The Shapley Value of 　 a Patent Licensing Game:

the Asymptotic Equivalence toNon-cooperative Results

by Yair Tauman and Naoki Watanabe

１ . Introduction

patent licensing games: non-cooperative mechanisms (policies)

upfront fee or royalty by Kamien and Tauman (1984, 1986)　　　　　 Sen (forthcoming in GEB) auction by Katz and Shapiro (1985, 1986)

the innovator has full bargaining power i.e., take-it-or-leave-it offers to potential

licensees

Other　 Papers

optimal licensing policy Cournot and Bertrand Kamien, Oren and Tauman (1992)

Bertrand with differentiated goods Muto (1993)

optimal combination of licensing policies Sen and Tauman (2002)

Why cooperative approach?

Macho-Stadler et al. (1996) licensing agreements are basically contract terms 　　 signed as negotiation results

a role of patent system may be facilitating the resolution of disputes in complicated bargaining procedures over licensing issues

analyze licensing agreements from the cooperative

point of view

a key problem

how to define the worth of a coalition of players

in the presence of strategic interactions? a new characteristic function appropriate for oligopolistic markets

von-Neumann and Morgenstern’s minmax …incredible threat!

the main proposition

(Proposition 4) In increasing the number of firms in the market, the Shapley value of the innovator approximates his payoff in the auction mechanism (even for general demand functions).

quite surprising: (1) the innovator does not have full bargaining power in cooperative games and (2) the Shapley value measures a fair contribution of him to the total industry profit

2. The Characteristic Functions

{ }

{ } coalition : players, ofset : 0=

producer) a(not entity outside 0,player

0>-- and c <<0 where ,-

technology reducing-cost :innovator

∞<< demand,market : )-,0(max=

pricemarket : cost, production :

commodity shomogeneou

firms of ...,,1=set the withindustry Cournot a

00 NSNN

εcaεεcc

acpaQ

pc

nN

⊆∪

→

{ }

SNjSilmnelqlmemqQ

Qap

εcplmemqlmS

neq

lmeq

lm

lN

mS

SSNS

- , ),,(+),(=

and -= where

),+-)(,(=),(Π :profit total the

firm licensee-non eachfor l) (m,

firm licensee eachfor ),( : outcome unique

Cournot la a compete firms +

itwithout firms operate and merge : 0S-0 in firms

tech new with firms operate and merge :0 in firms

.0=0let ,every For

0

∈∈

∪⊆

the minmax approach

)-)(,(=),(Π where

),,(Π||≤≤1

max ||-≤≤1

min=)(

-00 s.t. any for

),(Π||≤≤1

max ||-≤≤1

min=)0(

with 0any for

0

cpmlnelqmlS

mlSSlSnm

Sv

SNNS

lmSSmSnl

Sv

NSS

∈⊆

⊆

Propositions 1 and 2: minmax=maxmin

KS

KSεaSn

Sv

SεaSn

Scaε

KSn

εSncaSSv

nεcaK

≥ || if 0

< || if 2))|S|(n--c-()1+||+(4

1

=)(

1+ |S|n-≥K≥ || if 2))1+|S|(n-+c-()1+||+(4

1

K≥ ||≥K-n if )-(

< || if 2)1+

)1+||-(+-(||=)0(

Then, .2/)1+(≤/)-(=Let

3. The Shapley Value

{ }

{ }[ ]j

vjvn

v

vv

jjjNj

Njjj

jjj

Njj

j

n

player of oncontributi marginal expected the measures

, )( )( ! )1+(

1=)(Sh

,))(Sh(=)(Sh lueShapley va The

in preceding players ofset the : ′ | ′=

. in players of ordering an : ),,,(=

0

0

010

∑ ∪

∈

∈

R

RR

R

PP

RRP

R

－

Proposition 4 : lim Sho(v)=e(a-c)

{ }

)-(})1+||-(4

2)|)S|(n--c-(-2)

2+-

( {1+

1+

1- 1+-=||

} )1+|-(4

2)|)S|(n--c-(a-

)1+|-(4

2)1)+|S|(n-+c-(a{

1+1

+

- =||

)-(1+

1+2)

+1)1+||-(+-

(1- ||

||1+

1=

0=||

)0(-)00( 1+

1=)(0Sh

caεSn

εaεcan

nKnS Sn

εSn

εn

KnKS

caεnn

εSncaKS

Sn

nS

vvn

v

→

∪

∑

∑∑

∑ PPRP

4. The Emptiness of the Core

{ }

lueShapley va the of use thefor

ionjustificatbetter a provides of coreempty The

=)(Core then n, ≥ If : 5 nPropositio

}NS )( ≥ )(:)({=)(Core

} )( ≥ and

)(=)(:{=)(

2

0

0

001+

v

φvK

SvSwvIwv

Njjvw

NvNwwvI

j

n

⊂∀∈

∈ ∀

∈ R

5. Remarks

Kamien (1992) : if n>2K, then the innovator auctions off K licenses and extract the entire industry profit, which is

e(a-c)

This equivalence occurs even with general demand functions that are downward sloping and differentiable in price

Watanabe and Muto (2004, 2005) Imai and Watanabe (2005, forthcoming in JER)