Conference in honor of Claude d’Aspremont and Jean-François Mertens CORE, 23-25 June 2011

1. Per-unit royalty vs fixed fee: the case of weak patentsRabah Amir, University of Arizona David Encaoua, Paris School of Economics, University Paris I Yassine Lefouili, Toulouse School of Economics Conference in honor of Claude d’Aspremont and Jean-François Mertens CORE, 23-25 June 2011

2. How to sell a patent license? • Since Arrow’s contribution (1962), substantial literature comparing three contracts (upfront fee, auction and per-unit royalty) to sell a license to the members of an oligopoly. • Main conclusion: the best licensing scheme (licensor’s view) is very sensitive in the context of analysis. It varies according to the nature of the licensor, the competition regime and the informational setting (Sen, 2005). • Illustration: fixed fee and auction dominate per-unit royalty when the licensor is not active in the industry and licensees compete in quantities (Kamien & Tauman 1984, 1986), whereas the comparison is reversed when the licensor is an insider (Sen & Tauman, 2007)

3. But, patents are not certain property rights! • Common feature of all this literature: patents are viewed as “certain” or “ironclad” rights, the validity of which is “unquestionable”. This clearly contradicts what we observe: about half of the litigated patents in the US are invalidated (Allison & Lemley, 1998). • 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 can be invalidated by a potential infringer (Ayres and Klemperer, 1999; Shapiro, 2003; Lemley and Shapiro, 2005). • Uncertainty over patent’s validity is even stronger than many applications are granted a patent protection, even though they do not meet one or several of the statutory requirements: utility, novelty and non-obviousness or inventiveness. • Definition: patents that have a very questionable validity, i.e. that have a high probability to be invalidated by a court if challenged,are qualified as “weak patents” (ex: the patent sandwich of Mc Donald!)

4. Challenging patent’s validity • Even if challenging a patent’s validity is possible, it is nevertheless difficult. First, challenging may be directly prevented by the patent holder. Ex: in Japan, Microsoft (MS) forced its licensed OEM suppliers to pledge not to file lawsuits on the grounds that Windows infringes a patent right (Matsushima et al., 2011). • Second, the US standard required to prove invalidity (clear and convincing evidence)is very demanding for the challenger, especially for new patentable subject matters (ex: i4i vs. Microsoft). • Third, challenging has the dimension of a public good:each competitor benefits from a successful challenge initiated by one of them, since they all get freely the new technology  low individual incentives to challenge. • Main result on licensing a weak patent: licensing revenues under the shadow of patent litigation are greater than the revenues the patent holder could expect if licensing was made posterior to the patent’s assessment. (Farrell & Shapiro 2008, Encaoua & Lefouili 2009). This result raises serious diffusion & antitrust concerns.

5. I4i v. Microsoft • The Canadianfirm i4i (Infrastructures for Information Inc.) sued Microsoft (MS) for willful infringement of a patent about an XML markup language (words in boldface, centered, etc.) used in Word. At trial, MS contended that i4i’s patent was not valid because invention was already embodied in a software product sold in the market a year before the patent application. But the parties disagreed over whether the protected software in i4i’s patent was the same or was different from the earlier marketed software. The Court did not found clear and convincing evidence and condemned MS to pay $290 million. In appeal, the US Federal Circuit (CAFC) didn't accept the “preponderance of evidence” proof made by MS. • The Supreme Court (SC) rejected recently (June 2011) the contention made by MS that a burden of proof lower than the “clear and convincing evidence” should have been used. • This illustrates the difficulty for a challenger to invalidate a patent granted by a PO, especially in the US where the presumption of validity for a granted patent is very strong (§282 of the Patent Act of 1952).

6. The cost-quality trade-off • Is it possible to avoid the unjustified licensing rents associated to bad quality patents? One evident solution should be to reinforce the review process at the PO in order to improve the quality of the granted patents. • However, Lemley (2001) raises a significant cost-quality trade-off: improving the review process for each patent would be very expensive (monetary and non monetary costs such as increasing backlogs), with only small overall benefits, since many patents have no significant commercial value. • Lemley suggests that it should be rational and more efficient to maintain some “ignorance” at the PO’s level and to substitute a higher ex post opportunity to challenge a patent’s validity, to the ex ante improvedquality goal. According to Lemley, allowing the challenge of a patent’s validity is sufficient to correct the unavoidable mistakes at the PO’s level.

7. How to sell a weak patent license? • Lemley’s suggestion is not sufficient: licensing a weak patent in the shadow of patent litigation does not avoid unjustified rents (Farrell & Shapiro, 2008; Encaoua & Lefouili, 2009). But which licensing scheme is better for the weak patent’s holder? Is the nature of the best scheme for a weak patent robust to the features that affect the licensing scheme choice when the patent is ironclad? • The paper investigates these questions.

8. Paper’s contribution • Weak patent holder’s decision: charge a per-unit royalty or a fixed fee for licensing in the shadow of patent litigation?  The paper extends two strands of literature: • The (extensive) literature that focused on ironclad (perfect) patents. • The (burgeoning) literature on weak patents that focused on the inefficiencies stemming from the low private incentives to challenge patent’s validity • The paper gives a sufficient condition under which a weak patent holder prefers a per-unit royalty to a fixed fee: the strategic effect of a cost increase on the aggregate profit is positive. • This condition is satisfied regardless of whether: • The licensor is an outsider or an insider in the oligopoly, • The licensees compete à la Cournot or à la Bertrand with differentiated products (in general models of oligopoly).

9. Organization of the lecture • The model • The main result: sufficient condition under which a per-unit royalty dominates a fixed fee • Extension 1: Insider • Extension 2: Litigation cost • Application 1: Cournot with homogenous products • Application 2: Bertrand with differentiated products • One exception: Monopoly in the product market

10. The model • Patent owner P, not active in the industry, sells a license for a cost reduction technology: unit cost reduction from c+ to c = c+ - ε. Buyers are members of an oligopoly: n > 1 identical risk-neutral firms producing initially at marginal cost c+. • 3-stage game: • 1st stage: P offers a licensing contract to all firms, involving the payment of either a per-unit royalty r or a fixed fee F. • 2nd stage: The n firms, independently and simultaneously, decide whether to purchase or not a license. • 3rd stage: Competition occurs among the industry members, with the cost structure inherited from 2nd stage: c = c+ - ε for licensees and c+ for non-licensees.

11. Description of the 2nd stage • If a firm does not accept the license offer, it can challenge the patent's validity before a court. θ= probability that the patent is challenged and its validity upheld by the court θ [0,1]measures the patent’s quality. Low θ weak patent, while θ =1  ironclad patent. • If the patent is upheld by the court (θ), it becomes an ironclad right  a firm that does not purchase a license uses the old technology and produces at cost c+, while those firms who accept the offer use the new technology and pay the royalty r or the fixed fee F. • If the patent is invalidated by the court (1 - θ), all firms, including those which accepted the license offer, use freely the new technology.

12. More on the 3rd stage • The type of competition between licensees and non-licensees is not specified except to assume the existence and uniqueness of Nash equilibrium and some properties that hold under various settings. • Notations: π e (k, c) (resp. π i (k, c)) = equilibrium profit of an efficient (resp. inefficient) firm when k ≤ n efficient licensees produce at cost c < c+and n-k inefficient non-licensees produce at cost c+. qe(n, c) = equilibrium output (with n licensees) • Assumptions: A1 : π e (k, c),π i (k, c) and qe(n, c) class C2 in c over [0, c+]. • A2 : ∂ π i (k, c) / ∂ c > 0. • A3 : ∂ π e (k, c) / ∂ c < 0. • A4 : π j (k+1, c) < π j (k, c), for j = e, i. • A5: π i (k, c) < π e (k+1, c).

13. Results on the 2nd stage equilibria • Lemma 1. With a royalty r, two types of equilibria are possible: 1. all the n firms buy a license (no litigation) 2. only n-1 firms buy a license and one firm litigates. • Lemma 2. With a fixed fee F, many equilibria are possible according to the value of F (demand function of licenses): - n licenseesiffF ≤ F(θ)= θ[πe(n, c+-ε)-πi(n-1,c+-ε)] …. - k licensees(where 0 ≤ k< n) iff F k+1 ≤ F ≤ F k where F k= πe(k, c+-ε)-πi(k-1,c+-ε). • For both licensing schemes, we focus on equilibria involving all the n firms as licensees (i.e. equilibria in which litigation is deterred).

14. Optimal per-unit royalty deterring litigation • Definition: The optimal per-unit royalty that is accepted by all firms is the solution of the program max r [n r q e (n, c+- ε +r)] u.c. π e (n, c+-ε +r) ≥ θ π I (n-1, c+- ε +r)+(1-θ) π e(n, c+-ε). • Proposition 1: The optimal per-unit royalty that deters any litigation is the unique value r(θ) that binds the constraint. It satisfies the following properties, for low θ : • r(0)=0, r’(0) > 0; • if r ≤ r(θ), k=n is the only 2nd stage equilibrium; • If r > r(θ), k=n-1 and one firm challenges is the only 2nd stage equilibrium; • Pr(θ)= nr(θ)qe[n, c+- ε +r(θ)] > θ Pr(1)+(1-θ)Pr(0)=θ Pr(1), i.e. the revenues when licensing occurs prior to the patent’s assessment are higher than the revenues the patent holder would expect if the validity issue were resolved before licensing.

15. Optimal fixed fee deterring litigation • Proposition 2: The value F(θ) = θ[πe(n, c+-ε)-πi(n-1,c+-ε)] is the optimal fixed fee accepted by all the n firms. If θ is sufficiently small, it is the only second stage equilibrium • Note: Since F(θ) is proportional to θ (i.e. F(θ)= θF(1)), where F(1) is the fee corresponding to an ironclad patent, fixed fee licensing revenues per license are the same whether licensing occurs prior or posterior to the patent’s validity assessment: PF(θ)= F(θ)= θF(1)= θPF(1)+(1-θ)PF(0) = θPF(1) • Remark: Since Pr(θ) > θ Pr(1) and PF(θ) = θPF(1), there is a presumption that a per-unit royalty should be preferred to a fixed fee. We show that this presumption is correct under some condition.

16. Royaltydominates fixed fee: A SUFFICIENT CONDITION • Proposition 3: For low θ, if ∂πe (n, c) / ∂c > - q e(n, c)(1) the optimal per-unit royalty r(θ) deterring litigation provides higher licensing revenues for an outsider than the optimal fixed fee F(θ) deterring litigation. • Interpretationcondition (1):strategic effectof an increase in the marginal cost on a firms' equilibrium profits is positive:

17. Novelty of condition (1) • Condition (1) means that the positiveprice effectqe (n, c) ∂p e (n, c) / ∂c of a cost increase outweighs the negativequantity effect(pe (n, c) - c) ∂qe(n, c) / ∂c. • Literature on oligopoly focused on the overall effect(direct + strategic effect) of a cost change on profits, and showed that it could be positive (Kimmel, 1992, Février and Linnemer 2004). But what matters in our setting is the sign of the strategic effect. • Conclusion: even in a setting where a patent holder would have used the fixed fee licensing scheme if the patent was certain (θ=1), it could prefer the per-unit royalty scheme when the patent is weak (θ low), if the strategic effect of a common cost variation on the profits in the product market is positive (condition (1)).

18. Extension 1: the insider case • Suppose now that P is active in the industry and has at least two competitors (n > 2). Cost structure following the licensing stage subgame: • one firm - the patent holder P - that produces at unit cost c+ - ε • a number k < n of firms - the licensees indexed by l - that produce at a unit cost c = c + - ε + r (where c Є[c+ - ε, c+]) • and the remaining n-k-1 firms - the non-licensees indexed by n - that produce at unit cost c+ . • Optimal r(θ) deterring litigation: π l (n -1;c + - ε + r(θ)) = θ πn(n -2;c +- ε + r(θ)) + (1-θ ) πl(n -1; c +- ε) Pr (θ) = π p(n -1; c + - ε + r(θ) ) + (n -1) r (θ) ql(n -1; c + - ε + r(θ)) • Optimalfixed feeF(θ) deterring litigation: F(θ) = θ[π l(n -1; c + - ε) - πn(n -2; c + - ε )] PF (θ)= π p(n -1; c + - ε ) + (n- 1) F (θ)

19. Best licensing for an insider • Proposition 4: For an insider firm holding a weak patent, who has to choose a licensing scheme deterring litigation, the optimal per-unit royalty r(θ) provides higher revenues than the optimal fixed fee F(θ),if the strategic effect of an individual increase of the unit cost on the aggregate profit of the industry is positive (condition 2) ∂π(c,…, c)/ ∂ci > - qi (c,…, c), for i=1,…,n (2) πis the aggregate profit of the industry. • Condition (2) for the insider case generalizes condition (1) for the outsider case. • Interpretation of (2): From an equally efficient cost structure, the strategic effect of an increase in one firm’s unit cost on the aggregate profit is positive.

20. First robustness result • Under condition (2) stating that the strategic effect of an increase in one firm’s unit cost on the aggregate profit is positive, a per-unit royalty dominates a fixed fee for a weak patent, regardless of whether the patent holder is an outsider or an insider in the industry: ∂π(c,…, c)/ ∂ci > - qi (c,…, c), i=1,…,n (2)

21. Extension 2: litigation cost • Introduce a litigation cost C > 0 supported by a firm that challenges a patent’s validity before a court. The higher is C, the lower are the incentives to challenge the patent’s validity, and the higher are the licensing revenues (under both schemes) that the patent holder can extract without triggering litigation. • Proposition 5: Under condition (2), the dominance of the per-unit scheme over the fixed fee scheme remains true when the model is extended to include (small) legal costs incurred by a challenger.

22. A question • Are the assumptions made on the profit functions and the sufficient condition (2) under which a per-unit royalty dominates a fixed fee for licensing a weak patent, satisfied in standard oligopoly models? • Cournot oligopoly with homogenous goods and Bertrand oligopoly with differentiated goods, both with general demand functions, the main restrictions being those needed to ensure existence and uniqueness of a Nash equilibrium in pure strategies.

23. Cournot competition with homogenous goods. • Oligopoly composed by n > 1 firms competing à la Cournot. Cost structure: marginal costs c1,…cn. Thegame is said to be symmetric if c1=…=cn =c. • Inverse demand function P(⋅) satisfying the following assumptions: C1: P(⋅) is twice continuously differentiable and P′(⋅)<0 whenever P(⋅)>0. C2: P(0)> ci >P(Q) for Q sufficiently high, i=1,…,n C3: P′(Q)+QP′′(Q)<0 for all Q ≥ 0 with P(⋅) > 0 (ensures existence equilibrium, Novshek 1985 )

24. Cournot competition with homogenous products • Proposition 4: Under Assumptions C1-C3, • a/ There exists a unique Cournot equilibrium. Furthermore, if the game is symmetric, the equilibrium is symmetric, and the functions q*(c) and π*(c) decrease in c. • b/ Assumptions A1-A5 (or A’1-A’5 if P is insider) on the equilibrium profit functions are satisfied. • c/ Condition (2) holds: ∂π* (c,…,c) / ∂ci > - qi*(c,..,c), i=1,…,n • Conclusion: Under Cournot competition, the holder of a weak patent prefers the use of a per-unit royalty rather than a fixed fee, regardless whether the patent holder is active or not in the market. Recall that the reverse holds for an outsider holding an unquestionable patent (i.e. θ=1, Kamien, 1992).

25. A caveat on proposition 4 • Proposition 4 asserts that the profit π*(c) decreases in c when the game is symmetric. While intuitive, this result does not hold for a sufficiently convex demand function in a symmetric Cournot model. • Counter-example: n=2, c1=c2=c and P(Q)=Q-1/bwith ½<b<1. In this case P′(Q)+QP′′(Q) > 0, i.e. assumption C3 is not satisfied. Profit π(c)=[(2b-1)/c]b-1 /[bb21+b ] increases in c. Moreover,no moreuniqueness of Cournot equilibrium (two equilibria, one of which has each firm producing zero output). • By imposing assumption C3 to guarantee existence and uniqueness of Cournot equilibrium, one also obtains that the counter-intuitive result on the positive effect of a cost increase does not hold.

26. Bertrand competition with differentiated products • Industry composed by n single product firms competing à la Bertrand with marginal costs c1,…,cn and imperfect substitutes products. • Di (p₁, p₂,...,pn) = demand function for good i satisfying the following assumptions B1 - B4 : • B1: Diis twice continuously differentiable on the subspace where Di(p₁, p₂,...,pn) > 0. • B2: (i)(∂Di/∂pi)<0, (ii)(∂Di/∂pj)>0, (iii) ∑k (∂Di/(∂pk) <0, i=1,…,n: own price effect dominates cross-price effects( demand level).

27. Assumptions • B3: Di(∂²Di)/(∂pj∂pi)-(∂Di/∂pj)(∂Di/∂pi)>0,j ≠ i:the price elasticity of demand increases in any rival’s price (Amir, 2005). • B4: ∑k(∂²logDi/∂pi ∂pk))<0, i=1,..n :along the diagonal, own effects of price changes dominate cross effects, for the slope of demand. B4 guarantees the uniqueness of Bertrand equilibrium (Milgrom and Roberts, 1990, Vives, 1999). • Bertrand oligopoly model is said to be symmetric if the demand functions are symmetric and c1 =…= cn=c

28. Bertrand competition with differentiated products • Proposition 5: Under Assumptions B1-B4, the Bertrand game is log-supermodular and has a unique equilibrium which is symmetric if the oligopoly game is symmetric. • Firm i’s equilibrium profit π*i is decreasing in ci and, if Bertrand oligopoly is symmetric, the per-firm equilibrium profit π* is decreasing in c. • The assumptions on the equilibrium profit functions are satisfied. - In symmetric Bertrand oligopoly, two properties hold: • cp*(c) is increasing, with a slope ≤ 1 • Condition (2) is satisfied: ∂π*(c) / ∂ci > - qi*(c), i=1,…,n

29. Sum-up • Under Bertrand competition with differentiated products, the per-unit royalty scheme provides higher licensing revenues than the fixed fee scheme for a weak patent, whereas for an unquestionable patent, this result holds only when the products are close substitutes or when the size of the cost-reduction is small (Kamien and Tauman, 1986, Kamien, Oren and Tauman, 1987, Muto,1993). • Thus, the per-unit royalty scheme provides higher licensing revenues than the fixed fee scheme for a weak patent, regardless whether the downward competition is of the Cournot type or of the Bertrand type with differentiated products.

30. One exception: the monopoly case • Suppose now there exists only one potential licensee (n=1) • Proposition 6: If dqM (c)/dc < 0 (3), the weak patent holder obtains higher licensing revenues from a fixed fee than from a per-unit royalty. • Application: Under usual assumptions on the demand function leading to the existence and uniqueness of the monopoly solution, condition (3) is fulfilled. Therefore, whenthe seller of a weak patent isconfronted toonly one potential licensee, charging a fixed fee rather than a per-unit royalty is better. The main reason is that the incentive to challenge a weak patent is high in the monopoly case .

31. Conclusion • We have obtained a clear-cut result on the comparison of a weak patent holder's profits under two schemes: the per-unit licensing scheme dominates the fixed fee licensing scheme for a weak patent and the weak patent holder is able to capture some unjustified rents by using the former. This result is independent of the type of downstream competition, of the degree of product differentiation and whether the patent holder is active or not in the downstream market: • Recall that varying any of these features can overturn the outcome of the comparison when ironclad patents are considered.

32. A proposal • Facilitating the challenge of a patent’s validity improves certainly the effective functioning of the patent system. But, it is not sufficient. Even if the stringent standard of proof to invalidate a patent is relaxed, licensing a weak patent through a per unit royalty under the shadow of patent litigation still involves unjustified revenues. • A promising avenue to reduce the harmful impact of very questionable patents should be to replace the ex post reexamination procedure by an ex ante opposition procedure in which each third party could justify its opposition to the patent grant. Licensing would occur after such an opposition. The existence of such procedure in Europe may explain why the EPO has a higher reputation than theUSPTO in terms of quality of the granted patents.