Per-unit royalty vs fixed fee: the case of weak patents Rabah Amir, University of Arizona David Encaoua, Paris School of Economics, University Paris I Yassine Lefouili, Toulouse School of Economics

1
Per-unit royalty vs fixed fee the case of weak
patents Rabah Amir, University of Arizona
David Encaoua, Paris School of Economics,
University Paris I Yassine Lefouili,
Toulouse School of Economics

• Conference in honor of Claude dAspremont and
  Jean-François Mertens
• CORE, 23-25 June 2011

2
How to sell a patent license?

• Since Arrows 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
  (licensors 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 patents 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 patents validity

• Even if challenging a patents 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 patents assessment. (Farrell Shapiro
  2008, Encaoua Lefouili 2009). This result
  raises serious diffusion antitrust concerns.
  

5
I4i v. Microsoft

• The Canadian firm 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
  i4is 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 i4is 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 POs level and to substitute a higher ex post
  opportunity to challenge a patents validity, to
  the ex ante improved quality goal. According to
  Lemley, allowing the challenge of a patents
  validity is sufficient to correct the
  unavoidable mistakes at the POs level.

7
How to sell a weak patent license?

• Lemleys 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
  patents 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
Papers contribution

• Weak patent holders 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 patents
  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 - e. Buyers are
  members of an oligopoly n gt 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 - e 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. ? probability that the patent is
  challenged and its validity upheld by the court ?
  ? ?0,1 measures the patents quality. Low ? ?
  weak patent, while ? 1 ? ironclad patent.
• If the patent is upheld by the court (?), 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 -
  ?), 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 p e (k, c) (resp. p i (k, c))
  equilibrium profit of an efficient (resp.
  inefficient) firm when k n efficient licensees
  produce at cost c lt c and n-k inefficient
  non-licensees produce at cost c.
• qe(n, c) equilibrium output (with n licensees)
• Assumptions A1 p e (k, c), p i (k, c) and
  qe(n, c) class C2 in c over 0, c.
• A2 ? p i (k, c) / ? c gt 0.
• A3 ? p e (k, c) / ? c lt 0.
• A4 p j (k1, c) lt p j (k, c), for j e, i.
• A5 p i (k, c) lt p e (k1, 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 licensees iff F F(?) ?pe(n,
  c-e)-pi(n-1,c-e)
• .
• - k licensees (where 0 k lt n) iff F k1 F
  F k where
• F k pe(k, c-e)-pi(k-1,c-e).
• 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- e r)
• u.c. p e (n, c-e r) ? p I (n-1, c- e
  r)(1-?) p e(n, c-e).
• Proposition 1 The optimal per-unit royalty that
  deters any litigation is the unique value r(?)
  that binds the constraint. It satisfies the
  following properties, for low ?
• r(0)0, r(0) gt 0
• if r r(?), kn is the only 2nd stage
  equilibrium
• If r gt r(?), kn-1 and one firm challenges is the
  only 2nd stage equilibrium
• Pr(?) nr(?)qen, c- e r(?) gt ?
  Pr(1)(1-?)Pr(0) ? Pr(1), i.e. the revenues
  when licensing occurs prior to the patents
  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(?) ?pe(n,
  c-e)-pi(n-1,c-e) is the optimal fixed fee
  accepted by all the n firms. If ? is sufficiently
  small, it is the only second stage equilibrium
• Note Since F(?) is proportional to ? (i.e. F(?)
  ?F(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 patents validity
  assessment
• PF(?) F(?) ?F(1) ?PF(1)(1-?)PF(0) ?PF(1)
• Remark Since Pr(?) gt ? Pr(1) and PF(?)
  ?PF(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
Royalty dominates fixed fee A SUFFICIENT
CONDITION

• Proposition 3 For low ?, if ?p e (n, c) / ?c gt
  - q e(n, c) (1)
• the optimal per-unit royalty r(?) deterring
  litigation provides higher licensing revenues for
  an outsider than the optimal fixed fee F(?)
  deterring litigation.
• Interpretation condition (1) strategic effect
  of an increase in the marginal cost on a firms'
  equilibrium profits is positive

17
Novelty of condition (1)

• Condition (1) means that the positive price
  effect qe (n, c) ?p e (n, c) / ?c of a
  cost increase outweighs the negative quantity
  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 (?1), it could
  prefer the per-unit royalty scheme when the
  patent is weak (? 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 gt 2). Cost
  structure following the licensing stage subgame
• one firm - the patent holder P - that produces at
  unit cost c - e
• a number k lt n of firms - the licensees indexed
  by l - that produce at a unit cost c c - e
  r (where c ?c - e, c)
• and the remaining n-k-1 firms - the non-licensees
  indexed by n - that produce at unit cost c .
• Optimal r(?) deterring litigation
• p l (n -1c - e r(?)) ? p n(n -2c - e
  r(?)) (1-? ) p l(n -1 c - e)
• Pr (?) p p(n -1 c - e r(?) ) (n -1) r
  (?) ql(n -1 c - e r(?))
• Optimal fixed fee F(?) deterring litigation
• F(?) ?p l(n -1 c - e) - p n(n -2 c - e
  )
• PF (?) p p(n -1 c - e ) (n- 1) F (?)

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(?) provides higher revenues than the
  optimal fixed fee F(?), if the strategic effect
  of an individual increase of the unit cost on
  the aggregate profit of the industry is positive
  (condition 2)
• ? p (c,, c)/ ?ci gt - qi (c,, c), for
  i1,,n (2)
• p is 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 firms 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 firms 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
• ? p (c,, c)/ ?ci gt - qi (c,, c), i1,,n
  (2)

21
Extension 2 litigation cost

• Introduce a litigation cost C gt 0 supported by a
  firm that challenges a patents validity before a
  court. The higher is C, the lower are the
  incentives to challenge the patents 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 gt 1 firms competing à la
  Cournot. Cost structure marginal costs c1,cn.
  The game is said to be symmetric if c1cn c.
• Inverse demand function P() satisfying the
  following assumptions
• C1 P() is twice continuously differentiable
  and P'()lt0 whenever P()gt0.
• C2 P(0)gt ci gtP(Q) for Q sufficiently high,
  i1,,n
• C3 P'(Q)QP''(Q)lt0 for all Q 0 with P() gt 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 p(c) decrease in c.
• b/ Assumptions A1-A5 (or A1-A5 if P is
  insider) on the equilibrium profit functions are
  satisfied.
• c/ Condition (2) holds ?p (c,,c) / ?ci gt -
  qi(c,..,c), i1,,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.
  ?1, Kamien, 1992).

25
A caveat on proposition 4

• Proposition 4 asserts that the profit p(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 n2, c1c2c and P(Q)Q-1/b with
  ½ltblt1. In this case P'(Q)QP''(Q) gt 0, i.e.
  assumption C3 is not satisfied. Profit
  p(c)(2b-1)/cb-1 /bb 21b increases in c.
  Moreover, no more uniqueness 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 Di is twice continuously differentiable on
  the subspace where Di(p1, p2,...,pn) gt 0.
• B2 (i)(?Di/?pi)lt0, (ii)(?Di/?pj )gt0,
• (iii) ?k (?Di/(?pk) lt0, i1,,n own price
  effect dominates cross-price effects( demand
  level).

27
Assumptions

• B3 Di(?²Di)/(?pj?pi)-(?Di/?pj)(?Di/?pi)gt0, j ? i
  the price elasticity of demand increases in any
  rivals price (Amir, 2005).
• B4 ?k(?²logDi/?pi ?pk))lt0, i1,..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
  cnc

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 is equilibrium profit pi is decreasing in
  ci and, if Bertrand oligopoly is symmetric, the
  per-firm equilibrium profit p 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 ?p (c) / ?ci gt
  - qi(c), i1,,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 (n1)
• Proposition 6 If dqM (c)/dc lt 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, when the seller of
  a weak patent is confronted to only 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 patents 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 the USPTO in terms of quality of
  the granted patents.