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 d’Aspremont
and Jean-François Mertens
CORE, 23-25 June 2011
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),2
whereas the comparison is reversed when the
licensor is an insider (Sen & Tauman, 2007)
•
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
3
court if challenged, are qualified as “weak patents” (ex:
the patent sandwich of Mc Donald!)
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 raises4
serious diffusion & antitrust concerns.
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 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
5
presumption of validity for a granted patent is very strong (§282
of the Patent Act of 1952).
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 improved
quality goal. According to Lemley, allowing the challenge
of a patent’s validity is sufficient to correct the unavoidable
mistakes at the PO’s level.
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.

7
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:
1.
2.
•
•
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).
•
8
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

9
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.
10
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.
11
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).
12
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(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).
13
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.
14
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)
o 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.
15

ROYALTY DOMINATES FIXED FEE: A SUFFICIENT
CONDITION
•
•
Proposition 3: For low θ, if ∂π e (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.
Interpretation condition (1): strategic effect of an
increase in the marginal cost on a firms'
equilibrium profits is positive:

pe

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
e
e
e
e

n,c

q

n,c


q

n,c


n,c

p
n,c

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


c

c

c

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direct effect
strategic effect
16
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 (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
17
profits in the product market is positive (condition (1)).
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 π n(n -2;c +- ε + r(θ)) + (1-θ ) π l(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))

Optimal fixed fee F(θ) deterring litigation:
F(θ) = θ[π l(n -1; c + - ε) - π n(n -2; c + - ε )]
PF (θ)= π p(n -1; c + - ε ) + (n- 1) F (θ)
18
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)/ ∂c > i
qi (c,…, c), for i=1,…,n (2)
π 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
firm’s unit cost on the aggregate profit is positive.
19
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)
20
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.
21
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.

22
COURNOT COMPETITION WITH
HOMOGENOUS GOODS.
 Oligopoly
composed by n > 1 firms competing à la
Cournot. Cost structure: marginal costs c1,…cn.
The game 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 )
23
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 perunit 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
24
outsider holding an unquestionable patent (i.e.
θ=1, Kamien, 1992).
-
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/b with
½<b<1. In this case P′(Q)+QP′′(Q) > 0, i.e.
assumption C3 is not satisfied. Profit π(c)=[(2b1)/c]b-1 /[bb 21+b ] 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.
25
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) > 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). 26
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
27
symmetric and c1 =…= cn=c
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) / ∂c > - q *(c), i=1,…,n
i
i
28
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 29
type with differentiated products.

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, 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 to30
challenge a weak patent is high in the monopoly
case .
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.

31
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 32
higher reputation than the USPTO in terms of quality
of the granted patents.
