ect provides an incentive to choose the mechanism that leaves a rent as low as possible to the licensees, i.e. the …xed fee mechanism.

While the e¢ciency e¤ect is always strictly positive (under Condition (3)), the punishment e¤ect is only weakly negative. We will therefore distinguish between the case when it is absent (B( ) = 0)

and the case when it is at work (B( ) < 0).

Optimal mechanism when there is no punishment e¤ect

Assume here that B( ) = 0. Note …rst that this holds if and only if i (n 1; c + r( )) = 0

(by A3). In words, a necessary and su¢cient condition for the punishment e¤ect to be zero is that an unsuccessful challenger gets the harshest possible punishment under the optimal per-unit royalty contract (and a fortiori under its …xed fee counterpart): it gets driven out of the market. In that case, the marginal cost of the licensees is irrelevant for an unsuccessful challenger. This scenario is more likely to hold the more competitive the industry.

The next proposition follows from the analysis in Section 4.1.

Proposition 1 Assume that an unsuccessful challenger is driven out of the market under both mechanisms (i.e. i (n 1; c + r( )) = 0). Then the optimal per-unit royalty contract deterring litigation generates higher licensing revenues than its …xed fee counterpart if Condition (3) holds.

A simple intuition for this proposition can be provided through an alternative interpretation of Condition (3): the latter formalizes, in our general competitive setting, the intuitive idea that an increase in marginal cost relaxes competition. To see why notice …rst that Condition (3) also means that an "arti…cial" increase in all …rms' marginal costs has a positive e¤ect on the equilibrium pro…ts in a symmetric industry. The term "arti…cial" refers to a situation in which …rms manage to commit to the prices/outputs corresponding to a higher marginal cost while their e¤ective marginal cost remains unchanged. Moreover, we will show in Section 6 that, in the speci…c context of a symmetric Cournot oligopoly with homogenenous products (resp. Bertrand oligopoly with di¤erentiated products), a key condition for Condition (3) to hold is that the equilibrium output (resp. price) is decreasing (resp.

increasing) in the marginal cost of production.

With this interpretation in mind, it becomes clear that per-unit royalties allow the patent holder to relax competition between the licensees, which results in higher aggregate pro…ts. In the absence of the punishment e¤ect, the choice of a licensing mechanism is fully driven by the e¢ciency e¤ect and, therefore, the patent holder chooses to use a per-unit royalty contract.

Optimal mechanism when the punishment e¤ect is at work

Assume now that B( ) < 0. Then the condition that the e¢ciency e¤ect is positive (A( ) > 0) is no longer su¢cient for the per-unit royalty mechanism to dominate the …xed fee mechanism. Since the punishment e¤ect is negative, a per-unit royalty contract will be used only if the magnitude of the e¢ciency e¤ect is higher than that of the punishment e¤ect (i.e. A( ) > jB( )j). While this may not be the case on the whole parameter space in our general setting, it turns out that it is always true for su¢ciently weak patents. The reason is that the e¤ect of leaving more pro…ts to the licensees under the per-unit royalty mechanism on the patent holder's pro…t, which is captured by B( ), is at most, a second-order term for su¢ciently small (because B(0) = 0 and B 0 (0) = 0). In contrast, the e¢ciency e¤ect A( ) is a …rst-order term (because A(0) = 0 and A 0 (0) > 0 under Condition (3)).

The next result provides a condition, which is less restrictive than Condition (3), that ensures the superiority of the per-unit royalty mechanism for the licensing of (su¢ciently) weak patents.

Proposition 2 For a su¢ciently weak patent, the optimal per-unit royalty contract deterring litigation provides higher licensing revenues than the optimal …xed fee contract deterring litigation if

@ e @c (n; c ) > q e (n; c ): (4)
Condition ( 4) is a local version of Condition (3) and, therefore, has the same economic interpretation. Section 6 will show the mildness of Condition (4) but, because Proposition 2 provides a local result, it is not informative regarding the range of patent strength values for which the per-unit royalty mechanism is preferred by the licensor when an unsuccessful challenger remains active in the market. The following example shows that this range can be substantially wide.

Example Consider the special case of Cournot competition with linear demand p = a Q and assume that a c 2(n 1) . In that environment i (n 1; c ) > 0, i.e. a challenger would always remain active in the market should the patent be upheld. Long but straightforward computations

show that a threshold ^ exists such that the optimal per-unit royalty contract deterring litigation generates higher licensing revenues than its …xed fee counterpart if is greater than ^ : In this setting, the parameters a, c and a¤ect the comparison of Pr ( ) and PF ( ) only through the ratio a c : We compute ^ for the set of the 1000 pairs n; a c such that n = 2; 3; :::; 11 and a c = 2(n 1) + i where i = 1; 2; :::; 100 and …nd values that are all greater than 0:5: More speci…cally all values are within the interval [0:501; 0:678]. This shows that, even when the punishment e¤ect is at work, the superiority of the per-unit royalty scheme is not associated only with small values of .

Extensions

We now assess the robustness of Proposition 2 by considering three extensions of our baseline model.

Litigation costs

Let us assume in this section that a …rm that challenges the patent's validity before a court has to incur some legal costs C 0. 19 It is straightforward that the higher those costs the higher the licensing revenues the patent holder can extract from the licensees without triggering litigation. This qualitative observation holds for both mechanisms. However, we show in what follows that, on the quantitative side, the marginal e¤ect of litigation costs on the patent holder's licensing revenues is higher under the per-unit royalty mechanism than under the …xed fee mechanism if condition (4)

holds. This implies that the result in Proposition 2 remains true -and is actually strenghtened -if the model is extended to include (small) legal costs that any challenger must incur.

Suppose …rst that the patent holder makes a license o¤er involving the payment of a per-unit royalty r 2 [0; [ : Note that the inclusion of legal costs in our setting does not a¤ect the fact that the strategy "not buy a license and not challenge the patent's validity" is always dominated by the strategy "buy a license". Therefore, the only way a patent holder can deter litigation is to make a license o¤er that is accepted by all …rms. This will be the case if and only if

e (n; c + r) i (n 1; c + r) + (1 ) e (n; c ) C:
It is easily shown the latter constraint is met if and only if r r( ; C) where r( ; C) is the solution in r to the equation

e (n; c + r) = i (n 1; c + r) + (1 ) e (n; c ) C
and that, for and C su¢ciently small, the optimal per-unit royalty license deterring litigation involves the payment of the royalty r( ; C) (i.e. the constraint is binding). Note also that r( ; C) is strictly increasing in both its arguments.

Suppose now that the patent holder makes a license o¤er involving the payment of a …xed fee F 0: Such a license o¤er is accepted by all …rms if and only if

e (n; c ) F i (n 1; c ) + (1 ) e (n; c ) C
and, therefore, the optimal …xed fee license deterring litigation involves the payment of the fee

F ( ; C) = e (n; c ) i (n 1; c ) + C:
Let us now compare the licensing revenues derived by the patent holder under the two mechanisms.

Under the optimal per-unit royalty contract, they are given by Pr ( ; C) = nr( ; C)q e (n; c + r( ; C))

and under the optimal …xed fee contract, they are given by

PF ( ; C) = n F ( ; C) = n e (n; c ) i (n 1; c ) + nC:
Since Pr (0; 0) = PF (0; 0) ; a su¢cient condition for the existence of ~ > 0 and C > 0 such that the inequality Pr ( ; C) > PF ( ; C) holds for any < ~ and C < C is that @ Pr @ (0; 0) > @ PF @ (0; 0) and @ Pr @C (0; 0) > @ PF @C (0; 0)

The former inequality has already been shown to be equivalent to Condition (4). Surprisingly enough, the latter inequality is equivalent to Condition (4) too. Indeed, @ PF @C (0; 0) = n and @ Pr @C (0; 0) = n @r @C (0; 0) q e (n; c ):

Di¤erentiating with respect to C the equation de…ning r( ; C) at point ( ; C) = (0; 0) ; we get: @r @C (0; 0) = 1 @ e @c (n; c ) : Thus, @ Pr @C (0; 0) = nq e (n; c ) @ e @c (n; c ) :

Hence @ Pr @C (0; 0) > @ PF @C (0; 0) () @ e @c (n; c ) > q e (n; c ):

Therefore, the result in Proposition 2 is robust -and is actually strengthened 20 -in the presence of relatively small legal costs.

Two-part tari¤ contracts

Assume in this section that the patent holder can o¤er a two-part tari¤ licensing contract involving the payment of a …xed fee F 0 and a per-unit royalty r 0. The optimal two-part tari¤ contract F ( ) ; r ( ) deterring litigation is a solution to the following maximization program Since the objective function is increasing in F , the constraint has to be binding at the optimum.

Therefore, the optimal two-part tari¤ contract F ( ) ; r ( ) is such that F ( ) = e (n; c + r ( )) Note that the latter constraint could be rewritten as r r( ), and that F ( ) = 0 if and only if r ( ) = r( ), i.e. the constraint is binding. The next proposition shows that our …nding that 2 0 The free riding problem arising from the public good nature of patent invalidation is ampli…ed when a potential challenger needs to incur positive litigation costs. The fact that @ Pr @C (0; 0) > @ PF @C (0; 0) under Condition (4) implies that, under the latter, the per-unit royalty mechanism allows the patent holder to exploit the ampli…cation of the free riding problem better than the …xed-fee mechanism (for weak patents and small litigation costs).

the holder of a su¢ciently weak patent …nds it optimal to use a per-unit royalty contract whenever Condition (4) holds, extends to a setting where the larger set of two-part tari¤ contracts is considered.

Proposition 3 For su¢ciently weak patents, the optimal two-part tari¤ licensing contract deterring litigation is a pure per-unit royalty contract if:

@ e @c
(n; c ) > q e (n; c ):

Internal patent holder

Let us consider the case where the patent holder is active in the output (downstream) market. More speci…cally, we assume that one of the n …rms operating in the market, say …rm 1, gets a patent on a technology that lowers the unit production cost from c to c : We assume that n 3 (as we want

to have at least two potential licensees).

Here again, we assume that there exists a unique equilibrium to the competition game for any cost structure (with identical market pro…ts for …rms producing at the same unit cost) and we set some general assumptions on the equilibrium pro…t functions. We focus on industry cost structures that can emerge following the licensing game, that is, situations in which: one …rm -the patent holder -produces at unit cost c , a number k n 1 of …rms -the licensees -produce at a unit cost c 2 [ c ; c] and the remaining n k …rms -the non-licensees -produce at unit cost c:

We denote by p (k; c), l (k; c) and n (k; c) the equilibrium market pro…ts of the patent holder, a licensee producing at an e¤ective unit cost c and a non-licensee respectively.

Given the new environment we consider, we need to replace the assumptions A1-A5 made in our baseline model with the following alternative, but related, assumptions:

A1'. The equilibrium pro…ts p (k; c), l (k; c) and n (k; c) are continuously di¤erentiable in c over [0; c] over the subset of [0; c] in which n (c; k) > 0: Furthermore, the function c ! q l (n; c) is continuously di¤erentiable over the subset of [0; c] in which it is strictly positive.

A2'. An identical increase in the costs of all …rms but the patent holder decreases each one of those …rms' equilibrium pro…t: @ l @c (n 1; c) < 0. A3'. A non-licensee's equilibrium pro…t is increasing in the licensees' unit cost: If n (k; c) > 0 then @ n @c (k; c) > 0 and if n (k; c) = 0 then n (k; c 0 ) = 0 for any c 0 < c: A4'. A …rm's market pro…t is decreasing in the number of licensees in the industry: for any c < c and any k < n 1 it holds that p (k; c) > p (k + 1; c); l (k; c) > l (k + 1; c) and n (k; c) n (k + 1; c): A5'. A …rm's market pro…t increases as it moves from the subgroup of non-licensees to the subgroup of licensees: for any c < c and any k < n 1 it holds that n (k; c) < l (k + 1; c):

The comparison of the innovator's overall pro…t, i.e. the sum of its market pro…t and licensing revenues, under the two licensing mechanisms, yields the following result:

Proposition 4 For su¢ciently weak patents, the optimal per-unit royalty contract deterring litiga-tion generates higher overall pro…t for the patent holder than its …xed fee counterpart if @ p @c (n 1; c ) + (n 1) @ l @c (n 1; c ) > (n 1) q l (n; c ):

(5)

Moreover, the reverse holds if the reverse strict inequality is satis…ed.

To see how Condition (5) compares to its counterpart when the patent holder is not active on the market, i.e. Condition (4), let us rewrite both of them with the same notations. For that purpose, let us denote by (c 1 ; c 2 ; :::; c n ) the sum of all …rms' equilibrium market pro…ts, i.e. the equilibrium (downstream) industry pro…t, and q i (c 1 ; c 2 ; :::; c n ) …rm i's output when each …rm j = 1; 2; :::; n produces at unit cost c j : The su¢cient condition for a patent holder who is an industry outsider to prefer the per-unit royalty mechanism for su¢ciently weak patents can be rewritten as (replacing

c
with the generic variable c):

n X i=1 @ @c i (c; c; :::; c) > n X i=1 q i (c; c; :::; c) : (6)
The su¢cient condition for a patent holder who is an industry insider to prefer the per-unit royalty mechanism for su¢ciently weak patents can be rewritten as (replacing again c by the generic variable c, and denoting P the patent holder):

n X i=1 i6 =P @ @c i (c; c; :::; c) > n X i=1 i6 =P
q i (c; c; :::; c) :

The two inequalities have very close interpertations: Condition (6) means that the strategic e¤ect of an identical increase in all …rms' (common) unit cost on the industry pro…t is positive and Condition (7) means that the strategic e¤ect of an increase in the costs of all …rms but one on the industry pro…ts is positive (…rms being equally e¢cient initially). Note also that both conditions are implied by the following inequality when it holds for any i = 1; 2; :::; n : @ @c i (c; c; :::; c) > q i (c; c; :::; c) :

This condition means that when …rms are equally e¢cient initially, the strategic e¤ect of an increase in one …rm's unit cost on the industry pro…t 21 is positive.

We next show that Condition (8) holds (i) for standard Cournot competition under complete generality, and (ii) for Bertrand competition with di¤erentiated products under strategic complementarity, provided existence an uniqueness of a pure-strategy equilibrium hold in these two environments.

It then follows that both Conditions ( 6) and ( 7) hold since they are implied by Condition (8).

Two standard oligopoly applications

In this section, we provide su¢cient conditions of a general nature on the primitives of the two most widely used models of imperfect competition, which lead to Assumptions A1-A5 and A1'-A5' and Conditions (3), ( 4) and ( 5) being veri…ed. Since some of the results below are new to the oligopoly literature, and of some independent interest, we derive them for fully asymmetric versions of the Cournot and Bertrand oligopolies with linear costs. Accordingly, we also change the notation as needed, relative to the other parts of the paper.

Cournot competition with homogeneous products

Consider an industry consisting of n …rms competing in Cournot fashion. Firm i's marginal cost is denoted c i (…xed production costs are assumed to be zero or otherwise sunk). Suppose the …rms face an inverse demand function P ( ) satisfying the following minimal conditions:

C1 P ( ) is twice continuously di¤erentiable and P 0 ( ) < 0 whenever P ( ) > 0:

C2 P (0) > c i > P (Q) for Q su¢ciently high, i = 1; 2; :::; n.

C3 P 0 (Q) + QP 00 (Q) < 0 for all Q 0 with P ( ) > 0.
These assumptions are quite standard. C3 is the familiar condition used by [START_REF] Novshek | On the existence of Cournot equilibrium[END_REF] to ensure downward-sloping reaction curves (for any cost function). It states that each …rm's marginal revenue is decreasing in rivals' output (see [START_REF] Amir | Cournot oligopoly and the theory of supermodular games[END_REF] for an alternative condition).

Firm i's pro…t function and reaction correspondence are (here,

Q i = P j6 =i q j ) i (q i ; Q i ) = q i [P (q i + Q i ) c i ] and r i (Q i ) = arg max q i 0 i (q i ; Q i ):
The next proposition provides general conditions under which Assumptions A1-A5 and A1'-A5' hold for a Cournot oligopoly.

Proposition 5 Under Assumptions C1-C3, (a) There exists a unique Cournot equilibrium.

(b) Firm i's equilibrium output q i and pro…t i are di¤erentiable in c i and in c j for any j 6 = i:

(c) Firm i's equilibrium pro…t i is decreasing in c i and increasing in c j for any j 6 = i:

If in addition, the game is symmetric (with c denoting the unit cost), then (d) The unique Cournot equilibrium is symmetric.

(e) The equilibrium output q strictly decreases in c. Assumptions A4 and A4' follow from repeated applications of part (c), with one rival …rm's cost decreasing at a time. Assumptions A5 and A5' follow from part (c).

We now discuss the scope of these assumptions, focusing on parts (f) and (c), the other assumptions being well known for existence and uniqueness of Cournot equilibrium. Though intuitive, part (f) actually has a less universal scope than one might think. Indeed, there is an extensive literature dealing with taxation in oligopolistic industries and one of its key insights is that a common cost increase can lead to some …rms bene…ting at the expense of others [START_REF] Seade | Pro…table cost increases and the shifting of taxation[END_REF][START_REF] Kimmel | E¤ects of cost changes on oligopolists' pro…ts[END_REF][START_REF] Février | Idiosyncratic shocks in a asymmetric Cournot oligopoly[END_REF]. More surprisingly, in a symmetric setting, a cost increase may be bene…cial to all …rms, when the inverse demand function is su¢ciently convex. In light of this result, known as the cost paradox, part (f) may be viewed as giving su¢cient conditions for this counter-intuitive e¤ect of taxation not to arise.

Since the cost paradox literature considers cost increases that are common to all …rms, it does not deal with the pro…t e¤ects of a unilateral cost change, as in the two novel results of part (c). In light of this discussion, it emerges that ruling out the cost paradox and its plausible implications is one of the most restrictive assumptions of the present setting. Assumption C3 is then appropriate since it guarantees existence and uniqueness of Cournot equilibrium and the absence of the cost paradox.

Proposition 6 Under Assumptions C1-C3, (i) Condition ( 8) and, therefore, Conditions (4) and ( 5) are veri…ed.

(ii) Condition (3) holds whenever Q m ( c ) Q e (n; c) where Q m ( c ) is the monopoly output when the marginal cost is c .

As can easily be seen in the proof, this result requires only that total equilibrium output decreases with a unilateral unit cost increase, which holds universally in Cournot competition with linear costs.

Moreover, it can be easily shown that Q m ( c) < Q e (n; c) and that the monopoly output Q m (c) is decreasing in c, which allows to interpret the condition Q m ( c ) Q e (n; c) as being not too large.

To get a sense of how restrictive this condition is, consider the special case of a linear inverse demand

p = a Q. Then Q m ( c ) Q e (n; c) if and only if n 1 n+1 (a c).
To see why this condition is not very restrictive, note that in this particular setting an innovation is drastic 22 if (a c) :

Bertrand competition with di¤erentiated products

Consider an industry consisting of n single-product …rms, with constant unit costs c 1 ; c 2 ; :::; c n : Assume that the goods are imperfect substitutes. Denoting D i (p 1 ; p 2 ; :::; p n ) the demand for the good produced by …rm i; its pro…t function and reaction correspondence are de…ned as usual by

i (p i ; p i ) = (p i c i )D i (p i ; p i ) and r i (p i ) = arg max p i i (p i ; p i ):
The Bertrand oligopoly is symmetric if the demand functions are symmetric and c 1 = ::: = c n , c:

Let S i , (p 1 ; p 2 ; :::; p n ) 2 R n + j D i (p 1 ; p 2 ; :::; p n ) > 0 . We assume that for every …rm i:

B1 D i is twice continuously di¤erentiable on S i .
B2 (i) @D i @p i < 0, (ii) @D i @p j > 0 and (iii) P n k=1 @D i (p;p;:::;p) @p k < 0 over the set S i .

B3 D i @ 2 log D i @p j @p i @ log D i @p j @ log D i @p i > 0 over the set S i , for j 6 = i.

B4

P n j=1 @ 2 D i (p 1 ;p 2 ;:::;pn) @p i @p j < 0 over the set S i .

These conditions are quite general, and are commonly invoked for di¤erentiated-good demand systems. They have the following meanings and economic interpretations. For B2, part (i) is just the ordinary law of demand; part (ii) says that goods i and j are substitutes; and part (iii) is a dominant diagonal condition for the Jacobian of the demand system, which is required to hold only at equal prices (see e.g., [START_REF] Vives | Oligopoly pricing: old ideas and new tools[END_REF]. It says that, along the diagonal, own price e¤ect on demand exceeds the total cross-price e¤ects. B3 says that each demand has (di¤erentiably) strict log-increasing di¤erences in own price and any rival's price. The exact economic interpretation is that the price elasticity of demand strictly increases in any rival's price, which is a very natural assumption [START_REF] Milgrom | Rationalizability, learning, and equilibrium in games with strategic complementarities[END_REF]. B4 says that the Hessian of the demand system has a dominant diagonal, which is a standard assumption invoked to guarantee uniqueness of Bertrand equilibrium [START_REF] Milgrom | Rationalizability, learning, and equilibrium in games with strategic complementarities[END_REF]Roberts, 1990 or Vives, 1999). B2(iii) and B4 hold that own e¤ects of price changes dominate cross e¤ects, for the level and the slope of demand, respectively.

The following result provides su¢cient conditions for A1-A5 and A1'-A5' to hold here.

Proposition 7 Under Assumptions B1-B4, (a) The Bertrand game is of strict strategic complements, and has a unique Bertrand equilibrium.

(b) Firm i's equilibrium price p i is increasing in c j for any j:

(c) Firm i's equilibrium price p i and pro…t i are di¤erentiable in c i and c j for any j 6 = i:

(d) Firm i's equilibrium pro…t i is increasing in c j for any j 6 = i:

In addition, if the game is symmetric, then (e) the unique Bertrand equilibrium is symmetric.

(f ) the equilibrium price increases in c:

(g) per-…rm equilibrium pro…t i is di¤erentiable in c, and decreasing in c.

We leave to the reader the task of matching the di¤erent parts of Proposition 7 to Assumptions A1-A5 and A1'-A5', as this step is quite similar to the Cournot case. [START_REF] Anderson | Tax incidence in di¤erentiated product oligopoly[END_REF] extends the analysis of the e¤ects of taxation to Bertrand competition with di¤erentiated products, and report analogous …ndings on the cost paradox as in the Cournot case. Since Proposition 7 contains only intuitive results on the e¤ects of cost changes on pro…ts, one concludes that assumptions B2-B4, which are needed for existence and uniqueness of Bertrand equilibrium, also preclude any counter-intuitive e¤ects of exogenous cost increases. This result is more general than those in [START_REF] Anderson | Tax incidence in di¤erentiated product oligopoly[END_REF] due to the abscence of concavity assumptions.

Proposition 8 Under Assumptions B1-B4, (i) Condition ( 8) and, therefore, Conditions (4) and Condition (5) are veri…ed.

(ii) Condition (3) holds whenever p e (n; c) p m ( c ) where p m ( c ) is the monopoly price when the marginal cost is c .

As can be seen from the proof, this result requires only that each …rm's equilibrium price increase with a unilateral unit cost increase, which holds in Bertrand oligopoly with linear costs whenever the game is supermodular (i.e, B3 holds). As in the Cournot case, the condition p e (n; c) p m ( c ) may also be interpreted as being not too large (for a given level of product di¤erentiation) or, alternatively, as product di¤erentiation being su¢ciently low (for a given innovation size ).

Conclusion

The issue of patent quality is one of the most serious problems facing the patent system. One negative consequence of the issuance of weak patents on social welfare stems from the legal costs of patent dispute resolution in court. The use of licensing agreements to deter or settle patent litigation allows to avoid those direct costs. It however induces an indirect loss in welfare if the licensing contract that is preferred by the patent holder is socially suboptimal. In the case of an ironclad patent, it is unclear which type of contracts (royalty or …xed fee) is socially preferrable: under royalty licensing the licensees are less e¢cient than under …xed fee licensing but their number is possibly higher under royalty licensing (see e.g. [START_REF] Kamien | Patent licensing[END_REF]. However, such a tradeo¤ does not exist for uncertain patents whenever the patent holder prefers to deter litigation (as is very often the case in practice):

in this case the patent holder needs to license all potential users of the technology to deter litigation, which implies that the number of licensees will be the same under both licensing schemes. In that case, per-unit royalties are clearly dominated by …xed fees from a social perspective.

This paper provides an analysis of the licensing of uncertain patents in a general framework and derives conditions under which the licensor prefers to use a per-unit royalty contract. In particular, it is shown that will always be the case for (su¢ciently) weak patents under a mild condition which is broadly satis…ed under Cournot competition with homogeneous products and Bertrand competition with di¤erentiated products. A signi…cant di¤erence with respect to the literature on the licensing of ironclad patents is that we get a clear-cut result: per-unit royalties are preferred by the holder of a weak patent independently of the type of downstream competition, the degree of di¤erentiation between products and whether the patent holder is active or not in the downstream market, while varying any of these three features can overturn the result when ironclad patents are considered.

Our model generates some testable predictions that might be worth investigating. First, our results suggest that per-unit royalty licenses should be more prevalent in industries with a signi…cant proportion of …rms holding questionable patents, e.g., industries relying on some new patentable subject matter (biotechnology, software, business methods,...). Second, if the predictions of our model are correct then under the presumption that the EPO is more stringent in checking the

Appendix

Proof of Lemma 1

All …rms accepting the license o¤er is an equilibrium if and only if: 1) e (n; c ) which can be rewritten as: Since Pr (0) = PF (0) then Pr ( ) > PF ( ) for su¢ciently small if:

e (n; c + r) i (n 1; c + r) + (
g (r; ) e (n; c + r) i (n 1; c + r) (
d Pr ( ) d j =0 d PF ( ) d j =0 > 0 (9)
which can be rewritten as: Therefore, ( 9) is equivalent to:

n i (n 1; c ) e (n; c ) @ e @c (n; c ) q e (n; c ) > n e (n; c ) i (n 1; c )
which can be rewritten as: @ e @c (n; c ) > q e (n; c

) because i (n 1; c ) e (n; c ) < 0:
Proof of Proposition 3

To prove that F ( ) = 0 for small enough, it is su¢cient to show that the function h(r; ) = rq e (n; c + r) + e (n; c + r) i (n 1; c + r) (1 ) e (n; c ) is increasing for small values of r and (this follows directly from r (0) = 0 and r( ) being continuous in ). We have: @h @r (0; ) = q e (n; c ) + @ e @c (n; c ) @ i @c (n 1; c ):

Let us now assume that @ e @c (n; c ) > q e (n; c ). We need to distinguish between two cases:

-Case 1: @ i @c (n 1; c ) = 0 : in this @h @r (0; ) = @h @r (0; 0) > 0 for any 2 [0; 1] -Case 2: @ i @c (n 1; c ) 6 = 0: in this case @h @r (0; ) > 0 for any 2 0; min 1; q e (n; c )+ @ e @c (n; c ) @ i @c (n 1; c ) . Therefore, in both cases, @h @r (0; ) > 0 for su¢ciently small values of . Using the continuity of r ! @h @r (r; ), we can then state that @h @r (r; ) > 0 for su¢ciently small values of r and , which, as claimed before, is su¢cient to complete the proof.

Proof of Proposition 4

Under the per-unit royalty mechanism, the optimal royalty rI ( ) for su¢ciently weak patents is the solution in r to the following equation:

l (n 1; c + r) = n (n 2; c + r) + (1 ) l (n 1; c )
and the patent holder's overall pro…t is ~ r ( ) = p (n 1; c + rI ( )) + (n 1) rI ( ) q l (n 1; c + rI ( )):

Under the …xed fee mechanism, the optimal fee is given by FI ( ) = h l (n 1; c ) n (n 2; c ) i and the patent holder's overall pro…t is then

~ F ( ) = p (n 1; c ) + (n 1) h l (n 1; c ) n (n 2; c ) i : Since ~ r (0) = ~ F (0) then ~ r ( ) > ~ F ( ) for su¢ciently small if d ~ r ( ) d j =0 > d ~ F ( ) d j =0 (10)
which can be rewritten as

r0 I (0) @ p @c (n 1; c ) + (n 1) q l (n; c ) > (n 1) h l (n 1; c ) n (n 2; c ) i :
because rI (0) = 0: Moreover, di¤erentiating at = 0 the equation de…ning rI ( ), we get r0 I (0):

@ l @c (n 1; c ) = n (n 2; c ) l (n 1; c ) which yields r0 I (0) = n (n 2; c ) l (n 1; c ) @ l @c (n 1; c ) :
Hence, inequality ( 10) is equivalent to n (n 2; c ) l (n 1; c ) @ l @c (n 1; c ) @ p @c (n 1; c ) + (n 1) q l (n; c ) > (n 1) e (n 1; c ) i (n 2; c ) which can be rewritten as @ p @c (n 1; c ) + (n 1) q l (n; c ) > (n 1) @ l @c (n 1; c ) or, equivalently, as

@ p @c (n 1; c ) + (n 1) @ l @c (n 1; c ) > (n 1) q l (n; c ):
Proof of Proposition 5

(a) This follows from the key slope property that every selection of r i satis…es (see [START_REF] Amir | Cournot oligopoly and the theory of supermodular games[END_REF][START_REF] Amir | Cournot oligopoly and the theory of supermodular games[END_REF][START_REF] Amir | On the e¤ects of entry in Cournot markets[END_REF] for details)

1 < r i (Q 0 i ) r i (Q i ) Q 0 i Q i < 0 for all Q 0 i > Q i : (11)
(b) We …rst show that q i is continuously di¤erentiable in c i : Viewed as a correspondence in the parameter c i , q i is upper hemi-continuous (or u.h.c.), as a direct consequence of the well-known property of u.h.c. of the equilibrium correspondence for games with jointly continuous payo¤ func-tions (e.g., Fudenberg and Tirole, 1990). Since q i is also single-valued in c (part (b)), q i must be a continuous function. That q i is continuously di¤erentiable in c i follows from the Implicit Function Theorem applied to the …rst order conditions, and the smoothness of P ( ). That i is also continuously di¤erentiable in c i follows from the fact that q i has that same property for all i.

The proof for the parameter c j ; j 6 = i; follows along the same lines.

(c) Throughout part (c), …x i and denote …rm i's output, pro…t and its rivals' total outputs at equilibrium by q i ; i and Q i respectively when the cost vector is (c 1 ; c 2 ; :::; c n ). Denote the same three variables by b q i ; b i and b Q i after …rm i's cost alone changes to b c i > c i , all other …rms' unit costs remaining the same. Adding the n …rst order conditions at the Cournot equilibrium yields

nP (Q ) + Q P 0 (Q ) = n X k=1 c k : (12)
Since the LHS of ( 12) is strictly decreasing in Q , the increase in …rm i's cost from c i to b c i increases the RHS of ( 12), which causes the solution to (12) to decrease. In other words, b Q < Q .

We now show that for any …rm j 6 = i, we must have b

Q j < Q j . To this end, …rst observe that b Q j + r j ( b Q j ) = b Q < Q = Q j + r j (Q j ). Since (11) holds that Q j + r j (Q j ) is increasing in Q j , we must have b Q j < Q j . For …rm j, b j = b q j h P (b q j + b Q j ) c j i q j h P (q j + b Q j ) c j i by the Cournot property > q j P (q j + Q j ) c j since b Q j < Q j = j :
We now show that for …rm i, we must have b Q i > Q i . To this end, …rst observe that since for any j 6 = i, r j is strictly decreasing (cf. ( 11)) and b

Q j < Q j , we have b q j = r j ( b Q j ) > r j (Q j ) = q j , for every …rm j 6 = i. Then since b Q i = P j6 =i b q j and Q i = P j6 =i q j , we have b Q i > Q i . To show that i > b i , consider i = q i P (q i + Q i ) c i b q i P (b q i + Q i ) c i by the Cournot property > b q i [P (b q i + b Q i ) c i ] since b Q i > Q i : > b q i [P (b q i + b Q i ) b c i ] since b c i > c i = b i :
For the remaining parts, we consider the case of a symmetric Cournot oligopoly (c i = c for all i).

(d) Due to the symmetry of the game, asymmetric equilibria, if any, would come in n-tuples.

Hence, the conclusion follows from part (a) directly.

(e) Let q denote per …rm equilibrium output. Di¤erentiating the …rst order condition w.r.t. c; @q @c (n + 1) P 0 (nq ) + nq P " (nq ) = 1:

Using the …rst order condition and C3, the term in brackets is strictly negative, so @q @c < 0. We now show that per-…rm pro…t decreases in c. Denote the equilibrium variables by q i ; i and Q i when the unit cost is c, and by q 0 i ; 0 i and Q 0 i the same variables when the unit cost is c 0 > c: Di¤erentiating i = q [P (nq ) c] with respect to c yields @ i @c = @q @c [P (nq ) c] + q P 0 (nq )n @q @c 1

= @q @c (n + 1)q P 0 (nq ) q by (12)

= q 2P 0 (Q ) + Q P " (Q ) (n + 1) P 0 (Q ) + Q P " (Q )
by ( 13).

Clearly, C3 implies that 2P 0 (Q) + QP " (Q) < 0 for all Q, so the numerator in the above fraction is < 0. It is then easy to see that the denominator is also < 0. Hence @ i @c < 0: Proof of Proposition 6

Let us show that Condition (8) holds (which implies that both Conditions ( 6) and (7) hold ).

Total di¤erentiation w.r.t. c i in = (P (Q ) c i ) q i + X j6 =i (P (Q ) c j ) q j yields @ @c i = [P 0 (Q ) @Q @c i 1]q i + (P (Q ) c i ) @q i @c i + X

j6 =i

[P 0 (Q ) @Q @c i q j + (P (Q ) c j ) @q j @c i ]:

which can be rewritten as: @ @c i = q i + X j P 0 (Q ) @Q @c i q j + (P (Q ) c j ) @q j @c i :

When c i = c j = c, the latter becomes: @ @c i = q i + X j P 0 (Q ) @Q @c i q j + (P (Q ) c) @q j @c i = q i + P 0 (Q ) @Q @c i X j q j + (P (Q ) c) X j @q j @c i

= q i + P 0 (Q ) @Q @c i :Q + (P (Q ) c) @Q @c i = q i |{z} direct e¤ect + @Q @c i P 0 (Q ):Q + P (Q ) c | {z } strategic e¤ect : (15)
Using the Cournot …rst order condition yields Q P 0 (Q ) + P (Q ) c = n 1 n Q P 0 (Q ) < 0: From the proof of Proposition 6, @Q @c i < 0. Hence @Q @c i [P 0 (Q ):Q + (P (Q ) c)] > 0, so @ @c i > q i in (15). Let us now show the second part of this proposition. We have e (n; c) e (n; c 0 ) (c 0 c)q e (n; c 0 ) = (P (nq e (n; c)) c)q e (n; c) (P (nq e (n; c 0 ) c)q e (n;

Proof of Proposition 7

For …rm i, charging a price of c i strictly dominates any price below c i . Hence, we restrict attention to the price space [c i ; 1) as the action set for …rm i. Then log i (p i ; p i ) is well de…ned.

(a) This is well known, see [START_REF] Milgrom | Rationalizability, learning, and equilibrium in games with strategic complementarities[END_REF] or [START_REF] Vives | Oligopoly pricing: old ideas and new tools[END_REF].

(b) The equilibrium price p i is increasing in c i since (i) the price game is log-supermodular (part (a)), (ii) log i (p i ; p i ) = log(p i c i ) + log D i (p i ; p i ) has increasing di¤erences in (p i ; c i ), from @ 2 log i (p i ; p i )=@p i @c i = (p i c i ) 2 > 0, and (iii) the constraint set [c i ; 1) is ascending in c i . So the conclusion follows from [Milgrom and Roberts, 1990, Theorem 7]. That p i is also increasing in c j for any j 6 = i follows from a similar argument since @ 2 log i (p i ; p i )=@p i @c j = 0.

(c) We …rst show that every equilibrium price p i is continuously di¤erentiable in c j , for all i, j: Viewed as a correspondence in the parameter c j , p i is u.h.c., by the u.h.c. property of the equilibrium correspondence for games with continuous payo¤ functions (jointly in all actions), see e.g., Fudenberg and Tirole, 1990. Since p i is also single-valued in c j (from part (i)), p i is a continuous function. Then the fact that p i is continuously di¤erentiable in c j follows from the Implicit Function Theorem. Finally, continuous di¤erentiability of i follows from that of all the p 0 i s. (d) Di¤erentiating i = (p i c i )D i (p i ; p i ) with respect to c j , for i 6 = j, yields

@ i @c j = @p i @c j D i (p i ; p i ) + (p i c i ) X k @D i @p k @p k @c j : (16)
Using the …rst order condition D i (p i ; p i ) + (p i c i ) @D i @p i = 0, (16) reduces to @ i @c j = (p i c i ) X k6 =i @D i @p k @p k @c j 0 since @D i @p k > 0 (goods are substitutes) and @p k @c j 0 from part (b).

(e) When the Bertrand game is symmetric, the unique Bertrand equilibrium must be symmetric, for otherwise equilibria would come in pairs.

(f) The argument is the same as for part (b) since @ 2 log i (p i ; p i )=@p i @c = (p i c) 2 > 0.

(g) From an argument similar to the proof of part (c), p and thus i = (p c)D i (p ; p ; :::; p ) are di¤erentiable w.r.t. c. We derive an expression for @p @c : The FOC at a Bertrand equilibrium is D i (p ; :::; p ) + (p c)@D i (p ; :::; p )=@p i = 0:

Using the Implicit Function Theorem and di¤erentiating the FOC with respect to c yields 0 @ @D i @p i + X k6 =i @D i @p k 1 A @p @c + (p i c i ) @p @c X k @D i @p k @p i + ( @p @c 1) @D i @p i = 0:

Hence, using B2 and B4, @p @c = @D i =@p i 2 @D i @p i + P k6 =i @D i @p k + (p i c i ) P k @D i @p k @p i > 0:

(18)

We can di¤erentiate = (p c)D i (p ; p ; :::; p ) with respect to c to obtain @ @c = ( @p @c 1)D i (p ; :::; p ) + (p c) @p @c X k @D i @p k = D i (p ; :::; p ) " 1 P k6 =i @D i @p k @D i =@p i @p @c # from (17) = D i (p ; :::; p ) " 1 P k6 =i @D i @p k 2 @D i @p i + P k6 =i @D i @p k + (p i c i ) P k @D i @p k @p i # using (18) = D i (p ; :::; p ) " 2 P k @D i @p k + (p i c i ) P k @D i @p k @p i 2 @D i @p i + P k6 =i @D i @p k + (p i c i ) P k @D i @p k @p i # < 0 by B2 and B4:

Proof of Proposition 8

Let us show that Condition (8) holds, which implies that both Conditions ( 6) and ( 7) hold.

We have = (p i c i ) D i + X j6 =i p j c j D j . Then: @ @c i = @p i @c i 1 D i + (p i c i ) @D i @c i + X j6 =i @p j @c i D j + (p i c i ) @D j @c i

maxF 0 ,

 0 r2[0; ] n (F + rq e (n; c + r)) s:t: e (n; c + r) F i (n 1; c + r) + (1 ) e (n; c ):

  It is straightforward to relate the di¤erent parts of Proposition 5 to Assumptions A1-A5 and A1'-A5'. Part (a) is needed to avoid vacuous statements. Assumptions A1 and A1' are implied by part (b) and the proof of part (e). Assumption A2 follows from part (f) and Assumption A2' follows from combining part (f) with part (c). Assumptions A3 and A3' are implied by part (c).

  c ) > 0 (by A2 and A5) and g ( ; ) = e (n; c) i (n 1; c) (n; c) e (n; c )) < 0 (by A2): Combining this with g being continuous (by A1) and strictly decreasing in r (by A3) yields: i/ the existence and uniqueness of a solution in r to the equation g (r; ) = 0 (within the interval [0; [), which we denote by r( ); ii/ the equivalence between the inequalities g (r; ) 0 and r r( ): Proof of Proposition 2

  nr 0 (0) q e (n; c ) > n e (n; c ) i (n 1; c ) because r (0) = 0: Moreover di¤erentiating at = 0 the equation e (n; c + r( )) = i (n 1; c + r( )) + (1 ) e (n; c ) ;

  Q e (n; c)) c)Q e (n; c) (P (Q e (n; c 0 ) c)Q e (n; c 0 ) :It is easy to see from C3 and C1 that a monopolist's pro…t(P (Q) c)Q is concave. Denoting Q m (c)the monopoly output with marginal cost c, it follows that (P (Q) c)Q is decreasing over the interval[Q m (c); Q e (n; c)]. From Proposition 7(e), c < c 0 implies Q e (n; c 0 ) < Q e (n; c). So the strategic e¤ect e (n; c) e (n; c 0 ) (c 0 c)q e (n; c 0 ) is negative if Q m (c) Q e (n; c 0 ).Thus, for Condition (3) to hold it is su¢cient to have, besides C1-C3, that Q m ( c ) Q e (n; c).

For instance, it has been shown that under Cournot competition with homogeneous products, …xed fees dominate per-unit royalties when the licensor is an industry outsider[START_REF] Kamien | Patent licensing[END_REF] Tauman, 1984, 1986;[START_REF] Kamien | Optimal licensing of cost-reducing innovation[END_REF]. However, the reverse result holds if the licensor is an industry insider[START_REF] Shapiro | Patent licensing and R&D rivalry[END_REF][START_REF] Wang | Fee versus royalty licensing in a Cournot duopoly model[END_REF] Kamien and Tauman,

2002;[START_REF] Sen | Monopoly pro…t in a Cournot oligopoly[END_REF] Sen and Tauman, 2007, 2012). Furthermore, under price competition with di¤erentiated products, per-unit royalties dominate …xed fees when the products are close substitutes or, if not, when the size of the cost reduction is small, while the reverse holds if the products are weak substitutes and the innovation is large[START_REF] Muto | On licensing policies in Bertrand competition[END_REF]. The results are quite di¤erent if one considers the same di¤erentiated product environment but assumes that …rms compete in quantities instead of prices[START_REF] Wang | Fee versus royalty licensing in a di¤erentiated Cournot duopoly[END_REF].2 This concerns the patent disputes that are not settled prior to the court judgement.

We assume only that a unique equilibrium in the competition game exists and set very mild assumptions on the resulting equilibrium pro…ts (as functions of the unit costs).

See Bhattacharya et al. (2012) for a recent survey on the licensing of ironclad patents and other R&D arrangements.

0 This game is similar to the one in[START_REF] Encaoua | Licensing "weak" patents[END_REF]. However we set much weaker assumptions in the present paper and use this game to conduct a thorough analysis of the optimal licensing scheme from the patent holder's perspective.1 1 Following[START_REF] Farrell | How strong are weak patents?[END_REF] and[START_REF] Encaoua | Licensing "weak" patents[END_REF], we focus on take-it-or-leave-it license o¤ers. 1

[START_REF] Farrell | How strong are weak patents?[END_REF] recall that under patent law, a licensor can ask for royalties only for use of the patented technology. Under this rule the royalty r cannot exceed the innovation size . The reason is that a licensee would always prefer to use the "old" technology rather than pay r > for the use of the patented technology (see footnote 22 in[START_REF] Farrell | How strong are weak patents?[END_REF]. Moreover, setting a royalty higher than the innovation size can trigger an antitrust investigation. In particular, in Europe, the royalty imposed by the licensor has to be expressly linked to the patent and re ‡ect its actual value, and the European Commission carefully considers if excessive royalties are being charged.1

Section 5.2 extends the analysis to two-part tari¤ contracts. 1

For patents granted by the European Patent O¢ce (EPO), the timing is slightly di¤erent. Indeed, any patent issued by the EPO can be opposed by a third party and the notice of opposition must be …led in writing at the EPO within nine months from the publication of the mention of the grant of the European patent.1

We assume that the patent holder commits not to license an unsuccessful challenger. Note however that our results will not be a¤ected if post-litigation licensing occurs (because the patent holder cannot commit not to do that) as long as a patent upheld by the court is licensed in a way that does not leave any surplus to an unsuccessful challenger. This will always be the case under the …xed fee regime. It will also be the case under the per-unit royalty regime as long as the patent holder …nds it optimal to license an unsuccessful challenger at the maximum per-unit royalty it accepts to pay, i.e. r = . Under the speci…c setting of Cournot competition with homegeneous goods and linear demand, it can be checked that the latter holds if the innovation size is not too large.

6 The superscript "e" (respectively "i") stands for "e¢cient" (respectively "ine¢cient"). 1 7 The fact that the pro…t function for a …rm can be de…ned in terms of the number of …rms that use the innovation, and not which …rms use the innovation, relies on an implicit symmetry assumption. For instance, in the context of price competition with di¤erentiated products, demands must be symmetric (i.e., products are symmetrically di¤erentiated).

8 It can be easily shown that this will always be the case for su¢ciently small.

9 Assuming that the patent holder incurs positive legal costs as well does not alter the subsequent analysis. The reason is that we focus on the scenario in which the patent holder …nds its optimal to deter litigation. Note that including positive litigation costs for the patent holder actually make this scenario more likely.

1 What we call "industry pro…t" is the sum of all market pro…ts.

2 A process innovation is drastic if the cost reduction it leads to is so large that the user of that technology can act as an unconstrained monopolist.

which can be rewritten as:

We have already shown that @p k @c i > 0 for any k; i (see the proof for part (b) of Proposition 8). Moreover, we have