The Shapley Value as a Guide to FRAND Licensing Agreements

Pierre Dehez; Sophie Poukens

doi:10.1515/rle-2013-0016

Article

The Shapley Value as a Guide to FRAND Licensing Agreements

Pierre Dehez and Sophie Poukens

Published/Copyright: September 5, 2014

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From the journal Review of Law & Economics Volume 10 Issue 3

Abstract

We consider the problem faced by standard-setting organizations of specifying Fair, Reasonable and Non-Discriminatory (FRAND) agreements. Along with Layne-Farrar et al. (2007. “Pricing Patents for Licensing in Standard-Setting Organizations: Making Sense of FRAND Commitments.” 74 Antitrust Law Journal 671–706), 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.

Keywords: patent licensing; Shapley value; core

Acknowledgments

The authors are grateful to Paul Belleflamme, Agnieszka Kupzok, Pavitra Govindan and an anonymous referee for useful comments and suggestions on earlier drafts.

Appendix

Proof of Proposition 1 We will proceed in two steps. Let’s denote by (N,v) the game associated with the patent situation (N,M,p,π). It is easily verified that any core allocation x satisfies the following equivalent inequalities:

∑i∈Sxi≤v(N)−v(N∖S)for all S⊂N

i.e. no coalition can obtain more than its contribution to the grand coalition. Applied to an individual player i, it reads

xi≤v(N)−v(N∖i)

The first half of the proposition then follows from the fact that v(N)−v(N∖i)=πi+pi for all i∉M. Consider now an allocation x∈R+n satisfying xi≤πi+piforalli∉M. If M/⊂S, we have

v(S)=0≤∑i∈Sxi

If instead M⊂S, we have

∑i∈Sxi=∑i∈Nπi−∑i∉Sxi≥∑i∈Nπi−∑i∉S(πi+pi)=∑i∈Sπi−∑i∉Spi=v(S)

Hence x∈C(N,v). This completes the proof of the equivalence. ♦

Proof of Proposition 2 Let us define θi=πi+piandp0=p(N). Using eq. [2], v can be decomposed as v=v1−v2 where the games (N,v1) and (N,v2) are defined by:

v1(S)=∑i∈Sθiif M⊂S=0if not

v2(S)=p0if M⊂S=0if not

To compute the Shapley value of the game (N,v1), we only have to compute what any player outside M receives. Looking at players’ orderings, we observe that the marginal contribution of a player i ∉ M is either 0orθi. It is θiif and only if player i is preceded by all players in M. For a given ordering of the players in M∪{i} where player i is last, there are Cnm+1 ways to place them. Hence, the number of times i is preceded by the players in M is given by:

Cnm+1m!(n−m−1)!=n!(m+1)!(n−m−1)!m!(n−m−1)!=n!m+1

Using the first definition of the Shapley value, we get

SVi(N,v1)=1m+1θiforalli∉M

Combining symmetry and efficiency, we obtain

SVi(N,v1)=1m∑i∈Mθi+1m+1∑i∉Mθiforalli∈M

The game (N,v2) is a simple game in which players outside M are null and players in M are substitutable. Hence, its Shapley value is given by:

SVi(N,v2)=0for all i∉M=p0mfor all i∈M

By linearity of the Shapley value, SV(N,v)=SV(N,v1)−SV(N,v2) resulting in eq. [4]. ♦

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Published Online: 2014-9-5

Published in Print: 2014-11-1

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https://doi.org/10.1515/rle-2013-0016

Keywords for this article

patent licensing; Shapley value; core