Pool Formation with Three Patent Owners

Abstract

We analyze the endogenous formation of patent pools among three patent owners and the associated welfare effects. Under a condition of synergistic three-patent combination, either a unique symmetric equilibrium or infinitely many asymmetric equilibria can arise when patents are fragmented. By using the notion of equilibrium binding agreements, we show that (1) when there is a unique symmetric equilibrium under fragmented patents, the complete pool is both stable and welfare-maximizing; (2) fragmented patents are stable in the presence of infinitely many asymmetric equilibria; and (3) when considering only a single specific asymmetric equilibrium under fragmented patents, the complete pool is welfare-maximizing if it is stable, while fragmented patents can be both stable and welfare-maximizing under certain conditions. We also discuss an alternative version of synergism and an alternative bargaining protocol for patent pool formation.

1. Introduction

Cooperation among intellectual property owners, of which patent pools are a prominent example, is a common practice in a variety of industries. A patent pool is an agreement among multiple patent owners to license their patents as a package to external users (or share them internally). Given the practical significance of patent pooling, theoretical exploration of the topic has been initiated by, among others, Shapiro (2001), Kim (2004) and, remarkably, Lerner and Tirole (2004).

The purpose of this paper is to investigate the endogenous formation of patent pools and the associated welfare implications of stable pool structures in a setting with three patent owners. Rather than assigning a binary label of “substitutability” or “complementarity”, we focus on analyzing the relationship between stable pool structures and value accumulation as patents increase in a framework à la Lerner and Tirole (2004).1 To establish a solid strategic foundation for finding stable pool structures, we adopt the notion of equilibrium binding agreements, proposed by Ray and Vohra (1997).2 It is noted that while our analysis is centered on patent pools, it can be applied to other economic scenarios that involve similar endogenous pooling arrangements.

Our analysis builds upon Lerner and Tirole (2004)’s seminal work. There are two important features of their model, one being retained and the other being relaxed here. First, we maintain their assumption that patents are symmetric in importance, allowing each to be used individually or as a component of an innovative technology. This avoids the oversimplification of treating patents as perfect complements or substitutes [see Lerner and Tirole (2004) for further discussion] and reflects the cumulative nature of innovation (see, e.g., Scotchmer (1991)). Second, we relax the assumption of only two polar pooling options (a grand coalition or fragmented patents) from Lerner and Tirole (2004). Assuming the removal of all in-between pool structures is restrictive when we study how stable patent pools are endogenously formed, considering the diversity of pool structures observed in practice.3 In our three-patent setting, we allow for the possibility of one incomplete pool and explore when such a structure can be stable.

For the sake of full tractability, we conduct our analysis in a three-patent setting, which includes the scenario of one incomplete patent pool. This limitation serves two purposes. First, as Proposition 3 suggests, a patent pool of any size up to three members can be stable for certain pool formation games; i.e., there is already a full display of stable pool structures even with a very limited number of patents. Second, even for the simplest pool formation game which is numerically equivalent to a collection of Cournot oligopoly models,4 a complete characterization of stable pool structures, applicable to any number of patents, may not exist. As noted by Ray and Vohra (1997), the stability of coalition structures is highly sensitive to the number of players in Cournot oligopoly.5 Therefore, an extension of our analysis to an arbitrary number of patents would require further simplification of the basic framework, which is an interesting direction for future research but beyond the scope of this paper. Additionally, to avoid introducing excessive complexities, we limit our analysis to the innovative technology whose value satisfies a condition of synergism—i.e., the total value of the combined patents exceeds the sum of the individual patent values.

To model the strategic behavior of patent owners, we employ the notion of equilibrium binding agreements being proposed by Ray and Vohra (1997) as the protocol for patent pool formation. Although this involved solution concept applies to general multi-player settings, it remains intuitive and straightforward in the three-patent case (see Section 3 for details). In conjunction with a formulation of patent pool formation games, it aligns well with our research question, providing advantages over other game-theoretic models and solution concepts. Unlike “coalition-proof” solution concepts, such as coalition-proof Nash equilibrium (Bernheim et al., 1987), where the coalitional behavior by players is implicit, the notion of equilibrium binding agreements explicitly identifies the coalition structure that is formed endogenously. Additionally, unlike coalitional bargaining games, whose equilibria can be highly sensitive to modeling details, this notion provides a unified approach for studying the stability of various coalition structures in abstract pool formation games (see Section 5 for further discussion). Traditional cooperative game theory is also less applicable here, as patent pool formation games cannot be captured by characteristic functions due to externalities across patent pools, and the normal cooperative solutions typically focus on allocation rather than the endogenous formation of coalition structures.

An important finding of our analysis is that, when patents are fragmented, there may exist either a unique symmetric equilibrium (which happens if the value of the technology embodying all three patents is sufficiently high) or infinitely many asymmetric equilibria with only one patent owner endowed with market power. In the symmetric case, a straightforward patent pool formation game arises, and the stable pool structure can be either a complete pool or an incomplete pool, while fragmented patents are never stable (see Propositions 1 and 6). In the case of infinitely many asymmetric equilibria, we explore various classes of patent pool formation games regarding market power. First, we allow that any patent owner can be endowed with any level of market power, showing in Proposition 2 that fragmented patents are stable in such a setting. Second, we consider a class of pool formation games where the identity and level of market power is predetermined under fragmented patents. Proposition 3 states that a patent pool of any size can form, depending on the degree of asymmetry in the equilibrium under fragmented patents. We also identify stable pool structures for several other alternative pool formation games between these two extreme cases (Propositions 4 and 5).

Our welfare analysis reveals a pattern of a stable complete pool: it is generally welfare-maximizing, as it minimizes the total price of all three patents across all feasible pool structures (Propositions 1, 3, and 6; there is one exception, as outlined in Corollary 2). The fragmented patent structure can be stable and welfare-maximizing in certain pool formation games where the value of the technology embodying all three patents is sufficiently low (Proposition 3). The incomplete pool, however, is never welfare-maximizing, though it is possible for it to be stable in certain pool formation games (Proposition 3). Our findings suggest some policy implications, particularly regarding the socially optimal market power if it can be adjusted as a policy tool (Corollary 1).

There is a growing body of theoretical literature on patent pools, and we review some of the closely related papers.6 Aoki and Nagaoka (2006) tackle the problem of endogenous patent pool formation using Maskin (2003)’s solution concept in a partition function game. In contrast to Lerner and Tirole (2004), the basic model thereof exclusively involves essential patents that must be used as a whole to be valuable. Based on the model of Lerner and Tirole (2004), Brenner (2009) investigates the formation of patent pools in a class of games in which only one (perhaps incomplete) patent pool is allowed to be formed, adopting a sequential unanimity game as the protocol of pool formation. However, he only focuses on the scenario in which the equilibrium of fragmented patents is symmetric, and addresses the optimal formation rules for preventing a welfare-decreasing pool without explicitly characterizing the stable pool structure. In this sense, our paper is complementary to his. Also adopting Lerner and Tirole (2004)’s framework, Choi (2010) analyzes the incentives to form a patent pool in the shadow of patent litigation and shows the effect of pooling arrangements on sheltering invalid patents from challenges. There are only two patents, owned by two separate firms in Choi (2010)’s model, which does not involve a strategic stage of patent formation. Hence, our paper and Choi (2010)’s focus on different aspects of patent pools. Quint (2014) further distinguishes between essential and nonessential patents, and explores the welfare effect of different kinds of patent pools using a logit-demand model. Still, he does not study endogenous pool formation. Tesoriere (2019) is the only paper we are aware of that adopts the notion of equilibrium binding agreements when investigating the stability of patent pools. But the model specification (all patents are essential) and the research question (whether a pool’s sharing rule is stable against arbitrage) are very different from ours.

Three-player games offer a tractable yet insightful framework for studying coalition formation, and many important applications often involve only a small number of participants. A body of literature explores how coalitions form among three players in contexts such as abstract coalitional games [Binmore (1985), Moldovanu (1992), Ray (2007), Gomes (2020), and Shen and Qu (2025)], legislative bargaining [Baron and Herron (2003) and Kalandrakis (2004)], international trade [Riezman (1985), Krishna (1998), Aghion et al. (2007), and Seidmann (2009)], and horizontal mergers [Wang et al. (2024)]. Our work also contributes to this area of research.

This paper is organized as follows. Section 2 provides all the preliminaries of the model. Section 3 defines equilibrium binding agreements in our setup and investigates the stable pool structures and the associated welfare effects for two-patent pool formation games. Section 4 addresses the same questions for various pool formation games where the identity and level of market power are predetermined in specific ways. Section 5 discusses an alternative definition of synergism and a sequential bargaining game as an alternative pool formation protocol. Section 6 concludes. All proofs are relegated to Appendix A.

2. The Setup

Let $N=\left\{1,2,3\right\}$ denote the set of three patent owners, each possessing one patent with the same index as its owner. A (patent) pool structure is a partition of these three owners, and each element of this partition is a (patent) pool. Thus, the number of patents within a pool can be one, two, or three, depending on the ambient pool structure. We summarize all feasible patent pool structures in the following table (

Table 1. Patent pool structures.

Patent Pool Structure	Partitions	Notation
Complete pool	$\left\{\left\{1,2,3\right\}\right\}$	C
	$\left\{\left\{1\right\},\left\{2,3\right\}\right\}$	$I_1$
Incomplete pool structures	$\left\{\left\{2\right\},\left\{1,3\right\}\right\}$	$I_2$
	$\left\{\left\{3\right\},\left\{1,2\right\}\right\}$	$I_3$
Fragmented pool structure	$\left\{\left\{1\right\},\left\{2\right\},\left\{3\right\}\right\}$	F

).

We call an incomplete pool structure I-structure when there is no need to emphasize who the stand-alone owner is.

2.1. Lerner and Tirole (2004)’s Model

We consider Lerner and Tirole (2004)’s model but assuming that the technology embodies at most three patents. The cost of patent licensing is zero for owners. There is a continuum of licensees (who are not patent owners) uniformly distributed over an interval $[\underline{\theta },\overline{\theta }]$.7 For $\theta \in [\underline{\theta },\overline{\theta }]$, licensee $\theta$’s gross surplus of using k patents ($k\in \left\{1,2,3\right\}$) is $\theta +V\left(k\right)$, where $V\left(k\right)$ is strictly increasing in k.8 It is possible for licensees to adopt a technology that embodies a proper subset of N. In this surplus function, $\theta$ reflects heterogeneous licensees regardless of the number of patents to be used, while the component $V\left(k\right)$ is not licensee-specific. For notational convenience, let

$$u=V\left(1\right)+\overline{\theta },v=V\left(2\right)+\overline{\theta },\mathrm{and}w=V\left(3\right)+\overline{\theta }.$$

Following Lerner and Tirole (2004), we assume that the support $[\underline{\theta },\overline{\theta }]$ is sufficiently wide such that interior solutions always exist whenever necessary. In particular, $w>0$ (otherwise, the technology would never be adopted).

Once a pool structure P is formed, the timing of the game is as follows.

(a)

Pools compete by setting prices $p_i$ ($i\in N$) simultaneously, where $p_i$ is patent i’s price. (We assume that a pool’s profit is equally shared by all owners within this pool.9 Thus, owners within one pool aim to maximize their joint profit.)

(b)

Licensee $\theta \in [\underline{\theta },\overline{\theta }]$ selects a basket of patents according to the following maximization problem:

$$\underset{B\subseteq P,B\ne ⌀}{max}\left\{\theta +V\left(\left|\bigcup B\right|\right)-\sum _{i\in \bigcup B}{p}_i\right\}$$

where $\bigcup B$ is the number of patents in the basket B and ${\sum }_{i\in \bigcup B}{p}_i$ is the total price of the patents in B.10 We denote this solution by $B^{*}$. Clearly, all licensees make the same decision irrespective of $\theta$. Note that $B\subseteq P$ requires no independent licensing within a patent pool; i.e., a licensee buys one patent in a pool only if they buy all the patents in this pool.

(c)

Licensee $\theta \in [\underline{\theta },\overline{\theta }]$ employs the technology and pays the prices for $B^{*}$ if and only if $\theta +V\left(\left|\bigcup B^{*}\right|\right)\ge {\sum }_{i\in \bigcup B^{*}}{p}_i$. That is, without using any patent, the surplus of each licensee is zero. This decision is licensee-specific.

Following Lerner and Tirole (2004), we focus on the pure-strategy subgame-perfect equilibria in which all owners have positive sales.11

2.2. Synergism

We mainly investigate a case in which the technology embodying all three patents exhibits a synergistic effect (i.e., “the total effect is greater than the sum of the individual effects”). We say that the three-patent combination is synergistic if $w\ge 3v/2$; i.e., the value of the technology embodying three patents is no lower than the sum of the individual patent values measured by v.12 We assume this throughout the paper, except for in Section 5.

Indeed, by introducing a proper concept of synergism, the U.S. Supreme Court ruled whether a combination patent (a new arrangement of existing components in the prior art) meets the nonobviousness requirement (Section 103 of the Patent Act of 1952) necessary for patentability. In cases like Anderson’s Black Rock, Inc. v. Pavement Salvage Co., and Sakraida v. Ag Pro Inc., the Supreme Court defined synergism as a combined effect that is greater than the sum of the elements functioning separately, serving as an indication of nonobviousness. For further discussion on the role of synergism in the U.S. patent system, see Fenton (1980) and Baker (1981). While we do not address patentability directly, the concept of synergism is relevant to analyses of practical interest, especially as competition authorities often require patent pools to include only patents essential to the innovative technology.13

The following lemma characterizes the equilibrium profits under different pool structures when the three-patent combination is synergistic. Let $\Delta \equiv \overline{\theta }-\underline{\theta }$ and $\Lambda \equiv \left[v-u,\left(w-u\right)/2\right]$.

Lemma 1. 

Suppose that the three-patent combination is synergistic.

(1) 

The equilibrium profit of either pool under I-structure is $w^2/\left(9\Delta \right)$.

(2) 

If $w\le 2u$, there exist infinitely many equilibria under F-structure such that the profit of one owner is $\left(w-u-\lambda \right)\left(u-\lambda \right)/\Delta$ and the profit of either of the other two owners is $\lambda \left(u-\lambda \right)/\Delta$, where $\lambda \in \Lambda$. Otherwise, the equilibrium profit of each owner under F-structure is $w^2/\left(16\Delta \right)$.

We emphasize that there may exist a unique symmetric equilibrium or infinitely many asymmetric equilibria under F-structure, depending on whether the technology embodying all three patents is sufficiently valuable. Particularly, when it is not (i.e., $w\le 2u$), one owner can set a price equal to $w-u-\lambda$ (yielding a profit of $\left(w-u-\lambda \right)\left(u-\lambda \right)/\Delta$) which is higher than the price $\lambda$ (yielding a profit of $\lambda \left(u-\lambda \right)/\Delta$) set by the other two owners in the (asymmetric) equilibrium with $\lambda \in \Lambda$. Note that $\lambda$ measures the degree of symmetry for all possible asymmetric equilibria under F-structure: The smaller $\lambda$ is, the more asymmetric the equilibrium is, and the greater market power one owner possesses. The equilibrium is uniquely symmetric when $\lambda$ reaches its highest value in $\Lambda$. As $\lambda$ increases, the equilibrium profit of the owner endowed with market power decreases and that of the other two owners increases.14

Let the triple $\left(F,i,\lambda \right)$, $i\in N$, and $\lambda \in \Lambda$ denote F-structure under which owner i is endowed with market power and sets the equilibrium price equal to $w-u-\lambda$ (and the other two owners set the same equilibrium price $\lambda$). Let $\overline{\mathcal{F}}=\left\{\left(F,i,\lambda \right):i\in N,\lambda \in \Lambda \right\}$ be the set of all those triples.

2.3. Pool Formation Games

Finally, we introduce a pool formation game $G=\left(N,\mathcal{P},{\left\{{\pi }_i\right\}}_{i\in N}\right)$, with $\mathcal{P}=\left\{C,I_1,I_2,I_3\right\}\bigcup \mathcal{F}$, where the following hold:

• $\mathcal{F}=\left\{F\right\}$ if there is a unique equilibrium under F-structure; otherwise, $\mathcal{F}$ is a subset of $\overline{\mathcal{F}}$ (to be specified later).
• For $i\in N$, ${\pi }_i\left(P\right)$ is owner i’s equilibrium profit under $P\in \mathcal{P}$.

For $P,Q\in \mathcal{P}$, and $S\subseteq N$, we write ${\pi }_S\left(P\right)>{\pi }_S\left(Q\right)$ if ${\pi }_i\left(P\right)>{\pi }_i\left(Q\right)$ for all $i\in S$.

3. Stable Pool Structures: Equilibrium Binding Agreements

Given a pool formation game $G=\left(N,\mathcal{P},{\left\{{\pi }_i\right\}}_{i\in N}\right)$, we can recursively define the set of equilibrium binding agreements (written as ${\mathcal{P}}^E$), which is a subset of $\mathcal{P}$, as follows:

(C1)

$F^{\prime }\in {\mathcal{P}}^E$ for all $F^{\prime }\in \mathcal{F}$.

(C2)

For $i\in N$, $I_i\in {\mathcal{P}}^E$ if there are no $j\in N\setminus \left\{i\right\}$ and $F^{\prime }\in \mathcal{F}$ such that ${\pi }_j\left(F^{\prime }\right)>{\pi }_j\left(I_i\right)$.

(C3)

$C\in {\mathcal{P}}^E$ if neither of the following holds:

(C3.1)

There are $i\in N$ and $S\in \left\{i,N\setminus \left\{i\right\}\right\}$ such that $I_i\in {\mathcal{P}}^E$ and ${\pi }_S\left(I_i\right)>{\pi }_S\left(C\right)$;

(C3.2)

There are $i,j\in N$ ($i\ne j$) and $F^{\prime }\in \mathcal{F}$ such that ${\pi }_i\left(F^{\prime }\right)>{\pi }_i\left(C\right)$ and ${\pi }_j\left(F^{\prime }\right)>{\pi }_j\left(I_i\right)$. (Note that ${\pi }_j\left(F^{\prime }\right)>{\pi }_j\left(I_i\right)$ implies that $I_i\notin {\mathcal{P}}^E$.)

We say that a pool structure P is stable if P is the coarsest pool structure in ${\mathcal{P}}^E$.

The definition of ${\mathcal{P}}^E$ requires that all possible deviations be internal (the pool structure can only become finer) and farsighted (deviating owners must consider further deviations triggered by the current deviation). (C1) says that all $F^{\prime }\in \mathcal{F}$ must be in ${\mathcal{P}}^E$ because the F-structure is the finest one, making further internal deviations impossible. (C2) states that an I-structure is in ${\mathcal{P}}^E$ if no internal deviation leading to some $F^{\prime }\in \mathcal{F}$ makes the deviating owner better off. (C3) mandates that the complete pool belongs to ${\mathcal{P}}^E$ if neither any I-structure in ${\mathcal{P}}^E$ nor any $F^{\prime }\in \mathcal{F}$ (via some interim I-structure due to farsightedness) “blocks” the complete pool.

For an illustration, consider the situation in which there is a unique symmetric equilibrium under F-structure. Particularly, let $G_F$ be a pool formation game G with $\mathcal{F}=\left\{F\right\}$ (i.e., $w>2u$). Lemma 1 states that ${\pi }_i\left(I_i\right)=2{\pi }_j\left(I_i\right)=w^2/\left(9\Delta \right)$ for $i,j\in N$ ($j\ne i$), and ${\pi }_i\left(F\right)=w^2/\left(16\Delta \right)$ for $i\in N$. Clearly, $I_i\notin {\mathcal{P}}^E$ for all $i\in N$ because ${\pi }_j\left(F\right)>{\pi }_j\left(I_i\right)=w^2/\left(18\Delta \right)$ for $i\in N$, $j\ne i$ (i.e., either owner in the incomplete pool is better off deviating to F-structure). This further implies that (C3.1) does not hold. To see that (C3.2) does not hold, note that ${\pi }_i\left(F\right)<{\pi }_i\left(C\right)=w^2/\left(12\Delta \right)$ for all $i\in N$. That is, given the outcome of ending up in F-structure, no owner in the complete pool is willing to dissolve it. Therefore, ${\mathcal{P}}^E$ for $G_F$ is $\left\{C,F\right\}$ and the complete pool is stable.

Following Lerner and Tirole (2004), we define a pool structure as welfare-maximizing for a pool formation game G if it results in the lowest total price of all three patents compared to the total prices under any other feasible pool structures for G. The following proposition summarizes our findings for $G_F$.

Proposition 1. 

The stable pool structure for $G_F$ is C-structure and it is welfare-maximizing.

By Lemma 1 and Proposition 1, we derive an immediate corollary without directly invoking the concept of synergism: If $w>max\left\{3v/2,2u\right\}$, then the pool formation game is $G_F$, where the stable pool structure is the welfare-maximizing complete pool. In other words, when the technology embodying all three patents is sufficiently valuable, the stable pool structure will be the complete pool, which also maximizes welfare. (Therefore, in this case, a full coalition poses no concern for a social planner or competition authority.)

Next, we investigate the stable pool structure with infinitely many asymmetric equilibria under F-structure (i.e., $3v/2\le w\le 2u$). Let $G_{\overline{\mathcal{F}}}$ be G with $\mathcal{F}=\overline{\mathcal{F}}$ (i.e., all equilibria under F-structure are considered possible). The following lemma is useful.

Lemma 2. 

For $G_{\overline{\mathcal{F}}}$, there exist $\underline{\lambda }$ and $\overline{\lambda }$ in Λ with $\underline{\lambda }<\overline{\lambda }$ such that for $i,j\in N$, $i\ne j$, we have (1) ${\pi }_i\left(F,i,\lambda \right)>{\pi }_i\left(C\right)$ if $\lambda <\overline{\lambda }$, and (2) ${\pi }_j\left(F,i,\lambda \right)>{\pi }_j\left(I_i\right)$ if $\lambda >\underline{\lambda }$.

The values of $\underline{\lambda }$ and $\overline{\lambda }$ are provided in Proof of Lemma 2. In the case where $\mathcal{F}=\overline{\mathcal{F}}$, Lemma 2 highlights two facts: (1) if the equilibrium under F-structure is sufficiently asymmetric (i.e., $\lambda <\overline{\lambda }$), the owner with market power earns a higher profit under F-structure than in the complete pool; (2) if the equilibrium is sufficiently symmetric (i.e., $\lambda >\underline{\lambda }$), the owner without market power is better off under F-structure than in the incomplete pool. Therefore, it is clear that there exists $\lambda \in \Lambda$ for $G_{\overline{\mathcal{F}}}$ such that F-structure “blocks” all other pool structures simultaneously. As a result, we have the following proposition for $G_{\overline{\mathcal{F}}}$.

Proposition 2. 

The stable pool structure for $G_{\overline{\mathcal{F}}}$ is F-structure.

It is difficult to analyze the welfare effect of stable F-structure for $G_{\overline{\mathcal{F}}}$, because all elements in $\overline{\mathcal{F}}$ are in ${\mathcal{P}}^E$ and the total price of all patents depends on the realized $\lambda$. We postpone the welfare analysis regarding the stable F-structure to the next section.

4. Market Power and Stable Pool Structures

In this section, we focus on the scenario where there are infinitely many asymmetric equilibria under F-structure (i.e., $3v/2\le w\le 2u$). In the pool formation game $G_{\overline{\mathcal{F}}}$, this spectrum of market powers under F-structure allows for diverse behavioral modes of a deviating pool member. However, this flexibility may be overly optimistic in theory [see, e.g., Ray and Vohra (1997) and Diamantoudi (2003) for further discussion] and unrealistic in practice, especially when the possession of market power is influenced by historical, institutional factors, or policy favoritism.

To address this excessive optimism, we first investigate the stable pool structures and their associated welfare effects for a specific class of pool formation games, with the equilibrium under F-structure uniquely designated.15 For simplicity, we assume that patent owner 1 is the only one endowed with market power, setting an equilibrium price of $w-u-\lambda$ under F-structure.16 The following proposition identifies the stable pool structures for this class of pool formation games. Let $G_{\lambda }$ be a pool formation game G with $\mathcal{F}=\left\{\left(F,1,\lambda \right)\right\}$ for some $\lambda \in \Lambda$.

Proposition 3. 

If $\lambda \le \underline{\lambda }$, the stable pool structure for $G_{\lambda }$ is $I_1$-structure and not welfare-maximizing. If $\lambda \ge \overline{\lambda }$, it is C-structure and welfare-maximizing. If $\lambda \in \left(\underline{\lambda },\overline{\lambda }\right)$, it is F-structure; furthermore, it is welfare-maximizing when $w<18u/11$ and $\lambda \le u-w/2$.

Proposition 3 demonstrates that the stability of a pool structure for this class of pool formation games depends on the level of market power held by owner 1 under F-structure. Note that $I_2$- and $I_3$-structures, two incomplete pool structures under which owner 1 is a member of the incomplete pool, are never in ${\mathcal{P}}^E$ for any $G_{\lambda }$. This is because ${\pi }_1\left(F,1,\lambda \right)>{\pi }_1\left(I_i\right)$ for any $\lambda \in \Lambda$ and $i\in \left\{2,3\right\}$ (i.e., owner 1 is always better off to leave the incomplete pool). Any other feasible pool structure can be stable for some $G_{\lambda }$s.

An interesting observation is that the change in stable pool structure is not “monotonic” in the change in $G_{\lambda }$: when $\lambda$ increases, the stable pool structure for $G_{\lambda }$ changes from $I_1$-structure to F-structure, and to the complete pool. The rationale is as follows. F-structure is stable when neither $I_1$-structure nor the complete pool is in ${\mathcal{P}}^E$. By Lemma 2, $I_1\notin {\mathcal{P}}^E$ if $\lambda >\underline{\lambda }$ (i.e., $\lambda$ is large enough to make it profitable for either owner without market power in the incomplete pool to deviate to F-structure), and $C\notin {\mathcal{P}}^E$ if $\lambda <\overline{\lambda }$ (i.e., $\lambda$ is small enough to make owner 1 in the complete pool better off under F-structure). This trade-off makes the domain of $\lambda$, which induces stable F-structure, be $\left(\underline{\lambda },\overline{\lambda }\right)$, being located in a middle area of $\Lambda$.

Several remarks regarding welfare follow Proposition 3.

(1)

When the identity and level of market power are predetermined, stability is sufficient to ensure that the complete pool is welfare-maximizing, echoing our findings in Proposition 1.

(2)

The stable incomplete pool structure can never be welfare-maximizing. (Actually, regardless of whether the incomplete pool structure is stable, the total price of all three patents under this structure is consistently higher than that of the complete pool in any pool formation game analyzed in this paper; see proofs of Propositions 1 and 6.)

(3)

Stability is neither sufficient nor necessary for F-structure to be welfare-maximizing. However, F-structure can be both stable and welfare-maximizing for $G_{\lambda }$ with $\lambda \in \left(\underline{\lambda },u-w/2\right]$, a proper subset of $\left(\underline{\lambda },\overline{\lambda }\right)$, when the value of the technology embodying all three patents is sufficiently low (i.e., $w<18u/11$).

The following corollary compares the welfare effects of stable pool structures across different $G_{\lambda }$. This comparison is particularly relevant if the social planner aims to maximize welfare under the stable pool structure by influencing the market power of a specific owner under F-structure (i.e., treating $\lambda$ as a policy tool). For $\lambda \in \Lambda$, let $T_{G_{\lambda }}$ denote the total price of all three patents under the stable pool structure for $G_{\lambda }$.

Corollary 1. 

If $w<18u/11$, $inf\left\{T_{G_{\lambda }}:\lambda \in \Lambda \right\}=w-u+\underline{\lambda }$.17 Otherwise, $min\left\{T_{G_{\lambda }}:\lambda \in \Lambda \right\}=w/2$.

This corollary indicates that when w is sufficiently small, the appropriate level of market power for one specific owner can lead to stable F-structure, thereby enhancing welfare (with the total price of all three patents approximating $w-u+\underline{\lambda }$). However, to achieve this outcome, the social planner must ensure that $\lambda$ does not fall too low, because a $\lambda$ below $\underline{\lambda }$ induces a stable incomplete pool structure, which is not welfare-maximizing. Given that the stable complete pool is otherwise welfare-maximizing, maintaining a high $\lambda$ that induces the stable complete pool appears to be both robust and practical—particularly when the social planner lacks information about u, v, and w. Thus, the social planner simply needs to prevent any pricing advantage for owners under F-structure, without the complexities of estimating all these value parameters that influence $\underline{\lambda }$.

Alternative Games Regarding Market Power

In Proposition 2, we analyze a pool formation game $G_{\overline{\mathcal{F}}}$ where any owner can be endowed with any level of market power in $\Lambda$. In contrast, in Proposition 3, both the identity and degree of market power are predetermined for each $G_{\lambda }$. Between these two extremes, a variety of other pool formation games can be formalized, some of which we discuss here.

We begin by considering a class of pool formation games $G_{\lambda }^{\prime }$, defined by G with $\mathcal{F}=\left\{\left(F,i,\lambda \right):i\in N\right\}$ for some $\lambda \in \Lambda$. Clearly, given $\lambda \in \Lambda$, the only distinction between $G_{\lambda }^{\prime }$ and $G_{\lambda }$ is that the identity of market power under F-structure is not predetermined in $G_{\lambda }^{\prime }$. The following proposition identifies the stable pool structure for every $G_{\lambda }^{\prime }$.

Proposition 4. 

If $\lambda \in \left(\underline{\lambda },\overline{\lambda }\right)$, the stable pool structure for $G_{\lambda }^{\prime }$ is F-structure. Otherwise, it is C-structure.

In contrast to Proposition 3, Proposition 4 asserts that none of the incomplete pool structures are stable for $G_{\lambda }^{\prime }$, regardless of the value of $\lambda$. Particularly, no incomplete pool structure belongs to ${\mathcal{P}}^E$, as the undetermined identity of market power allows any owner to anticipate gaining market power by deviating to F-structure. Interestingly, as the following corollary reveals, the stable complete pool may not be welfare-maximizing for certain $G_{\lambda }^{\prime }$ games if the value of the technology embodying all three patents is sufficiently low. This phenomenon—stability does not guarantee that the complete pool is welfare-maximizing—is not observed in any other pool formation games analyzed in this paper.

Corollary 2. 

If $w<18u/11$ and $\lambda \le \underline{\lambda }$, the stable C-structure for $G_{\lambda }^{\prime }$ is not welfare-maximizing.

For a given $\widehat{\lambda }\in \Lambda$, we next analyze two pool formation games with undetermined but restricted market power: $G_{\widehat{\lambda }}$, defined by G with $\mathcal{F}=\left\{\left(F,1,\lambda \right):\lambda \ge \widehat{\lambda }\right\}$; and $G_{\widehat{\lambda }}^{\prime }$, defined by G with $\mathcal{F}=\left\{\left(F,i,\lambda \right):i\in N,\lambda \ge \widehat{\lambda }\right\}$. The identity of market power is undetermined in $G_{\widehat{\lambda }}^{\prime }$, while owner 1 is solely endowed with market power in $G_{\widehat{\lambda }}$. However, the following proposition shows that both games share the same stable pool structure. Particularly, the complete pool becomes stable when market power is sufficiently limited (i.e., $\widehat{\lambda }\ge \overline{\lambda }$). Moreover, as in $G_{\lambda }^{\prime }$, none of the incomplete pool structures are in ${\mathcal{P}}^E$ for either $G_{\widehat{\lambda }}$ or $G_{\widehat{\lambda }}^{\prime }$.

Proposition 5. 

If $\widehat{\lambda }\ge \overline{\lambda }$, the stable pool structure for both $G_{\widehat{\lambda }}$ and $G_{\widehat{\lambda }}^{\prime }$ is C-structure. Otherwise, it is F-structure.

Note that $G_{\widehat{\lambda }}^{\prime }=G_{\overline{\mathcal{F}}}$ when $\widehat{\lambda }=v-u<\overline{\lambda }$. Thus, Proposition 2 is a corollary of Proposition 5.

5. Discussion

In this section, we first explore the stable pool structures when the two-patent combination is synergistic and then discuss an alternative protocol for patent pool formation.

5.1. Synergistic Two-Patent Combination

We can similarly define the synergism of a two-patent combination by $v\ge 2u$ (i.e., the value of the two-patent combination is no lower than the sum of the individual patent values, measured by u). The following proposition identifies the stable pool structures and the associated welfare effects under this alternative assumption, without assuming that the three-patent combination is synergistic. It turns out that this characterization is much simpler than in the case of the synergistic three-patent combination, as there is always a unique symmetric equilibrium under F-structure (and hence $\mathcal{F}=\left\{F\right\}$) when the two-patent combination is synergistic.

Proposition 6. 

Suppose that the two-patent combination is synergistic. If

$$w\in \left[{\displaystyle \frac{\sqrt{6}}{2}}v,\left(3-\sqrt{3}\right)v\right]\bigcup \left(\sqrt{2}v,\infty \right),$$

the stable pool structure for G (with $\mathcal{F}=\left\{F\right\}$) is C-structure and it is welfare-maximizing. Otherwise, I-structure is stable but not welfare-maximizing.

Two points arise when comparing Proposition 6 with Proposition 1. First, as in Proposition 1, the complete pool can be stable, and when it is, it is also welfare-maximizing. Also, F-structure is never stable. However, unlike Proposition 1, incomplete pool structures may be stable under this alternative assumption.

5.2. Sequential Bargaining as an Alternative Protocol

Another widely studied coalition formation protocol, based on sequential bargaining, is the infinite-horizon unanimity game, analyzed by Bloch (1996) and generalized by Ray and Vohra (1999).18 In this protocol, following some order of moves, the first player proposes a coalition consisting of some players including themselves (opting out to form a singleton coalition is also an option). If the proposal is accepted sequentially by all members of the suggested coalition, this coalition is formed and the remaining players continue the game. If one potential member rejects, the rejector initiates the next proposal. The game ends once all players are in a coalition, and a coalition structure is formed. If this game continues forever, all players receive a payoff of zero. The key results in Bloch (1996) and Ray and Vohra (1999) show that for a symmetric game—where the worth of a coalition in a given coalition structure depends only on the number of players in every coalition in that coalition structure—a focal prediction on the coalition structure in (stationary-perfect) equilibrium is a subgame-perfect equilibrium of a simple sequential game of choosing coalition sizes.

In our context, a pool formation game is symmetric if there is a unique symmetric equilibrium under F-structure (i.e., $\mathcal{F}=\left\{F\right\}$).19 The sequential game of choosing coalition (or pool) sizes proceeds as follows. The first owner chooses $k_1\in \left\{1,2,3\right\}$. If $k_1$ is 2 or 3, the game ends with the formation of I-structure or C-structure, respectively. If $k_1=1$, the second owner chooses $k_2\in \left\{1,2\right\}$. If $k_2=1$, F-structure is formed; if $k_2=2$, I-structure is formed.

The following proposition indicates that these two pool formation protocols provide a common prediction for the pool structure formed for any symmetric game.20

Proposition 7. 

Consider a symmetric pool formation game with three patent owners. The stable pool structure for this game is a subgame-perfect equilibrium of the sequential game of choosing pool sizes.

This proposition demonstrates the robustness of a stable pool structure by using the notion of equilibrium binding agreements for a symmetric pool formation game with three owners. In contrast, it is expected that if the bargaining approach is employed, the resulting pool structure for an asymmetric pool formation game is highly sensitive to modeling choices. By sidestepping the formalization of these nuanced details, the notion of equilibrium binding agreements offers a synthesized approach to solving pool formation games, irrespective of whether the game is symmetric or not.

6. Conclusions

In this paper, we analyze stable patent pool structures using the notion of equilibrium binding agreements (Ray & Vohra, 1997), with the results determined by the value parameters of one, two, and three patents. Several key observations emerge. First, when the technology embodying all three patents is sufficiently valuable, a unique symmetric equilibrium exists under fragmented patents, and the complete pool is both stable and welfare-maximizing. Second, for a pool formation game with a particular asymmetric equilibrium under fragmented patents, patent pools of any size, up to three members, can be stable in certain pool formation games. Moreover, the stable complete pool often maximizes welfare, while the fragmented structure can also be stable and welfare-maximizing under specific conditions. Third, the stable incomplete pool is the least desirable pool structure, as it is never welfare-maximizing for any pool formation game analyzed in this paper.

It is worth emphasizing that this paper contributes to the broader literature of coalition formation with applications [see, e.g., Ray (2007) and Ray and Vohra (2015) for reviews]. While much of the existing literature on coalition formation focuses on symmetric games, our work extends beyond this constraint. By studying both symmetric and asymmetric games, we offer a more comprehensive analysis of how stable patent pools can form endogenously. This departure from the standard assumption of symmetry allows us to explore a wider range of strategic behaviors, providing insights into how varying degrees of market power influence the stability of different pool structures.

To conclude our paper, we point out some of the potential directions for further research. It could be interesting to study the effect of different licensing policies on endogenous pool formation, sharpening our understanding of issues related to patent licensing; e.g., the competition authority may enforce provision of independent licensing as part of the pool arrangement (Lerner et al., 2007). Additionally, applying alternative solution concepts—such as stability definitions that allow for broader forms of deviation [e.g., Diamantoudi and Xue (2007)]—could also provide deeper insights into the relationship between market power and stable pool structures.