Pay-for-Delay with Settlement Externalities
†
∗
‡
Emil Palikot and Matias Pietola
March 27, 2017
Abstract
This paper studies settlements between an incumbent patent holder and multiple potential
entrants to the market in the shadow of patent litigation. We show that there exists litigation in
equilibrium for patents with intermediate strength, whereas suciently weak or strong patents are
not taken to court. The incumbent pays to delay some entrants while accommodates the others,
either through licensing or litigation. This identies a new source of sequentiality in equilibrium
entry to markets governed by patents. Settlement externalities between the entrants are the
driving force behind our results: when one more entrant is delayed from entering the market,
there is less competition and the litigation threat from the other entrants is increased. Our results
bring new important insights for the hotly debated topic of pay-for-delay agreements, witnessed in
the pharmaceutical industry and sanctioned by antitrust authorities both in the US and Europe.
1
Introduction
Patents are probabilistic rights: while they grant a legal monopoly to the incumbent patent holder,
potential entrants can engage in a costly and lengthy litigation to challenge patent validity already
before the patent is set to expire (Lemley and Shapiro, 2005; Shapiro, 2003). In the shadow of such
litigation, incumbents have resorted to settlement agreements with potential entrants, either to delay
or advance entry to the market.
Pay-for-delay agreements, where the incumbent makes a reverse
payment to compensate an entrant for her entry delay, have been subject to both academic and antitrust
∗ We are grateful to Yassine Lefouili, Patrick Rey, Marc Ivaldi, Jorge Padilla, Bruno Jullien, Carl Shapiro for their
valuable comments on various stages of the paper.
† Ph.D candidate at Toulouse School of Economics, [email protected]
‡ Ph.D candidate at Toulouse School of Economics, www.pietola.com, [email protected]. Matias Pietola
gratefully acknowledges the nancial support from the Yrjö Jahnsson Foundation and the Finnish Cultural Foundation.
1
scrutiny, especially in the pharmaceutical industry.
1 But licensing agreements, where the incumbent
provides a license for early entry, have received little attention in this regard, although both types
of settlement deals appear in practice and often simultaneously with dierent entrants. The question
why incumbents discriminate against similar entrants has not been answered in the current economic
literature, to our knowledge. This paper aims to ll the gap.
We show that, facing a credible litigation threat from more than one entrant, the incumbent
implements a divide and conquer equilibrium strategy where it pays to delay some entrants, while
accommodates the others, although all entrants are completely identical in our model. This identies a
new source of sequentiality in equilibrium entry to markets governed by patents. Entry accommodation
happens either through licensing or litigation, depending on total litigation costs and patent strength.
There is licensing when the patent is either suciently weak or strong, whereas intermediate patents
are litigated. Litigation is more likely to occur when total litigation costs are low. Finally, we show
that conditional settlement agreements, where a delayed entrant is allowed to enter early in the event
of patent invalidation, always lead to full entry delay and should therefore be banned as restrictions
to competition.
To better understand the economic logic behind patent settlements, it is necessary to allow for
multiple entry to the market.
This is because a settlement agreement between the incumbent and
one entrant imposes an externality to the other entrants, a strategic feature not present when the
incumbent has only a single entrant to deal with: under single entry the incumbent can use a reverse
payment to share industry prots with the entrant and full entry delay maximizes these joint prots.
However, under multiple entry, the litigation threat from other entrants becomes stronger when one
more entrant is delayed. The reason is that an entrant faces less competition after successful litigation
against the incumbent, and hence obtains higher prots after a successful entry.
secure full entry delay, the incumbent must compensate
all
Consequently, to
entrants with expected duopoly prots.
This may render a full entry delay strategy too costly for the incumbent, so that she accommodates
entry to the market. Depending on total litigation costs and patent strength, such accommodation
happens either through licensing or litigation, as our results indicate.
1 On
17 June 2013, the US Supreme Court gave its decision in the FTC v. Actavis case and ruled that reverse payment
settlements should be subject to a rule of reason analysis instead of being per se illegal or legal. In the European
Union, the European Commission has been monitoring reverse payment settlements based on its sector inquiry into the
pharmaceutical industry; the First Report on monitoring was published in July 2010. According to the Commission, the
aim has been to identify settlements that delay generic entry to pharmaceutical markets, to the detriment of consumers
in the single market.
2
Relation to the Literature
This paper relates to two strands of economic literature: licensing and
litigating uncertain patents (Farrell and Shapiro, 2008; Lemley and Shapiro, 2005; Kamien and Tauman, 1986; Katz and Shapiro, 1987) and pay-for-delay settlement agreements (Shapiro, 2003; Meunier
and Padilla, 2015; Elhauge and Krueger, 2012; Edlin et al., 2015; Manganelli, 2014; Gratz, 2012). The
main contribution is to connect these areas of economic research.
To our knowledge, pay-for-delay
agreements and licensing have not been captured in the same economic model before. Yet assuming
that dierent agreements are made independently from each other obfuscates an important economic
mechanism, triggered by settlement externalities between the entrants. Therefore, we believe, studying dierent settlement agreements together is essential for building an economic theory for patent
settlements.
The notion of a probabilistic patent as established in the economic literature is central to our
analysis. A patent does not grant an absolute right to exclude others from practicing the invention
at hand (Lemley and Shapiro, 2005).
Patents can be, and in reality frequently are, challenged in
2
court. Therefore, a patent gives to its owner a right to exclude others only with some probability.
Yet an important aspect of patent disputes is that individual rms have sub-optimal incentives to
challenge the patents asserted against them (Lemley and Shapiro, 2005).
This has a resemblance
with private incentives in the public good provision problem: rivals to the litigator will benet from
a judgment that declares the patent invalid. This feature of free-riding is also present in our model.
Insucient incentives to challenge patent validity may lead to an anti-competitive outcome, unless
there are antitrust rules limiting the scope of patent settlements. At the limit, when patent strength
approaches zero, a weak patent can be used as a leaf to cover an agreement no to compete (Lemley
and Shapiro, 2005).
The canonical pay-for-delay model is introduced by Shapiro (2003). He considers a setup with a
single entrant who may challenge an incumbent patent holder in a court of law.
The parties have
the opportunity to settle their dispute instead of going to court, by agreeing on a reverse payment
from the incumbent to the entrant as well as an entry date for the entrant.
In this case parties
will agree on extending the monopoly period and divide the resulting prots. Our model builds on
2 Resulting
uncertainty of the property rights is an unavoidable feature of the patent system. The fact that patents
are invalidated neither is an accident nor a mistake; it is impossible to perfectly review hundreds of thousands of patents
applications. Inventors le over 350.000 patent applications a year with the U.S. Patent and Trademark Oce, and spend
over USD 5 billion a year just on the process of obtaining those patents (Lemley, 2000). Economic literature, often,
focuses on litigation, despite the fact that a great majority of patents are never asserted in litigation. Only 1.5 percent
of all patents are ever litigated, and only 0.1 percent are litigated to trail (Lemley, 2000; Lanjouw and Schankerman,
2001). The quality of the patent application review system has to be weighted against the associated costs. The patent
system is usually skewed towards granting patents, based on a brief process (Lemley and Shapiro, 2005).
3
this model with the novelty of allowing for multiple entry to the market. This important extension
reveals the connection between pay-for-delay agreements, licensing and litigation, and allows us to
capture a source of sequentiality commonly observed in market entry. Previous work assumes only one
entrant can credibly threaten the patent holder with litigation, while entrants with weaker incentives
to challenge the patent holder will free ride on the litigation eorts taken by the more aggressive one
(Meunier and Padilla, 2015). Instead of incorporating this economic logic to our model exogenously,
we show that in equilibrium at most one entrant follows through with litigation. Others will free ride,
even if all entrants are symmetric and have the ability to litigate. Thus we show that the assumption
taken in some models is actually an outcome of strategic interaction between the entrants.
Shapiro (2003) proposes a general rule for evaluating patent settlements in antitrust policy: consumers should not be worse o from the settlement as compared to no settlement and hence litigation.
This welfare analysis simplies to comparing the duration of exclusion resulting from the settlement
(entry date agreed) to the expected entry date when there is no settlement and entry may occur
through patent invalidation only. When the reverse payment exceeds the litigation cost incurred by
the incumbent, the settlement exclusion period exceeds the expected exclusion period from litigation
(Shapiro, 2003). Elhauge and Krueger (2012) argue that all pay-for-delay settlements with a reverse
payment higher than litigation costs should be per se illegal, independently from patent strength. Our
results have implications for this policy debate: focusing solely on pay-for-delay agreements is not
sucient, but the incentives to license and litigate should also be taken into account. Banning reverse
payments may have the side-eect of removing the incentives for early licensing.
The paper is organized as follows. The next section is devoted to case studies in the pharmaceutical
industry, which shed light on the environment of patent settlements in practice. Section 3 introduces
our baseline model for two entrants, which is then generalized in Section 4 to allow for more than two
entrants. Section 5 considers dierent policy implications and extensions of our model.
2
Pay-for-Delay in the Pharmaceutical Industry
Contracts between an incumbent patent holder and potential entrants to the market is a case of general
interest. One of the applications, and the main motivation for our work, stems from the pharmaceutical
industry. In this sector, branded pharmaceutical companies hold patents on original drugs they have
invented, but face competitive pressure from generic producers of bio-equivalent medicines. Generic
entrants can simultaneously contest the brand once the primary patent, protecting the main chemical
4
compound, has expired. Since then, usually several generic producers race to the market neck-to-neck.
Yet often, despite the expiry of the main patent, the legal situation is unclear: the brand has applied for
a secondary patent protection and sometimes holds patents on alternative ways of manufacturing the
medicine. Therefore, even if the chemical compound itself is no longer protected, a generic entrant may
still infringe some other patents owned by the brand. A natural way to clarify the legal environment is
to resolve uncertainty in a court of law. However, as such litigation is both lengthy and costly, parties
tend to enter into settlement agreements. These agreements settle the patent dispute, provide an entry
date for the generic producer and involve money transfers.
Cases
Pay-for-delay settlement agreements have raised a major attention both in the United States
and in Europe. These are agreements with a long settlement exclusion period and a reverse payment
from the branded company to the generic entrant, in return to her entry delay. The European Commission has deemed reverse payments as restrictions to competition by object, pursuant to the Article
101 of the Treaty on the Functioning of the European Union. The Commission imposed large nes to
both branded and generic rms in its landmark decisions, the Lundbeck decision in June 2013 and the
3
Perindopril decision in July 2014.
The Lundbeck case considers a number of agreements between a Danish pharmaceutical
company Lundbeck and several generic drugs producers.
In the 1970's and the 1980's
Lundbeck developed an antidepressant drug citalopram that started being marketed in the
1990's. The medicine was largely successful and constituted a major product for Lundbeck;
for example 80-90% of the company's revenues in 2002.
At the time of the settlements,
years 2002 and 2003, patents related to the chemical compound and the original process
had expired. In principle, the market was free for generic producers to enter. However,
Lundbeck still had a number of patents related to more ecient or alternative ways of
manufacturing the drug. Lundbeck implemented a so-called generic strategy that involved
dierent kinds of agreements with several entrants. For example, in UK it allowed one rm
to enter the market, but oered a reverse payment to another one. In Iceland, Lundbeck
allowed a market entry without litigation.
Overall, the generic strategy was a mixture
of reverse payments, takeovers, licensing, accommodation and even introduction of own
authorized generic producers.
3 See
also the Fentanyl decision in December 2013.
5
The Perindopril case involves a French pharmaceutical manufacturer Servier and generic
producers of perindopril, a medicine for treating high blood pressure developed by Servier
in the 1980's. Perindopril became Servier's most successful product with annual global sales
in years 2006 and 2007 exceeding USD 1 billion with average operating margins beyond
90%. Generic entry started to impose a credible threat to Servier once the patent governing
the main compound expired in May 2003. Anticipating this, Servier had started to design
and implement a generic strategy from the late 1990's. This strategy included acquiring
new patents and resulted in ve settlement agreements with dierent generic producers,
between the years 2005 and 2007. Four of these agreements were considered pay-for-delay
settlements, while the fth one was a licensing deal.
Servier considered litigation and
licensing as alternative tools in its strategy.
In the US, the Drug Price Competition and Patent Term Restoration Act of 1984 (the so-called
Hatch-Waxman Act) shapes the regulatory approval of generic drugs. The aim of this legislation is
to promote entry of generics by guaranteeing the rst one of them a duopolistic position.
When a
generic producer of a bio-equivalent drug les an Abbreviated New Drug Application (ANDA) to the
Food and Drug Administration (FDA), the Hatch-Waxman Act requires to declare a relationship to
a patent mentioned in the Orange Book, a list of approved drug products together with a catalog of
patents related to each of them. If a generic producers states that the relevant patents are no longer
valid, or that it is not infringing, a certication is granted. The Hatch-Waxman Act provides 180 days
of exclusivity for a generic producer who makes such a certication: no other generic producer can
obtain approval from the FDA during this time. The aim of the exclusivity is to promote litigation
of weak patents, but many patent disputes between pharmaceutical companies have been settled with
reverse payments.
The authorities in the US had many views on how to deal with reverse payments, until the US
Supreme Court gave its decision in the FTC v. Actavis Inc. case in June 2013. The Department of
Justice and the Federal Trade Commission presumed reverse payments illegal, but rebuttable.
The
courts of lower instances adopted dierent approaches, including per se illegality, a scope of the patent
test and a quick rule of reason analysis, depending on the court. The Supreme Court rejected all
these legal standards, and held that the reverse payments are subject to a complete rule of reason
analysis.
The FTC vs. Actavis case was brought to the US Supreme Court by the Federal Trade
6
Commission (FTC) in 2013.
The case considers a deal made between a Belgian phar-
maceutical company Solvay Pharmaceuticals and a generic producer Actavis, Inc. Solvay
was granted a new patent for AndroGel in 2003. Later on, Actavis led an ANDA to the
FDA and stated that Solvay's new patent was invalid and the generic version produced by
Actavis did not infringe upon the AndroGel patent. Solvay settled the case with Actavis:
the settlement agreement included a reverse payment from Solvay to Actavis in return for
an exclusion period during which Actavis agreed to stay out of the market. The agreed
entry date was 65 months before the AndroGel patent expired. The FTC considered the
arrangement between Solvay and Actavis as an antitrust violation and brought a lawsuit
against them. The District Court and the appellate court, the Eleventh Circuit, dismissed
the case. The Eleventh Circuit applied the scope of the patent test to conclude that the
reverse payment settlement was not anti-competitive, since the settlement exclusion period
did not exceed the scope of the AndroGel patent. According to the Court, the parties were
not obliged to continue litigation for the sole purpose of avoiding antitrust liability. The US
Supreme Court rejected this analysis and held that it is not sucient to base the legal analysis on patent law policy and that the antitrust question must be addressed. The Supreme
Court argued that: (1) FTC's complaint could not have been dismissed without analyzing
the potential justications for such decision; (2) the patentee is likely to have enough power
to implement antitrust harm in practice; (3) the antitrust action is likely to prove more
feasible administratively than the Eleventh Circuit believed: a large, unexplained payment
from the patentee to the generic producer can provide a workable surrogate for a patent's
weakness; (4) the parties could have made another type of settlement agreement, by allowing the generic producer to enter the market before the patent expires without a need for
a reverse payment.
Main Lessons
The basic logic behind reverse payments is the following. On one hand, the brand
would like to postpone generic entry until patent expiration, as generic entry leads to two major
4 On the other hand,
changes in the market: prices decrease and the brand loses its market share.
generic entrants also care about the settlement exclusion periods, because priority to enter the market
results in higher prots than late entry under established competition.
Thus, to provide the right
incentives to settle with late entry, the brand must compensate the generic producer, typically with
4 See
paragraphs 3 and 2133 of the Perindopril decision.
7
money. Otherwise the generic producer takes its chances in court and enters early in the event of patent
invalidation. The odds in court have been quite good for the generic producers. The generic producers
have won 62% of all patent litigation cases that resulted in a ruling, according to the Pharmaceutical
Sector Inquiry conducted by the European Commission between the years 2000 and 2007.
the term weak patent has even entered the business jargon.
5 In fact,
6
But even a weak patent can be useful for the incumbent brand, due to free-riding between the
generic companies in their litigation eorts.
Success in trial opens the market for everybody, not
merely for the generic producer who took litigation eort and incurred, often a signicant, litigation
cost. The generic entrants have expressed their concern to win the battle, but lose the war due to
follow-up entry to the market.
7 The incentives to settle litigation are pronounced when other generic
producers lie in wait in the shadow of the litigation. All in all, we can identify three strategies available
to generic producers:
8
1. Actively seek settlement with the patent holder;
2. Continue litigation and seek to launch at earliest possibility;
3. Pull litigation and wait for the market to open and ensure readiness for launch.
Several other observations can be made from the case studies in the pharmaceutical industry. Firstly,
the entry game starts after the expiry of a certain patent and this date is common knowledge. Secondly,
the entrants arrive to the market simultaneously and their subsequent sequential entry is an outcome
of an interplay between the patent holder and the entrants. Some entrants may be accommodated,
while the others are delayed.
Such divide and conquer strategies are achieved through a myriad of
decisions: pay-for-delay agreements, licensing deals, litigation and take-overs.
3
A Model with Two Entrants
We consider the following environment as our baseline model. There is one incumbent patent holder
and two symmetric entrants to the market. The incumbent has a legal monopoly unless a court of
law declares the patent invalid. The patent is valid with probability
5 For
θ ∈ [0, 1]
(patent strength) and
process patents, the corresponding gure was 74%. For litigations initiated by branded companies, 32% of the
judgments found the invoked patent to be infringed, and in an additional 12% of the cases the court annulled the invoked
patent. See Pharmeceutical Sector Inquiry by the European Commission.
6 See paragraph 280 of Lundbeck decision
7 See paragraph 493 of the Perindopril decision by the European Commission.
8 See paragraph 1020 of the Perindopril decision.
8
invalid otherwise; patent strength is common knowledge. The entrants can challenge the incumbent
at date zero and the patent expires at date one.
Each entrant chooses from three strategies: litigate over patent validity, wait for the market to open
or settle with the incumbent. An entrant who litigates incurs a litigation cost
ruling at date
l ∈ [0, 1);
the incumbent has a litigation cost
C ≥ 0.
c ≥ 0 and obtains a court
Litigation outcomes are assumed
perfectly correlated: the court declares the patent invalid with probability
1−θ
if at least one entrant
litigates. An entrant who waits saves on the litigation cost and is free to enter the market if the other
entrant litigates and wins in court. An entrant who settles makes a payment
and agrees on a future entry date
t ∈ [0, 1].
p∈R
to the incumbent
A settlement agreement includes a non-challenge clause
9
and remains in force even if the other entrant litigates and wins in court (agreements must be kept).
The following game formalizes the interaction between the incumbent and the entrants. The incumbent moves rst and makes a public take-it-or-leave-it settlement oer
(p, t) to each entrant.
Then,
after observing these oers, the entrants simultaneously but independently choose to litigate, wait or
settle.
The equilibrium concept is subgame perfect Nash equilibrium in pure strategies and we use
backwards induction to solve the game. The game is played at date zero and time runs continuously
afterwards.
Competition is modeled in reduced form, with instantaneous prot functions
and
π (·)
prot.
for an entrant.
Π (·) for the incumbent
These are mappings from the number of rms in the market to rm-level
The individual prots, as well as the industry prot, are assumed weakly decreasing in the
number of rms in the market.
Payos from Litigate, Wait or Settle
Each rm has an objective to maximize her own expected
payo from date zero until patent expiration. The settlement payo of an entrant, given the strategy
taken by the other entrant, writes
S = −p + (1 − t) π (2) − [π (2) − π (3)]




(1 − max {t, l}) (1 − θ)




0






(1 − max {t, x})
By accepting the settlement oer from the incumbent, the entrant pays
if litigate
(1)
if wait
if settle with date
p
x
and enters the market at
date t. If the other entrant waits, the entrant obtains the duopoly prot between her date of entry and
9 We
consider conditional settlement terms as one extension of our model below.
9
the patent expiration date. If the other entrant litigates, there is a triopoly after the expected entry
date of litigation. If the other entrant settles, there is a triopoly after the latest settlement entry date.
The entrant has a best response to settle if and only if the settlement payo exceeds the maximum of
the litigation and waiting payos:
S ≥ max {L, W } ≥ 0
where
W =



(1 − l) (1 − θ) π (3)
if litigate


0
if wait or settle
(2)
(3)
is the waiting payo of the entrant, which is at least zero, and
L = −c + (1 − θ) (1 − l) π (3) + (1 − θ) [π (2) − π (3)]



0
if litigate or wait


max {x − l, 0}
if settle with date
(4)
x
is the litigation payo of the entrant. Observe that litigate is never a best response to litigate, because
waiting allows to save on the litigation cost. By litigating, the entrant pays a litigation cost
c and enters
the market at the expected entry date of litigation. She obtains the triopoly prot when competing
against both the incumbent and the other entrant, but receives the duopoly prot if the other entrant
is delayed from entering the market due to a settlement agreement with the incumbent. A settlement
with the other entrant imposes a positive externality (a settlement externality) to the entrant, whose
litigation and settlement payos are higher when the other entrant is delayed longer.
Credibility of the Litigation Threat
Intuitively, there is no credible litigation threat for a su-
ciently strong patent: the cost of litigation is too high compared to the probability patent invalidation.
Since settlement externalities are positive, the litigation payo of an entrant satises the inequality
L ≥ −c + (1 − l) (1 − θ) π (3)
(5)
where the lower bound is achieved when the other entrant does not settle (litigates or waits) or settles
with an entry date weakly below l. The sign of the lower bound on litigation payo is crucial for the
10
outcome of the game. We shall say that the litigation threat is credible for a suciently weak patent,
θ ≤ θ̂ ≡ 1 −
c
(1 − l) π (3)
(6)
in which case the lower bound on litigation payo is positive. For a suciently strong patent,
there is no credible litigation threat from the entrants. To see this, recall that
condition for a settlement.
and an entry date
because
t = 1,
S = −p < 0 ≤ W
S ≥0
θ > θ̂,
is a necessary
By making a settlement oer with a strictly positive payment
p > 0
the incumbent ensures that an entrant always rejects the settlement oer,
(settle is dominated by wait). If the incumbent makes such oers to both
entrants, the entrants play a litigation game (litigate or wait) with the lowest possible litigation payo,
which is strictly negative when
θ > θ̂.
Thus, there exists a unique subgame equilibrium in which the
entrants have a dominant strategy to wait until patent expiration. The following result summarizes
this reasoning.
Proposition 1. Both entrants wait in equilibrium if and only if θ > θ̂.
Proof.
and
Suppose
θ > θ̂ and that the incumbent oers t = 1 and p > 0 to both entrants.
W ≥ 0 ≥ L
Then
W ≥0>S
no matter what strategy the other entrant takes, so each entrant has a dominant
strategy to wait. The incumbent gets the highest possible monopoly payo
Suppose (wait,wait) is played in equilibrium. Then
θ > θ̂
I = Π (1)
must hold, because
θ ≤ θ̂
in equilibrium.
implies
L≥0
no
matter what strategy the other entrant takes. Thus wait is never a best response to wait.
Proposition 1 shows that for a suciently strong patent,
from the entrants.
θ > θ̂, there is no credible litigation threat
By making unacceptable settlement oers, the incumbent ensures that there is
never litigation, monopolizes the market and obtains the monopoly payo
The entrants receive
W =0
I = Π (1)
with zero cost.
from waiting until patent expiration.
The game becomes interesting when the patent is suciently weak,
θ ≤ θ̂
(litigation is not too
costly), which is assumed for now on. Then the lower bound on the litigation payo is positive. If
both settlement oers are rejected, the entrants play a chicken game in going to court: the litigator
is worse o than the free-rider who waits for the market to open and saves on litigation eort. Yet,
there is no incentive for the litigator to wait, because the patent is never challenged if both entrants
wait. Indeed, the incumbent must settle with both entrants to avoid litigation, because wait is never
a best response to wait or settle. Furthermore, as litigate is never a best response to litigate, there
11
are essentially three potential equilibria of the game under a credible litigation threat: (litigate,wait),
(litigate,settle) and (settle,settle). Table 1 summarizes these strategies.
Settle
Litigate
×
×
×
Settle
Litigate
Wait
×
×
Wait
Table 1: Potential equilibria under a credible litigation threat
Licensing and Pay-for-Delay
Let us now consider patent settlements in more detail.
At rst
sight, the incumbent can choose from a large set of dierent settlement strategies: a settlement oer
may specify any date of entry
t ∈ [0, 1]
and any transfer of value
p∈R
(positive or negative) between
the incumbent and the entrant. Importantly, the following lemma greatly simplies the analysis by
showing that only two types of settlement agreements can occur in equilibrium.
Lemma 1. There are only two types of settlements in equilibrium:
1. Licensing agreements with an entry date t = l and a licensing fee
p = (1 − l) θ




π (2)




π (2)






π (3)
if delay



0
if litigate

c
otherwise
if litigate + 
if license
(7)
from the entrant to the incumbent.
2. Pay-for-delay agreements with an entry date t = 1 and a reverse payment
− p = (1 − l) (1 − θ)




π (2)




π (3)






π (3)
if delay



0
if litigate

c
otherwise
if litigate − 
if license
(8)
from the incumbent to the entrant.
Proof.
There are no settlements if
θ>θ
(Proposition 1). Assume
θ ≤ θ.
Let us rst characterize best
responses in stage two of the game. An entrant's best response to litigate is settle if
12
S≥W
and wait
otherwise. The inequality
S≥W
p≤
is equivalent to



(t − l) π (2) + (1 − l) θπ (2)
if
t<l


(1 − t) θπ (2) − (t − l) (1 − θ) π (3)
if
t≥l
An entrant's best response to settle is settle if
S≥L
and litigate otherwise. The inequality
S≥L
is
equivalent to
p≤c+



(x − t) π (2) + (1 − x) π (3)
if
t<x


(1 − t) π (3)
if
t≥x
− (1 − θ)



(1 − l) π (3)
if
x<l


(x − l) π (2) + (1 − x) π (3)
if
x≥l
Consider now the rst stage of the game. If there is one settlement in equilibrium, the incumbent's
payo writes
I = −C + p + tΠ (1) + (1 − t) Π (2) − (1 − l) (1 − θ) [Π (2) − Π (3)]
It is never optimal for the incumbent to leave an entrant strictly better o from a settlement, as
otherwise she could deviate by strictly increasing
p
while still ensuring that the entrant accepts the
settlement oer. Thus an entrant who settles must be indierent between accepting or rejecting the
settlement oer. Using this indierence, in equilibrium
p=
must hold. Suppose
t < l.



(l − t) π (2) + (1 − l) θπ (2)
if
t<l


(1 − t) θπ (2) − (t − l) (1 − θ) π (3)
if
t≥l
Then by increasing
t
innitesimally, the incumbent gains
Π (1) − Π (2) − π (2) ≥ 0
since industry prots are weakly decreasing. Thus
t ≥ l.
Again, by increasing
t
innitesimally, the
incumbent gains
Π (1) − Π (2) − θπ (2) − (1 − θ) π (3) ≥ 0
Thus
t = 1.
If there are two settlements in equilibrium, the incumbent has the payo
I = P + min {t, x} Π (1) + (max {t, x} − min {t, x}) Π (2) + (1 − max {t, x}) Π (3)
13
where
P
denotes the sum of the two individual payments.
Using a similar argument as above, in
equilibrium, each entrant must be indierent between accepting or rejecting the settlement oer, so
p=c+



(x − t) π (2) + (1 − x) π (3)
if
t<x


(1 − t) π (3)
if
t≥x
− (1 − θ)



(1 − l) π (3)
if
x<l


(x − l) π (2) + (1 − x) π (3)
if
x≥l
Summing the two payments yields
P =2c +



(x − t) π (2) + (1 − x) 2π (3)


(t − x) π (2) + (1 − t) 2π (3)



(1 − l) π (3)
− (1 − θ)


(t − l) π (2) + (1 − t) π (3)
By increasing
min {t, x}
if
t<x
if
t≥x
− (1 − θ)
if
t<l
if
t≥l
max {t, x}
if
x<l


(x − l) π (2) + (1 − x) π (3)
if
x≥l
innitesimally, the incumbent gains
Π (1) − Π (2) − π (2) − (1 − θ)
By increasing



(1 − l) π (3)



0
if
min {t, x} < l


π (2) − π (3)
if
min {t, x} ≥ l
innitesimally, the incumbent gains
Π (2) + π (2) − Π (3) − 2π (3) − (1 − θ)
Since industry prots are weakly decreasing,
constant in time, we have a corner solution:



0
if
max {t, x} < l


π (2) − π (3)
if
max {t, x} ≥ l
min {t, x} ≥ l.
t, x ∈ {l, 1}.
Furthermore, since the changes are
The payments follow by inserting these entry
dates to the expressions for the payments.
Lemma 1 shows that there are only two types of settlements in equilibrium. A licensing agreement
allows the entrant to enter the market without infringing the patent, but the entrant must pay a
licensing fee in return. The licensing fee accounts for the prots the entrant earns before the expected
entry date of litigation. A pay-for-delay agreement involves a commitment from the entrant to stay
out of the market even if the patent has been declared invalid in the court of law.
In return, the
incumbent makes a reverse payment to compensate for the lost prots after the expected entry date
14
of litigation. Furthermore, when the incumbent settles with both entrants, she is able to capture the
litigation cost of both entrants. If there is only one settlement, no litigation costs are saved.
To glean intuition for the proof of the lemma, recall that an entrant has a best response to settle
if and only if
S ≥ max {L, W }.
This means that, in equilibrium, any entrant who settles must receive
at least her deviation payo of rejecting the settlement oer and going to court or waiting instead.
However, the incumbent will never leave an entrant strictly better o from a settlement: otherwise,
the incumbent has a protable deviation to strictly increase the payment in the settlement oer, equal
to the amount
4p = S − max {L, W } > 0
while still ensuring that the entrant accepts the settlement oer.
(9)
Thus, an entrant who settles in
equilibrium must be indierent between accepting or rejecting the settlement oer:
S = max {L, W }
(10)
This indierence pins down equilibrium payments given the entry dates. There is no reason for the
incumbent to allow entry during the litigation period. By refusing to settle, an entrant stays out of
the market for sure during the litigation period, and obtains zero prots. Therefore, the incumbent
can ask an entrant to pay the prot she makes and gets the entire industry prot from the litigation
period. Since industry prot is weakly decreasing in the number of rms, the incumbent monopolizes
the market during the litigation period.
After the litigation period, there is a possibility of entry through litigation and the incumbent
cannot monopolize the market with zero cost.
On one hand, an entrant must be compensated for
missed opportunities in the market due to entry delay after the expected entry date of litigation. On
the other hand, an entrant is willing to pay only the prot she makes before the expected entry date
of litigation. Furthermore, the innitesimal net benet of entry delay is constant in time. Therefore,
if an entrant is delayed slightly beyond the litigation period, it is optimal for the incumbent to delay
her until patent expiration.
15
Litigation for Intermediate Patents
Using Lemma 1 to account for the payments, the incumbent
has the following net payo in equilibrium with two settlements:
I = 2c + lΠ (1) + (1 − l)




Π (1) − 2 (1 − θ) π (2)




Π (2) + θπ (2) − (1 − θ) π (3)






Π (3) + 2θπ (3)
if (delay,delay)
if (licence,delay)
(11)
if (licence,license)
A licensee is willing to pay a lower licensing fee when the other entrant obtains a license: there is a
negative settlement externality between the licensees that reduces individual licensing fees, harming
the incumbent.
Similarly, a delayed entrant must be compensated more when the other entrant is
delayed as well: there is a positive settlement externality between the delayed entrants that increases
the individual reverse payments, again harming the incumbent. However, litigation has the benet of
eliminating these settlement externalities: the incumbent's payo under litigation writes
I = −C + lΠ (1) + (1 − l)



θΠ (1) + (1 − θ) Π (2) − (1 − θ) π (3)




θΠ (2) + (1 − θ) Π (3) + θπ (2)






θΠ (1) + (1 − θ) Π (3)
if (litigate,delay)
if (litigate,license)
(12)
if (litigate,wait)
A licensee is still willing to pay a high licensing fee when the other entrant litigates, because the
litigator does not enter the market when the court is upheld in court. Still, the incumbent still gets
away with a small reverse payment, as the waiting payo equals the expected triopoly prot.
The
optimal settlement design compares the benets of litigation to the cost of litigation, because the
incumbent eectively loses the total litigation costs
2c + C
in court. We have the following result.
Proposition 2. For any patent strength θ ≤ θ̂, the equilibrium of the game is (delay,delay) if the gain
from monopolization is suciently high,
Π (1) ≥ Π ≡ 2π (2) + max {Π (3) , Π (2) − π (3)}
(13)
Otherwise, there exists an interval Θ ⊆ 0, θ̂ of patent strength such that there is litigation in equilibh
i
rium if θ ∈ Θ and no litigation otherwise. The interval shrinks in total litigation costs C + 2c and is
empty for costs large enough. In particular, for zero litigation costs, Θ = [0, 1].
Proof.
See the Appendix.
16
If the gain from monopolizing the market is suciently high,
Π (1) ≥ Π,
the incumbent will
always delay both entrants, regardless of patent strength. Otherwise patent strength plays a role in
determining the equilibrium strategies: there exists an interval of patent strength such that benets
from litigation exceed the total cost of going to court; in particular, for zero litigation costs, this is true
for any patent strength. The intuition why litigation occurs for intermediate patent strength is that,
by going to court, the incumbent has the chance of monopolizing the market, but spends money in
litigation. For a suciently weak patent, the chance of winning in court and monopolizing the market
is too low compared to the litigation cost. For a suciently strong patent, the cost of litigation is too
high compared to the small reverse payments needed to delay entry with settlements.
Patent Strength vs. Pay-for-Delay
An interesting question is how patent strength aects the
equilibrium number of delayed entrants.
Since a reverse payment accounts for a share
1−θ
of the
prots a delayed entrant loses after the litigation period, the cost of entry delay decreases in patent
strength. Thus, intuitively, entry delay is more likely to occur when the patent is strong. However, the
licensing revenue also increases in patent strength, because the incumbent is able to charge a share
θ
of the prots a licensee makes in the market.
When there are two settlements in equilibrium, (delay,delay) provides the highest (settle,settle)
payo for an incumbent with a suciently strong patent,
min Π (1) − Π (2) − π (2) , 21 [Π (1) − Π (3) − 2π (3)]
θ >θ ≡1−
π (2) − π (3)
(14)
By contrast, the incumbent has the highest payo in (license,license) when the patent is suciently
weak,
θ ≤θ ≡1−
max
1
2
[Π (1) − Π (3) − 2π (3)] , Π (2) + π (2) − Π (3) − 2π (3)
π (2) − π (3)
(15)
Both thresholds of patent strength are decreasing in the gain from monopolization and increasing in
the dierence between duopoly and triopoly entry prot, which captures the intensity of the settlement
externality between the two entrants. If the gain from monopolization is high and entry prot decreases
little in subsequent entry, even a weak patent can be enough to monopolize the market. Finally, the
divide and conquer strategy of (license,delay) provides the highest payo for the incumbent when the
patent has intermediate strength:
θ < θ ≤ θ.
In particular, when subsequent entry decreases industry
17
prot less than the rs entrant, there is no scope of such a divide and conquer strategy:
Π (1) − Π (2) − π (2) ≥ Π (2) + π (2) − Π (3) − 2π (3) =⇒ θ = θ
(16)
Thus the divide and conquer strategy (license,delay) is an equilibrium strategy for intermediate patent
strength only if subsequent entry reduces industry prot more than initial entry. The following result
summarizes how patent strength aects entry delay in equilibrium without litigation.
Proposition 3. Suppose
Π (1) < Π, θ ∈
/ Θ
and θ ≤ θ̂. Then, the equilibrium of the game is (de-
lay,delay) if θ > θ, (license,delay) if θ < θ ≤ θ and (license,license) if θ ≤ θ. Furthermore, θ = θ if
the the change in industry prot weakly decreases in entry and θ < θ otherwise.
Proof.
Suppose
Π (1) < Π, θ ∈
/Θ
and
θ ≤ θ̂.
Then, there is no litigation in equilibrium (Proposition
2). Suppose the incumbent oers two pay-for-delay agreements if
and one licensing agreement if
θ<θ≤θ
θ > θ,
and two licensing agreements if
one pay-for-delay agreement
θ ≤ θ.
Then each entrant has
a best response to settle if the other entrant settles or waits. Since litigate is never a best response to
litigate, there exists a unique subgame perfect equilibrium in which both entrants settle.
Looking at the equilibrium with litigation more closely, we observe that licensing and litigation are
substitutes for the incumbent. Comparing the payos between (litigate,license) and (litigate,wait) we
immediately see that (litigate,license) is never an equilibrium.
Since monopoly maximizes industry
prot, the incumbent is always better o by not oering a licensing agreement. Then both entrants
will enter the market only if the incumbent loses in court. If the incumbent wins, she monopolizes
the market.
This means that in equilibrium with litigation, a settlement is always a pay-for-delay
agreement.
Proposition 4. Suppose Π (1) < Π and θ ∈ Θ. Then, the equilibrium of the game is (litigate,delay) if
Π (2) ≥ Π (3) + π (3)
(17)
and (litigate,wait) otherwise.
Proof.
Suppose
Π (1) < Π and θ ∈ Θ.
17 holds and the incumbent oers
Then, there is litigation in equilibrium (Proposition 2). Suppose
t=1
to both entrants, but one entrant is oered a reverse payment
−p = (1 − l) (1 − θ) π (3) while the other entrant is charged p > 0.
18
Then, the entrant who is oered the
reverse payment has a best response to settle unless the other entrant settles. But the other entrant
never settles, because
S < 0 ≤ W , and her best response to settle is litigate, since θ ≤ θ̂.
Consequently,
there is a unique subgame perfect equilibrium in which one entrant accepts a pay-for-delay agreement
while the other entrant litigates. Suppose 17 does not hold and the incumbent oers
to both entrants. Then
S<0≤W
t=1
and
p>0
for both entrants and there are two subgame perfect equilibria; in
both of these equilibria one entrant litigates while the other one waits.
Proposition 4 shows that entry delay is independent of patent strength when there is litigation in
equilibrium. Given that there is litigation, the incumbent monopolizes the market with probability
θ.
Otherwise there is entry to the market, and the incumbent can either compete in a triopoly or
duopoly, but needs to compensate the delayed entrant with a reverse payment that is proportional to
1 − θ.
Therefore, the incumbent chooses to delay one of the entrants independently of patent strength.
Only the mode of competition matters. If the triopoly prot of an entrant is less than the change in
incumbent's prot when moving from duopoly to triopoly, the incumbent delays the entrant.
Example 1
To provide an illustration, suppose that the instantaneous prot functions are deter-
mined by a textbook Cournot quantity-setting game with a simple inverse demand
1 − Q,
where
Q
denotes the industry output. All rms are symmetric and marginal costs are normalized to zero. The
equilibrium rm-level outputs are
π (1) =
1
4,
π (2) =
1
9 and
π (3) =
q (1) =
1
2,
q (2) =
1
3 and
q (3) =
1
4 , resulting in equilibrium prots
1
16 . The thresholds of patent strength are
prot decreases more in subsequent entry). Furthermore,
π (2) − 2π (3) =
1
9
θ=
2
7 and
θ=
3
7 (industry
− 81 < 0 so the equilibrium
with litigation is (litigate,wait).
4
A Model with
n
Entrants
Let us now consider a more general environment with
n
symmetric entrants to the market.
The
aim is to show how the optimal settlement design changes when there are more potential entrants to
the market. As in the model with two entrants, for a suciently strong patent, there is no credible
litigation threat from the entrants.
Proposition 5. All entrants wait in equilibrium if and only if
θ > θ̂ (n) ≡ 1 −
c
(1 − l) π (1 + n)
19
(18)
Proof.
The proof follows similar steps as in the proof of Proposition 1.
For the details, see the
Appendix.
Observe that the relevant threshold
θ̂ (n) is weakly decreasing in the number n of potential entrants
to the market. Even a weak patent may shield the incumbent from entry if competition between the
entrants is intensive enough. For a suciently weak patent, though, the litigation threat is credible
and the incumbent cannot monopolize the market with zero cost. Indeed,
θ ≤ θ̂ (n)
implies that the
litigation payo is always positive. Thus wait can only be a best response if at least one other entrant
litigates. But if at least one other entrant litigates, waiting saves on the litigation cost. There is never
more than one entrant who litigates in equilibrium, because otherwise each litigator has a protable
deviation to wait, keeping the strategies of the other entrants xed.
who litigates, unless the incumbent settles with everybody.
There is always one entrant
Therefore strategies where all entrants
wait, more than one entrant litigates or all entrants either wait or settle can be ruled out from the
equilibrium path. There is no litigation if and only if all entrants settle. Otherwise, under litigation
there is one entrant who litigates while the others either settle or wait.
Licensing and Pay-for-Delay
As in two entrants model, the indierence condition pins down the
payments in equilibrium. The following lemma is a generalization of Lemma 1 to the environment of
n
entrants.
Lemma 2. There are only two types of settlements in equilibrium:
1. Licensing agreements with an entry date t = l and a licensing fee
p = (1 − l) θπ (1 + e) +



c
if all settle


0
otherwise
(19)
from the entrant to the incumbent, where e is the number of licensees.
2. Pay-for-delay agreements with an entry date t = 1 and a reverse payment10
− p = (1 − l) (1 − θ) π (2 + n − d) −



c
if all settle


0
otherwise
(20)
10 Notice that the number two in the reverse payment accounts for the increase in the number of players in the market
when the pay-for-delay agreement is rejected.
20
from the incumbent to the entrant, where d is the number of delayed entrants.
Proof.
See the Appendix.
The licensing fee paid by an individual licensee is decreasing in the number of other licensees,
whereas an individual reverse payment is increasing in the number of delayed entrants. The general
model shows that these settlement externalities are more pronounced when there are many potential
entrants to the market. If the incumbent settles with all entrants, she obtains the following payo:






I = nc + lΠ (1) + (1 − l) Π (1 + n − d) + (n − d) θπ (1 + n − d) − d (1 − θ) π (2 + n − d)

|
{z
} |
{z
}


licensing revenue
(21)
cost of entry delay
Using a similar argument as in the baseline model, we can show that there is no licensing under
litigation, yielding the following payo of the incumbent under litigation:






I = −C + lΠ (1) + (1 − l) (1 − θ) Π (1 + n − d) − d (1 − θ) π (2 + n − d)

{z
}
|


(22)
cost of entry delay
Importantly, the cost of entry delay grows quickly in the number of delayed entrants
individual reverse payment as well as the number of reverse payments is increased.
when
d = n,
d,
since each
In particular,
the incumbent needs to compensate each entrant with the expected duopoly prot from
litigation, amounting to a large total reverse payment:
− pn = n (1 − l) (1 − θ) π (2) − nc
(23)
We have the following result, which generalizes Proposition 2.
Proposition 6. For any patent strength θ ≤ θ̂ (n), there are n delayed entrants in equilibrium if the
gain from monopolization is suciently high,
Π (1) ≥ Π (n) ≡ nπ (2) + max
d<n








Π (1 + n − d) − dπ (2 + n − d)
|
{z
}

cost of entry delay
Otherwise, there exists an interval Θ ⊆ 0, θ̂ of patent strength such that there is litigation in equih
i
librium when θ ∈ Θ and no litigation otherwise. The interval shrinks in total litigation costs and is
empty for costs large enough. In particular, for zero litigation costs, Θ = [0, 1].
21
Proof.
See the Appendix.
The underlying economic intuition behind Proposition 6 is the same as in the model with two
entrants.
We shall oer an intuition for the proof here.
The proof of the result has the following steps.
We rst show that the equilibrium payo of an incumbent who settles with everybody is increasing
and convex in patent strength, whereas the incumbent's payo in equilibrium with litigation is linear
and increasing in patent strength. Therefore, the dierence between the litigation and no litigation
equilibrium payos is concave in patent strength.
Furthermore, when the incumbent settles with
everyone, she saves on the litigation costs, which means that for costs high enough, litigation will not
occur in equilibrium. At the other extreme, when total litigation costs go to zero, there will always be
litigation in equilibrium.
The main take-away from the general result is that the gain from monopolization must be very
large for
n pay-for-delay agreements to be concluded in equilibrium.
For suciently large
n, the cost of
entry delay is too high and the incumbent accommodates entry to the market, either through licensing
or litigation. The following result shows how the number of delayed entrants depend on patent strength
when the incumbent settles with all entrants.
Proposition 7. Suppose
Π (1) < Π, θ ∈
/ Θ
and θ ≤ θ̂. Then, in equilibrium, there are n − d (θ)
licensees and






d (θ) ∈ arg max Π (1 + n − d) + (n − d) θπ (1 + n − d) − d (1 − θ) π (2 + n − d)

|
{z
} |
{z
}
d


(24)
reverse payments
licensing fees
delayed entrants, where d (θ) is weakly increasing in patent strength θ.
Proof.
See the proof of Proposition 6.
Proposition 7 generalizes Proposition 3 to the case of
n
entrants.
The settlement externalities
are pronounced when the number of potential entrants to the market is large. This means that the
extreme strategies of licensing to all or delaying everyone are less likely to constitute an equilibrium.
On one hand, the incumbent needs to pay a high total reverse payment for delaying many entrants,
because more entrants need to be compensated and individual reverse payments go up when there
is less competition in the market.
On the other hand, licensing becomes more protable when the
22
number of delayed entrants is higher. Therefore, the optimal settlement strategy of the incumbent is
a divide and conquer strategy, where some entrants receive a license while the others are delayed.
Proposition 8. Suppose Π (1) < Π and θ ∈ Θ. Then, in equilibrium, one entrant litigates, n − dˆ − 1
entrants wait and there are






dˆ ∈ arg max Π (1 + n − d) − dπ (2 + n − d)

{z
}
|
d


(25)
reverse payments
s.t d < n
pay-for-delay agreements, where dˆ does not depend on patent strength θ.
Proof.
See the proof of Proposition 6.
Proposition 8, as a generalization of Proposition 4, shows that when litigation is observed in
equilibrium, all settlement agreements are made to delay entry to the market. There is no licensing
under litigation. Given that there is litigation in equilibrium, the incumbent never oers a licensing
agreement, because the gain in reducing the cost of entry delay is equal to having an additional entrant
who waits, but there is an additional cost of forgoing the chance of monopolizing the market. Since
the industry prots are weakly decreasing, the additional licensing revenue is always less than the loss
of monopolization. The equilibrium number of delayed entrants does not depend on patent strength,
because both the gain and cost of entry delay is proportional to the probability of patent invalidation.
5
Policy Implications
Conditional Settlement Terms
Divide and conquer strategies are optimal for the incumbent due
to settlement externalities between the entrants: when one more entrant is delayed from entering the
market, the others experience an increase in their expected payo from pursuing litigation against
the incumbent. Imagine now that the incumbent can condition settlement agreements so that entry
dates are applicable only when the patent stays valid.
If the patent is invalidated, then the entry
date in the settlement agreement are no longer binding, and otherwise delayed entrant is free to enter
the market.
As before, the entrant who settles must be indierent between accepting or rejecting
the settlement agreement in equilibrium. However, since settlement agreements no longer hold if the
patent is invalidated in court, the litigation payo of an entrant does not depend on settlements with
23
other entrants:
L = (1 − l) (1 − θ) π (1 + n) − c
(26)
Suppose the incumbent oers a pay-for-delay agreement to each entrant, with an entry date
t=1
and
a reverse payment
− p = (1 − l) (1 − θ) π (1 + n) − c
(27)
This is the minimum reverse payment that satises the settlement indierence condition of an entrant
when all other entrants accept the settlement oer. If one entrant would reject the settlement oer and
litigate against the incumbent, the best response would be to wait. However, this cannot constitute
an equilibrium, because accepting the settlement agreement weakly dominates litigation. Thus, the
incumbent can monopolize the market with a total reverse payment equal to
− np = n (1 − l) (1 − θ) π (1 + n) − nc
(28)
Proposition 9. Suppose conditional settlement terms are allowed and θ ≤ θ̂. Then, in equilibrium,
there are n pay-for-delay agreements.
Proof.
See the Appendix.
Proposition 9 has an important policy implication: if conditional settlement terms are allowed,
in equilibrium the incumbent will conclude full entry delay settlement agreements with all entrants
and entry to the market occurs only after patent expiration. The intuition for Proposition 9 is simple:
conditional settlement terms eliminate settlement externalities between the entrants, because litigation
payos are not aected by entry delay: a successful litigator always faces full competitive pressure after
patent invalidation.
Antitrust Limits to Patent Settlements
To analyze welfare implications of patent settlements,
economic literature (Shapiro, 2003; Elhauge and Krueger, 2012) as well as antitrust policy (Lundbeck
and Servier cases) have focused on environments where there is only one entrant or a single settlement.
Yet, dierent types of settlements occur in practice. Therefore, the overall eect of patent settlements
can be ambiguous.
To study the eects of patent settlements to consumer welfare, we shall model consumer surplus in
reduced form, using a weakly increasing instantaneous function
24
σ (·)
that maps the number of rms in
the market to the surplus of the consumers. In the benchmark case of no patent settlements, there is
entry to the market only when the litigation threat is credible,
θ ≤ θ̂,
in which case all entrants enter
the market after the expected entry date of litigation. The consumer surplus writes
CS = lσ (1) + (1 − l) [θσ (1) + (1 − θ) σ (1 + n)]
(29)
Proposition 10. Suppose θ ∈ Θ. Then, settlements decrease consumer surplus. Suppose θ ∈/ Θ. Then,
settlements increase consumer surplus before the expected entry date from litigation, θ + l (1 − θ), and
reduce consumer surplus afterwards; taking into account both eects, the consumers strictly benet
from settlements if and only if
θ [σ (1 + n) − σ (1)] > σ (1 + n) − σ (1 + n − d (θ))
Proof.
Suppose
θ ∈ Θ.
(30)
Then Proposition 6 implies that there is never licensing and the change in
consumer surplus when settlements are banned is positive
h
i
(1 − l) (1 − θ) σ (1 + n) − σ 1 + n − dˆ ≥ 0
since
σ (·)
is weakly increasing. Suppose
θ∈
/ Θ.
(31)
Then Proposition 6 implies that there is no litigation
and settlements strictly improve consumer welfare when
(1 − l) (1 − θ) [σ (1 + n) − σ (1 + n − d (θ))] + (1 − l) θ[σ (1) − σ (1 + n − d (θ))]
|
{z
}
|
{z
}
≥0
(32)
≤0
is strictly positive.
Proposition 10 shows that consumers are strictly better o from patent settlements whenθ
∈ Θ
(litigation costs are high) and the inequality
θ [σ (1 + n) − σ (1)] > σ (1 + n) − σ (1 + n − d (θ))
is satised. This is always satised when
n
(full delay).
d (θ) = 0
(33)
(full licensing) and never satised when
d (θ) =
However, the number of delayed entrants is weakly increasing in patent strength.
Considering our baseline model with two entrants, for which
25
θ>θ
implies
d (θ) = 2
and
θ≤θ
implies
d (θ) = 0.
For intermediate patent strength,
θ < θ ≤ θ implies d (θ) = 1, and the consumers are strictly
better o from settlements when
σ (3) − σ (2)
σ (3) − σ (1)
θ > θ̃ ≡
Example 2
(34)
Let us come back to our illustration with the quantity-setting game in Example 1, with
the rm-level outputs
q (1) =
1
2,
q (2) =
1
3 and
q (3) =
1
4 in equilibrium. The instantaneous consumer
surpluses can be calculated by integrating the demand function from zero to the equilibrium industry
output:
σ (1) =
3
7 , where
θ =
3
8,
σ (2) =
θ > θ̃.
4
9 and
σ (3) =
15
32 . Thus
θ̃ =
7
27 . Recall from Example 1 that
θ=
2
7 and
Consequently, consumers are strictly better o from settlements if and only if
there is no litigation and the patent is suciently weak:
θ≤θ=
3
7 . Only if the patent is strong enough
to delay both entrants from entering the market, consumers are worse o from patent settlements. The
positive eect of one licensing agreement outweighs the negative eect of one pay-for-delay agreement.
Consumer Surplus Analysis
Let us now specify a particular utility function of consumers to
see under which conditions consumers are strictly better o from settlements.
We will consider an
oligopoly model of Dixit (1979) and focus on the environment with two entrants.
11 The rms have
constant marginal costs normalized to zero and we assume xed costs are sunk. The consumer utility
is dened as
U=
X
[α − pi ] qi −
β
2
i=1,2,3 qi
P
+ γ [q1 q2 + q2 q3 + q1 q3 ]
2
i=1,2,3
where
and
qi
is the quantity purchased from rm
α [β − γ] > 0.
i,
who charges a price
Maximization with respect to
qi
pi .
We assume
α, β > 0, β 2 > γ 2
allows us to compute a linear demand structure,
where:
pi = α − βqi −
X
qj γ
j
where
ratio
j 6= i.
γ/β 2
Notice that the goods are substitutes (complements) when
γ
is positive (negative). The
gives the level of product dierentiation and the market size is captured by
α.
Proposition 11. The threshold of patent strength θ̃ is increasing in elasticity of demand
β
and in-
creasing (decreasing) in substitutability γ if goods are complements, γ < 0 (substitutes, γ > 0).
11 We
consider an economy with three single-product rms and a competitive numeraire sector. There is a continuum
of consumers characterized by the same utility function. The utility function is separable and linear in the numeraire
good, so there are no income eects, which allows us to perform a partial equilibrium analysis. In our specication the
Cournot equilibrium is unique.
26
Proof.
See the Appendix
Proposition 11 provides some rules when consumers are likely to benet from settlements. Exact
conditions are derived in the Appendix. One can note that the question whether consumer benets
from allowing settlements depends on level of product dierentiation and elasticity of demand. In the
markets where consumers have high elasticity patent settlements are more likely to benet consumers.
Moreover, the magnitude of the eect increases in the market size
α,
and decreases in the length of
litigation.
6
Concluding Remarks
Since consumer welfare depends largely on the date of entry to the market governed by a patent,
settlements aimed at delaying or accelerating entry have a serious economic impact.
a model of patent settlements in an environment with multiple entrants.
We propose
This allows us to study
an important phenomenon, which is new to the literature on patent disputes, namely settlement
externalities between potential entrants to the market. So far mainly single entry has been a topic of
economic scrutiny, and it has been shown, that an incumbent patent holder can agree with a potential
entrant to delay her entry with a reverse payment that compensates for the missed opportunities.
Allowing for a multiple entry highlights an important externality of such an agreement, as it makes
entry for the other entrants more attractive.
In fact, for every entrant to accept full entry delay
settlements, the incumbent must oer expected duopoly prots, as these are attainable by rejecting
the settlement oer and pursuing litigation instead.
There are two types of potential equilibria: with and without litigation. Both types of equilibria are
characterized by a divide and conquer strategy implemented by the incumbent, who divides entrants
into two groups: some entrants are accommodated either through licensing or litigation while the rest
are delayed from entering the market. This is so even if all entrants are completely identical in our
model.
We show that litigation occurs for patents with intermediate strength, since the settlement
externalities are most pronounced.
At the extremes, when the patent is either suciently weak or
strong, the cost of litigation is too high compared to the small gain in letting the court decide on
patent validity.
Our results shed light on the discussion about economic consequences of pay-for-delay settlements.
We develop two extensions that seek to clarify the most important policy dilemmas. First, we address
27
the questions whether there should be limits to patent settlements.
We nd that when there is no
licensing in equilibrium, allowing for settlements is disadvantageous to consumer welfare. On the other
hand, when there is licensing, the overall eect is ambiguous.
Secondly, we prove that settlements
which are conditional on patent validity should not be allowed, because the elimination of settlement
externalities leads to full entry delay: the incumbent makes pay-for-delay settlement agreements with
all potential entrants to the market.
A particular feature of the pharmaceutical market in the US is the Hatch-Waxman Act. This law
aims to promote entry of generics by guaranteeing the rst entrant a duopoly position.
In light of
our results, such a policy response should not be eective. Once an exclusivity for the rst entrant is
granted, the incumbent has to just delay her entry. There are no settlement externalities, because the
other entrants are excluded from the market by law. The likely eect of the regulation is monopolization
instead of the prospect of accommodation.
We believe to have important results for the discussion on pay-for-delay and patent settlements
more generally. Our modeling choice was driven by a clarity of the settlement externalities argument.
Several interesting extensions remain for further research: a detailed analysis of consumer surplus,
licensing with royalties, entry at risk and potential damages from such entry when the patent is upheld
in court. Finally, we assume perfect correlation of litigation outcomes, which implies that there is at
most one lawsuit against the incumbent in equilibrium. Relaxing the assumption of perfect correlation
of litigation outcomes would probably weaken the settlement externalities between the entrants, but
not eliminate them.
28
Appendix
Proof of Lemma 2
entry date
t
Stage 2.
We adopt the following notation:
st
denotes the number of settlements with an
or later. The game is solved using backwards induction. We assume
Each entrant decides to litigate, wait or settle. A settlement
(p, t)
θ ≤ θ.
grants the settlement
payo
Z
max{t,l}
S = −p+
Z
1
π (2 + s̃0 − s̃x ) dx + θ
t
π (2 + s̃0 − s̃x ) dx
max{t,l}
Z
1
+ (1 − θ)
π (2 + s̃0 − s̃x + h) dx
max{t,l}
where
s̃x
x
h = n − 1 − s̃0
and
other
denotes the number of
entrants who settle with an entry date equal to or later than
if at least one entrant litigates and
h = 0
otherwise.
If the entrant rejects
the settlement oer, she can choose from two strategies: litigate or wait. If she litigates, she gets the
litigation payo
Z
L = (1 − θ)
1
π (1 + n − s̃x ) dx − c
l
If she waits, she gets
W =L+c
if at least one entrant litigates and
W =0
otherwise.
Suppose no other entrant litigates. Then the entrant has a best response to settle or litigate, since
θ ≤ θ =⇒ L ≥ 0 = W .
The entrant settles if and only if
S ≥ L,
1
Z
p≤p=c+
Z
p
π (1 + n − s̃x ) dx
or stays constant in
if
p≤p
l
is weakly decreasing in
is weakly increasing in
s̃0
s̃0 .
and
n − s̃x
s̃0
(number of other entrants who settle), because
is weakly decreasing in
s̃0 .
That is, the upper bound
p
s̃0 − s̃x
decreases
Therefore, there exists a subgame perfect equilibrium without litigation only
when all other entrants settle,
Z
p≤p=c+
1
Z
s̃0 = n − 1.
1
π (1 + n − s̃x ) dx − (1 − θ)
t
where
1
π (2 + s̃0 − s̃x ) dx − (1 − θ)
t
The upper bound
which is equivalent to:
π (1 + n − s̃x ) dx
l
Otherwise the entrant litigates, because her litigation payo strictly exceeds the
settlement payo. Changing notation to
s̃x = sx − 1
29
if
x≤t
and
s̃x = sx
if
x > t,
all entrants have a
best response to settle if and only if
max{t,l}
Z
p ≤ p = c+
Z
1
Z
π (1 + n − sx ) dx+θ
max{t,l}
π (1 + n − sx ) dx−(1 − θ)
t
π (2 + n − sx ) dx
max{t,l}
l
holds for all entrants. If one entrant would deviate by rejecting and going to court, all other entrants
still have a best response to settle, because
p is weakly decreasing in the number of entrants who settle.
Suppose there is at least one entrant who litigates. Then the entrant has a best response to settle
or wait, since
W = L + c ≥ L.
The entrant settles if and only if
max{t,l}
Z
Z
1
l
max{t,l}
t
and waits otherwise. Since
p
best response to settle when more than
s̃x = sx
x > t, s0
if
p≤p=
s̃0
s̃0
of other entrants who settle with
entrant settle if the entrant has a
entrants settle. Changing notation to
s̃x = sx − 1
if
x≤t
entrants have a best response to settle when
max{t,l}
Z
s̃0
is weakly increasing in the number
the incumbent, the entrant has a best response to settle when
and
π (1 + n − s̃x ) dx
π (2 + s̃0 − s̃x ) dx−(1 − θ)
π (2 + s̃0 − s̃x ) dx+θ
p≤p=
max{t,l}
Z
Z
1
max{t,l}
Z
π (1 + s0 − sx ) dx+θ
π (1 + s0 − sx ) dx−(1 − θ)
t
π (2 + n − sx ) dx
max{t,l}
l
If one entrant would deviate by rejecting and waiting, accepting a settlement agreement is even more
protable, because
Stage 1.
Z
p
is weakly decreasing in the number of entrants who settle.
Considering the rst stage of the game, the incumbent's problem writes:
l
Z
Π (1 + s0 − st ) dt + θ
max
st
0
1
Z
Π (1 + s0 − st ) dt + (1 − θ)
l
1
Π (1 + n − st ) dt + P − C1{s0 <n}
l
s.t s0 ≤ n, st nonincreasing
P ∈R
where
denotes the total payment from the entrants. There is no reason for the incumbent to
make settlement oers that are not accepted or to leave an entrant strictly better o from a settlement.
Thus
Z
max{t,l}
t
where
Z
1
π (1 + s0 − sx ) dx + θ
p=
n − s0
Z
π (1 + s0 − sx ) dx − (1 − θ)
max{t,l}
max{t,l}
π (2 + n − sx ) dx
l
entrants are oered unacceptable settlement deals, for example
30
t = 1 and p < 0 to ensure
that they never settle. Similarly
Z
max{t,l}
Z
1
max{t,l}
Z
π (1 + n − sx ) dx + θ
p=c+
π (1 + n − sx ) dx − (1 − θ)
t
π (2 + n − sx ) dx
max{t,l}
l
Combining the two expressions for the payments, we have
max{t,l}
Z
1
Z
Z
π (1 + s0 − sx ) dx + θ
pt =
t
max{t,l}
π (1 + s0 − sx ) dx − (1 − θ)
max{t,l}
π (2 + n − sx ) dx
l
+c1{s0 =n}
This equality implies
ṗt =



−π (1 + s0 − st )
if
t<l


−θπ (1 + s0 − st ) − (1 − θ) π (2 + n − st )
if
t>l
and
Z
p1 = − (1 − θ)
1
π (2 + n − st ) dt + c1{s0 =n}
l
The total payment can be expressed as the number of settlements times the payment associated with
full entry delay, subtracted with changes in payments due to early entry dates:
Z
1
P =p1 s0 −
(s0 − st ) ṗt dt
0
l
Z
Z
(s0 − st ) π (1 + s0 − st ) dt +
=
0
1
Z
θ (s0 − st ) π (1 + s0 − st ) dt − (1 − θ)
l
1
st π (2 + n − st ) dt
l
+nc1{s0 =n}
Plugging this into the incumbent's payo, we obtain:
Z
l
[Π (1 + s0 − st ) + (s0 − st ) π (1 + s0 − st )] dt
I=
0
Z
1
{θ [Π (1 + s0 − st ) + (s0 − st ) π (1 + s0 − st )] + (1 − θ) [Π (1 + n − st ) − st π (2 + n − st )]} dt
+
l
+nc1{s0 =n} − C1{s0 <n}
The incumbent gets the industry prot at any time during the litigation period. There is no reason for
31
the incumbent to allow entry to the market during the litigation period: it keeps its monopoly position
in order to maximize industry prots, so we set
s0 = st
for all
t < l.
However, after the litigation
period the payo depends on the outcome of possible litigation on patent validity.
Notice that the
expected instantaneous prot depends on time only through entry to the market. Consequently, the
patent holder's problem reduces to selecting an entry schedule
st =



s
0
for all
t ∈ [0, l]


s1
for all
t ∈ (l, 1]
that maximizes its expected payo, subject to
Proof of Propositions 2 and 6
s1 ≤ s0 ≤ n.
Suppose the incumbent settles with all entrants.
Then, using
Lemma 2, the incumbent's payo equals
IS (d) = nc + lΠ (1) + (1 − l) [Π (1 + n − d) + (n − d) θπ (1 + n − d) − d (1 − θ) π (2 + n − d)]
This denes a family of ane functions of
θ
parametrized by
d.
The epigraph of an ane function is
a half-space and any intersection of half-spaces is convex set. Dene
d (θ) ∈ arg max {Π (1 + n − d) + (n − d) θπ (1 + n − d) − d (1 − θ) π (2 + n − d)}
d
The value function
IS (d (θ))
is convex if and only if its epigraph is convex. The epigraph of the value
function is convex, therefore the value function is convex too. Notice that
∂IS (d)
= (1 − l) [(n − d) π (1 + n − d) + dπ (2 + n − d)] > 0
∂θ
Therefore
IS (d (θ))
is strictly increasing, convex function of
θ.
Suppose now that there is litigation in
equilibrium. Then, using Lemma 2, the incumbent's payo equals
IL = −C + lΠ (1) + (1 − l) [θΠ (1 + e) + θeπ (1 + e) + (1 − θ) (Π (1 + n − d) − dπ (2 + n − d))]
32
Since industry prot is weakly decreasing in the number of rms in the market,
e = 0
is optimal.
Dene
dˆ ∈ arg max {Π (1 + n − d) − dπ (2 + n − d)}
d
s.td
Clearly
θ.
dˆ does
Since
not depend on
IS (d (θ))
θ
<n
and therefore the value function
is convex and
IL dˆ
IL dˆ
is linear and nondecreasing in
is linear, the dierence
4I (θ) = IL dˆ − IS (d (θ))
is concave in
θ.
Notice that
n ∈ arg max {Π (1 + n − d) − dπ (2 + n − d)}
d
d (θ) = n and IS (n) ≥ IL dˆ . Otherwise, 4I (1) = I dˆ − I (d (1)) = −C − nc ≤ 0 and
4I (0) = I dˆ − I (d (0)) = −C − nc ≤ 0. Therefore, if C + nc > 0, {0, 1} ∈
/ Θ. In particular,
implies
C + nc = 0
implies
1∈Θ
Then by concavity of
and
4I (·)
0 ∈ Θ.
Now take any
θ, θ0 ∈ [0, 1]
such that
4I (θ) ≥ 0
and
4I (θ0 ) ≥ 0.
we have that
4I (tθ + (1 − t) θ0 ) ≥ t4I (θ) + (1 − t) 4I (θ0 ) ≥ 0
for all
t ∈ [0, 1].
Proof of Proposition 5
p>0
Suppose
to all entrants. This implies
θ > θ̂ (n).
Suppose further that the incumbent oers
S < 0 ≤ W.
t=1
and
Thus settling is dominated by waiting and there is
never a settlement. The litigation payo of an entrant is strictly negative
L = (1 − θ) (1 − l) π (1 + n) − c < 0
while the waiting payo is
W = L+c ≥ 0
if at least one entrant litigates and
W =0
otherwise. Thus
all entrants have a dominant strategy to wait, so the incumbent has the monopoly payo
in equilibrium. Suppose now that all entrants wait in equilibrium. Then
33
θ > θ̂
I = Π (1)
must hold, because
θ ≤ θ̂ (n)
implies
L≥0
so that wait is not a best response to all others wait.
Proof of Proposition 9
As before, entrants who settle must be indierent between accepting the
settlement oer and rejecting it. Hence, the settlement payo:
Z
max{t,l}
Z
1
π (1 + s0 − sx ) dx + θ
S=
t
π (1 + s0 − sx ) dx + (1 − θ) (1 − l) π (1 + n) − p
max{t,l}
and the litigation payo:
L = (1 − θ) (1 − l) π (n) − c1s0 =n
The payment indierence condition derived previously now takes the following form:
Z
max{t,l}
Z
1
π (1 + s0 − sx ) dx + θ
p=
π (1 + s0 − sx ) dx + c1s0 =n
t
max{t,l}
Summing over all individual payments we obtain an expression for the total payment:
Z
l
Z
(s0 − st ) π (1 + s0 − st ) dt + θ
P =
0
1
(s0 − st ) π (1 + s0 − st ) dt + nc1s0 =n
l
Plugging the total payment into the incumbent's prot we derive her objective
Z
l
[Π (1 + s0 − st ) + (s0 − st ) π (1 + s0 − st )] dt
I=
0
Z
1
[Π (1 + s0 − st ) + (s0 − st ) π (1 + s0 − st )] dt
+θ
l
+ (1 − θ) (1 − l) Π(1 + n) + nc1s0 =n
Since industry prot is maximized at zero entry,
st = s0
for all
t
and therefore
s1 = s0 = n
maximizes
the incumbent's prot.
Consumer Surplus Analysis (Proposition 11, to be completed)
one rm, therefore there is no parameter
γ,
inverse demand is given by:
p = α − βq
for monopoly
34
In monopoly there is only
When there are two or three rms, dierentiation starts to matter,
pi = α − βqi − γqj
for duopoly
and
pi = α − βqi − γ (qj + qk )
for triopoly
Each rm maximizes price times quantity. The Nash equilibrium of the Cournot game(s) is unique.
The equilibrium quantities and prots are (set
Π = π)
α
α2
,π (1) =
for monopoly
2β
4β
α2 2β 2 + γ 2 − β − γ
α
q (2) = 2
,π (2) =
2
γ + 2β 2
(2β 2 + γ 2 )
q (1) =
for duopoly
α2 (β − γ) β 2 − 2γ + 3βγ − 2γ 2
α (β − γ)
q (3) =
,π
(3)
=
2
2 (β 2 − γ)
4 (β 2 − γ)
for triopoly
The consumer surpluses are
σ (1) =
α2
8β
for monopoly
α2
for duopoly
+ γ2
α2 2β 3 − 5β 2 γ + [4 − 5γ] γ 2 + 4βγ [2γ − 1]
σ (3) =
4 (β 2 − γ)
σ (2) =
2β 2
for triopoly
Now, using these expressions for consumer surplus, we obtain
θ̃ ≡
β 2 −γ
2β 2 +γ 2
2 −γ
− β 2β
2β 3 − 5β 2 γ + [4 − 5γ] γ 2 + 4βγ [2γ − 1] −
2β 3 − 5β 2 γ + [4 − 5γ] γ 2 + 4βγ [2γ − 1]
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