Abstract
, we model the problem as a cooperative game with transferable utility, allowing for patents that have substitutes. Assuming that a value has been assigned to these "weak" patents, we obtain a formula for the Shapley value that gives an insight into what FRAND agreements could look like.

gathered together in a patent pool and licensed as a single package to prevent excessive independent pricing and to reduce transaction costs through the possibility of one-stop shopping.
Augustin Cournot established in 1838 that a merger between two monopolists producing complementary goods generates both a higher joint profit and a lower final price by preventing the accumulation of margins. Shapiro (2001) points out that Cournot's complements problem arises when several firms own intellectual property rights required for the development of a product. If they do not coordinate, each patent holder will charge an excessive price in exchange for access to its intellectual property right, without taking into account the impact of its decision on the level of sales of the final product and on the revenue of all patent holders. The complements problem is particularly crucial in the context of standard-setting since access to all complementary intellectual property rights must be granted in order to ensure full compatibility with the standardized technology. This situation is referred to as the "tragedy of the anticommons" by Heller and Eisenberg (1998) because, in contrast with the tragedy of the commons, the fragmentation of the intellectual property rights leads to an underutilization of the associated standard.
While there is no doubt about the global profitability of pooling complementary intellectual property rights, patent holders have to overcome substantial obstacles in order to ensure the pool formation: finding agreement on the amount to be charged as a licensing fee and on the way to split the collected revenues among patent holders. In their case study of the MPEG-2 video patent pool, Lerner et al. (2007) illustrate the difficulty in reconciling heterogeneous objectives.
1 One of the most debated issues during the pool formation was the licensing fee to be charged to licensees. Some members, like Columbia University, wanted to maximize the licensing revenue they would receive, in contrast with other companies whose main purpose was to accelerate the adoption of the standard. This was the case for Sony, both licensor and licensee of MPEG-2 patents, which focuses on maximizing the sales of its standard-based products. Many SSOs, such as the 3G Patent Platform Partnership 2 and the RFID Consortium, 3 require their members to stick to a particular commitment: the licensing of the intellectual property rights composing the standardized technology under Fair, Reasonable And Non-Discriminatory (FRAND) terms. This requirement mainly aims at avoiding abusive licensing terms that result from the monopoly power given to pool members. Indeed, once a standard has become popular in an industry, it becomes very costly for a firm to produce a non-compliant good and patent owners are thus tempted to charge excessive licensing fees. Such opportunistic behavior -referred to as a hold-up in the literature -must be avoided since it is likely to have two adverse effects on economic efficiency: an increase in the deadweight loss resulting from market power and the selection of inferior technologies (Llanes and Poblete, 2014) . According to Shapiro (2001) , if standard-setting bodies such as the International Telecommunications Unions (ITU), the European Telecommunications Standard Institute (ETSI) or the American National Standards Institute (ANSI) require their participants to accept the licensing of any essential patent under FRAND terms before adopting a standard, this is primarily to address the risk of hold-up. The guarantee that no licensee will be subject to discriminatory treatment and that the fee received from them must stay within a reasonable interval strongly reduces the possibility of monopoly power abuses. It should be noted that, while FRAND predominates in Europe, the terms are usually restricted to RAND (Reasonable and Non-Discriminatory) in the United States. With the requirement for reasonableness and non-discrimination mostly oriented toward the hold-up problem, the American RAND commitment mainly focuses on ensuring economic efficiency issues, leaving aside fairness considerations. Despite being often referred to, the notion of FRAND agreement suffers from a lack of definition. Several methods, beyond simple proportionality rules, have been advocated to give content to this ambiguous normative concept, but none of them has ever gained universal approval. Baumol and Swanson (2005) have suggested using the Efficient Component Pricing Rule (ECPR). This approach assumes that multiple technologies compete during the development phase and that the one winning an auction eventually becomes the effective standard. This auction mechanism requires that intellectual property right holders submit potential licensing fees to the downstream users who then elect the winner. Baumol and Swanson show that the issue of the auction provides a benchmark for a reasonable licensing fee as it reflects the state of competition before the adoption of the standard. They specify the non-discriminatory component by requiring licensing fees to be competitively neutral, i.e. such that patent holders are indifferent between licensing their technology to rivals and producing the final product themselves. For that purpose, they adapt the ECPR in order for the license fee to compensate the licensor for the incremental licensing costs as well as for the opportunity cost of licensing the technology. Layne-Farrar et al. (2007) have instead proposed to use the Shapley value, a sharing rule that compensates members of a cooperative project on the basis of their marginal contributions. These two approaches rely on very different driving forces. The first one, based on market competition and efficient pricing, relies on non-cooperative principles
The Shapley Value as a Guide to FRAND Licensing Agreements while the second one is rooted in fairness principles, regardless of market conditions or efficiency. This led Layne-Farrar, Padilla and Schmalensee to consider the Baumol-Swanson method as a possible benchmark for RAND agreements in the United States, where economic efficiency is typically the foremost concern, and the Shapley value as a possible benchmark for FRAND agreements in the European Union, where fairness is viewed to be as important as efficiency.
We here follow the proposal of Layne-Farrar, Padilla and Schmalensee to use cooperative game theory to analyze the compensation problem faced by SSOs. For this purpose, we translate the problem into a cooperative game with transferable utility (or side payments) and consider two major solution concepts, namely the core and the Shapley value. The core identifies the set of socially stable allocations, and the Shapley value provides an axiomatic method for a fair division. By considering ex-ante available patents as the players of their game, these authors place themselves at the outset of the pooling process. The immediate result of such a game where firms with substitutable patents are allowed to compete is that any attempt to form a pool is due to fail. The Shapley value indeed equally remunerates substitutable patents while the core only retains allocations that exclusively remunerate patents without competitors. In other words, there will always be a coalition of firms in a position to challenge Shapley value allocations. Our analysis instead assumes that the choice has been made regarding the firms that form the pool and that a value has been assigned to the patents that have substitutes.
We construct a surplus sharing game -patent game -on the basis of the modeling framework introduced by Muto et al. (1989) who analyze the cooperative behavior of players willing to share a technology covered by a single patent. 4 We extend their model to the case where the technology relies on multiple patents. We follow Layne-Farrar, Padilla and Schmalensee by allowing standards to be composed of two types of patents: those facing competition due to the existence of economically interchangeable alternatives and those being the unique contribution to a particular component of the standard. The latter are strong patents that are essential for the technology to stand up and cannot be invented around. The former are weak patents that, while being also essential, can be invented around, at a known cost, through some alternative solution. Firms owning strong patents have the ability (in essence equivalent to a veto right) to prevent the other players from realizing a profit, independent of the number of patents they own and of their relative importance. Firms detaining only weak patents do not have that blocking potential. The identification of the strong patents is typically straightforward since, in most technological standards, there are only a few strong patents around which the standard is organized together with a (possibly large) number of weak patents. The assignment of a value to weak patents is the difficult part. LayneFarrar et al. (2007) recommend evaluating the worth of an essential patent on the basis of the number of competing alternatives it admits before the implementation of the standard. Linking the value of a patent to the availability of substitutable technologies relies on a specific valuation approach, known as option pricing. If an alternative can easily be chosen, the incremental contribution of a patent is likely to be small. This finds an echo in Shapiro (2001) who states that if a patent can be easily invented around, "the patented technology contributes little if anything to the final product, and any reasonable royalty would be bsmith@example.net." It is indeed quite intuitive that, if an essential patent admits technological alternatives, it can be considered less valuable than other essential patents that do not face competing technologies. In other words, offering the same compensation to the owner of a weak patent and to the holder of a strong patent cannot be considered as fair and if the cost of inventing around such a patent is small, so should be the compensation. In what follows, we assume that weak patents are well identified and that there is an agreement on their value.
5
In patent games, firms are of different types. On the one hand, there is a fixed subset of patent owners that agree to license their intellectual property rights on FRAND terms. They are the members of the standard-setting pool. They may be vertically integrated in the sense that they are not only active in the research field but also in the downstream product market. On the other hand, there are firms that want to be granted access to the whole set of essential patents in order to commercialize the standard-based product. We assume perfect patent protection. It implies that the only way to acquire the information covered by a patent is to be granted a license. Following Kamien and Tauman (1986) , we distinguish patent licensing through a fixed fee and patent licensing through a royalty. The former consists in a fixed payment for granting access to the standard while the latter is a payment proportional to the quantity sold of the standard-based product. In the present setting, we only consider fixed licensing fees.
The worth of a coalition of firms is defined on the basis of the profits it can secure on its own. We exclude strategic considerations by assuming that markets are sufficiently partitioned: even if all firms can commercialize the same standardbased product, they operate on different markets. In other words, there are no competition issues as licensors and licensees do not threaten each other's profit. We also exclude the exchange of information and knowledge that could take place within the patent pool. As a consequence, profits can simply be added. A coalition is in a position to generate profit only if it has been granted access to the whole set of required patents. It means that only coalitions that include all strong patent owners are able to generate a profit. This profit is simply the sum of the individual profits that its members can obtain by commercializing a standard-based product, from which the value of the missing weak patents has been removed.
Using the axioms that underlie the Shapley value, these assumptions lead to a fairly simple formula. It specifies the share of each participant in the total profit, from which one deduces who pays what and to whom. Beyond the numerical outcome, it suggests a number of properties that a FRAND agreement should have. Strong patent owners share equally an amount made of their total profit, augmented by the licensing fees they receive from the other firms and reduced by the licensing fees they pay to the weak patent owners. Hence, what a strong patent owner receives does not depend on the number of its strong patents nor on the value of the weak patents it possibly owns. Furthermore, non-profit organizations like universities or research institutions that own strong patents are entitled to the same share as any other strong patent owner. Weak patent owners receive licensing fees for an amount that is proportional to the value of the patents they own. This proportion is the same for all firms and only depends on the number of strong patent owners. It is at most one half and decreases with the number of strong patent owners. Weak patent owners, as well as any other firm willing to use the standard, give up an amount that is proportional to their profits. Again, the proportion depends only on the number of strong patent owners. It is at least one half and increases with the number of strong patent owners. Furthermore, the licensing fee paid by outside users only goes to strong patent owners: outside users do not pay for weak patents. Because the proportions that enter into the Shapley formula only depend on the number of strong patent owners, bringing in additional firms, either additional outside users or new weak patent owners (as in the case of an extension or improvement of the technology), only induces an increase in the income of strong patent owners. Adding a firm with a strong patent instead negatively affects these proportions, reducing the income of weak patent owners and increasing the licensing fee paid to the strong patent owners.
The remainder of our paper is organized as follows. Section 2 introduces cooperative games with transferable utility and defines patent games. The core is defined in Section 3. It is shown that only a few inequalities are needed to characterize it for a patent game. The Shapley value is defined in Section 4. It is applied to patent games and then particular cases are analyzed. The last section offers concluding remarks and an appendix gathers the proofs of the two main propositions.
Patent games
Cooperative games cover situations in which a group of individuals consider cooperating on a common project with the objective of maximizing collective welfare and allocating it between its members. It is assumed that utility is transferable through some commodity-money that allows for transfers (side payments) among players. A cooperative game with transferable utility is defined by a characteristic function v that associates with each coalition S Ì N a real number v(S) that represents the minimum gain that coalition S can realize without the participation of the others. As a consequence, the marginal contribution of a player is maximal at the grand coalition N. Furthermore, convexity implies superadditivity.
The minimal requirement to be imposed on an allocation is individual rationality in the sense that players receive at least what they are worth alone. This defines imputations, the allocations ðx 1 ; . . . x n Þ 2 R n such that X i2N x i ¼ vðNÞ and xðiÞ ! vðiÞ for all i 2 N Superadditivity ensures that this set of imputations is non-empty.
We denote by Å N the set of all players' orderings. The marginal contributions vector is the imputation μðσÞ associated with the players' ordering σ ¼ ði 1 ; . . . ; i n Þ in Å N is given by:
There are n! marginal contribution vectors not necessarily distinct. An example of a computation of marginal contribution vectors is given in Section 4. A patent situation is defined by a set N ¼ {1, . . . , n} of n firms, a subset M Ì N of m firms owning strong patents, the values of the weak patents owned by each firm, p i ! 0 for firm i, and the additional net sales (profit) that each firm plans to obtain from operating the standard, π i ! 0 for firm i. We assume that all patents have been recognized to be essential. That means that there are four types of firms: firms owning strong patents and possibly also weak patents ði 2 M; p i ! 0Þ; firms owning only weak patents ði ‚ M; p i > 0Þ and firms owning no patent at all ði ‚ M; p i ¼ 0Þ: The last are those who are willing to use the standard covered by the patent pool formed by the other firms. As we allow for π i ¼ 0; there is room for non-profit organization such as universities or research institutions to be included in the set of patent owners. We assume that there is at least one firm owning strong patents: m ! 1: The case where there is no strong patent will be considered separately.
The patent game ðN; vÞ associated with the patent situation ðN; M; p; πÞ is defined by the following characteristic function:
½2
8 Convex games form an important class of games on which solution concepts tend to converge. In particular, the core of a convex game is non-empty and contains the Shapley value. See Shapley (1971) .
Here v(S) is the total profit that coalition S can ensure for itself if it forms, net of the value of the weak patents needed to meet the standard. The implicit assumption is that firms operate on separate markets so that profits can simply be added. In particular v(N) is the total profit that has to be divided among the n firms. The following lemma has important consequences.
Lemma Patent games are convex.
9
We observe that the value of the weak patents held by strong patent owners does not enter into the definition of the patent game: as far as patents are concerned, the only data actually needed are the identity of the firms owning strong patents and the value of the patents owned by firms without strong patents. Furthermore, vðiÞ ¼ 0 for all i whenever m ! 2. To ensure that vðSÞ ! 0 for all S & N; we introduce the following additional assumption:
It says that the coalition formed by the strong patent owners can generate enough profit to cover the cost of all missing weak patents. The marginal contributions of player i to coalitions containing M are given by:
They are equal to zero for coalitions not containing M. Hence, firms owning strong patents are substitutable players. There is indeed no difference between strong patent owners: each of them has a veto right, independent of the number and relative importance of the patents it owns. Consider the following patent situation involving four firms: firm 1 owns only strong patents (p 1 ¼ 0) firm 2 owns strong patents and weak patents for a value p 2 > 0 firm 3 owns only weak patents for a value p 3 > 0 firm 4 is an outside user (p 4 ¼ 0)
Here M ¼ f1; 2g and the "winning" coalitions are those involving the first two firms:
3 The core of a patent game
The concept of core extends individual rationality to coalitions (Gillies, 1953) . Applied to a transferable utility game ðN; vÞ; it is the set of allocations that no coalition can improve upon:
Alternatively, no coalition is in a position to formulate an objection against a core allocation. Clearly, core allocations are imputations. Geometrically, the core is a polyhedron whose dimension is at most n -1 and it may be empty. In the case of a convex game, the core is the non-empty polyhedron whose vertices are precisely the marginal contribution vectors. Furthermore, there is no ambiguity: two convex games coincide if and only if they have the same core (Shapley, 1971) . The core of a patent game is actually defined by at most 2n À m inequalities. Beyond non-negativity, an allocation is in the core of a patent game if (and only if) the firms without strong patents do not obtain more than their stand-alone profit augmented by the value of the weak patents they own. This is the content of the following proposition.
Proposition 1
The core of the patent game defined by ðN; M; p; πÞ is given by:
½3
Two opposite forces are present in the definition of the core of a patent game. On the one hand, the core contains the allocations that give the total value of the game to any one of the strong patent owners: for any given j 2 M, the allocation x satisfying x i ¼ P i2N π i and x i ¼ 0 for all i 6 ¼ j belongs to the core. The core considers all strong patents as equivalently indispensable: if a strong patent owner withdraws from the pool, no profit is bsmith@example.net. Every strong patent owner has a veto power and can claim the entire value of the game since its marginal contribution to the grand coalition is the whole cooperative surplus. On the other hand, each intellectual property right owner may end up with nothing since the dissemination of essential patents across multiple owners implies that no one is able to realize a profit by standing alone. All patent holders are competing with each other to capture the greatest possible share of the cooperative surplus. While being a cooperative concept, the core highlights the competitive forces that may exist within the game.
Consider the case of three firms where firm 1 alone owns strong patents, firm 2 owns only weak patents, and firm 3 is an outside user. The associated characteristic function is given by:
and vðSÞ ¼ 0 for all other coalitions. There are only two active inequalities in the definition of the core, which are 0 x 2 π 2 þ p 2 0 x 3 π 3 while the inequalities π 1 À p 2 x 1 π 1 þ π 2 þ π 3 are redundant. In the case where M ¼ {1,2}, the characteristic function is defined by:
and vðSÞ ¼ 0 for all other coalitions. Here only one inequality defines core allocations:
The Shapley value of a patent game
The Shapley value is an allocation rule defined as the average marginal contribution vector, considering all n! players' orderings: Shapley (1953) has shown that it is the only additive allocation rule that is symmetric and satisfies the null player property. Additivity is an independence property that guarantees that, when confronted with several games, a player evaluates them independently of each other. This property gives the Shapley value its linear structure. The symmetry and null player properties require that substitutable players are treated equally and null players get zero.
There are alternative axiomatizations. The most remarkable is due to Young (1985) who shows that the Shapley value is the only symmetric and marginalist
The Shapley Value as a Guide to FRAND Licensing Agreements allocation rule in the sense that what a player receives will depend exclusively on the value of his or her marginal contributions, independent of the contributions of the other players.
Applied to a convex game, the Shapley value defines a core allocation. 10 The core's vertices are indeed the marginal contribution vectors. The Shapley value can equivalently be defined as the expected marginal contribution to a coalition chosen at random, given that coalitions of the same size are equally likely and that all coalition sizes have the same probability 1=n to occur. This gives rise to the following formulation: Proposition 2 The Shapley value of the patent game defined by ðN; M; p; πÞ is given by: Figure 1 illustrates the Shapley formula [4]. Strong patent owners divide equally their total profits as well as the licensing fees they receive from the other firms, an amount that is proportional to their profits:
Together, strong patent owners will pay to weak patent owners a compensation that is proportional to the total value of the weak patents:
By the nature of strong patents, the number of strong patent owners and their relative importance plays no role: the Shapley value treats them equally. Furthermore, they receive no compensation for the weak patents they own. Any other firm i‚M pays a part βðmÞ π i of its profit as a licensing fee to the owners of strong patents and, if it owns weak patents, it receives licensing fees from the strong patent owners equal to βðmÞp i : Hence, the strong patent owners collect a proportion αðmÞ of the profit generated by weak patent owners and outside users, a proportion that is greater than 1/2 and increases with the number of strong patent owners. When m is large, they actually collect most of their profit.
As a consequence, if firms know that the Shapley value will be used as a compensation scheme, individual firms have a strong incentive to submit strong patents. Actually, it is enough for a firm to convince its partners that it owns one strong patent that is essential for the standard, whatever the quality of the other patents it possibly owns.
The situation of non-profit organizations depends on the kind of patent they own. If they own strong patents, they get the same income as other strong patent owners. If they only own weak patents, they get licensing fees proportional to the value of their patents.
We could have started with the game restricted to the strong patent owners and then added weak patent owners and outside users, as illustrated in Figure 1 . Indeed the proportions αðmÞ and βðmÞ only depend on m, and outside users do not pay for weak patents. Adding weak patent owners or outside users does not affect these proportions and, as far as licensing fees are concerned, weak patent owners and outside users are treated in the same way: the strong patent owners reap at least one half of the profits they collect. Adding firms owning strong patents instead modifies the proportions.
By convexity, the Shapley value belongs to the core i.e. the licensing fees structure it recommends is stable: neither individual firms nor coalitions of firms have an interest in challenging it. In the four-firm example, there are eight distinct marginal contribution vectors out of n! ¼ 24. They are given by the following table together with their multiplicity:
The corresponding Shapley value is given by:
To conclude this section, we will consider now particular patent situations. No weak patent (p ¼ 0). The Shapley value then reduces to:
Every firm owns strong patents (m ¼ n). In that case, firms are substitutable and the Shapley value is the equal division:
Only one player holds strong patents (m ¼ 1). 11 In that case, the Shapley value coincides with the average of the core's vertices: the strong firm (say
firm 1) receives half of the net profits of the other firms and returns to them half of the value of the weak patents they own:
The amount allocated to the weak patent owners is given by:
The patent owner collects half of the profit obtained by the other firms and compensates the firm owning weak patents: all firms except the strong patent owner keep half of their profit augmented by the value of their weak patent. In the case of three firms, the Shapley value is given by the following allocation:
Only weak patents (m ¼ 0). This situation is not covered by the preceding analysis. If M ¼˘; the patent game can be written as:
In the absence of strong patents, it is always possible for a coalition to realize profit since all missing patents can be bypassed as long as the required investment is made. This is an additive characteristic function up to a constant. By additivity, the Shapley value is simply given by 12 :
It recommends that each firm keeps the profit that it can realize by commercializing the standard-based product. All players uniformly support the value of the whole patent set, and the value of each patent is completely redistributed to the firm holding it. Therefore, a patent holder is compensated in proportion to the value of the patents in its possession and effectively perceives a compensation if the value of its patents is superior to the per capita value of the whole patent set.
12 It corresponds to a particular case of the compensation problem studied by Dehez and Tellone (2013) in a data sharing context.
The Shapley Value as a Guide to FRAND Licensing Agreements

Concluding remarks
In order to use cooperative games with side payments, it is necessary to know what each coalition is worth, whatever the solution concept under consideration. When applied to patent games, this requires that firms have reached an agreement on the level of net sales that any coalition can raise by selling the standard-based product. Assuming that markets are segmented simplifies this requirement. There must also be an agreement on which patents are strong and on the value of the other patents included in the pool. In view of the recommendations of the Shapley value, identifying the strong patents is the most crucial bsmith@example.net. Being able to convince your partners that a patent you own admits no close substitute indeed makes a huge difference in terms of profit. Specifying the whole range of possible options requires a considerable amount of information.
13
What if markets are not segmented? First of all, the simple characterization of the core given in Proposition 1 is lost. Numerically, computing the Shapley value is easy once we know the profit that a coalition containing the strong patent owners can secure for itself. However, no general formula can be obtained. Assuming that πðSÞ is known for all S ' M; the patent game is again defined by
where this time we do not impose additivity of the set function π. It is quite natural to assume the profit function π to be superadditive as merging coalitions can only be beneficial. Convexity is nevertheless not granted, and the Shapley value then does not necessarily belong to the core. It however turns out that the licensing fees paid to the weak patent owners remain the same. The Shapley value is indeed defined as the sum of the value associated with the game
and the value associated with the patent game for which π i ¼ 0 for all i.
Referring to the proof of Proposition 2 and using the second Shapley formula, we obtain for i‚M: SV i ðN; π; pÞ ¼
½5
Strong patent owners remain substitutable players and, as a consequence, are entitled to an equal amount. While the second part of eq.
[5] defines the licensing fees paid by the strong patent owners to the weak patent owners, without further information on the profit function π, the formula gives no more clues about the licensing fees that accrue to strong patent owners.
Cooperative game theory offers other allocation rules beyond the Shapley value. The most well-known is the nucleolus introduced by Schmeidler (1969) . It is concerned with the minimization of coalitions' dissatisfaction and is a core selection when the latter is non-empty. It shares properties with the Shapley value (symmetry and null player) but fails to be additive.
14 Another point: it is worth mentioning that in some cases it may be justified to treat players asymmetrically, for instance because they differ in size as measured by their market share or production volume. Such situations can be accommodated by using the asymmetric version of the Shapley value, referred to as the weighted Shapley value, obtained by assigning exogenous weights to players. 

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