Patent package structures and sharing rules for royalty revenue

Abstract

The role of patent pools—one-stop systems that gather patents from multiple patent holders and offer them to users as a package—is gaining research attention. To bolster the scarce stream of the literature that has addressed how a patent pool agent distributes royalty revenues among patent holders, we conduct an axiomatic analysis of sharing rules for royalty revenue derived from patents managed by a patent pool agent. In our framework, the patent pool agent organizes the patents into some packages, which we call a package structure. By using the hypergraph formulation developed by van den Nouweland et al. (Int J Game Theory 20:255–268, 1992), we analyze sharing rules that consider the package structure. In our study, we propose a sharing rule and show that it is the unique rule that satisfies efficiency, fairness, and independence requirements. In addition, we analyze sharing rules that enable a patent pool agent to organize a revenue-maximizing and objection-free profile.

1 Introduction

Against the background of the rapid increase in the number of patents, the role of patent pools—one-stop systems that gather patents from multiple patent holders and offer them to users as a package—is gaining research attention, as the fragmentation of patent ownership increases the costs of negotiation between patent owners and users. Shapiro (2001) and Lerner and Tirole (2004) showed that by forming patent pools and collectively determining license fees, patent holders can achieve lower prices compared with setting competitive license fees individually.

Most economics research on patent pools has focused on their impact on economic efficiency and relationship with competition policy (Layne-Farrar and Lerner 2011). By contrast, few studies have addressed how a patent pool agent distributes royalty revenues among patent holders. Among the studies conducted on sharing rules,

Aoki and Nagaoka (2004) analyzed patent pool participation by assuming that patent holders share pool revenues equally.

Kim (2004) examined the impact of vertically integrated firms that act as licensors upstream and licensees downstream. The author assumed that pool members aim to maximize the pool’s revenue and agree on an (unspecified) sharing rule.¹ Specifically, the research focused on the economic effects such as the final product price resulting from the presence of integrated firms in patent pools.

Layne-Farrar and Lerner (2011) empirically investigated the relationship between sharing rules and participation in nine large patent pools such as MPEG and Bluetooth. They found that these nine pools adopted one of the following three sharing rules: the numeric proportional sharing rule, the value-based sharing rule, and royalty-free. They showed that pools with numeric proportional sharing rules attract fewer joiners and that patent holders with higher “values” are less likely to accept numeric proportional sharing rules. They also found that none of the nine pools adopted the equal sharing rule assumed by Aoki and Nagaoka (2004).

Tesoriere (2019) theoretically studied whether a sharing rule is stable against arbitrage, meaning that pool members have no incentive to trade their patents. The study showed that the numeric proportional sharing rule is the only such rule that achieves the stability concept and that it induces a firm with few patents to stay outside the patent pool.

In our study, we conduct an axiomatic analysis of sharing rules, with a focus on the common practice of patent pool agents offering their patents in the form of multiple packages. One example from Japan is ULDAGE, a patent pool agent that manages over 800 patents necessary for television broadcasting. These 800 patents are organized into five patent packages: ARIB (for receivers of 2 K digital broadcasting), CATV (for 2 K digital cable television), Satellite-UHDTV (for 4 K/8 K), CATV-UHDTV (for cable 4 K/8 K), and IPTV (for optical fiber cable IP broadcasting). Since certain core patents have multiple technological purposes, some of these 800 patents belong to two or more packages. In addition to this agent, other prominent patent pool agents organize their patents into multiple packages. For example, Via LA (a merger of Via Licensing and MPEG LA) offers packages necessary for electronic and information technologies such as AAC, MPEG-4, and Display Port, while SISVEL provides packages such as Wi-Fi 6 and Cellular IoT. In our study, we call such a profile of packages offered by a patent pool agent a package structure and examine sharing rules that address the complexities arising from such package structures.

A package structure defines the assignment of patents to specific packages. Table 1 provides an example of a package structure, demonstrating how seven patents are organized into four packages by a (hypothetical) patent pool agent. For instance, patent 1 is contained in packages A and C. A package structure can also be formulated as a hypergraph, which is an extension of the concept of network structures. Figure 1 illustrates the package structure in Table 1 as a hypergraph. By formulating a package structure as a hypergraph, we can leverage existing research on allocation rules involving hypergraphs in the field of cooperative game theory. This allows us to apply these studies to analyze sharing rules that consider package structures.

Table 1 An example of a package structure

Fig. 1

A hypergraph $\{\{1,2\},\{3,4\},\{1,3,5\},\{6,7\}\}$

van den Nouweland et al. (1992) introduced hypergraphs to model communication structures among individuals and studied sharing rules for the profits generated by coalitions of individuals. They generalized the Myerson value (Myerson 1977), a sharing rule for cooperative games with network structures, to the class of games with hypergraph structures.²

Our objective is to provide axiomatic foundations for the numerical proportional rule among the sharing rules empirically investigated by Layne-Farrar and Lerner (2011). While Tesoriere (2019) characterized the numerical proportional rule based on its robustness against patent exchanges among patent holders, we are motivated to demonstrate that the numerical proportional rule can be characterized by changes in the structure of patent packages and their corresponding worth through the framework of cooperative games. Specifically, we adapt the framework developed by van den Nouweland et al. (1992) by replacing individuals with patents and communication structures with package structures. However, directly applying the approach of van den Nouweland et al. (1992) to our framework poses certain challenges, as the interpretation of a “pie” to be distributed among members differs between their framework and ours. To illustrate this gap, consider the following distributions of payoffs among elements 1–7 in the example shown in Fig. 1.

In the framework of van den Nouweland et al. (1992), the seven elements represent individuals, and the hypergraph structure specifies the communication relationships between them. To understand their discussion, we denote the profit generated by the members of S as v(S) and the share assigned to an individual i as $\psi _i$. As shown in Fig. 1, individuals 1 to 5 collectively generate their profit $v(\{1,2,3,4,5\})$, and individuals 6 and 7 yield $v(\{6,7\})$. In this sense, a sharing rule, $\psi$, is assumed to satisfy

$$ \psi _1+\psi _2+\psi _3+\psi _4+\psi _5=v(\{1,2,3,4,5\}){\text {, }}\psi _6+\psi _7=v(\{6,7\}). $$

However, in our framework, the hypergraph structure represents a set of patent packages, indicating that patents 1 to 5 belong to three packages: $\{1,2\}$, $\{3,4\}$, and $\{1,3,5\}$. These packages generate royalty revenue $v(\{1,2\}) + v(\{3,4\}) + v(\{1,3,5\})$. Similarly, patents 6 and 7 yield $v(\{6,7\})$. In this context, $\psi _i$ represents the share assigned to a patent i. Therefore, in our framework, a sharing rule $\psi$ is required to satisfy

$$ \psi _1+\psi _2+\psi _3+\psi _4+\psi _5+\psi _6+\psi _7=v(\{1,2\}) + v(\{3,4\}) + v(\{1,3,5\})+v(\{6,7\}). $$

We propose sharing rules that satisfy this requirement and proceed to axiomatically characterize them.

Furthermore, we address substitutability among packages. In the aforementioned example, the value, or royalty revenue, generated from a package depends only on the patents included in that package. However, when a patent pool agent introduces a new package that implements a technology related to an existing package, the introduction of the new package may impact the revenue of the existing package. Therefore, by extending the aforementioned model, we also introduce a model in which the value of each package is not only determined by its constituent patents but also influenced by the structure of the other packages offered by the agent.

As shown in the subsequent sections, when substitutability is considered, selecting a package profile that maximizes overall revenue may result in reduced earnings for some packages. This could lead to objections from patent holders whose patents are included in the affected packages. In this paper, we show that the sharing rule that equally divides the total revenue may serve as an option for patent pool agents to organize a package profile that maximizes the total revenue and prevents patent holders from deviating from the pool.

The remainder of this paper is organized as follows. Section 2 introduces the model and proposes our sharing rule. Section 3 introduces and analyzes the sharing rule that reconciles revenue maximization and objection-freeness. Section 4 provides a summary and additional remarks. Section 5 provides all the proofs and counterexamples for the independence of the axioms.

2 Model and sharing rule

2.1 Preliminary

Let $N=\{1,\ldots ,n\}$ be the set of all patents gathered by a patent pool agent, where the agent may represent an organization, manager, or administrator handling a set of patents. We call each $S\subseteq N$ a package. For every $S\subseteq N$, let v(S) denote the worth of package S, where $v(\emptyset )=0$. We suppose that the worth of a package represents the royalty revenue earned by the package. We assume that $v(S)\ge 0$ for every $S\subseteq N$.

The patent pool agent selects which packages to offer for sale from the set of all possible packages $2^N{\setminus } \{\emptyset \}$. We use $\mathcal {H}\subseteq 2^N{\setminus } \{\emptyset \}$ to denote the set of packages S selected by the patent pool agent. We call $\mathcal {H}$ a package structure or a profile of packages. For a given N, we call $(v,\mathcal {H})$ a patent package game, and let ${\text {PP}}(N)$ denote the set of all patent package games for patent set N.

Moreover, for every $S\subseteq N$ and $\mathcal {H}\subseteq 2^N{\setminus } \{\emptyset \}$, let $\mathcal {H}(S)=\{T\in \mathcal {H}| T\subseteq S\}$. Let $N^*_\mathcal {H}\subseteq N$ denote the set of patents that belong to at least one package in $\mathcal {H}$, i.e., $N^*_\mathcal {H}=\cup _{T\in \mathcal {H}}T$.

Let $\psi :{\text {PP}}(N)\rightarrow \mathbb {R}_+^N$ denote a sharing rule, where $\psi _i(v,\mathcal {H})$ represents the share assigned to patent i,³ Therefore, a sharing rule is required to divide the total revenue $\sum _{T\in \mathcal {H}}v(T)$ earned by profile $\mathcal {H}$.

Axiom 1

A sharing rule $\psi$ satisfies pool efficiency (PE) if for every $(v,\mathcal {H})\in {\text {PP}}(N)$,

$$ \sum _{j\in N}\psi _j(v,\mathcal {H})=\sum _{T\in \mathcal {H}}v(T). $$

Next, we introduce the fairness requirement, which was initially proposed by Myerson (1980) for games with network structures and later extended by van den Nouweland et al. (1992) to games with communication structures. Although the term “fairness” generally carries a wide range of meanings, we use this term for the sake of terminological consistency.

Axiom 2

A sharing rule $\psi$ satisfies fairness (F) if for every $(v,\mathcal {H})\in {\text {PP}}(N)$, $T\in \mathcal {H}$, and $i,j\in T$,

$$ \psi _i(v,\mathcal {H})-\psi _i(v,\mathcal {H}{\setminus }\{T\}) = \psi _j(v,\mathcal {H})-\psi _j(v,\mathcal {H}{\setminus }\{T\}). $$

The fairness requirement states that all patents in a package T gain or lose the same amount if T is removed from profile $\mathcal {H}$. More specifically, this axiom stipulates that patents forming a package, especially when indispensable to its functionality, be treated with equal weight. This is because each patent holds veto power over the formation of the package. Moreover, even in cases where the patents may not be essential, the axiom requires a sharing rule to treat them equally within the package if exogenous values, such as the number of citations and remaining years of validity, are not considered. Note that the absence of external values is also assumed by Tesoriere (2019) and Aoki and Nagaoka (2004). Layne-Farrar and Lerner (2011) discovered that two DVD pools implemented value-based sharing rules, while six pools utilized sharing rules unaffected by external value weights.

The following axiom requires the share of patent i to be independent of the packages that do not involve i.

Axiom 3

A sharing rule $\psi$ satisfies package independence (PI) if for every $(v,\mathcal {H})\in {\text {PP}}(N)$, $T\in \mathcal {H}$, and $i\in N{\setminus } T$,

$$ \psi _i(v,\mathcal {H})=\psi _i(v,\mathcal {H}{\setminus } \{T\}). $$

This axiom requires that the share allocated to patent i depends only on the packages in which i is included as an element. This stipulation asserts that if the contribution of i to the profile $\mathcal {H}$ remains unchanged (with v remaining constant), then the share received by i should not change. In other words, it ensures that if neither v nor the collection of packages containing i is affected by the change in profile $\mathcal {H}$, the agent does not have to recalculate the allocation to each patent i. The package independence requirement can be seen as a complementary counterpart of the fairness requirement. A sharing rule that incorporates exogenous value weights, such as those employed by the DVD pools, may not necessarily adhere to this requirement because of the exogenous weights.

The following result shows that our sharing rule is the unique rule that satisfies PI along with PE and F.

Proposition 2.1

A sharing rule $\psi$ satisfies PE, PI, and F if and only if $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T)}{|T|}$ for every $i\in N$.

We call this sharing rule the package-wise equal sharing rule. In addition to this characterization, by applying Theorem 2.3 of van den Nouweland et al. (1992), which characterizes the Myerson value, we can associate the package-wise equal sharing rule with the Shapley value⁴ For every $(v,\mathcal {H})\in {\text {PP}}(N)$ and $i\in N$, ${\text {Sh}}_i(v^\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T)}{|T|}$, where for every $(v,\mathcal {H})\in {\text {PP}}(N)$ and $S\subseteq N$, let $v^\mathcal {H}(S)=\sum _{T\in \mathcal {H}(S)}v(T)$ denote the total revenue obtained from the packages in $\mathcal {H}$ that consist only of patents in S. This equivalence with the Shapley value suggests that the package-wise equal sharing rule assigns to patent i the expected value of the marginal contributions of patent i.

2.2 Sharing rule and substitutability among packages

We introduce substitutability among packages. When the agent opts to introduce a new package T along with the current packages S in $\mathcal {H}$, the addition of the new package may influence the revenue of the existing ones in the presence of substitutability among packages. In other words, the revenue of package S in $\mathcal {H}$ can be different from that of S in $\mathcal {H}\cup \{T\}$.

Therefore, we now use $v(S,\mathcal {H})$, instead of v(S), to denote the worth of package $S\in \mathcal {H}$. This notation states that the revenue of package S may depend not only on package S but also on entire package structure $\mathcal {H}$. For every $\mathcal {H}\subseteq 2^N{\setminus }\{\emptyset \}$ and $S\in \mathcal {H}$, we call $(S,\mathcal {H})$ an embedded package.⁵ Let ${\text {EM}}_N$ be the set of all embedded packages and $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$. Moreover, let ${\text {PP}}^*(N)$ denote the set of all patent package games $(v,\mathcal {H})$ with substitutability.⁶ Let $\psi :{\text {PP}}^*(N)\rightarrow \mathbb {R}^N_+$ denote a sharing rule.

By replacing v(T) with $v(T,\mathcal {H})$, we can immediately generalize the package-wise equal sharing rule:

$$ \psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}. $$

Moreover, by setting $v_*^\mathcal {H}$ as follows, the equivalence with the Shapley value also holds: For every $\mathcal {H}\subseteq 2^N{\setminus } \{\emptyset \}$ and $S\subseteq N$, $v_{*}^\mathcal {H}(S)=\sum _{T\in \mathcal {H}(S)}v(T,\mathcal {H})$; For every $(v,\mathcal {H})\in {\text {PP}}^*(N)$ and $i\in N$, ${\text {Sh}}_i(v_{*}^\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$.

In view of these generalizations, one may conjecture that the preceding preliminary result could also be smoothly generalized to the class with substitutability. However, the following example demonstrates that the sharing rule no longer satisfies F and PI (redefined by replacing ${\text {PP}}(N)$ with ${\text {PP}}^*(N)$) in the presence of substitutability. Let $N=\{1,2\}$ and consider v satisfying Table 2, where for simplicity let $v(S,\mathcal {H}')=0$ for every $\mathcal {H}'\subseteq 2^N{\setminus } \{\emptyset \}$ with $\mathcal {H}'\not \in \{\mathcal {H}_1, \mathcal {H}_2, \mathcal {H}_3\}$ and every $S\in \mathcal {H}'$.⁷

Table 2 $\psi$ violates F and PI

The rightmost column of the table describes the shares $\psi _i=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$. If $\psi$ satisfied F, then we would obtain $\psi _1(v,\mathcal {H}_3)-\psi _1(v,\mathcal {H}_1)=\psi _2(v,\mathcal {H}_3)-\psi _2(v,\mathcal {H}_1)$. However, we have $3-1\ne 3-2$. Moreover, if $\psi$ satisfied PI, then we would have $\psi _1(v,\mathcal {H}_3)=\psi _1(v,\mathcal {H}_2)$. However, we have $3\ne 4$.

Therefore, Proposition 2.1 does not hold in the presence of substitutability. This gap arises because once $\mathcal {H}$ changes to $\mathcal {H}'$, the worth of each package $S\in \mathcal {H}\cap \mathcal {H}'$ may also change. However, by extending F and PI in a way that they restrict a sharing rule for changes of the worth v of packages, instead of the structures $\mathcal {H}$ of packages, we can overcome this gap.

Axiom 4

A sharing rule $\psi$ satisfies fairness for worth (FW) if for every $\mathcal {H}\subseteq 2^N{\setminus }\{\emptyset \}$, $T\in \mathcal {H}$, and $v,v'$ with $v'(T,\mathcal {H})>v(T,\mathcal {H})$ and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(T,\mathcal {H})\}$, we have

$$ \psi _i(v',\mathcal {H})-\psi _i(v,\mathcal {H}) = \psi _j(v',\mathcal {H})-\psi _j(v,\mathcal {H}). $$

for every $i,j\in T$.

FW requires that if the worth of package T in $\mathcal {H}$ changes with keeping the worth of the other packages unchanged, then all the patents in T should obtain or lose the same amount. In this version of the fairness requirement, it is similarly stated that if all patents in a package are essential to the functionality provided by the package, these patents should be considered equally weighted when there are no exogenous value indices.

Axiom 5

A sharing rule $\psi$ satisfies package independence for worth (PIW) if for every $\mathcal {H}\subseteq 2^N{\setminus }\{\emptyset \}$, $T\in \mathcal {H}$, and $v,v'$ with $v'(T,\mathcal {H})>v(T,\mathcal {H})$ and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(T,\mathcal {H})\}$, we have

$$ \psi _i(v',\mathcal {H})=\psi _i(v,\mathcal {H}) $$

for every $i\in N{\setminus } T$.

PIW states that if the worth of package T in $\mathcal {H}$ changes with keeping the worth of the other packages unchanged, then the change in package T does not affect the shares allocated to the patents outside of T. In other words, it ensures that if the change in v does not impact the worth of packages containing i, the allocation to i remains unchanged.

While F and PI constrain a sharing rule applied to the domain of v without considering substitutability, FW and PIW constrain a sharing rule applied to the domain of v with substitutability. In this sense, FW and PIW serve as generalizations of F and PI. Due to the different domains, there is no direct strong or weak relationship between F and FW (and, PI and PIW).⁸

For the sake of completeness, by replacing v(T) with $v(T,\mathcal {H})$, we redefine PE in the presence of substitutability.

Axiom 6

A sharing rule $\psi$ satisfies PE if for every $(v,\mathcal {H})\in {\text {PP}}^*(N)$,

$$ \sum _{j\in N}\psi _j(v,\mathcal {H})=\sum _{T\in \mathcal {H}}v(T,\mathcal {H}). $$

The following proposition shows that these three requirements axiomatically characterize the package-wise equal sharing rule in the presence of substitutability.

Proposition 2.2

A sharing rule $\psi$ satisfies PE, FW, and PIW if and only if $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$ for every $i\in N$.

Proposition 2.2 can be seen as an extension of Proposition 2.1, where the axioms F and PI are replaced by FW and PIW. However, the proofs of Propositions 2.2 and 2.1 are significantly different because FW and PIW impose constraints on changes in the worth v of packages, while F and PI focus on changes in package structure $\mathcal {H}$.

These results provide an axiomatic rationale for using the package-wise equal sharing rule and, equivalently, (an extension of) the numeric proportional sharing rule empirically analyzed by Layne-Farrar and Lerner (2011).

3 Objection-freeness and revenue maximization

In addition to ensuring a fair distribution of royalty revenue, the patent pool agent faces the task of selecting profile $\mathcal {H}$ that maximizes royalty revenue $\sum _{T\in \mathcal {H}}v(T,\mathcal {H})$. However, even if the agent chooses a revenue-maximizing profile, it may not be accepted by all patent holders who contribute to the profile. This is because some patent holders may object to the chosen profile if another profile offers them higher profits under the sharing rule.

Table 3 provides a numerical example that demonstrates the conflict between revenue-maximizing profiles and objection-free profiles.⁹ In this example, we consider two patents: patent 1 owned by patent holder 1 and patent 2 owned by patent holder 2. We test the sharing rule $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$ in this example. The rightmost column of the table shows the shares based on the package-wise equal sharing rule.

Table 3 All the profiles allow some patent holders to object

The revenue-maximizing profile is $\mathcal {H}_4$ as the total revenue is 8. However, patent holder 1 has an incentive to object to $\mathcal {H}_4$ by demanding that the agent includes the singleton package of patent 1 in $\mathcal {H}_4$, as $\psi _1(\mathcal {H}_6)=5>4=\psi _1(\mathcal {H}_4)$. Patent holder 1 can propose the singleton package $\{1\}$ without requiring the support of patent holder 2. In this sense, the revenue-maximizing profile allows some patent holders to object.

To further analyze the relationship between revenue-maximizing profiles and objection-free profiles, we formally define the concept of an objection. Let $\mathcal {H}\subseteq N{\setminus }\{\emptyset \}$ and $\psi$ be an arbitrary sharing rule. An objection to profile $\mathcal {H}$ for $\psi$ is $(S, \mathcal {H}')$ that satisfies the following three conditions:

• (a) $\psi _j(v,\mathcal {H}')>\psi _j(v,\mathcal {H})$ for every $j\in S$,

• (b1) $\mathcal {H}'{\setminus } \mathcal {H}\subseteq 2^S{\setminus } \{\emptyset \}$,

• (b2) $\mathcal {H}{\setminus } \mathcal {H}' \subseteq \{T\in \mathcal {H}|S\cap T\ne \emptyset \}$.

Condition (a) is an incentive requirement, which states that the proposed profile $\mathcal {H}'$ provides higher payoffs to the owners of the patents in S. Conditions (b1) and (b2) are feasibility requirements: (b1) states that the newly proposed packages, $\mathcal {H}'{\setminus } \mathcal {H}$, consist only of patents in S and can be formed without involving patents in $N{\setminus } S$; and (b2) states that the packages proposed for deletion, $\mathcal {H}{\setminus } \mathcal {H}'$, contain a patent in S. In other words, (b1) states that for a new package to be formed, it requires the agreement of all patent holders of S, and (b2) states that a package is canceled if at least one member disagrees. Profile $\mathcal {H}$ is objection-free for $\psi$ if no objections are raised against $\mathcal {H}$ for $\psi$.

The concept of objection is close to that of deviation used to define the core for TU-games and that of an objection proposed by Aumann and Maschler (1964) for the bargaining set. Moreover, the feasibility conditions, (b1) and (b2), are consistent with that of the formation and elimination of links in a network. For a link between two nodes (or individuals) to be formed, the agreement from both nodes is needed, whereas the elimination of a link only requires disagreement from either of the nodes.

Given the definition of an objection, we reexamine the example in Table 3. The example also shows that the set of objection-free profiles can be empty. Patent holder 2 has an incentive to withdraw its patent from package $\{1,2\}$ in $\mathcal {H}_6$ and instead propose package $\{2\}$, which leads to $\mathcal {H}_5$. In $\mathcal {H}_5$, both patent holders have an incentive to jointly offer package $\{1,2\}$ instead of the two separate packages, which results in $\mathcal {H}_4$. Therefore, we have a cycle: $\mathcal {H}_4\rightarrow (\mathcal {H}_6{\text { or }}\mathcal {H}_7)\rightarrow \mathcal {H}_5\rightarrow \mathcal {H}_4$. For $\mathcal {H}_1$, $\mathcal {H}_2$, $\mathcal {H}_3$, and $\mathcal {H}_8$, both patent holders have a common incentive to cancel the singleton packages and jointly offer the two-patent package.

This analysis suggests that the package-wise equal sharing rule (i) does not necessarily make a revenue-maximizing profile objection-free in the presence of substitutability¹⁰ and (ii) does not guarantee the existence of objection-free profiles. If (i) is resolved, then the established revenue-maximizing profile is objection-free and resolves (ii). Therefore, we explore a sharing rule that resolves (i), namely, a sharing rule that makes revenue-maximizing profiles objection-free.

The set of objection-free profiles can vary depending on v and the choice of a sharing rule, while that of revenue-maximizing profiles depends only on v. Therefore, for every $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$, let RM(v) denote the set of revenue-maximizing profiles: $RM(v)=\{\mathcal {H}\subseteq 2^N{\setminus } \{\emptyset \} | \sum _{T\in \mathcal {H}}v(T,\mathcal {H})\ge \sum _{T\in \mathcal {H}'}v(T,\mathcal {H}'){\text { for every }}\mathcal {H}'\subseteq 2^N{\setminus } \{\emptyset \}\}$. Moreover, let $OF^\psi (v)$ denote the set of objection-free profiles for the sharing rule $\psi$.

This notation allows us to formulate our objective as follows: What sharing rule $\psi$ achieves $RM(v) \subseteq OF^\psi (v)$? The following axiom plays a key role in answering this question. It states that if the total revenue either increases or is unchanged, then the shares do not decrease.

Axiom 7

A sharing rule $\psi$ satisfies pool monotonicity (PM) if for every $i\in N$ and $(v,\mathcal {H})$, $(v',\mathcal {H}')\in {\text {PP}}^*(N)$, $\sum _{T\in \mathcal {H}'}v'(T,\mathcal {H}')\ge \sum _{T\in \mathcal {H}}v(T,\mathcal {H})$ $\Rightarrow$ $\psi _i(v',\mathcal {H}') \ge \psi _i(v,\mathcal {H})$.

Moreover, we redefine F in the presence of substitutability by replacing ${\text {PP}}(N)$ in Axiom 2 with ${\text {PP}}^*(N)$.

Axiom 8

A sharing rule $\psi$ satisfies F if for every $(v,\mathcal {H})\in {\text {PP}}^*(N)$, $T\in \mathcal {H}$, and $i,j\in T$,

$$ \psi _i(v,\mathcal {H})-\psi _i(v,\mathcal {H}{\setminus }\{T\}) = \psi _j(v,\mathcal {H})-\psi _j(v,\mathcal {H}{\setminus }\{T\}). $$

The following proposition shows that a single sharing rule, which is different from the rule $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$, satisfies F and PM along with PE.

Proposition 3.1

A sharing rule $\psi$ satisfies PE, F, and PM if and only if $\psi _i(v,\mathcal {H})=\sum _{T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|N|}$ for every $i\in N$.

Proposition 3.1 suggests that PE, F, and PM yield the sharing rule that distributes the total revenue equally among all patents. This rule directly divides the total revenue among all patents, while the rule $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$ divides the revenue of each package among the constituent patents of the package. The following proposition shows that the rule characterized in Proposition 3.1 makes revenue-maximizing profiles objection-free for an arbitrary v.

Proposition 3.2

If a sharing rule $\psi$ satisfies:

• PM, then $RM(v) \subseteq OF^\psi (v)$ for every $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$.

• PM, F, and PE, then $RM(v)=OF^\psi (v)$ for every $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$.

Proposition 3.2 highlights the central role of PM in reconciling revenue maximization and objection-freeness. The first statement suggests that a pool monotonic sharing rule ensures the existence of a revenue-maximizing and objection-free profile since RM(v) is always nonempty for an arbitrary v. Therefore, regardless of the situation v the pool agent faces, pool monotonicity enables the agent to choose a revenue-maximizing and objection-free profile. In other words, assuming that the pool agent chooses a revenue maximizing profile, pool monotonicity is a sufficient condition of a sharing rule for the selected profile to be stable in the sense of the objection-freeness. However, it also suggests that PM may allow for the possibility of $OF^\psi (v){\setminus } RM(v)$ to be nonempty. We can consider profile $\mathcal {H}\in OF^\psi (v){\setminus } RM(v)$ to be an inefficient “stuck” profile, as it fails to maximize the total revenue, while all potential objections are not feasible.

The second statement shows that F and PE eliminate these stuck profiles and that a profile becomes revenue-maximizing if and only if it is objection-free. Moreover, F and PE are “tight” requirements because removing either could allow for the possibility of a sharing rule that makes $OF^\psi (v){\setminus } RM(v)$ nonempty, as described below.

Consider the following sharing rule: $\tilde{\psi }_i(v,\mathcal {H})=\omega _i \sum _{T\in \mathcal {H}} v(T,\mathcal {H})$ with $\omega _1=1$ and $\omega _j=0$ for $j\in 2,...,n$. This rule satisfies PM and PE but not F. We apply this sharing rule to the example in Table 4, where let $v(S,\mathcal {H})=0$ for every $\mathcal {H}\subseteq 2^N{\setminus } \{\emptyset \}$ with $\mathcal {H}\not \in \{\mathcal {H}_1,\mathcal {H}_2\}$ and $S\in \mathcal {H}$.

Table 4 The combination of PM and PE does not imply $RM(v)=OF^\psi (v)$

In this example, $\mathcal {H}_2$ is the unique revenue-maximizing profile, while both $\mathcal {H}_1$ and $\mathcal {H}_2$ are objection-free profiles because without the involvement of patent 2, the holder of patent 1 cannot change $\mathcal {H}_1$ to $\mathcal {H}_2$.¹¹

Now, consider the sharing rule $\hat{\psi }_i(v,\mathcal {H})=c$ for every $i\in N$, where c is an arbitrary constant. This rule readily violates PE but satisfies PM and F since it always assigns c to all patents. As a result, no one has any incentive to object to any profile, and $OF^{\hat{\psi }}$ includes all profiles. However, the revenue maximization requirement still selects $\mathcal {H}_2$.

The combination of Propositions 3.1 and 3.2 implies that the sharing rule $\psi _i(v,\mathcal {H})=\sum _{T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|N|}$ guarantees the existence of objection-free profiles and that each objection-free profile maximizes the total revenue. In this sense, our result suggests that this sharing rule serves as a mediator between the patent pool agent, which aims to maximize revenue, and patent holders, which may have an incentive to object.

4 Conclusion

In this study, we conducted an axiomatic analysis of sharing rules for royalty revenue by incorporating package structures. The main findings can be summarized as follows.

• The package-wise equal sharing rule, which assigns to each patent i the summation of the average revenues of the packages containing patent i, is the unique sharing rule that satisfies the pool efficiency, fairness, and package independence requirements. Moreover, this sharing rule incorporates the contributions of each patent to the packages and can be seen as an extension of the numeric proportional sharing rule empirically analyzed by Layne-Farrar and Lerner (2011) (Sect. 2).

• The package-wise equal sharing rule does not guarantee that revenue-maximizing profiles are objection-free. The sharing rule that equally distributes the total revenue among all patents ensures that revenue-maximizing profiles are objection-free. This rule is characterized by the pool efficiency, fairness, and pool monotonicity requirements (Sect. 3).

In this study, a sharing rule is formulated as a function of v and $\mathcal {H}$ (i.e., royalty revenues and a patent package structure). However, other factors can contribute to the evaluation of patents, such as total citation counts and the remaining years until expiration, as highlighted by Layne-Farrar and Lerner (2011). To incorporate citation counts into a sharing rule, it would be necessary to consider a directed graph that represents the citation network among patents. This graph would capture the relationships between patents based on citations, allowing us to incorporate citation scores as an additional criterion in the patent evaluation process. The remaining years until patent expiration can be represented as an n-dimensional vector. However, further research is needed to propose a sharing rule that integrates all these factors and remains practical and explainable.

Moreover, we conjecture that the entire set of $v\in {\text {PP}}^*(N)$ forms the class of “hypergraph function form games,” akin to partition function form games. To analyze this class of games, it is needed to establish key properties such as superadditivity and convexity, similar to what Hafalir (2007) has done for partition function form games. Investigating these properties is part of our ongoing and future research.

5 Proofs and independence of axioms

5.1 Proof of Proposition 2.1

A sharing rule $\psi$ satisfies PE, PI and F if and only if $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T)}{|T|}$ for every $i\in N$.

Proof

The sharing rule $\psi$ straightforwardly satisfies the three axioms. Hence, we focus on the uniqueness. Suppose that there are two sharing rules $\psi$ and $\psi '$ such that both satisfy the three axioms. In the same manner as Myerson (1980), van den Nouweland et al. (1992), and Casajus (2009), there is $(v,\mathcal {H})\in {\text {PP}}(N)$ satisfying the following (1) and (2)¹²:

$$\begin{aligned}{} & {} \psi (v,\mathcal {H}) \ne \psi '(v,\mathcal {H}), \end{aligned}$$

(1)

$$\begin{aligned}{} & {} \psi (v,\mathcal {M}) = \psi '(v,\mathcal {M}){\text { for every }}\,\mathcal {M}\subseteq \mathcal {H}. \end{aligned}$$

(2)

Moreover, by (1), we have

$$\begin{aligned} {\text { there is}}\, k\in N\, {\text {such that}}\, \psi _k(v,\mathcal {H}) > \psi '_k(v,\mathcal {H}). \end{aligned}$$

(3)

If the patent k belongs to exactly one package that is a singleton, i.e., $\{T\in \mathcal {H}|k\in T\}=\{\{k\}\}$, or belongs to no package, then PI and PE result in a contradiction by removing all packages from $\mathcal {H}$ except for $\{k\}$. Therefore, we consider $T\in \mathcal {H}$ with $|T|\ge 2$ and $k\in T$. For every $i\in T\in \mathcal {H}$ with $i\ne k$, we have

$$\begin{aligned}{} & {} \psi _i(v,\mathcal {H}) - \psi _k(v,\mathcal {H}) \overset{{\text {F}}}{=} \psi _i(v,\mathcal {H}{\setminus }\{T\}) - \psi _k(v,\mathcal {H}{\setminus }\{T\})\\{} & {} \quad \overset{(2)}{=}\psi '_i(v,\mathcal {H}{\setminus }\{T\}) - \psi '_k(v,\mathcal {H}{\setminus }\{T\})\\{} & {} \quad \overset{{\text {F}}}{=}\psi '_i(v,\mathcal {H}) - \psi '_k(v,\mathcal {H}). \end{aligned}$$

By (3), we obtain $\psi _i(v,\mathcal {H})>\psi '_i(v,\mathcal {H})$. Letting $C_k\in \mathcal {C}(\mathcal {H})$ be the component containing patent k,¹³ we repeat this for every $i\in C_k$ and then obtain

$$ \sum _{i\in C_k} \psi _i(v,\mathcal {H}) > \sum _{i\in C_k} \psi '_i(v,\mathcal {H}). $$

For the component $C_k\in \mathcal {C}(\mathcal {H})$, PE implies that

$$ \sum _{i\in N{\setminus } C_k} \psi _i(v,\mathcal {H}) < \sum _{i\in N{\setminus } C_k} \psi '_i(v,\mathcal {H}). $$

However, for an arbitrary $T\in \mathcal {H}(C_k)$, we have

$$\begin{aligned}{} & {} \sum _{i\in N{\setminus } C_k} \psi _i(v,\mathcal {H}) \overset{{\text {PI}}}{=} \sum _{i\in N{\setminus } C_k} \psi _i(v,\mathcal {H}{\setminus } \{T\})\\{} & {} \quad \overset{(2)}{=} \sum _{i\in N{\setminus } C_k} \psi '_i(v,\mathcal {H}{\setminus } \{T\})\\{} & {} \quad \overset{{\text {PI}}}{=} \sum _{i\in N{\setminus } C_k} \psi '_i(v,\mathcal {H}), \end{aligned}$$

which is a contradiction. $\square$

5.2 Independence of axioms for Proposition 2.1

[A sharing rule that satisfies PI and F but not PE]

Sharing rule $\psi ^1$

$$ \psi ^1_i(v,\mathcal {H})=\sum _{T:i \in T\in \mathcal {H}}v(T). $$

satisfies PI and F but not PE.

[A sharing rule that satisfies PE and PI but not F]

Let $\psi ^2_i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}Z(i,T,v)$, where

$$ Z(i,T,v)=\left\{ \begin{array}{ll}v(T) &{} {\text {if index}}\, i\, {\text {is the minimum in}}\, T, \\ 0 &{} {\text {otherwise.}}\end{array}\right. $$

This rule satisfies PE and PI but not F.

[A sharing rule that satisfies PE and F but not PI]

Let $\psi ^3$ be

$$ \psi ^3_i(v,\mathcal {H})=\frac{\sum _{T\in \mathcal {H}}v(T)}{|N|}. $$

This sharing rule straightforwardly satisfies PE and F. However, it violates PI: Let $N=\{1,2\}$, $\mathcal {H}=\{\{1\},\{2\}\}$, and $v(\{1\})=v(\{2\})=1$. We have $\psi ^3_1(v,\mathcal {H})=1\ne 0.5 = \psi ^3_1(v,\mathcal {H}{\setminus } \{\{2\}\})$.

5.3 Proof of Proposition 2.2

A sharing rule $\psi$ satisfies PE, FW, and PIW if and only if $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$ for every $i\in N$.

Proof

Sufficiency: We show that $\psi _i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$ satisfies PE, FW, and PIW. For every $(v,\mathcal {H})\in {\text {PP}}^*(N)$, we have

$$\begin{aligned} \sum _{j\in N}\psi _j(v,\mathcal {H}) =\sum _{j\in N}\sum _{\begin{array}{c} T\in \mathcal {H}\\ j\in T \end{array}}\frac{v(T,\mathcal {H})}{|T|} =\sum _{T\in \mathcal {H}}\sum _{j\in T}\frac{v(T,\mathcal {H})}{|T|} =\sum _{T\in \mathcal {H}}v(T,\mathcal {H}). \end{aligned}$$

Therefore, $\psi$ satisfies PE.

Next, let $\mathcal {H}\subseteq 2^N{\setminus }\{\emptyset \}$, $T^*\in \mathcal {H}$, and $v,v'$ with $v'(T^*,\mathcal {H})>v(T^*,\mathcal {H})$ and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(T^*,\mathcal {H})\}$. For every $i,j\in T^*$, we have

$$\begin{aligned} \psi _i(v',\mathcal {H})-\psi _i(v,\mathcal {H})= & {} \sum _{T:i\in T\in \mathcal {H}}\frac{v'(T,\mathcal {H})}{|T|} - \sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}\\= & {} \frac{v'(T^*,\mathcal {H})}{|T^*|} - \frac{v(T^*,\mathcal {H})}{|T^*|}\\= & {} \sum _{T:j\in T\in \mathcal {H}}\frac{v'(T,\mathcal {H})}{|T|} - \sum _{T:j\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}\\= & {} \psi _j(v',\mathcal {H})-\psi _j(v,\mathcal {H}). \end{aligned}$$

Therefore, $\psi$ satisfies FW.

Moreover, for every $i\in N{\setminus } T^*$,

$$ \psi _i(v,\mathcal {H})= \sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}=\sum _{T:i\in T\in \mathcal {H}}\frac{v'(T,\mathcal {H})}{|T|} =\psi _i(v',\mathcal {H}). $$

Hence, $\psi$ satisfies PIW.

Uniqueness: Let $\psi$ be a sharing rule that satisfies PE, FW, and PIW. Let $(v,\mathcal {H})\in {\text {PP}}^*(N)$. For every $i\in N{\setminus } N^*_\mathcal {H}$, we have $\psi _i(v,\mathcal {H})\overset{{\text {PE,PIW}}}{=}0=\sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$ as $\{T:i\in T\in \mathcal {H}\}=\emptyset$. Henceforth, we consider $i\in N^*_\mathcal {H}$. Let $\tau _i(\mathcal {H})$ denote the set of packages in $\mathcal {H}$ that contain i:

$$ \tau _i(\mathcal {H})=\{T\in \mathcal {H}| i\in T\}=\{T_1,\ldots , T_k,\ldots ,T_m\}. $$

First, consider $v_0$ with $v_0(S,\mathcal {H})=0$ for every $(S,\mathcal {H})\in {\text {EM}}_N$. By PE and the nonnegativity of $\psi$, $\psi _i(v_0,\mathcal {H})=0$.

Next, for every $k=1,\ldots ,m$, define $v_k$ as follows:

$$\begin{aligned} v_k(T_k,\mathcal {H})=v(T_k,\mathcal {H}) \end{aligned}$$

(4)

and

$$\begin{aligned} v_k(S,\mathcal {M})=v_{k-1}(S,\mathcal {M}){\text { for every }}(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(T_k,\mathcal {H})\}. \end{aligned}$$

(5)

By PIW, we have

$$\begin{aligned} \psi _j(v_k,\mathcal {H})=\psi _j(v_{k-1},\mathcal {H}) {\text { for every }}j\in N{\setminus } T_k. \end{aligned}$$

(6)

Moreover, FW enables us to define

$$\begin{aligned} a_k := \psi _j(v_k,\mathcal {H})-\psi _j(v_{k-1},\mathcal {H}) {\text { for every }}j\in T_k. \end{aligned}$$

(7)

By PE, for every $k=1,\ldots ,m$, we obtain

$$\begin{aligned} \sum _{j\in N}\psi _j(v_k,\mathcal {H})-\sum _{j\in N}\psi _j(v_{k-1},\mathcal {H}) =\sum _{h=1}^k v_k(T_h,\mathcal {H})-\sum _{h=1}^{k-1} v_{k-1}(T_h,\mathcal {H}). \end{aligned}$$

(8)

The left-hand side of (8) is

$$\begin{aligned} \sum _{j\in N}\psi _j(v_k,\mathcal {H})-\sum _{j\in N}\psi _j(v_{k-1},\mathcal {H})\overset{(6)}{=} & {} \sum _{j\in T_k}\psi _j(v_k,\mathcal {H})-\sum _{j\in T_k}\psi _j(v_{k-1},\mathcal {H})\\\overset{(7)}{=} & {} |T_k|\cdot a_k. \end{aligned}$$

The right-hand side of (8) is

$$\begin{aligned} \sum _{h=1}^k v_k(T_h,\mathcal {H})-\sum _{h=1}^{k-1} v_{k-1}(T_h,\mathcal {H}) \overset{(5)}{=}v_k(T_k,\mathcal {H}) \overset{(4)}{=}v(T_k,\mathcal {H}). \end{aligned}$$

Hence, we obtain $a_k=\frac{v(T_k,\mathcal {H})}{|T_k|}$.

From $i\in T_k$ and (7), it follows that

$$\begin{aligned} \psi _i(v_k,\mathcal {H})= & {} a_k+\psi _i(v_{k-1},\mathcal {H})\\= & {} a_k+a_{k-1}+\psi _i(v_{k-2},\mathcal {H})\\= & {} a_k+a_{k-1}+\cdots +a_1. \end{aligned}$$

Therefore, we have $\psi _i(v,\mathcal {H})\overset{{\text {PIW}}}{=}\psi _i(v_m,\mathcal {H})=a_m+a_{k-1}+\cdots +a_1= \sum _{T:i\in T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|T|}$. $\square$

5.4 Independence of axioms for Proposition 2.2

[A sharing rule that satisfies PIW and FW but not PE]

Let $\psi ^4$ be

$$ \psi ^4_i(v,\mathcal {H})=\sum _{T:i \in T\in \mathcal {H}}v(T,\mathcal {H}). $$

This sharing rule satisfies FW and PIW by replacing $\frac{v(T,\mathcal {H})}{|T|}$ with $v(T,\mathcal {H})$ in the proof of the sufficiency-part of Proposition 2.2. However, $\psi ^4$ violates PE as it is not divided by |T|.

[A sharing rule that satisfies PE and FW but not PIW]

Let $\psi ^5$ be

$$ \psi ^5_i(v,\mathcal {H})=\frac{1}{|C|}\sum _{T\in \mathcal {H}(C)}v(T,\mathcal {H}), $$

where C is the component of $\mathcal {H}$ that contains i. This sharing rule straightforwardly satisfies PE. It also satisfies FW as follows: Let $\mathcal {H}\subseteq 2^N{\setminus }\{\emptyset \}$, $T^*\in \mathcal {H}$, and $v,v'$ with $v'(T^*,\mathcal {H})>v(T^*,\mathcal {H})$ and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(T^*,\mathcal {H})\}$. For every $i,j\in T^*$, we have

$$\begin{aligned} \psi ^5_i(v',\mathcal {H})-\psi ^5_i(v,\mathcal {H}) = \frac{1}{|C|}v'(T^*,\mathcal {H}) - \frac{1}{|C|}v(T^*,\mathcal {H}) = \psi ^5_j(v',\mathcal {H})-\psi ^5_j(v,\mathcal {H}). \end{aligned}$$

However, it violates PIW: Consider, for example, $N=\{1,2\}$, $\mathcal {H}=\{\{1\},\{1,2\}\}$, $v(\{1,2\},\mathcal {H})=1$, and $v(S,\mathcal {M})=0$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(\{1,2\},\mathcal {H})\}$. Let $v'(\{1\},\mathcal {H})=5$, and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(\{1\},\mathcal {H})\}$. Then, we have $\psi ^4_2(v',\mathcal {H})=\frac{1+5}{2} \ne \frac{1}{2}=\psi ^4_2(v,\mathcal {H})$.

[A sharing rule that satisfies PE and PIW but not FW]

Let $\psi ^6_i(v,\mathcal {H})=\sum _{T:i\in T\in \mathcal {H}}Z(i,T,v)$, where

$$ Z(i,T,v)=\left\{ \begin{array}{ll}v(T,\mathcal {H}) &{} {\text {if index}}\, i\, {\text {is the minimum in}}\, T, \\ 0 &{} {\text {otherwise.}}\end{array}\right. $$

This sharing rule satisfies PE because

$$\begin{aligned} \sum _{j\in N}\psi ^6_j(v,\mathcal {H}) =\sum _{j\in N}\sum _{\begin{array}{c} T\in \mathcal {H}\\ j\in T \end{array}}Z(j,T,v) =\sum _{T\in \mathcal {H}}\sum _{j\in T}Z(j,T,v) =\sum _{T\in \mathcal {H}}v(T,\mathcal {H}). \end{aligned}$$

Moreover, it satisfies PIW as follows: Let $\mathcal {H}\subseteq 2^N{\setminus }\{\emptyset \}$, $T^*\in \mathcal {H}$, and $v,v'$ with $v'(T^*,\mathcal {H})>v(T^*,\mathcal {H})$ and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(T^*,\mathcal {H})\}$. For every $i\in N{\setminus } T^*$, we have $\psi ^6_i(v',\mathcal {H}) = \sum _{T:i\in T\in \mathcal {H}}Z(i,T,v')$. Since $i\not \in T^*$, we have $Z(i,T,v')=Z(i,T,v)$ for every $T:i\in T\in \mathcal {H}$. Therefore, $\sum _{T:i\in T\in \mathcal {H}}Z(i,T,v')=\sum _{T:i\in T\in \mathcal {H}}Z(i,T,v)= \psi ^6_i(v,\mathcal {H})$. However, $\psi ^6$ violates FW: Consider $N=\{1,2\}$, $\mathcal {H}=\{\{1,2\}\}$, $v(\{1,2\},\mathcal {H})=1$, and $v(S,\mathcal {M})=0$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(\{1,2\},\mathcal {H})\}$. Let $v'(\{1,2\},\mathcal {H})=5$, and $v'(S,\mathcal {M})=v(S,\mathcal {M})$ for every $(S,\mathcal {M})\in {\text {EM}}_N{\setminus } \{(\{1,2\},\mathcal {H})\}$. Then, we have $\psi ^6_1(v',\mathcal {H})-\psi ^6_1(v,\mathcal {H})=5-1 \ne 0-0=\psi ^6_2(v',\mathcal {H})-\psi ^6_2(v,\mathcal {H})$.

5.5 Proof of Proposition 3.1

A sharing rule $\psi$ satisfies PE, F, and PM if and only if $\psi _i(v,\mathcal {H})=\sum _{T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|N|}$ for every $i\in N$.

Proof

The sharing rule $\sum _{T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|N|}$ immediately satisfies the three axioms. We show that the function is the unique sharing rule satisfying PE, F, and PM.

Let $(v,\mathcal {H})\in {\text {PP}}^*(N)$ and arbitrarily fix $i\in N$. Let $\mathcal {H}'=\mathcal {H}\cup \{N\}$. Moreover, define $v'$ as follows:

$$\begin{aligned} v'(N,\mathcal {H}'):= & {} \sum _{T\in \mathcal {H}}v(T,\mathcal {H}),\nonumber \\ v'(S,\mathcal {M}):= & {} 0 {\text { for every}}\, (S,\mathcal {M})\in {\text {EM}}(N){\setminus } \{(N,\mathcal {H}')\}. \end{aligned}$$

(9)

We have $\sum _{T\in \mathcal {H}'{\setminus } \{N\}}v'(T,\mathcal {H}')=0$. From F, it follows that $\psi _i(v',\mathcal {H}')-\psi _i(v',\mathcal {H}'{\setminus } \{N\})=\psi _j(v',\mathcal {H}')-\psi _j(v',\mathcal {H}'{\setminus } \{N\})$ for every $j\in N$. PE and the nonnegativity of $\psi$ imply that

$$\begin{aligned} \psi _j(v',\mathcal {H}'{\setminus }\{N\})=0 {\text { for every}}\, j \in N. \end{aligned}$$

Hence, $\psi _i(v',\mathcal {H}')=\psi _j(v',\mathcal {H}')$ for every $j\in N$. By PE, $\psi _i(v',\mathcal {H}')=\frac{1}{|N|}\sum _{T\in \mathcal {H}'}v'(T,\mathcal {H}')$. Moreover, by PM, we have $\psi _i(v,\mathcal {H})=\psi _i(v',\mathcal {H}')$. Hence, we obtain

$$ \psi _i(v,\mathcal {H})=\psi _i(v',\mathcal {H}')=\frac{1}{|N|}\sum _{T\in \mathcal {H}'}v'(T,\mathcal {H}')\overset{(9)}{=}\frac{1}{|N|}\sum _{T\in \mathcal {H}}v(T,\mathcal {H}). $$

$\square$

5.6 Independence of axioms for 3.1

[A sharing rule that satisfies PE and F but not PM]

Let

$$ \psi ^7_i(v,\mathcal {H})=a_i(v) +\frac{1}{|N|}\left[ \sum _{T\in \mathcal {H}}v(T,\mathcal {H})-\sum _{j\in N}a_j(v)\right] , $$

where $a_i(v)=\omega _i \cdot \min _{\mathcal {H}':\emptyset \ne \mathcal {H}'\subseteq 2^N{\setminus }\{\emptyset \}} \sum _{T\in \mathcal {H}'}v(T,\mathcal {H}')$, and $\omega _1=1$, $\omega _j=0$ for every $j=2,\ldots ,n$. We show that $\psi ^7$ satisfies PE and F. For PE, we have $\sum _{j\in N}\psi _j^7(v,\mathcal {H})=\sum _{j\in N}a_j(v) + [\sum _{T\in \mathcal {H}}v(T,\mathcal {H})-\sum _{j\in N}a_j(v)]=\sum _{T\in \mathcal {H}}v(T,\mathcal {H})$. For F, letting $(v,\mathcal {H})\in {\text {PP}}^*(N)$, $T^*\in \mathcal {H}$, and $i,j\in T^*$, we have

$$\begin{aligned} \psi ^7_i(v,\mathcal {H})-\psi ^7_i(v,\mathcal {H}{\setminus }\{T^*\})= & {} \frac{1}{|N|}\left[ \sum _{T\in \mathcal {H}}v(T,\mathcal {H})-\sum _{T\in \mathcal {H}{\setminus }\{T^*\}}v(T,\mathcal {H}{\setminus }\{T^*\})\right] \\= & {} \psi ^7_j(v,\mathcal {H})-\psi ^7_j(v,\mathcal {H}{\setminus }\{T\}). \end{aligned}$$

However, it violates PM as follows. Let $N=\{1,2\}$, $\mathcal {H}_1=\{\{1\}\}$, $\mathcal {H}_2=\{\{2\}\}$, and $\mathcal {H}_3=\{\{1\},\{2\},\{1,2\}\}$. Moreover, let

$v(\{2\},\mathcal {H}_2)=7$, $v(\{1\},\mathcal {H}_3)=2$, $v(\{2\},\mathcal {H}_3)=2$, $v(\{1,2\},\mathcal {H}_3)=4$,

and $v(S,\mathcal {M})=99$ for other $(S,\mathcal {M})$. Now, let

$v'(\{1\},\mathcal {H}_1)=2$, $v'(\{1\},\mathcal {H}_3)=2$, $v'(\{2\},\mathcal {H}_3)=2$, $v'(\{1,2\},\mathcal {H}_3)=6$,

and $v'(S,\mathcal {M})=99$ for other $(S,\mathcal {M})$. We have $\sum _{T\in \mathcal {H}_3}v'(T,\mathcal {H}_3)= 2+2+6 =10 > 8= 2+2+4 =\sum _{T\in \mathcal {H}_3}v(T,\mathcal {H}_3)$. However, $\psi _1(v',\mathcal {H}_3) = 2+ \frac{1}{2}[10-2]=6 < 7.5 = 7+ \frac{1}{2}[8-7] = \psi _1(v,\mathcal {H}_3)$.

[A sharing rule that satisfies PE and PM but not F]

Let $\psi ^8_i(v,\mathcal {H})=\omega _i\cdot \sum _{T\in \mathcal {H}}v(T,\mathcal {H})$, where $\sum _{j\in N}\omega _j=1$ and $\omega _j\ne \frac{1}{n}$ for some $j\in N$. This rule straightforwardly satisfies PE and PM but violates F as follows. Let $N=\{1,2\}$ and $\mathcal {H}=\{\{1\},\{2\},\{1,2\}\}$. Moreover, let $v(\{1,2\},\mathcal {H})=10$, $v(\{1\},\mathcal {H})=v(\{1\},\mathcal {H}{\setminus }\{\{1,2\}\})=0$, and $v(\{2\},\mathcal {H})=v(\{2\},\mathcal {H}{\setminus }\{\{1,2\}\})=0$. We consider, for example, $(\omega _1,\omega _2)=(0.6,0.4)$. We have $\psi ^8_1(v,\mathcal {H})-\psi ^8_1(v,\mathcal {H}{\setminus }\{\{1,2\}\})=6-0>4-0=\psi ^8_2(v,\mathcal {H})-\psi ^8_2(v,\mathcal {H}{\setminus }\{\{1,2\}\})$.

[A sharing rule that satisfies PM and F but not PE]

Sharing rule $\psi ^9_i(v,\mathcal {H})=\frac{2}{n}\cdot \sum _{T\in \mathcal {H}}v(T,\mathcal {H})$ satisfies F and PM. However, it violates PE as the summation is doubled.

5.7 Proof of Proposition 3.2

If a sharing rule $\psi$ satisfies:

• PM, then $RM(v) \subseteq OF^\psi (v)$ for every $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$.

• PM, F, and PE, then $RM(v)=OF^\psi (v)$ for every $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$.

Proof

Let $\psi$ satisfies PM. Fixing an arbitrary $v:{\text {EM}}_N \rightarrow \mathbb {R}_+$, we prove that $RM(v)\subseteq OF^\psi (v)$. Assume that there is $\mathcal {H}\in RM(v)$ such that $\mathcal {H}\not \in OF^\psi (v)$. By $\mathcal {H}\not \in OF^\psi (v)$, there is $(S,\mathcal {H}')$ such that $\mathcal {H}'$ satisfies (a), (b1), and (b2). It follows from (a) that $\psi _j(v,\mathcal {H}')>\psi _j(v,\mathcal {H})$ for every $j\in S$. Then, we have $\sum _{T\in \mathcal {H}'}v(T,\mathcal {H}') > \sum _{T\in \mathcal {H}}v(T,\mathcal {H})$ because if $\sum _{T\in \mathcal {H}'}v(T,\mathcal {H}') \le \sum _{T\in \mathcal {H}}v(T,\mathcal {H})$, PM implies that $\psi _j(v,\mathcal {H}')\le \psi _j(v,\mathcal {H})$ for every $j\in N$, which is a contradiction. Therefore, $\sum _{T\in \mathcal {H}'}v(T,\mathcal {H}') > \sum _{T\in \mathcal {H}}v(T,\mathcal {H})$. However, $\mathcal {H}\in RM(v)$ implies that $\sum _{T\in \mathcal {H}}v(T,\mathcal {H})\ge \sum _{T\in \mathcal {H}'}v(T,\mathcal {H}')$, which contradicts the assumption.

Now, let $\psi$ satisfies PM, F, and PE. By Proposition 3.1, $\psi _i(v,\mathcal {H})=\sum _{T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|N|}$. Since $\psi$ satisfies PM, it suffices to prove that $OF^\psi (v)\subseteq RM(v)$. We assume that there is $\mathcal {H}\in OF^\psi (v)$ such that $\mathcal {H}\not \in RM(v)$. Since $\mathcal {H}$ is not a revenue-maximizing profile, consider $\mathcal {H}'\in RM(v)$ with $\mathcal {H}'\ne \mathcal {H}$. We have $\psi _j(v,\mathcal {H}')=\sum _{T\in \mathcal {H}'}\frac{v(T,\mathcal {H}')}{|N|} > \sum _{T\in \mathcal {H}}\frac{v(T,\mathcal {H})}{|N|}=\psi _j(v,\mathcal {H})$ for every $j\in N$. Hence, $(N,\mathcal {H}')$ satisfies (a). Moreover, $(N,\mathcal {H}')$ satisfies $\mathcal {H}'{\setminus } \mathcal {H}\subseteq 2^N{\setminus } \{\emptyset \}$, i.e., (b1); and every package $T\in \mathcal {H}{\setminus } \mathcal {H}'$ has a nonempty intersection with N, i.e, (b2). Hence, $(N,\mathcal {H}')$ is an objection to $\mathcal {H}$ for $\psi$. This contradicts $\mathcal {H}\in OF^\psi (v)$. $\square$