Farsighted Stability in Patent Licensing
Toshiyuki Hirai1 , Naoki Watanabe2 , Shigeo Muto3
1 Toyama U., 2 Keio U., 3 Tokyo U. Sci

December 20, 2018 at Hitotsubashi University

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

An Abstract Game Approach:We define absolute maximality for abstract games, which was
originally defined by Ray and Vohra (2018) for TU games, andinvestigate whether we can refine farsightedly stable sets in patent
licensing.
Absolute maximality plays a role of something like subgame
perfection in extensive form games.
Some discussions are needed for it. (Today’s goal)

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction
We consider a negotiation on patent licensing among farsighted
one (external) patent holder (PH),(symmetric) firms.
The PH and firms (players) negotiate on
which firms are licensed,
how much fees for the license are.
Whether an agreement on patent licensing can be reached.
What properties the agreement has, e.g.
how much profit the patent holder can obtain,
how many firms are licensed.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction
Patent licensing problem
Non-cooperative game approach
Kamien and Taumann (1984,1986)
Licensing agreements are contract terms.
→ Cooperative game approach (ref. Tauman and Watanabe, 2007)
Watanabe and Muto (2008).
Watanabe and Muto (2008): Core; Bargaining set.
Kishimoto et al. (2011): Shapley value.
Kishimoto and Watanabe (2017): Kernel; Nucleolus.
Hirai and Watanabe (2018): vNM stable set.

We incorporate players’ farsightedness into the model of Watanabe
and Muto (2008) with some modification.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder
1.invite

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder
1.invite
2.Negotiate on fees

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder
1.invite
2.Negotiate on fees

Firms
3.All firms compete in a market,
knowing which firms are licensed.
No cartel is allowed.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder
1.invite
2.Negotiate on fees
(TU-game with coalition structure)
Firms
3.All firms compete in a market,
knowing which firms are licensed.
No cartel is allowed.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder
1.invite

Based on a foreseenmarket outcome.

2.Negotiate on fees
(TU-game with coalition structure)
Firms
3.All firms compete in a market,
knowing which firms are licensed.
No cartel is allowed.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction (WM2008 model)

Patent holder
1.invite
2.Negotiate on fees
(TU-game with coalition structure)

Based on a foreseenmarket outcome.
However, the players’
farsightedness is notconsidered in
the negotiation.

Firms
3.All firms compete in a market,
knowing which firms are licensed.
No cartel is allowed.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction

We incorporate farsightedness of the players in the licensing
negotiation.
An analogous motivation of Diamantoudi (2005).
We modify the model of Watanabe and Muto (2008) so that
1 PH and firms negotiate
not only on contracts of licensing fees,
but also which firms are licensed.
2

Firms compete in a market after patent licensing.
Firms commonly know which firms are licensed.
No cartel is allowed.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction

PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction

PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction

PH

PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Firms

Farsighted Stability in Patent Licensing

Introduction

PH

PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Firms

Farsighted Stability in Patent Licensing

Introduction

PH

PH

Firms

Firms
PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction

PH

PH

Firms

Firms
PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Introduction

PH

PH

Firms

Firms
PH

Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

PH

Firms

Farsighted Stability in Patent Licensing

Introduction

PH

PH

Firms

Firms
PH

PH

so on...
Firms

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Firms

Farsighted Stability in Patent Licensing

Introduction
We investigate farsighted stable sets (Harsanyi, 1974; Chwe, 1994)
in an abstract game describing such a negotiation.
In particular, we focus on symmetric farsighted stable sets.
A farsighted stable set where at each outcome in the set,
licensee firms obtain symmetric payoﬀs.

Main results
We characterize symmetric farsighted stable sets under certain
conditions.
The number of licensee firms that maximizes the profit of PH,
provided that each licensee firm receives an exogenously given
positive net profit.
The existence result follows from this characterization.
Maximality problem is also considered.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingModel: players

0: the patent holder (PH).
The PH has no production technology.

N = {1, ..., n}: the set of firms.
Potential users of the patented technology.
Firms are symmetric if they are not licensed.
Licensee firms become symmetric.

{0} ∪ N: the set of players.
A nonempty subset of {0} ∪ N: a coalition.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingModel: profits at market competition

Firms compete in a market after the patent licensing negotiation.
When s (0 ≤ s ≤ n) firms are licensed,
each licensee firm obtains W (s);each non-licensee firm obtains L(s).
L(0)(> 0): the profit of firms when no firm is licensed.
W (0) = L(n) = 0 by convention.

Assumption(a) W (s) > L(0) for all s = 1, ..., n;(b) L(0) > L(s) for all s = 1, ..., n − 1.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingModel: outcomes (1)An outcome of the negotiation is represented bya set of licensee firms;
a contract of profit sharing.
The set of feasible contracts when S ⊆ N is licensed: (s = |S|)}{∑xi = |S|W (|S|),S
n+1 x0 +i∈S
X = x = (x0 , x1 , ..., xn ) ∈ R
xj = L(|S|) for all j ∈ N \ S
X ∅ = {(0, L(0), ..., L(0))} ≡ {x ∅ }.
x ∅ : null contract.
(∅, x ∅ ): null outcome.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingModel: outcomes (2)The set of all outcomes:
X =∪({S} × X S ).

S⊆N

The set of symmetric outcomes:X̄ = {(S, x) ∈ X | xi = xj for all i, j ∈ S}.

The licensee firms pay an uniform fee at a symmetric outcome.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingModel: eﬀectiveness relation
An eﬀectiveness relation describes what a coalition can induce an
outcome from an outcome.
(S, x) →T (S ′ , x ′ );Coalition T can induce (S ′ , x ′ ) from (S, x).
Assumption
Let (S, x), (S ′ , x ′ ) ∈ X with S ′ ̸= ∅.
(i) (S, x) →T (∅, x ∅ ) if and only if ∅ ̸= T ⊆ {0} ∪ S;(ii) (S, x) →T (S ′ , x ′ ) if and only if T = {0} ∪ S ′ .
(i) Any participant of a contract can cancel it unilaterally.
(ii) A new contract is available with consents from all of PH
and licensee firms.
We inherit the spirit of the coalitional game model by Watanabe
and Muto (2008).
Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Indirect dominance relation
Definition
Let (S, x), (S ′ , x ′ ) ∈ X .
(S ′ , x ′ ) indirectly dominates (S, x), denoted by (S ′ , x ′ ) ≻ (S, x),
iﬀ there exist(S 0 , x 0 ), ..., (S m , x m ) andT 1 , ..., T msuch that (S 0 , x 0 ) = (S, x), (S m , x m ) = (S ′ , x ′ ), and for all
h = 1, ..., m,(S h−1 , x h−1 ) →T h (S h , x h );xi′ > xih−1 for all i ∈ T h .
We sometimes write the paths like(S 0 , x 0 ) →T 1 (S 1 , x 1 ) →T 2 · · · →T m−1 (S m−1 , x m−1 ) →T m (S m , x m )Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Indirect dominance relation
Definition
Let (S, x), (S ′ , x ′ ) ∈ X .
(S ′ , x ′ ) indirectly dominates (S, x), denoted by (S ′ , x ′ ) ≻ (S, x),
iﬀ there exist(S 0 , x 0 ), ..., (S m , x m ) andT 1 , ..., T msuch that (S 0 , x 0 ) = (S, x), (S m , x m ) = (S ′ , x ′ ), and for all
h = 1, ..., m,(S h−1 , x h−1 ) →T h (S h , x h );xi′ > xih−1 for all i ∈ T h .
We sometimes write the paths like(S 0 , x 0 ) →T 1 (S 1 , x 1 ) →T 2 · · · →T m−1 (S m−1 , x m−1 ) →T m (S m , x m )<T 1(S m , x m )<T 2(S m , x m )Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

<T m
(S m , x m )Farsighted Stability in Patent Licensing

Farsighted stable set

Definition
K ⊆ X is a farsighted stable set iﬀ internal and externalstabilities are satisfied.
Internal stability: for any (S, x), (S ′ , x ′ ) ∈ K ,(S ′ , x ′ ) ≻ (S, x) does not hold.
External stability: for any (S, x) ∈ X \ K ,(S ′ , x ′ ) ≻ (S, x) for some (S ′ , x ′ ) ∈ K .
K ⊆ X is a symmetric farsighted stable set iﬀ
K is a farsighted stable set and K ⊆ X̄ .

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Main results: preparation
DefineE = {ε ∈ R++ |s(W (s) − L(0) − ε) > 0 for some s = 1, ..., n } .
The set of net profits of licensee firms from patent licensingthat are strictly acceptable for PH.
E ̸= ∅ by W (s) > L(0) for all s = 1, ..., n.
For each ε ∈ E , define
B(ε) = arg max s(W (s) − L(0) − ε).
s=1,...,nThe optimal number(s) of licensee firms for PH under ε ∈ E .

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Main results: suﬃciency
For any ε ∈ E , defineX̄ (ε) = (S, x) ∈ X̄


|S| ∈ B(ε)
x0 = |S|(W (|S|) − L(0) − ε).
xi = L(0) + ε for all i ∈ S

Theorem 1
For each ε ∈ E , X̄ (ε) is a symmetric farsighted stable set.
In a farsighted stable set X̄ (ε),
net profit ε > 0 of licensee firms is exogenously given.
An established order of society or accepted stable standard of
behavior.
(von Neumann and Morgenstern, 1944).
Corollary 1
There exists a symmetric farsighted stable set. (By E ̸= ∅.)Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Main results: necessity
Theorem 2
Assume that L(s) is nonincreasing in s. If K is a symmetricfarsighted stable set, then K = X̄ (ε) for some ε ∈ E .

Corollary 2
Assume that L(s) is nonincreasing in s. Then, K is a symmetricfarsighted stable set iﬀ K = X̄ (ε) for some ε ∈ E .

Nonincreasingness of L(s) holds, when e.g.
the patented technology is a cost reduction technology, andthe market competition is a Cournot competition with linear
demand and cost functions.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingComparison with other solutions for the TU gameIn the present paper, the profits of PH x0 supported by symmetricfarsighted stable sets are
0 < x0 < max s(W (s) − L(0)).
s=1,...,nThe profits of PH supported by symmetric farsighted stablesets cover the ones supported by the bargaining set.
The supremum is consistent with the bargaining set by
Watanabe and Muto (2008).
The infimum is consistent with the bargaining set only ifn ∈ arg maxs=1,...,n s(W (s) − L(0)).

No clear relationship with vNM stable set: farsightedness vs.
myopia? Let x0 be the profit of PH in vNM stable set.
sW (s) + (n − s)L(s) − nL(0) ≤ x0
≤ maxt=0,...,n−s t(W (t) − L(s)).
Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Sketch of proof: Theorem 1
Let ε ∈ E . We show the “external stability” of X̄ (ε).
Fix an arbitrary (T , y ) ∈ X \ X̄ (ε).
Let s ∗ ∈ B(ε).

s ∗ (W (s ∗ ) − L(0) − ε)(> 0) is the profit of PH in X̄ (ε).

Recall x ∅ = (0, L(0), ..., L(0)).
Case 1. T = ∅. (⇒ (T , y ) = (∅, x ∅ ).)(∅, x ∅ ) →{0}∪S (S, x)
yields (S, x) ≻ (∅, x ∅ ) for any (S, x) ∈ X̄ (ε) byx0 > 0 = x0∅

xi = L(0) + ε > L(0) = xi∅ for all i ∈ S.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Sketch of proof: Theorem 1

Case 2. T ̸= ∅ and y0 < s ∗ (W (s ∗ ) − L(0) − ε).
(T , y ) →{0} (∅, x ∅ ) →{0}∪S (S, x)
yield (S, x) ≻ (T , y ) for any (S, x) ∈ X̄ (ε) byx0 = s ∗ (W (s ∗ ) − L(0) − ε) > y0 ,x0 > 0 = x0∅ ,xi = L(0) + ε > L(0) = xi∅ for all i ∈ S.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Sketch of proof: Theorem 1
Case 3. T ̸= ∅ and y0 ≥ s ∗ (W (s ∗ ) − L(0) − ε).
By (T , y ) ∈/ X̄ (ε), there exists j ∈ T with yj < L(0) + ε.
Otherwise,y0 ≥ s ∗ (W (s ∗ ) − L(0) − ε),yi ≥ L(0) + ε for all i ∈ T , ands ∗ ∈ B(ε)
imply (T , y ) ∈ X̄ (ε), a contradiction.
Let (S ′ , x ′ ) ∈ X̄ (ε), where j ∈ S ′ .
(T , y ) →{j} (∅, x ∅ ) →{0}∪S ′ (S ′ , x ′ )yield (S ′ , x ′ ) ≻ (T , y ) by
xj′ = L(0) + ε > yj ,x0′ > 0 = x0∅ ,xi′ = L(0) + ε > L(0) = xi∅ for all i ∈ S ′ .
Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Maximality problem
Recent literature discusses the credibility of indirect dominancerelation.
Ray and Vohra (2014, 2018), Dutta and Ray (2017), Dutta
and Vartiainen (2017).
Among others, we examine the absolute maximality by Ray
and Vohra (2018) in our farsighted stable sets.
(S 0 , x 0 )T1(S 1 , x 1 )Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

T2

(S 2 , x 2 )Farsighted Stability in Patent Licensing

Maximality problem
Recent literature discusses the credibility of indirect dominancerelation.
Ray and Vohra (2014, 2018), Dutta and Ray (2017), Dutta
and Vartiainen (2017).
Among others, we examine the absolute maximality by Ray
and Vohra (2018) in our farsighted stable sets.
(S 0 , x 0 )T1(S 1 , x 1 )T2(S 2 , x 2 )T̄ 2(S̄ 2 , x̄ 2 )Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Maximality problem
Recent literature discusses the credibility of indirect dominancerelation.
Ray and Vohra (2014, 2018), Dutta and Ray (2017), Dutta
and Vartiainen (2017).
Among others, we examine the absolute maximality by Ray
and Vohra (2018) in our farsighted stable sets.
(S 0 , x 0 )T1(S 1 , x 1 )T2(S 2 , x 2 )T̄ 2(S̄ 2 , x̄ 2 )T̄ 3(S̄ 3 , x̄ 3 )T̄2 prefers (S̄ 3 , x̄ 3 ) to (S 2 , x 2 ).

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

On Maximality

Not for the farsighted stable set, but for the players’ payoﬀscredibility of indirect domination
Ray and Vohra (2018) showed that in TU games,a necessary and suﬃcient condition for a farsighted stable set
to be absolutely maximal.
We confirmed the absolute maximality for abstract games.
Is X̄ (ε) absolutely maximal?

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

History
A history h is ((S 0 , x 0 ), (T 1 , (S 1 , x 1 )), ..., (T m , (S m , x m ))) suchthat
if (S ℓ−1 , x ℓ−1 )̸=(S ℓ , x ℓ ), then (S ℓ−1 , x ℓ−1 ) →T ℓ (S ℓ , x ℓ ),where T ℓ ̸= ∅,if (S ℓ−1 , x ℓ−1 )=(S ℓ , x ℓ ), then T ℓ = ∅.
A history consisting of a single outcome (m = 0) is called an initialhistory.
An initial history is unnecessarily consisting of (∅, x ∅ ).
For any history h, let (S(h), x(h)) denote the last outcome in h.
When h = ((S 0 , x 0 ), (T 1 , (S 1 , x 1 )), ..., (T m , (S m , x m ))), then(S(h), x(h)) = (S m , x m ).

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

(Negotiation) process

For any history h, define a function σ that gives
σ(h) = (R(h), (Q(h), z(h))) such that (S(h), x(h)) →R(h) (Q(h), z(h)).
A process prescribes
an outcome (Q(h), z(h)) that follows h,a coalition that induces (Q(h), z(h)) from (S(h), x(h)).
An outcome (S, x) is absorbing under a negotiation process σ,
if for any history h with (S(h), x(h)) = (S, x),(Q(h), z(h)) = (S(h), x(h)) = (S, x).

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingAbsorbing process

For any history h, define σ 1 (h) = σ(h) andσ k (h) = σ(h, σ 1 (h), ..., σ k−1 (h)) inductively for any k > 1.
σ is an absorbing process if for any history h, σ k reaches to someabsorbing outcome for a suﬃciently large k.
(Q k (h), z k (h)) is an absorbing outcome where
σ k (h) = (R k (h), Q k (h), z k (h)).
Denote (S σ (h), x σ (h)) the absorbing outcome that reaches
from h under σ.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing

Coalitionally acceptable absorbing process

An absorbing process is coalitionally acceptableif for each history h,R(h) ̸= ∅ implies xiσ (h) ≥ xi (h) for all i ∈ R(h).
No incentive for every member of any coalitions appearing inan absorbing process to stop the process.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingAbsolutely maximal absorbing process

An absorbing process σ is absolutely maximal
if for any history h, there exist no (T + , (S + , x + )) such that
(S(h), x(h)) →T + (S + , x + ),xiσ (h+ ) > xiσ (h) for all i ∈ T + ,where h+ = (h, (T + , (S + , x + ))).
No coalition has incentive to intervene the process in order toinduce another absorbing outcome.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingAbsolutely maximal farsighted stable set
A farsighted stable set K is absolutely maximal if there is anabsorbing, coalitionally acceptable, and absolutely maximal process
σ such that
K is the set of all absorbing outcomes of σ,
for any initial history h = (S 0 , x 0 ) such that (S 0 , x 0 ) ∈/ K,
a history (h, σ 1 (h), ..., σ k (h)) constitutes(S σ (h), x σ (h)) ≻ (S 0 , x 0 );for any history h = ((S 0 , x 0 ), (T 1 , (S 1 , x 1 ))) such that
(S 0 , x 0 ) ∈ K and (S 1 , x 1 ) ∈/ K,a history (h, σ 1 (h), ..., σ k (h)) constitutes(S σ (h), x σ (h)) ≻ (S 1 , x 1 ).
(In both histories, an outcome in σ k (h) is the first absorbing
outcome appeared in (h, σ 1 (h), ..., σ k (h)). )Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent LicensingAbsolute maximality of X̄ (ε)

Theorem 3
For any ε ∈ E , X̄ (ε) is an absolutely maximal farsighted stable set.
The sketch of the proof of Theorem 1 constructs the absorbing,coalitionally acceptable, and absolutely maximal process for
Theorem 3 with some additions.

Toshiyuki Hirai, Naoki Watanabe, Shigeo Muto

Farsighted Stability in Patent Licensing