Competition in Standard-Setting with Network Effects∗

Xiao Fu†

January 20, 2017

Job Market Paper

For the most recent version, please visit http://xiao-fu.weebly.com/

Abstract

Many technology products are based on standards that require the use of patents. This

paper studies the design of competing standards in industries with positive network effects

where the relevant network can be either industry-wide when standards are compatible (e.g.,

mobile phone standards) or standard-specific when standards are incompatible (e.g., video game

consoles). In a three-stage model, standards choose how many patent holders to include and

then compete in the marketplace. I find that competing standards have incentives to soften

competition through fragmenting their patent rights. Nevertheless, the degree of fragmentation

is lessened when competition among standards becomes more intense. Moreover, compatibility

among standards also affects the incentives to fragment patent rights. These results provide

alternative explanations to as why standards in many high-technology industries adopt highly

fragmented patent rights structure.

Keywords and phrases: Standard-Setting Process, Standards War, Standard-Essential

Patents, Compatibility

JEL classification codes: D2; L4; L13; L15; O3

∗I am extremely indebted to Guofu Tan for his continuing guidance and support. I am very thankful to Laura

Doval, Junjie Zhou, and Haojun Yu for their detailed advices. I also would like to thank Simon Wilkie, Yilmaz Kocer,

Isabelle Brocas, Juan Carrillo, and Yanhui Wu. I have benefitted from comments by seminar participants at USC,

as well as conference participants in 2016 Southern California Symposium on Network Economics and Game Theory.†Ph.D. Candidate, Department of Economics, University of Southern California; xiaofu@usc.edu.

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1 Introduction

“In several key industries, ..., our patent system is creating a patent thicket: a set of patent rights

requiring that those seeking to commercialize new technology obtain licenses from multiple patentees.”

— Carl Shapiro, Navigating the Patent Thicket

This Standards are technical specifications that regulate the common design of products and

services.1 Standardized industries such as mobile phones, computers, and digital media share

the following key features: (i) standards promote interoperability so that differently designed

products can work together, (ii) competing standards are developed in different organizations,

and (iii) conforming to a standard requires the use of patents and paying licensing fees to multiple

patent holders.2 For example, in the case of mobile telecommunications, (i) standards ensure

compatibility so that mobile phones from different carriers and/or makers can work together, (ii)

several independent standard-setting organizations have developed dozens of competing standards

over the last two decades (e.g., GSM and CDMA in the 2nd era of mobile networks), and (iii) each

standard is made up of hundreds or even thousands of patented technologies owned by dozens

of different firms.3 Moreover, for each of the aforementioned industries, the economic effects

of standard setting are significant because ex-ante (before a standard is designed) non-essential

patents may become essential ex-post (after a standard is designed).4 Economists and participants

in high-technology industries have raised concerns about there potentially being too many essential

patents, typically termed “patent thicket” phenomenon.5

This paper analyzes the formation of competing standards in industries that satisfy features (i)-

(iii) above. The research question is how do the certification decisions of standards interact with

1Spulber (2016) comments that “standards are specifications that affect product quality and interoperability ofparts and components. Standards apply to inputs, components, and final-products that affect production costs andconsumer benefits.”

2Baron and Spulber (2015) comment that in most cases, standards themselves do not grant users a right to use thepatented technologies described in the standards. This right must be negotiated in private agreements with patentholders. The overall royalty burden is determined by adding all the different claims for royalties together.

3For instance, Layne-Farrar and Lerner (2011) suggest that the W-CDMA standard covers 34 different patentowners holding 348 essential patents.

4By definition, an essential patent is strictly necessary for the standard. That is, an end user needs licenses toeach of these patents in order to comply with the standard.

5As defined by Shapiro (2001), the “patent thicket” phenomenon describes a situation in which the users of amajor invention must obtain licenses from multiple patent holders. Allison and Lemley (2002) suggest that thispattern seems to be consistent over time at least from the 1980s onwards.

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market competition. In the model, standards affect competition by choosing how many patent

holders to certify, and patent holders then independently set licensing royalties. The key insight of

the analysis is that competing standards have incentives to soften competition through fragmenting

their patent rights, that is, in the presence of competition, each standard always certifies multiple

patent holders. Nevertheless, the degree of fragmentation is lessened when competition becomes

more intense. Additionally, I show that compatibility among standards also affects the incentives

to fragment patent rights. Specifically, I find that fragmentation incentives strictly increase with

intra-standard network sensitivities and strictly decrease with inter-standard network sensitivities.

The present analysis helps explain the choice of technologies by standards, provides an alternative

explanation for the “patent thicket” phenomenon, and has interesting public policy implications.

Overview of model

The focus of the analysis is on the interaction between standard setting and market competition.

The model is feasible enough to accommodate any number of standards. I posit that each

standard performs a certification function, designing its technical specification based on a particular

combination of patented technologies, and it is member-friendly in the sense that its objective is

to maximize the total licensing revenue of its patent holders.6 After the certification decisions

have been made, patent holders independently set licensing royalties for their own patents. The

certification decision of each standard thus involves a basic trade-off: covering more patent holders

can increase royalties but makes the users less likely to adopt the standard. I solve for the

equilibrium number of patent holders within each standard, under the constraint that patent holders

are required to commit to grant licenses on nondiscriminatory terms.7

I consider a discrete-choice model of standard implementation in a market characterized by

positive network effects. The payoff a user derives from implementing a standard depends on the

number of users implementing the same or compatible standard(s). For example, when a user faces

a choice among competing standards, it tends to take into account the decisions of other users,

6I refer to “members” as the patent holders whose inventions are included in the standard, as opposed to “users”referring to the implementers.

7As acknowledged by Schimidt (2014), Baron and Spulber (2015), and Lerner and Tirole (2015), standards oftenrequire the patents included in the standards be made available by their owners with royalties that are fair, reasonable,and non-discriminatory, referred to as FRAND. The FRAND terms are the most common rules by which standardsnow deal with patent ownership, although standards usually do not indicate how to calculate FRAND royalties.

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as a more popular standard can have a better chance to survive or win a standards war. The

relevant network can be either industry-wide when standards are completely compatible (i.e., the

networks of different standards are connected with each other) or standard-specific when standards

are completely incompatible (i.e., each standard has its own proprietary network). I compare the

equilibria for the standards with complete compatibility and complete incompatibility.

Overview of results

The main results of the analysis are as follows. First, I show that fragmentation of patent rights

unambiguously raises licensing royalties for all the standards. The intuition behind this finding is

that by delegating pricing decisions to independent owners holding strictly complementary patents,

the standards credibly commit not to set royalties coordinately, therefore inducing less competition

from competitors. I then show that competing standards have incentives to soften competition

through certifying multiple patent holders. In other words, competition among the standards leads

to the fragmented structure of patent rights, as compared to the case of a single monopoly standard.

Second, I show that the degree of fragmentation is lessened when competition becomes more

intense, that is, either the number of standards increases or the intrinsic payoff of the standards

decreases. The intuition for this result is that increased competition would give each standard an

additional incentive to cut royalties, and in consequence, the incentive to raise royalties through

fragmentation is weakened. In addition, I show that both the equilibrium royalties and licensing

revenue will drop dramatically once the number of competing standards exceeds two, that is, having

three or more standards in the marketplace is much less profitable than two.

Third, I consider the situation in which standards are able to coordinate their certification

decisions. The equilibrium royalties under the aforementioned noncooperative standard-setting

process are found to be typically lower than the joint-profit maximizing level. I then show that

if standards can cooperate on certification, they will choose to cover more patent holders, and

by so doing, they will be able to induce the joint-profit maximizing royalties in the market for

patent licenses. This result suggests that cooperative certification can provide an alternative way

for competing standards to collude.

Finally, I incorporate network effects into the adoption of standard such that the payoff a user

derives from implementing a standard also depends on the number of users implementing the same

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or compatible standard(s). As mentioned earlier, the relevant network can be either industry-wide

when standards are completely compatible (e.g., mobile phone standards) or standard-specific when

standards are completely incompatible (e.g., video game consoles). I first explore the stable choice

probabilities under such models and then characterize the equilibria.

More specifically, with network effects, multiple equilibria can easily arise, even when network

sensitivities are homogenous, and the equilibria may be either symmetric or asymmetric. I provide

a sufficient condition under which there exists a unique symmetric equilibrium. This condition is

powerful since it works for both compatible and incompatible standards. I also characterize some

key properties of the asymmetric equilibria and show that weak network sensitivities can preclude

the possibility of asymmetric equilibria.

For the symmetric equilibria, I show that better compatibility reduces the incentives to fragment

patent rights. Specifically, I find that fragmentation incentives strictly increase with intra-standard

network sensitivities and strictly decrease with inter-standard network sensitivities. These results

indicate that, all else being equal, “patent thickets” are denser when standards are incompatible

(i.e., there are no inter-standard network effects) than when they are compatible (i.e., there exist

positive inter-standard network effects).

The remainder of this paper is organized as follows. Section 2 discusses the related literature.

In Section 3, I set up a basic model of standard setting and competition without network effects,

where the main assumptions and notations are introduced. In Section 4, I characterize the unique

equilibrium of the basic model and establish comparative statics results. In Section 5, I incorporate

network effects and different types of compatibility into the basic model. Section 6 concludes and

all omitted proofs and details are presented in the Appendix.

2 Related literature

This paper is closely related to several strands of literature in the fields of economics. The theoretical

framework contributes to a long literature on Cournot’s complementary monopolies problem and

its application in vertically structured industries. While I discuss the analysis in the context of

standards and patents, the model can generally be seen as a description of interaction between input

monopolies in the upstream market and oligopolies in the downstream market, and therefore it can

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apply to a number of different situations. The economics literature provides many examples of

complementary input monopolies selling to downstream manufacturers, including copper and zinc

monopolists selling to downstream producers of brass (Cournot (1838)), jet engines and avionics in

aircraft component markets (Choi (2008)), and Microsoft’s Windows operating system and Intel’s

microprocessors (Casadesus-Masanell and Yoffie (2007)). The baseline model of this paper most

nearly resembles that of Tan and Yuan (2001). In their paper, two competing firms independently

choose ex ante their divestiture strategies and delegate pricing decisions to the divisions. In their

theory, such a divisionalization softens the price competition between the two firms. As a result, the

prices and profits increase, but the total surplus is reduced. More recently, Quint (2014a) analyzes

a similar environment in which each products has its components supplied by different firms. He

shows sufficient conditions on a random utility model under which the pricing game has a unique

equilibrium. Together, this line of literature considers a world with differentiated products that

are made of nonoverlapping sets of essential inputs, each supplied by a different firm. To draw an

important distinction between this paper and this line of existing works, I note that none of the

papers mentioned above considers network effects. This paper also makes substantial contributions

to the analysis of standardization. Many of the papers in the literature focus only on the design of a

single standard (see, e.g., Farrell and Saloner (1988), Farrell and Simcoe (2012), Lerner and Tirole

(2015)).8 An important exception is Llanes and Poblete (2015). They model standard setting as

a coalition-formation problem in which two groups of technology developers compete to have their

own standards adopted in the market. The patent ownership in their setup is thus unconcentrated,

and they allow developers to differ in two dimensions: the number of patents they hold and the

value of those patents. The profits of a coalition of developers thus depend on the allocation of

developers into coalitions. My paper combines these approaches and the contribution is twofold.

First, I introduce a theoretical framework that is flexible enough to accommodate any number of

competing standards. I show that with competition, fragmentation of patent rights has a broader

interpretation. For example, in many models of a single standard, having one additional patent

holder in the standard will reduce licensing revenue and the reduced revenue has to be shared with

8Lerner and Tirole (2015) model standard setting as choosing patented technological functionalities and provideconditions under which a user-friendly standard and users form the efficient combination of technologies. Farrelland Saloner (1988) model consensus standard setting as a war of attrition with complete information, that is, twodevelopers argue for their preferred technological solutions until one side concedes. Farrell and Simcoe (2012) extendtheir early work by introducing private information on the quality of solutions.

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all the patent holders, e.g., see Schmidt (2014) and Quint (2014b). If that is the case, then from the

standard’s perspective, patent ownership must be the more concentrated the better. However, my

analysis suggests that the insights we have learned from these models can be incomplete. I show

that competing standards have incentives to soften competition through covering multiple patent

holders. As a result, both total royalties and licensing revenue can increase for all the standards.

Second, I evaluate the interaction between the certification decisions of standards and market

competition. As acknowledged by Lerner and Tirole (2006), the economics literature analyzing

standards has largely focused on just one role: that of a meeting room where technology developers

can negotiate; therefore the strategic roles played by standards are often ignored.

The final related literature is on oligopolies with network effects. In their pioneering work on

markets with network effects, Katz and Shapiro (1985) consider oligopolies with positive network

effects and different types of compatibility.9 Amir and Lazzati (2011) generalizes the theory in

Katz and Shapiro (1985) on oligopolistic markets with complete compatibility. They show that

Cournot oligopolies with fully compatible networks will never have an asymmetric equilibrium, only

symmetric equilibria are guaranteed. More recently, Amir and Gama (2015) show that oligopolies

with incompatible networks may have asymmetric equilibria, even if all the firms face the same

demand and costs of production. They center their attention to non-trivial symmetric equilibria

and set aside the existence of asymmetric equilibria for future study. Since all the aforementioned

papers focus on quantity competition, the model and many results of this paper are distinct

from these papers. Network effects are also discussed in the logit choice model by Anderson

et al. (1992), but only the case of symmetric equilibria under incompatible networks is briefly

considered. Starkweather (2003) further modifies the logit choice model to incorporate network

effects and compatibility issues. Following the theory of Miyao and Shapiro (1981), he provides a

sufficient condition on network sensitivities parameters under which there exists a unique stable

choice probability when products are fully incompatible with each other. The network effects in his

model can be either positive or negative. However, he does not characterize equilibrium prices or

provide comparative statics analysis. So the research question and analysis of this paper are still

much different from Starkweather.

9Their first model is when all the products are compatible; in the second one, all the goods supplied by differentfirms are incompatible; and lastly, in the case of partial compatibility, there are groups of goods that are compatiblewithin own group but incompatible with the goods outside the group.

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3 The basic framework

I consider a discrete choice model of standard implementation in which users face a choice over

a finite set k ∈ K = 1, 2, ...,K of distinct standards. I assume that the standard has full

flexibility in defining which technologies are essential. That is, even though some technologies may

not be essential for a standard ex-ante (before the standard is designed), if they are certified by

the standard, they become essential ex-post (after the standard is designed). I describe next the

preferences of the technology developers, end users, and standards.

Technology developers (certified patent holders)

There is a finite set t ∈ M = 1, 2, ...,M of symmetric technology developers, each can

potentially form a standard.10 For k ∈ K, each standard chooses members Mk ⊂ M and then

determines the set of technical specifications based on technologies developed by its members. I first

consider a situation where standards do not have any common patent holder (i.e., Mk⋂Mk′ = ∅). I

will relax this assumption later by analyzing licensing with overlapping patent ownership in section

4.1.

As I explain below, each end user makes a discrete choice of which standard to adopt, and

therefore users single-home. This means that the only way to license patents to a particular user

is to make the patents be covered by the standard she is using.

I further assume that patent holders are required to commit to license their inventions at

nondiscriminatory royalties. Let bkt denote the (linear) licensing royalties set by the tth patent

holder of standard k for t ∈ Mk. The total royalties for implementing standard k are given by

bk ≡∑

t∈Mkbkt.

End users

Demand for the standards comes from a market of total size N . I deploy a discrete-choice

model of standard implementation where each user adopts the standard (or the no-adoption option

denoted by 0) that gives her the highest payoff.

The payoff a representative user obtains from implementing standard k is given by

10This setting indicates that the efficient standard only certifies a single technology developer.

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uk = τk + εk, (1)

where τk is the deterministic payoff from adopting standard k (τ0 is normalized to 0) and

ε0, ε1, ..., εK reflect user-specific opportunity costs of implementing the standards (or the non-

adoption option), which are assumed to be i.i.d. Gumbel random variables. For k ∈ K, we have

τk = π− bk, where π is the intrinsic payoff of the standards and bk is the overall royalty burden for

implementing standard k.11

Let b = b1, ..., bK. The above formulation yields that the choice probability for standard k

takes the following fractional form

sk(b) =exp(τk)

1 +∑K

l=1 exp(τl), for k ∈ K. (2)

Standards

The total licensing revenue of standard k takes the following form

Rk = bksk(b)N, for k ∈ K. (3)

The objective of each standard is to maximize its licensing revenue Rk by choosing how many

patent holders to include.

Timing

The timing of the game is as follows:

1. Certification: standards independently choose how many patent holders to include, requiring

patent holders to make a commitment to grant licenses on nondiscriminatory terms.

2. Licensing: patent holders simultaneously and noncooperatively set licensing royalties, under

the constraint of the nondiscriminatory terms; Users then choose which standard to implement (or

the non-adoption option) and pay licensing fees to all the relevant patent holders.

My main interest is in how patent ownership is designed in stage 1 under different market

structures.

11In order to simplify the presentation and emphasize the effects of the nature of market competition onfragmentation, I assume that the intrinsic payoff is symmetric across standards and users. This simplification doesnot affect the qualitative results in the paper.

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4 Analysis

This section characterizes the equilibrium outcomes of the basic model. For an arbitrary allocation

of patent holders within each standard, the equilibrium royalties will be formalized. I will then

stylize the equilibrium number of patent holders and conduct comparative statics studies for

certification and royalties.

4.1 Characterization

Without loss of generality, I further normalize N to 1. For m = m1, ...,mK, the profit of the tth

patent holder in standard k is given by

Rkt = bktsk(b), for t ∈Mk and k ∈ K. (4)

Therefore, in the second stage, the first-order condition for an interior solution is

sk + bkt∂sk∂bk

= 0, for t ∈Mk and k ∈ K, (5)

which suggests that in equilibrium the royalties set by the independent patents holders are

identical within each standard. The equilibrium total royalties for implementing each standard

b∗ = b∗1, ..., b∗k must solve the following equation:

mksk + bk∂sk∂bk

= 0, for k ∈ K. (6)

From the above equation, one can see that b∗ depends on the number of patent holders within each

standard.

Given b∗ as induced by (6), the licensing revenue function for each standard can be rewritten

as

Rk = b∗ksk(b∗), for k ∈ K. (7)

Therefore, the equilibrium number of patent holders within each standard m∗ = m∗1, ...,m∗K

must solve the following equation:

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(sk(b∗) + b∗k

∂sk(b∗)

∂b∗k)∂b∗k∂mk

+∑l 6=k

b∗k∂sk(b

∗)

∂b∗l

∂b∗l∂mk

= 0, (8)

for k, l ∈ K and k 6= l. Using (6), the above equation can be rewritten as

mk − 1

mk

∂sk(b∗)

∂b∗k

∂b∗k∂mk

+∑l 6=k

∂sk(b∗)

∂b∗l

∂b∗l∂mk

= 0, (9)

for k, l ∈ K and k 6= l.

Now I am ready to describe the equilibrium of the two-stage game and the implications the

game has for total royalties. I pay particular attention to the case of K > 2. In the Appendix I

show that some desirable properties of the Logit choice model leads to an easy proof of equilibrium

existence and uniqueness.

Proposition 1 Suppose K > 2. Then,

(i) For any m, the total royalties increase with the number of certified patent holders within each

standard, and they are more sensitive to their own standards’ sizes than to the sizes of other

standards. That is,∂b∗k∂mk

>∂b∗l∂mk

> 0 for k, l ∈ K and k 6= l.

(ii) The two-stage game has a unique equilibrium with m∗k = m∗ > 1 for all k ∈ K.

The results in Proposition 1 rely on the complements effect within each standard and can be

generally seen as an extension of Tan and Yuan (2001) to the market with an arbitrary number of

competing firms. It stems from the fact that when firms sell strictly complementary goods, their

combined price will be higher than if all the goods were provided by a single monopolist. This

situation was originally studied by Cournot (1838) and later termed the complements effect. In

my setting, by delegating pricing decisions to independent owners holding strictly complementary

patents, standards credibly commit not to set royalties coordinately, therefore inducing less

competition from competing standards. As a result, both total royalties and licensing revenue

can increase for all the standards.

Proposition 1 suggests that considering competition among standards can shed new light on

how does fragmentation of patent rights affect licensing revenues. The existing works on standard-

essential patents often focus on the case with a single standard. Therefore, in many models, having

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one additional patent holder in the standard will reduce licensing revenue and the reduced revenue

has to be shared with all the patent holders, e.g., see Schmidt (2014) and Quint (2014b). If that is

the case, then from the standard’s perspective, patent ownership must be the more concentrated

the better. Proposition 1 implies that the insights we have learned from these models can be

incomplete: in the presence of competition (i.e., K > 2), standards have incentives to fragment

patent rights, and in consequence, such fragmentation can increase licensing revenues for all the

standards.

Cooperative certification

Denote the licensing revenue function specified in (7) by R(m, ...,m) when standards are all

symmetrically designed (i.e., mk = m > 1 for all k ∈ K). The following proposition provides an

interesting property of this function.

Proposition 2 For any K > 2, if mk = m for all k ∈ K, the licensing revenue of each standard

R(m, ...,m) is singled-peaked of m and reaches the maximum at m = m > m∗, where m∗ is defined

as in Proposition 1.

Proposition 2 demonstrates that when standards are all symmetrically designed, fragmentation

can increase their licensing revenue as long as the number of patent holders within each standard

falls into a certain range (i.e., 1 6 m < m). This result also suggests that the equilibrium total

royalties for implementing each standard is typically below the joint-profit maximizing level. An

immediate implication of Proposition 2 is that standards will achieve the joint-profit maximizing

total royalties if they are able to coordinate their certification decisions and set mk = m for

all k ∈ K. Therefore, cooperative certification can provide an alternative way for competing

standards to collude. To the extent that standards can achieve tacit collusion in licensing through

coordinated certification, the competition policy implication of the present analysis is that antitrust

agencies should consider the potential competitive effects in permitting formal cooperation between

competing standard-setting bodies.

Next, to illustrate the contrast between concentration and fragmentation of patent rights, I

describe a set of numerical examples. I demonstrate that in the presence of competition, it is

always profitable for standards to fragment their patent rights. Otherwise, a significant portion

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of licensing revenue may be lost. By the following lemma, I can get analytical results for the

competition among any K > 1 standards under the logit choice model.

Lemma 1 For any K > 1, the equilibrium m∗ and b∗ in the two-stage game must satisfy

m∗ = (1− s(b∗))b∗

m∗−1m∗ (1− s(b∗))[1− s(b∗) + b∗s(b∗)− (K − 1)b∗s(b∗)2] = (K − 1)b∗s(b∗)3

where s(b) = eπ−b

1+Keπ−b.

In what follows, I compare three certification strategies:

1. Cooperative certification: I consider the optimal symmetric certification decision. That is, I

let mk = m for all k ∈ K and solve the aforementioned joint-profit maximization problem for m.

Let the optimal licensing revenue be denoted by R(m).

2. Noncooperative certification: I solve for the equilibrium number of certified patent holders

m∗ by lemma 1, which generates the equilibrium licensing revenue R(m∗).

3. Concentration: I ignore fragmentation and compute the licensing revenue with mk = 1 for

all k ∈ K.

The comparison results are illustrated in Figure 1. We can see that when K = 1, the optimal

certification strategy is to concentrate patent rights (i.e., m∗k = m∗ = 1 for all k ∈ K); and therefore,

the licensing revenues are identical across the three cases. However, when K > 2, a concentrated

patent ownership is no longer optimal and case 3 always generates a smaller licensing revenue than

the other two; additionally, case 2 always generates a smaller licensing revenue than case 1, as

suggested by Proposition 2.

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Figure 1: Comparison of licensing revenues under different certification strategies. π is fixed at 8.

Integration of patent holders

I now consider the situation in which given m, some patent holders may integrate, that is,

either a patent holder buys up some other essential patents, or some patent holders form a patent

pool to coordinate licensing. In either case, the integrated patent holders bundle their patents and

license them at a joint licensing fee to users. The effects of such integration on total royalties are

summarized in the following proposition.

Proposition 3 Suppose K > 2. Then, fix any m,

(i) If some (or all) patent holders from the same standard integrate, the total royalties are reduced

for all the standards.

(ii) If some (or all) patent holders from different standards integrate, the total royalties are raised

for all the standards.

Proposition 3 shows that integration between patent holders from the same standard unam-

biguously reduces royalties for all the standards since the intra-standard complements effect is

weakened. By contrast, integration across standards has a unilateral effect: it eliminates the

competition between the merging firms, raising royalties for all the standards.

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The result in Proposition 3 part (i) is obvious because by Proposition 1 part (i) we already

know that the total royalties monotonically increase with the number of certified patent holders

within each standard. This result is consistent with the existing literature on how patent pools

of complementary patents affect royalties.12 It then immediately follows from Proposition 1 part

(ii) that the grand patent pool containing all the patents of a standard cannot be profitable for

participants because in the presence of competition, a concentrated patent ownership must be

undesirable (i.e, we have ∂Rk(m)∂mk

|mk=1 > 0 for any arbitrary ml with k, l ∈ K and k 6= l). This result

sheds some light on why in real-world markets many standard-related patent pools are incomplete

pools, typically containing a small percentage of eligible patent owners, e.g., see Layne-Farrar and

Lerner (2011).13

4.2 Comparative statics on royalties and certification

In real-world markets, there are many factors that can affect standards’ incentives to certificate

patent rights, including the number of competing standards, litigation cost and regulation on

intellectual property protection, and asymmetry across technology developers and users. In what

follows, I provide a formal analysis of two of these factors, i.e., how do the number of competing

standards and the symmetric intrinsic payoff from implementing the standard affect the certification

decisions and incentives to set royalties? I establish:

Proposition 4 For any K > 2,

(i) The introduction of an independent standard lowers both total royalties b∗ and licensing revenue

R∗. Moreover, it also leads to a smaller number of certified patent holders within each standard m∗

if π >√

7− 1.

(ii) An increase in π raises b∗, R∗, and m∗. Moreover, it holds that limπ→∞

b∗ = K−1K−2 and lim

π→∞R∗ =

K−1K(K−2) .

12A patent pool is an agreement by patent holders to license their patents as a single package to end users. Theeconomics literature generally argues that patent pools can help solve the complements problem. For example,Shapiro (2001) shows that patent pools decrease royalties when all patents are perfect complements. However, inpractice there is a lack of patent pools, in particular complete pools.

13For example, Layne-Farrar and Lerner (2011) suggest that patent pools usually adopt either numericproportionality rule or value-based rule for dividing royalty earnings among participants. They study the pertainingpractice of nine active patent pools from different industries, which on average cover 38 percent of all standard-essential patent holders eligible for pool.

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Figure 2 below illustrates the comparative-statics results for m∗. The intuition for these results

is that as market competition among standards becomes more intense, it gives each standard an

additional incentive to cut royalties, and consequently, the incentives to raise royalties through

fragmenting patent rights are reduced.

Proposition 4 part (ii) also suggests that the upper limits of R∗ and b∗ will experience a

significant drop once the number of standards exceeds two. In my setting, introducing a new

standard does not have to incur an entry cost, but in real-world markets such a cost cannot be

trivial. This result implies that if there are already two independent standards competing in the

marketplace, an outside standard may not want to enter the market for the fact that having three

standards is much less profitable than two. This result may shed some light on why in many high-

technology industries, such as mobile telecommunications and consumer electronics, “standards

(formats) wars” usually happen between exactly two differently designed standards.14

Figure 2: An example of equilibrium number of patent holders under different values of K and π

14For a comprehensive list of “standards war” and participants since the 1880s, see https://en.wikipedia.org/

wiki/Format_war.

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5 Standard setting with network effects

In this section, I modify the basic logit choice model to incorporate network effects into the user’s

payoff. A standard is said to exhibit network effects if users value it more when greater numbers

of other users adopt that same or compatible standard. That is, each standard’s network can be

either compatible or incompatible with that of its competitors. The relevant network can be either

industry-wide when standards are completely compatible or standard-specific when standards are

completely incompatible. I first characterize the stable choice probability under such models and

then conduct studies on properties of the equilibria. Finally, I compare the equilibria for standards

with complete compatibility and complete incompatibility. I center my attention to the effect of

compatibility on the incentives to fragment patent rights.

5.1 The logit choice model with network effects

Consider a market of total size N where users make a discrete choice over K distinct standards.

The payoff a representative user obtains from implementing standard k is given by

uk = τk + εk, for k ∈ K,

where the deterministic part τk is the expected payoff from implementing standard k and εdk is an

i.i.d Gumbel random variable that captures the heterogeneity in the user’s opportunity cost. More

specifically, we have

τk = π − bk + αkxk +∑

l∈K,l 6=kβklxl, for k ∈ K, (10)

where π is the intrinsic payoff of the standard, which is assumed to be identical across standards

and users; bk represent the (linear) total royalties for implementing standard k; xk is the user’s

perceived demand for standard k; αk is the intra-standard network effect sensitivity parameter;

and βkl is the inter-standard one. I assume βkl > 0 if standards k and l are compatible with each

other, otherwise βkl = 0. This specification indicates that the users perceived network value for

the standard is taken to be a continuous and strictly increasing function of her perceived market

coverage of that standard and other compatible standards. More precisely, for k, l ∈ K and k 6= l,

we have

17

αk > βkl > 0: complete compatibility,

αk > βkl = 0: complete incompatibility,

αk = βkl = 0: no network effects.

Complete compatibility refers to the case in which all the standards on the technology market

can work together. An example of such industries is mobile phone networks. As shown in Figure

3, carriers often build their cellular networks by different wireless technologies and use bilateral

agreements to complete a call made to one of its subscribers by a caller from a different carrier’s

network. A carrier can benefit from the expansion of its competitors’ networks because it would

be able to collect more revenue from providing termination services.

Figure 3: Mobile networks based on fully compatible technology standards

In contrast, complete incompatibility refers to the case in which each standard is incompatible

with that of its competitor. Common examples of industries with incompatible standards include

digital compact cassette, digital media formats, and video game consoles. Many observations

suggest that when a producing firm faces a choice among incompatible standards, it tends to

consider the number of firms which have adopted each standard, as a more popular standard will

have a better chance to survive or win a standards war.

Next, given the representative user’s expectations x = x1, ...xK and total royalties b =

b1, ...bK, the probability that she adopts standard k is given by15

15The payoff of the non-adoption option is normalized to 0, and I assume that the network effects on τk does notdepend on the choice of the outside option.

18

sk(x; b, α, β, π) =exp(τk)

1 +∑

i∈K exp(τi)=

exp(π − bk + αkxk +∑

l∈K,l 6=k βklxl)

1 +∑

i∈K exp(π − bi + αixi +∑

j∈K,j 6=i βijxj). (11)

Therefore, the above choice probability is a function of royalties and user expectations. I do not

explicitly model the process through which users’ expectations are formed. Instead, I impose the

requirement that in equilibrium users’ expectations are fulfilled. Self-fulfilling expectations imply

that at equilibrium xk = skN . Without loss of generality, we can normalize the total number of

users N to 1, and (11) then can be rewritten as

Fk(s) =exp(π − bk + αksk +

∑l∈K,l 6=k βklsl)

1 +∑

i∈K exp(π − bi + αisi +∑

j∈K,j 6=i βijsj), for k ∈ K. (12)

For any s = s1, ..., sK with sk > 0 and∑K

k=1 sk < 1, there is a unique vector b = b1(s), ..., bK(s)

that satisfies (12).

The profit maximization problem of the tth patent holder in standard k becomes

maxbkt

Rkt = bkt(s)sk, for t ∈Mk and k ∈ K,

where s is implicitly defined by (12).

The licensing revenue of standard k is given by

Rk = bk(s)sk, for k ∈ K. (13)

where bk =∑

t∈Mkbkt. Each standard maximizes the licensing revenue of its members by choosing

how many patent holders to include.

The timing of the game is as follows:

1. Users form expectations about the size of the network with which each standard is

associated.

2. Standards independently make decision on how many patent holders to include, and

patent holders then noncooperatively and simultaneously announce licensing royalties for their

own inventions.

3. Users make their implementation decisions, given their expectations and royalties.

19

5.2 Completely incompatible standards

In this section, I suppose that the standards are incompatible with each other, that is, I assume

βkl = 0 for k, l ∈ K and k 6= l. In what follows, I will pay particular attention to the case with

αk 6 1 for all k ∈ K since this case gives a lot of interesting comparative static results. The

following lemma describes some interesting features of the logit choice model with positive network

effects where each standard has its own proprietary network.

Lemma 2 (i) Suppose αk 6 1 for all k ∈ K. Then, the stable choice probability sk is increasing

in αk and bl and decreasing in αl and bk for k, l ∈ K and k 6= l. Furthermore, choice probabilities

are more sensitive to royalties than to the intra-standard network effect parameters. That is,

∂sk∂αk

+ ∂sk∂bk

< 0 and ∂sk∂αl

+ ∂sk∂bl

> 0 for k, l ∈ K and k 6= l.

(ii) The logit specification with network effects does not exhibit the independence of irrelevant

alternatives (IIA) property. That is, for any three standards j, k and l, the ratio of choice

probabilities of standard k and l will change as the parameters of standard j changes.

Next I look into the existence and uniqueness of the stable choice probability as defined in (12).

Denote S = s ∈ RK |∑

k∈K sk < 1, sk > 0. Thus, F is a continuos mapping from S to S. By

Brouwer fixed point theorem, there exists at least one solution s ∈ S to (12). However, unlike

the basic framework without network effects, multiple equilibria can exist. In other words, given a

vector of total royalties b, there could be multiple s that satisfy (12). Lemma 3 below provides a

simple sufficient condition which ensures that for any b there exists a unique s that satisfies (12).

Lemma 3 For any values of (αk, bk) : k ∈ K, there exists at least one s that satisfies (12).

Moreover, if 0 < αk 6 2 for all k ∈ K, then the solution to (12) is unique and can be achieved by

iteratively applying the mapping from any starting point.

Lemma 3 suggests that fix b and any starting point s0 = (s01, ..., s

0K), st converges to the unique

stable solution by the dynamics st = F (st−1) if 0 < αk 6 2 for all k ∈ K. The logic behind this result

is first acknowledged by Miyao and Shapiro (1981) under a more general setup than the present

model, and it is also closely related to the network Logit specification studied by Starkweather

(2003). These conditions are very powerful since they work for both incompatible and compatible

standards.

20

By Lemma 3, we know that at least one symmetric equilibrium exists. However, the above

conditions are still not sufficient for guaranteeing a unique equilibrium. An interesting feature of

the present model is that multiple equilibria can easily arise, even when the network sensitivities

parameters are homogeneous, and the equilibria may be either symmetric or asymmetric.

Let s = (s1, ..., sK) denote stable choice probabilities that satisfy (12), we have the following

proposition regarding the characteristics of equilibria for incompatible standards with symmetric

patent rights structure.

Proposition 5 Suppose standards are incompatible with each other and are all symmetrically

designed (i.e., mk = m > 1 for all k ∈ K).

(i) When network sensitivities are identical across standards (i.e., assume αk = α for all k ∈ K),

• There are at most three distinct entries in s.

• There exists no asymmetric equilibria if 2 6 m 6 1−αα and α 6 1

3 .

• There exists α > 0, which depends on K and π, such that there exists a unique symmetric

equilibrium s1 = ... = sK = s if 0 < α 6 minα, 1.

(ii) When network sensitivities differ across standards (i.e., assume 0 < α1 < ... < αK 6 1), for

any equilibrium, s must satisfy s1 < ... < sK .

Proposition 5 explores some interesting properties of the asymmetric equilibria. In my setting,

the asymmetric equilibria can easily arise when some standards are more widely adopted because

the users expect them to be so. For example, during the formats war between Blu-ray and HD DVD,

some major film studios (Fox, 20th Century, Warner Bros, etc.) announced that they were willing to

declare exclusive support for one format, but they wanted the format to have another studio partner

before they did so. In my setting, asymmetric equilibria can arise when demand is concentrated on

a single standard or a set of standards. In the case of homogeneous network sensitivities, I find that

there are at most three distinct entries in s and that weak network sensitivities can preclude the

possibility of asymmetric equilibria. In the case of heterogenous network sensitivities, I show that

there is no symmetric equilibrium and that in equilibrium the standard associated with a larger

value of network effects must have a higher market share. The existence of asymmetric equilibria

is set aside for future research.

Moreover, Proposition 5 part (i) suggests that in the case of homogeneous network sensitivities,

21

a unique symmetric equilibrium can be guaranteed when α does not exceed a certain threshold.

In the Appendix, I show that the threshold on α is positive, which is strictly increasing in K

and decreasing in π. Interestingly, this condition works without modification for standards with

completely compatible networks.

I now center my attention to the special case of symmetric equilibria and consider the situation

in which standards are able to coordinate their certification decisions. For the symmetric equilibria,

let s denote the equilibrium market share for each standard. Then, the equilibrium total royalties

b are given by

b = π + αs− log[s] + log[1− sK], (14)

and the equilibrium licensing revenue for implementing each standard becomes

R = αs2 + (π + log[1− sK

s])s. (15)

I still usem to denote the cooperative certification outcomes but add a subscript “I” that denotes

complete incompatibility. I have the following proposition describing the cooperative certification

decisions of incompatible standards.

Proposition 6 Suppose 0 < αk = α 6 1 and mk = m for all k ∈ K. Then, for any symmetric

equilibrium, R(m, ...,m) is single peaked in m and reaches its maximum at m = mI , which strictly

increases with α.

The results above extend those in Proposition 2, which shows that if standards are able to

cooperate on certification decisions, they will cover more patent rights when each standard has its

own proprietary network than when there are no network effects.

The impact of network effects on cooperative certification under complete incompatibility will

be discussed in detail in section 5.4.

5.3 Completely compatible standards

In this section, I incorporate inter-standard network effects into the user’s payoff and focus on the

homogenous case in which βkl = β > 0 for k, l ∈ K and l 6= k. The stability condition (ow becomes

22

Fk(s) =exp(π − bk + αksk + βs−k)

1 +∑

i∈K exp(π − bi + αisi + βs−i), for k ∈ K, (16)

where s−k =∑

i∈K,i 6=k si.

It turns out that many results in the case of incompatible standards still hold here. First, I find

that Lemma 3 holds with a minor modification, that is, for any values of (αk, β, bk) : k ∈ K, if

0 < β < αk 6 2 for all k ∈ K, we can conclude that the solution to (16) is unique (See A.6. Proof

of Lemma 3). When the above condition holds, there exists (at least) one symmetric equilibrium.

Second, the sufficient condition for the existence of a unique symmetric equilibrium holds

without modification in the present setting. Again, the existence of asymmetric equilibrium is

set aside for future research.

In what follows, I further assume that the inter-standard network sensitivities parameters are

homogenous. For the symmetric equilibria, let s denote the equilibrium market share for each

standard. Then, the equilibrium total royalties b are given by

b = π + αs+ β(K − 1)s− log[s] + log[1− sK], (17)

and the licensing revenue function becomes

R = [α+ β(K − 1)]s2 + (π + log[1− sK

s])s. (18)

The following proposition describes the cooperative certification decisions of compatible stan-

dards. I add a subscript “C” into m to denote complete compatibility.

Proposition 7 Suppose 0 < β < α 6 1 and mk = m for all k ∈ K. Then, for any symmetric

equilibrium, R(m, ...,m) is single peaked in m and reaches the maximum at m = mC , which strictly

decreases with β.

The results above further extend those in Proposition 2, which implies that better compatibility

among standards leads to lower incentives to fragment patent rights when standards are able to

coordinate their certification decisions. Since the two network sensitivities parameters α and β are

found to affect m in opposite ways, the net effect matters.

The impact of network effects on cooperative certification under complete compatibility will be

23

formalized in the next section.

5.4 Complete incompatibility versus complete compatibility

In this section, I compare certification decisions for standards with positive network effects under

complete incompatibility and complete compatibility. In particular, I want to see how does

compatibility affect the equilibrium number of patent holders. With a particular emphasis on

symmetric equilibria and homogenous network sensitivities, I first summarize the impact of network

effects and compatibility on cooperative certification decisions and then center my attention to the

case of noncooperative certification.

Cooperative certification

We already know that if standards are able to coordinate their certification decisions, there

exists a unique value of m such that the joint-profit maximizing royalties can be supported as a

noncooperative equilibrium outcome at the licensing stage. I denote such a value of m by m, which

was already found to strictly increase with α and decrease with β at any symmetric equilibrium.

Again, I use subscripts to distinguish complete incompatibility and complete compatibility. Some

early results of the previous sections are summarized in the following proposition.

Proposition 8 Suppose 0 < β < α 6 1. Then, competing standards can achieve the maximum

joint profit by setting mk = m for all k ∈ K. Moreover, m strictly increases with α and decreases

with β.

It is straightforward that Proposition 8 implies that mI is always larger than m and mC . I now

present a simple sufficient condition on π under which mC can be always larger than m.

Corollary 1 For 0 < β < α 6 1, mC > m if π > 2+log[2].

To finish this line of discussion, I present a figure that illustrates the above proposition. For

any K ∈ [1, 4], I set π = 60 and plot the cooperative certification outcomes under three cases:

(i) no network effects: m with α = β = 0, (ii) complete incompatibility: mI with α = 0.5 and

β = 0, and (iii) complete compatibility: mC with α = 0.5 and β = 0.1. Figure 4 illustrates how m

changes with three of the four aforementioned parameters, namely, α, β, and K. Since π satisfies

the condition in the above Corollary, we can see that for any K, it holds that mI > mC > m.

24

Figure 4: An example of the impact of network effects and compatibility on cooperative certification

Noncooperative certification

Next, I compare certification decisions for standards with positive network effects under

complete incompatibility and complete compatibility. I use m∗ to denote the noncooperative certifi-

cation outcomes at the symmetric equilibrium with homogenous network sensitivities. Proposition

9 is a central result of the paper.

Proposition 9 Suppose 0 < β < α 6 1. Then, at the symmetric equilibrium, m∗ strictly increases

with α and decreases with β.

Proposition 9 shows that compatibility among standards affects the incentives to fragment

patent rights. It implies that, all else being equal, fragmentation incentives are higher when

standards are completely incompatible (i.e, β = 0) than when they are completely compatible

(i.e, β > 0).

The intuition for Proposition 9 is as follows. First, consider an industry with incompatible

standards. In contract to the basic model without network effects, the introduction of network

effects increase the user’s payoff for implementing each standard; thus, the price competition among

25

the standards is softened in the sense that each standard has its own network and can collect more

licensing revenue if it keeps total royalties unchanged. It then follows from the implication of

Proposition 4 that fragmentation incentives will increase when market competition is weakened.

Second, consider an industry-wide network where standards are compatible. Since each standard

can benefit from the expansion of its competitors’ networks, such a situation is like the standards

are partially integrated with each other, which gives each patent holder an additional incentive to

set royalties cooperatively. Therefore, the incentives to soften competition through fragmenting

patent rights can become redundant.

To end this section, I present a figure that illustrates Proposition 9. For any α ∈ [0, 1], I set

π = 60, K = 2, and plot the noncooperative certification outcomes under different values of β.

Figure 5 shows that in the case of noncooperative standard-setting, better compatibility can reduce

fragmentation incentives.

Figure 5: An example of the impact of network effects and compatibility on noncooperativecertification

26

6 Conclusions

In many high-technology industries, more and more standards have incorporated not a single

invention but a combination of many different functionalities, each of which may be the subject of

multiple patents. In a highly fragmented technology market, this phenomenon has made the use

of other firms’ inventions costlier, due to higher transaction costs, licensing fees, and risk of hold-

up.16 This paper has offered a general model to study the interaction between standard setting and

market competition. The analysis has provided an alternative explanation for the “patent thicket”

phenomenon and identified several important factors that may contribute to this phenomenon. The

main insight is that competing standards have incentives to fragment patent rights because such

fragmentation softens the price competition among the standards. As a result, both total royalties

and licensing revenues can increase for all the standards, as compare to the case of concentrated

patent ownership. Additionally, I show that the degree of fragmentation can be lessened when

competition becomes more intense. In the network effects part, I further show that incompatible

standards have higher incentives to fragment patent rights than compatible standards.

The present analysis provides some interesting public policy implications for standardization

and compatibility. I conclude that the current shift in European standardization policy to pro-

mote interoperability among European standard-setting organizations is beneficial because better

compatibility among competing standards can reduce the degree of fragmentation.17 However, one

has to be careful in stating this result for at least two reasons. First, the present study focuses

on the certification decisions at symmetric equilibria. However, with network effects, asymmetric

equilibria can easily arise when some standards are more widely adopted because the users expect

them to be so. Second, although promoting compatibility can reduce fragmentation, patent holders

may have additional incentives to raise royalties when standards are completely compatible. Users

may thus bear heavier royalty burdens. The formal analysis of the impacts of network effects and

compatibility on user surplus is set aside for future study.

There are several other interesting directions for future research. One assumption in the paper

16For example, using a panel data on 2,441 publicly traded manufacturing firms from 1976 to 2002, Entezarkheir(2016) quantifies costs of the fragmentation and find that “patent thickets” lower firms profit.

17The recent plan includes a European Interoperability Framework, which is an agreed approach to promoteinteroperability for European organizations that want to collaborate in standard-setting process.

27

is that the end users do not adopt more than one standard, that is, the users “single-home”. Such

a restriction is without loss of generality if standards are incompatible with each other. However,

if standards are completely compatible, it is possible that sometimes the users are interested in

“multi-homing”. For example, in the mobile telecommunications industry, implementing multiple

standards is not uncommon for carriers. Future research may also look towards this direction.

Another interesting direction is to introduce vertical integration between patent holders and end

users into the technology market, which will change both the incentives to set royalties and to

fragment patent rights.

28

Appendix A. Omitted proofs

A.1. Proof of Proposition 1

(i) The first-order condition for the tth patent holder in standard k can be simplified as

∂Rkt∂bkt

= sk + bkt∂sk∂bk

= sk[1− bkt(1− sk)] = 0, for t ∈Mk and k ∈ K.

The equilibrium total royalties for implementing each standard b∗ = b∗1, ..., b∗K thus must

solve the following equations

mk = b∗k(1− sk(b∗)), for k ∈ K.

Therefore, for any m = m1, ...,mK, the noncooperative royalty-setting game in stage 2 has a

unique equilibrium.

In what follows, I show that an increase in mk for any k ∈ K will cause an increase in every

entry of b∗. For the sake of clear expression, I pick k = 1 (i.e., consider the impact of an increase in

m1 on the total royalties for all the standards) and the analysis can naturally extend to any other

k ∈ K and k 6= 1.

Applying standard comparative-static techniques to the royalty-setting rule, we obtain a matrix

equation of the form

Ax = y,

where A is a K ×K matrix, x and y are column vectors with K entries:

A =

(1− s1)(1 + b∗1s1) −b∗1s1s2 . . . −b∗1s1sK

−b∗2s2s1 (1− s2)(1 + b∗2s2) . . . −b∗2s2sK

. . . .

. . . .

. . . .

−b∗KsKs1 −b∗KsKs2 . . . (1− sK)(1 + b∗KsK)

,

29

x =

∂b∗1∂m1

∂b∗2∂m1

.

.

.

∂b∗K∂m1

, y =

1

0

.

.

.

0

.

Establishing that∂b∗k∂m1

> 0 for k ∈ K is equivalent to showing that the determinants |A| and

|Ak| are positive, where Ak is the matrix formed by replacing the kth column of A by the column

vector y.

The eigenvalues of A are 1− s1 + b∗1s1, ..., 1− sK−1 + b∗K−1sK−1, ZK , where

ZK = (1− sK)(1 + b∗KsK)− (1− sK + b∗KsK)K−1∑i=1

b∗i s2i

1−si+b∗i si

> (1− sK + b∗KsK)(1−K−1∑i=1

si)− b∗Ks2K > (1− sK + b∗KsK)sK − b∗Ks2

K = (1− sK)sK > 0.

It is then easy to check that

|Ak| = b∗ks1sk

K∏i=2,i 6=k

(1− si + b∗i si) > 0, for k ∈ K and k 6= 1,

|A1| = (ZK +1− sK + b∗KsK1− s1 + b∗1s1

b∗1s21)

K∏i=2

(1− si + b∗i si) > 0,

and

|A1||Ak|

=(ZK +

1−sK+b∗KsK1−s1+b∗1s1

b∗1s21)(1− sk + b∗ksk)

b∗ks1sk>

(1− sK + b∗KsK)s1 + (1− sK)sKs1

1− sk + b∗kskb∗ksk

> 1.

Therefore, we obtain∂b∗1∂m1

= |A1||A| >

∂b∗k∂m1

= |Ak||A| > 0 for k ∈ K and k 6= 1. By symmetry across

the standards, the above inequality implies that∂b∗k∂mk

>∂b∗l∂mk

> 0 for k, l ∈ K and k 6= l.

(ii) As suggested by (9), we can rewrite the first-order optimality conditions for the standards’

problem (in stage 1) as

30

∂Rk(m)

∂mk=mk − 1

mk

∂sk(b∗)

∂b∗k

∂b∗k∂mk

+∑l 6=k

∂sk(b∗)

∂b∗l

∂b∗l∂mk

= 0, for t ∈Mk and k, l ∈ K.

The claim that m∗k > 1 then follows from ∂Rk(m)∂mk

|mk=1 > 0 for any ml with k, l ∈ K and k 6= l.

Since the licensing revenue function is symmetric in m, I suppose that there exists (at least)

one symmetric equilibrium such that m∗1 = ... = m∗K = m∗ and m∗ = b∗(1− s(b∗)). The existence

and uniqueness of the equilibrium is shown below:

The equilibrium number of patent holder within each standard m∗ and equilibrium total

royalties b∗ must solve the following equations:

m∗ = b∗(1− s(b∗))m∗−1m∗ (1− s(b∗))[1− s+ b∗s(b∗)− (K − 1)b∗s2(b∗)] = (K − 1)b∗s3(b∗)

where s(b) = exp(π−b)1+Kexp(π−b) .

Plugging the first equation into the second one gives that for any equilibrium b∗ must solve

H(b∗) = −s(b∗)(1− s(b∗)K)b∗2 − (1− 3s(b∗) + s2(b∗)K)b∗ + 1− s(b∗) = 0. (19)

Now consider the derivative of H(b∗) with respect to b∗, we get

∂H(b∗)

∂b∗= −1 + eπ−b

∗(3K − 4 + 5b∗ − b∗2) + e2(π−b∗)(3K − 6 + 3b∗ + b∗2)K + e3(π−b∗)(K − 2)K2

(1 + eπ−b∗K)3.

It then follows that H(b∗) monotonically decreases with b∗ when b∗ > 0 from

1 + eπ−b∗(3K − 4 + 5b∗ − b∗2) + e2(π−b∗)(3K − 6 + 3b∗ + b∗2)K + e3(π−b∗)(K − 2)K2

> 1 + eπ−b∗(2 + 5b∗ − b∗2) + 2e2(π−b∗)(3b∗ + b∗2)

> 1 + e−b∗(2 + 5b∗ − b∗2) + 2e−2b∗(3b∗ + b∗2) > 0.

Then, from H(1) = s(1) > 0 and H(K−1K−2) = − e

KK−2 [e

KK−2 (K−2)+e

π+ KK−2 (K2−K−1)]

(K−2)2(eKK−2 +e

π+ KK−2K)

< 0, I conclude

that (19) has a unique solution such that 1 < b∗ < K−1K−2 .

31

Therefore, the two-stage game has a unique equilibrium with m∗ > 1. Q.E.D.

A.2. Proof of Proposition 2

To prove that under the symmetric patent ownership the licensing revenue function R(m, ...,m)

is single peaked and reaches the maximum at m = m > m∗, it suffices to establish the following

result: R(b, ..., b) = b exp(π−b)1+Kexp(π−b) is single peaked and reaches the maximum at b = b > b∗, where b∗

denotes equilibrium total royalties for implementing each standard. Take derivative for R(b, ..., b)

with respect to b, we have

∂R(b, ..., b)

∂b=eπ−b(1 +Keπ−b − b)

(1 +Keπ−b)2,

which gives us that 1 +Keπ−b − b = 0 from ∂R(b)∂b

∣∣b=b = 0.

Plugging b = 1 +Keπ−b into (19), we obtain H(b) = (1−K)eπ−b

1+Keπ−b< 0 = H(b∗). Immediately, the

above inequality indicates that b > b∗. Q.E.D.

A.3. Proof of Proposition 3

The first part of the result follows from Proposition 1 part (i) immediately. To prove the second

part, I refer to the patents required for more than one standards as common patents. I simplify

without loss of generality by setting K = 2 and assuming the two standards share a single common-

patent holder (denoted by t = 1). Let bk1 denote the royalties set by the common-patent holder

for implementing standard k for k = A,B. The licensing revenue of the common-patent owner is

given by

R1 = bA1sA(b) + bB1sB(b).

The equilibrium total royalties for the common patent, (bcA1, bcB1), must satisfy the following

first-order condition:

sk + bk1∂sk∂bk

+ bl1∂sl∂bk

= 0, for k, l = A,B and k 6= l.

32

Therefore, the equilibrium total royalties, (bcA, bcB), must satisfy the following condition:

mksk + bk∂sk∂bk

+ bl∂sl∂bk

= 0, for k, l = A,B and k 6= l.

In the presence of the common patent, the condition determining equilibrium total royalties

contains an extra positive term bl∂sl∂bk

. Thus, the reaction curves determining (bcA, bcB) must intersect

at a point that is higher and to the right of the intersection point of the reaction curve determining

(b∗A, b∗B), implying bcA > b∗A and bcB > b∗B. This analysis naturally extends to the case with more

than two standards and multiple common-patent holders. Q.E.D.

A.4. Proof of Proposition 4

(i) I first show that the introduction of an independent standard leads to lower b∗, while an increase

in the intrinsic payoff π leads to higher b∗. In what follows, superscript ∗ is suppressed in order to

simplify the presentation.

Let y = exp(π − b) and plug s = y1+Ky into (19), we have

K[(K − 2)b+ 1−K]y2 + [2K(b− 1) + 1− 3b+ b2]y + b− 1 = 0, (20)

Since for any equilibrium we have 1 < b < K−1K−2 , we get the left side of the above equation is strictly

decreasing in y when y > 0.

Applying standard comparative-statics techniques yields

∂b

∂π= −2K[(K − 2)b+ 1−K]y + 2K(b− 1) + 1− 3b+ b2

Λ,

and∂b

∂K= − [2K(b− 1) + 1− 2b]y + 2(b− 1)

Λ,

where Λ = [K(K − 2)y2 + (b2 − b+ 2bK − 2)y + 2b− 1]y−1 > 0.

By rearranging the above equations, we obtain

∂b

∂π=

[2K(b− 1) + 1− 2b+ b(b− 1)]y + 2(b− 1)

Λ=b(b− 1)y

Λ− ∂b

∂K.

33

I now show that ∂b∂K < 0 holds because [2K(b−1)+1−2b]y+2(b−1) > 0. The above inequality

obviously holds when 2K(b− 1) + 1− 2b > 0. Thus, I suppose 2K(b− 1) + 1− 2b < 0 and define y

such that [2K(b− 1) + 1− 2b]y + 2(b− 1) = 0. Plugging y back into the left side of (20), we have

−[1 + 4b3(K − 1) + 2b2(5− 4K) + 2b(2K − 3)]b− 1

(1 + 2b(K − 1)− 2K)2< 0.

The above inequality holds because 1 + 4b3(K − 1) + 2b2(5− 4K) + 2b(2K − 3) = 1 + 2b[2b2(K −

1) + b(5− 4K) + (2K − 3)] > 1 when b > 1 and K > 2.

Therefore, when (20) holds, we must have y < y since the left side of (19) strictly decreases

with y. We thus get [2K(b− 1) + 1− 2b]y+ 2(b− 1) > [2K(b− 1) + 1− 2b]y+ 2(b− 1) = 0. It then

holds that ∂b∂K < 0, and immediately, we also have ∂b

∂π > 0.

(ii) I now prove a simple sufficient condition under which the introduction of an independent

standard reduces the equilibrium number of patent holders. Rearranging (20) gives

yb2 + [1 + (2K − 3)y + (K − 2)Ky2]b− (1 +Ky)(Ky − y + 1) = 0. (21)

By Proposition 1, we know that there exists a unique solution to (21). Since the left side of (21)

is monotonically increasing in b when b > 0, we can conclude that b < π if and only if the left side is

positive when b = π, which yields the following sufficient condition: π > 12(2−K2+

√4 + 4K +K4).

I obtain the above inequality by plugging b = π back into (21). As the right side of the above

inequality is strictly decreasing in K, a stricter condition is thus π >√

7− 1 ≈ 1.6458.

Next, I suppose the above condition on π holds, which implies that b < π for any equilibrium.

By Lemma 1, take derivative for m with respect to K, we have

∂m

∂K=by2 + [K(K − 1)y2 + (2K − 1 + b)y + 1] ∂b∂K

(1 +Ky)2.

Therefore, ∂m∂K < 0 if and only if by2 + [K(K − 1)y2 + (2K − 1 + b)y + 1] ∂b∂K < 0.

Plug the expression of ∂b∂K into the above inequality and further simplify it by (21), we get

∂m

∂K< 0⇐⇒ yb2 + (1 +Ky)b− 2(1 +Ky)(1 + y) < 0.

34

Lastly, plugging the closed-form expression for b into the above inequality, I find that it holds

for any y > 1 and K > 2.

Moreover, take derivative for m with respect to π, we have

∂m

∂π=−by + [K(K − 1)y2 + (2K − 1 + b)y + 1] ∂b∂π

(1 +Ky)2.

Therefore, ∂m∂π > 0 if and only if −by + [K(K − 1)y2 + (2K − 1 + b)y + 1] ∂b∂π > 0.

Plugging the expression of ∂b∂π and the closed-form expression for b into the above inequality, I

find that it holds for any y > 0 and K > 2.

(iii) I now prove that the introduction of an independent standard leads to smaller licensing

revenue R, while an increase in the intrinsic payoff leads to larger R.

Consider the derivative of R with respect to K, we have

∂R

∂K=y[ ∂b∂K (1− b+ yK)− by]

(1 + yK)2< 0

⇐⇒ ∂b

∂K(1− b+ yK)− by < 0

⇐⇒ K(2K − 3bK + 4b− 1)y2 + (b3 + b2 + 6bK − 5b− 4K + 1)y + 2− 3b

= K(1− bK)y2 + (1− b+ b2 − b3 − 2bK)y − b < 0.

The last equality holds from (20), and the last inequality holds because b > 1 and K > 2.

Consider the derivative of R with respect to π, we have

∂R

∂π=y[ ∂b∂π (1− b+ yK) + b]

(1 + yK)2> 0

⇐⇒ ∂b

∂π(1− b+ yK) + b > 0

⇐⇒ K(b2 + 3bK − 5b− 2K + 1)y2 + (3b2 + 6bK − 6b− 4K + 1)y + 3b− 2

= K(b2 + b+K − 2)y2 + (3b+ 2K − 2)y + 1 > 0.

The last inequality holds because b > 1 and K > 2.

(iv) Lastly, I will derive the limit of b and R. We already know that both of them monotonically

35

increases with π. It is easy to see s→ 1K as π approaches infinity. Then, by solving H(b) = 0 for b

under s = 1K , we obtain lim

π→∞b = K−1

K−2 , and in consequence, limπ→∞

R = K−1K(K−2) . Q.E.D.

A.5. Proof of Lemma 2

(i) Taking derivatives for s0 = skexp(uk) with respect to αk and αl respectively yields,

∂s0

∂αk=

1− sk ∂uk∂sk

exp(uk)

∂sk∂αk

− skexp(uk)

∂uk∂αk

,

and∂s0

∂αl=

1− sk ∂uk∂sk

exp(uk)

∂sk∂αl− sk

exp(uk)

∂uk∂αl

.

By rearranging the equation above, we have

∂sk∂αk

=exp(uk)

1− sk ∂uk∂sk

∂s0

∂αk+

sk

1− sk ∂uk∂sk

∂uk∂αk

=exp(uk)

1− αksk∂s0

∂αk+

s2k

1− αksk,

and∂sk∂αl

=exp(uk)

1− sk ∂uk∂sk

∂s0

∂αl+

sk

1− sk ∂uk∂sk

∂uk∂αl

=exp(uk)

1− αksk∂s0

∂αl.

Since ∂s0∂αk

+∑

j∈K∂sj∂αk

= 0, the closed-form expression for ∂s0∂αk

is given by

∂s0

∂αk= −

∑j∈K sj

∂uj∂αk

/(1− sj ∂uj∂sk)

1 +∑

j∈K exp(uj)/(1− sj ∂uj∂sj)

= −s2k

1−αksk

1 +∑

j∈Kexp(uj)1−αjsj

.

Plug it back to the above equations, we have

∂sk∂αk

=s2k

1− αksk(1−

exp(uk)1−αksk

1 +∑

j∈Kexp(uj)1−αjsj

) =s2k

1− αksk(1 +

∑l∈K,l 6=k

exp(ul)1−αlsl

1 +∑

j∈Kexp(uj)1−αjsj

) > 0,

and

∂sk∂αl

= −s2l

1− αlsl

exp(uk)1−αksk

1 +∑

j∈Kexp(uj)1−αjsj

< 0,

for k, l ∈ K and k 6= l.

36

Similarly, the closed-form expressions for the derivatives for sk with respect to royalties are

given by

∂sk∂bk

= − sk1− αksk

(1 +

∑l∈K,l 6=k,

exp(ul)1−αlsl

1 +∑

j∈Kexp(uj)1−αjsj

) < 0,

and

∂sk∂bl

=sl

1− αlsl

exp(uk)1−αksk

1 +∑

j∈Kexp(uj)1−αjsj

> 0,

for k, l ∈ K and k 6= l.

It follows from the preceding expressions that ∂sk∂αk

+ ∂sk∂bk

< 0 and ∂sk∂αl

+ ∂sk∂bl

> 0.

(ii) Consider three different standards j, k and l, next I show that ∂(sk/sl)∂bj

does not exhibit the

IIA property.

∂(sk/sl)

∂bj=sl∂sk/∂bj − sk∂sl/∂bj

s2l

=sj

Ω(1− αjsj)s2l

(slexp(uk)

1− αksk− skexp(ul)

1− αlsl)

=sksls0

(1

1− αksk− 1

1− αlsl)∂s0

∂bj=

sk(αksk − αlsl)(1− αksk)(1− αlsl)sls0

∂s0

∂bj.

The above expression suggests that ∂(sk/sl)∂bj

may dependent on the parameters of standard j,

moreover, the dependence can be either positive or negative. Q.E.D.

A.6. Proof of Lemma 3

Let F : S→ S be defined as

Fk(s) =exp(π − bk + αksk +

∑l∈K,l 6=k βklsl)

1 +∑

i∈K exp(π − bi + αisi +∑

j∈K,j 6=i βijsj), for k ∈ K,

and

F0(s) = 1−∑k∈K

Fk(s).

I first show that for completely incompatible standards, if 0 < αk 6 2 for all k ∈ K, then F (s) is a

37

contraction mapping. From the above equation, we have

∂Fk(s)

∂sk= αkFk(1− Fk),

and∂Fl(s)

∂sk= −αkFkFl,

for k ∈ K. Therefore, with 0 < αk 6 2, we have

∑i∈K

∣∣∣∣Fi(s)∂sk

∣∣∣∣ =

∣∣∣∣∂Fk(s)∂sk

∣∣∣∣+∑

l∈K,l 6=k

∣∣∣∣∂Fl(s)∂sk

∣∣∣∣= αkFk(1− Fk +

∑l∈K

Fl)

< 2αkFk(1− Fk)

6 1.

Therefore, the magnitude of any eigenvalue of the Jacobian of F is less than 1. F (s) is a

contraction mapping. By the contraction mapping theorem, F has at most one fixed point.

Now, I extend the results to the case for compatible standards with homogenous inter-standard

network sensitivities, that is, assume βkl = β > 0 for k, l ∈ K and l 6= k. In this case, we have

∂Fk(s)

∂sk= αkFk(1− Fk)−

∑l∈K,l 6=k

βFkFl,

and∂Fl(s)

∂sk= βFl(1− Fl)− αkFkFl −

∑j∈K,j 6=k,l

βFlFj

= FkFl(β − αk) + βFlF0,

for k, l ∈ K and l 6= k

Next I consider two polar cases. The analysis can naturally extend to any hybrid case between

the two polar ones. First, if ∂Fl(s)∂sk

< 0 for all l ∈ K and l 6= k, we have,

38

∑i∈K

∣∣∣∣∂Fi(s)∂sk

∣∣∣∣ < αkFk(1− Fk) + (αk − β)Fk∑l∈K

Fl

< 2αkFk(1− Fk)

< 1.

The first inequality holds because∣∣∣∂Fl(s)∂sk

∣∣∣ < (αk − β)FkFl when ∂Fl(s)∂sk

< 0.

Second, if ∂Fl(s)∂sk

> 0 for all l ∈ K and l 6= k, we have

∑i∈K

∣∣∣∣Fi(s)∂sk

∣∣∣∣ < αkFk(1− Fk) + β(F0 − Fk)∑l∈K

Fl

< αkFk(1− Fk) + β |F0 − Fk| (1− Fk − F0)

< αk[Fk(1− Fk) + maxF0, Fk(1− Fk − F0)]

< αk[Fk(1− Fk) + maxF0(1− F0), Fk(1− Fk)]

< 1.

The first inequality holds because∣∣∣∂Fl(s)∂sk

∣∣∣ < βFlF0 when ∂Fl(s)∂sk

> 0, and the third inequality

holds because 0 < β 6 αk 6 2 for all k ∈ K. Q.E.D.

A.7. Proof of Proposition 5

(i) I first study the equilibria for incompatible standards with homogenous network effects. Let

Ψ = s0 +∑

k∈Ksk

1−αsk . For any equilibrium, s must satisfy

mΨ =bk

1− αsk(Ψ− sk

1− αsk), for k ∈ K, (22)

where bk = π + αsk − log[sk] + log[s0].

Since the left side does depend on k, the right side of (22) must be equal for all k ∈ K. Let the

right side be denoted by G(sk) and take derivative of G with respect to sk, we have

39

∂G(sk)

∂sk= −bk[1− α(Ψ− sk(1 + αΨ)]

(1− αsk)3− Ψ− sk(1 + αΨ)

sk(1− αsk)2

=−(1 + αΨ)[1 + (m+ 1)αΨ]s2

k + Ψ[(m+ 2)αΨ + 2−m]sk −Ψ2

sk(1− αsk)[Ψ(1− αsk)− sk].

The last equality holds from plugging bk = mΨ(1−αsk)2

Ψ−sk(1+αΨ) into the first equality. Fix any Ψ > 0

and 0 < α 6 1, the numerator either exhibits an inverted U-shape or is monotonically decreasing

in sk for sk > 0. This indicates that given Ψ, G(sk) may exhibit an inverted-N shape, and only if

this is the case, there can exist (at most) three distinct entries in s.

Now, by rearranging the numerator, we can see that for any Ψ > 0 and 0 < sk < 1, ∂G(sk)∂sk

< 0

if the following condition holds:

(1− αsk)[(m+ 1)αsk − 1]Ψ2 − sk[m− 2 + (m+ 2)αsk]Ψ− s2k < 0,

The above inequality holds when 2 6 m 6 1−αα and α 6 1

3 . If this is the case, G(sk) is

monotonically decreasing in sk, and therefore, fix any Ψ, G(sk) = Ψ has no multiple solutions.

Hence, there exists no asymmetric equilibria.

Next I show that there exists α > 0 such that there is a unique symmetric equilibrium if

0 < α 6 α. For any symmetric equilibrium, s must satisfy

mΨ =b

1− αs(Ψ− s

1− αs),

or equivalently,

mΨ(1− αs)Ψ− s(1 + αΨ)

=b

1− αs, (23)

where b(s) = π + αs− log s+ log(1−Ks) and Ψ = 1−Ks+ Ks1−αs .

It is easy to check that the left side of the above equation is continuous and strictly increasing

in s, and it equals m when s = 0 and mKK−1 when s = 1

K , respectively.

On the right side, we have lims→0

b1−αs = ∞ and lim

s→ 1K

b1−αs = −∞. Therefore, the symmetric

equilibrium solution is stable and unique if b1−αs never increases with s when s ∈ (0, 1

K ).

40

Now consider the derivative of b1−αs with respect to s, we get ∂( b

1−αs)/∂s 6 0 if and only if

α 61

s(1− sK)(2 + sK1−sK + π − Log s+ Log(1− sK))

.

Thus, we will get a sufficient condition for equilibrium uniqueness if the above inequality holds for

any s ∈ (0, 1K ).

The right side of the above inequality exhibits a U-shape when s ∈ (0, 1K ) and reaches its

local minimum 4K3+π+logK at s = 1

2K . Therefore, there exists a unique symmetric equilibrium

s1 = ... = sK = s if α < min 4K3+π+logK , 1. The threshold on α is thus derived.

(ii) Now I consider the case for heterogeneous network effects. For any two standards with

αk > αl, suppose in equilibrium sk 6 sl, then, by (22), we have

bk

bl=

1− αksk1− αlsl

(Ψ− sl

1−αlslΨ− sk

1−αk sk

) =1− αksk1− αlsl

(s0 + sk

1−αk sk + Φk,l

s0 + sl1−αlsl + Φk,l

) =s0 + Φk,l + sk(1− αks0 − αkΦk,l)

s0 + Φk,l + sl(1− αls0 − αlΦk,l)< 1,

where Φk,l ≡∑

j 6=k,lsj

1−αj sj .

By Lemma 2, when we fix b, any starting point s0 = (s01, ..., s

0K) will converge to s by iteratively

applying the mapping. However, given αl < αk and bk < bl, we can easily see that the dynamics of

the system can never converge to sk 6 sl for the two standards. This is a contradiction. Q.E.D.

A.8. Proposition 6

Due to the monotonicity between b and s, to establish that for any symmetric equilibrium b increases

with m, it suffices to show that s decreases with m. By (23), we already know that its left side

is strictly increasing in s, and the right side is non-increasing in s if 0 < α 6 α. Therefore, as m

goes up, the left side will move upwards, while the right side will not be affected. Consequently,

the new intersection point will be left to the original one, implying the equilibrium level of s will

become smaller.

Next, I will show that R, which is defined as in (15), is single peaked in s; then, since s is

monotonically decreasing in m, we can conclude that R is single peaked in m. Take derivative for

41

R with respect to s, we obtain

∂R

∂s= − 1

1− sK+ αs+ b(s),

∂2R

∂s2= −1− 2αs+ 2αs2(2− sK)K

s(1− sK)2< 0,

where b is defined as in (14). The last inequality holds because α 6 1 and 1− sK > 0.

With lims→0

∂R∂s =∞ and lim

s→ 1K

∂R∂s = −∞, we can see that R is single peaked in s when s ∈ (0, 1

K ).

It then follows from the above first-order conditions that for any joint-profit maximizing solution,

mI and sI must solve the following equations:

b(sI) = 1

1−sIK − αsI

mI(1− αsI) = 1−sI1−sIK − αsI

The first equation comes from the first-order optimality condition ∂R∂s |s=sI = 0 and the second one

comes from plugging the first equation into (23). The closed-form expression for mI is then given

by

mI = 1 +sI(K − 1)

(1− sIK)(1− αsI)> 1.

Finally, taking derivatives for mI and sI with respect to α respectively yields

∂sI∂α

=2s2I(1− s2

IK)2

1− 2αsI + 2αK(2− sIK)s2I

> 0.

and∂mI

∂α=

(K − 1)[(1− sIK)s2I + (1− αKs2

I)∂sI∂α ]

(1− sIK)2(1− αsI)2> 0,

for any K > 2 and 0 < α 6 1. This confirms that mI strictly increases with α. Q.E.D.

A.9. Proposition 7

By Lemma 2, for any symmetric equilibrium, the close-form expression for the derivative of sk with

respected to its own royalties when network sensitivities parameters are homogenous is given by

42

∂sk∂bk|sk=s = − s

1− (α− β)s

1− [α+ (K − 2)β]s+ s(K−1)1−sK

1− [α+ (K − 1)β]s+ sK1−sK

.

Suppose mk = m for all k ∈ K. It then follows from the licensing rule msk + bk∂sk∂bk

= 0 that at

the symmetric equilibrium s must satisfy

m− b(s)

1− (α− β)s

1− [α+ (K − 2)β]s+ s(K−1)1−sK

1− [α+ (K − 1)β]s+ sK1−sK

= 0,

where b is defined as in (17).

Therefore, the joint-profit maximizing solution mC and sC under complete compatibility must

satisfy b(sC) = 1

1−sCK − [α+ β(K − 1)]sC

mC [1− (α− β)sC ] = 1−sC1−sCK − [α+ (K − 2)β]sC

The first equation comes from the first-order optimality condition ∂R∂s |s=sC = 0.

Taking derivatives with respect to β yields

∂sI∂β

=(K − 1)(1− sK)2s

K − [α+ (K − 1)β](1− sK)2> 0,

and

∂mC

∂β= −(K − 1)[1− α(1− sIK)2]sI

(1− sIK)[1− (α− β)sI ]21− [α+ (K − 1)β](1− sIK)sI

K − α+ (2− sIK)αsIK + (K − 1)(1− sIK)2β< 0,

for any K > 2 and 0 < β < α 6 1. This confirms that mC strictly decreases with β. Q.E.D.

A.10. Proof of Corollary 1

To establish the result, it suffices to show that for any α > 0, mC(α, β) > mC(α, α) > mC(0, 0) = m.

The first inequality and the last equality are obvious, so we only need to study the second inequality,

that is, mC(α, α) > mC(0, 0).

43

Suppose β = α, then at the symmetric equilibrium, m′C and s

′C must satisfy

π + 2αs

′CK − log[s

′] + log[1− s′K]− 1

1−s′CK= 0

m′C =

1−s′C1−s′CK

− αs′C(K − 1)

It then turns out that

∂m′C

∂α=

(K − 1)(2s′CK − 1)s

′C

(1− αs′CK)2 + α(4− α− 2s′CK)s

′2CK

2> 0 iff s

′C >

1

2K.

The last inequality holds if π + α − log[ 12K ] + log[1

2 ] − 2 > 0 for any K > 2. Thus, a stronger

sufficient condition is π > 2+log[2]. Q.E.D.

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