A Stackelberg game under heterogeneous technology

3. A Stackelberg game under heterogeneous technology

Theory of strong Stackelberg reasoning is an improved version of an earlier theory [35], which provides an explanation of coordination in all dyadic (two- player) common interest games. It provides an explanation of why players tend to choose strategies associated with a payoff-dominant Nash equilibrium. Its distinc- tive assumption is that players behave as though their co-players will anticipate any strategy choice and invariably choose a best reply to it. Stackelberg strategies resulting from this form of reasoning do not form Nash equilibria. The theory makes no predictions, because a non-equilibrium outcome is inherently unstable, leaving at least one player with a reason to choose differently and thereby achieve a better payoff. Strong Stackelberg reasoning is a simple theory, according to which players in dyadic games choose strategies that would maximize their own payoffs if their co-players could invariably anticipate their strategy choices and play counter- strategies that yield the maximum payoffs for themselves. The key assumption is relatively innocuous, first because game theory imposes no constraints on players’

beliefs, apart from consistency requirements, and second because the theory does not assume that players necessarily believe that their strategies will be anticipated, merely that they behave as though that is the case, as a heuristic aid to choosing the best strategy. Strong Stackelberg reasoning is, in fact, merely a generalization of the minorant and majorant models introduced by [36] and used to rationalize their solution of strictly competitive games.

To promote the sharing of hybrid-enabling technology, the Player II (the leader) determine an optimal sharing effort sharing level and an optimal subsidy scheme.

Then the Player I (the follower) choose his/her optimal sharing level according to the optimal sharing effort level and subsidy. This leads to a Stackelberg equilibrium.

Proposition 5. If above conditions are satisfied, the feedback Stackelberg leader (Player II)-follower (Player I) and equilibria is given as:

L^R_S ¼α₁ð2�θÞðρ₂þξÞðρ₁þξÞ þϑ^R₁ðΓþδÞðð2�2θÞðρ₁þξÞ þθ ρð ₂þξÞÞ

2β^Rðρ₂þξÞðρ₁þξÞ , (29) L^F_S ¼α₂ð2�θÞðρ₂þξÞðρ₁þξÞ þϑ^F₁ðΓþδÞðð2�2θÞðρ₁þξÞ þθ ρð ₂þξÞÞ

2β^Fðρ₂þξÞðρ₁þξÞ , (30)

~L^R_S ¼ð1�θÞðβ₁ðρ₂þξÞ þðΓþδÞÞϑ^R₂

~β^Rðρ₂þξÞ , (31)

L~^F_S ¼ð1�θÞðβ₂ðρ₂þξÞ þðΓþδÞÞϑ^F₂

~β^Fðρ₂þξÞ : (32)

where L^R_S, L^F_S are the optimal effort level of hybrid-enabling technological improvements shared on renewable sources and fossil fuel at time t by Player I, respectively.~L^R_S,~L^F_Sare the optimal effort level of technological improvements shared on renewable sources and fossil fuel at time t by Player II, respectively.

The optimal level of subsidy for sharing hybrid-enabling on renewable sources is given by

ω₁¼

α₁ð2�3θÞ þϑ^R₁½2a₂�a₁�

α₁ð2�θÞ þϑ^R₁½2a2þa1� , 0≤ θ ≤2 3

0: otherwise

8>

<

>: (33)

Similarly, the optimal level of subsidy for sharing hybrid-enabling technology on fossil fuel is given by:

ω₂¼

α₂ð2�3θÞ þϑ^F₁½2a2�a1�

α₂ð2�θÞ þϑ^F₁½2a2þa1� , 0≤ θ ≤2 3

0: otherwise

8>

><

>>

:

(34)

The optimal sharing payoff functions under hybrid-enabling technology on renewable sources and on fossil fuel for Player I and Player II are given below

V^{ð Þ}_S^Ið Þ ¼K θ�Γþδ_Þ� ρ₁þξ

ð Þ Kþb1, V^{ð Þ}_S^IIð Þ ¼K ð1�θÞðΓþδÞ ρ₂þξ

ð Þ Kþb2, (35) where a1, a2, b1and b2are given in the proof.

Proof. We define the optimal revenue functions for Player I and Player II under hybrid-enabling technology as V^{ð Þ}_S^Ið ÞK and V^{ð Þ}_S^IIð Þ, respectively, which areK continuously differentiable. Applying HJB equation to V^{ð Þ}_S^Ið Þ, for Player I, weK obtain

ρ₁V^{ð Þ}_S^Ið Þ ¼K max

L^R_S, L^F_S

f g^≥0nhθ α� ₁L^R_Sð Þ þt α₂L^F_S þβ₁~L^R_Sþβ₂~L^F_SþðΓþδÞKÞi

�1

2β^Rð1�ω₁Þ� �L^R_S 2

�1

2β^Fð1�ω₂Þ� �L^F_S 2

þ∂V^{ð Þ}_S^Ið ÞK

∂K ϑ₁�L^R_S, L^F_S�

þϑ₂�L~^R_S,L~^F_S�

�ξK

h i

þ1 2

∂²V^{ð Þ}_S^Ið ÞK

∂K² φ²ð ÞK )

: (36)

Via the first order conditions, we obtain the optimal values L� ^R_S, L^F_S� for Player I as:

L^R_S ¼θα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁

β^Rð1�ω₁Þ , (37)

L^F_S ¼θα₂þV^{0ð Þ}_S^Ið ÞK ϑ^F₁

β^Fð1�ω₂Þ , (38)

where^∂V∂^{ð ÞSI}K^{ð ÞK} �V^{0ð Þ}_S^Ið ÞK :The optimal sharing revenue function, V^{0ð Þ}_S^IIð Þ, forK Player II and the associated HJB equation is

ρ₂V^{ð Þ}_S^IIð Þ ¼K max

L~^RS,~L^FS

� �

≥0

1�θ

ð Þ�α₁L^R_Sþα₂L^F_Sþβ₁L~^R_S þβ₂L~^F_SþðΓþδÞK�

h i

n

�1

2~β^R�L^R_Sð Þt �2

�1

2~β^F� �~L^F_{S 2}

�1

2ω₁β^R� �L^R_S 2

�1

2ω₂β^F� �L^F_S 2

þ∂V^{ð Þ}_S^IIð ÞK

∂K ϑ₁�L^R_S, L^F_S�

þϑ₂�L~^R_S,L~^F_S�

�ξK

h i

þ1 2

∂²V^{ð Þ}_S^IIð ÞK

∂K² φ²ð ÞK )

: (39)

Substituting the results of Eqs. (37) and (38) into Eq. (39), obtain

ρ₂V^{ð Þ}_S^IIð Þ ¼K max

~L^R_S,L~^F_S

� �

≥0

1�θ

ð Þ

α₁�θα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁� β^Rð1�ω₁Þ þ

α₂�θα₂þV^{0ð Þ}_S^Ið ÞKϑ^F₁� β^Fð1�ω₂Þ 0

@ 2

4 8<

:

þβ₁L~^R_Sþβ₂L~^F_SþðΓþδÞK�

�1

2~β^R�~L^R_Sð Þt �2

�1

2~β^F� �L~^F_S 2

�1

2ω₁β^R θα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁ β^Rð1�ω₁Þ

!2

�1

2ω₂β^F θα₂þV^{0ð Þ}_S^Ið ÞK ϑ^F₁ β^Fð1�ω₂Þ

!23 5

þ∂V^{ð Þ}_S^IIð ÞK

∂K

ϑ^R₁hθα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁i

β^Rð1�ω₁Þ þϑ^F₁hθα₂þV^{0ð Þ}_S^Ið ÞK ϑ^F₁i

β^Fð1�ω₂Þ þϑ₂�~L^R_S,L~^F_S�

�ξK 2

4

3 5

þ1 2

∂²V^{ð Þ}_S^IIð ÞK

∂K² φ²ð ÞK )

:

(40) Via the first order conditions of (Eq. (40)), we obtain the optimal values

~L^R,~L^F

� �

for Player II as:

L~^R_S ¼ð1�θÞβ₁þV^{0ð Þ}_S^IIð ÞK ϑ^R₂

β~^R , (41)

L~^F_S ¼ð1�θÞβ₂þV^{0ð Þ}_S^IIð ÞK ϑ^F₂

~β^F : (42)

And the optimal value forðω₁,ω₂Þ

ω₁¼

α₁ð2�3θÞ þϑ^R₁h2V⁰Sð ÞIIð Þ �K VS0ð ÞIð ÞK i

α₁ð2�θÞ þϑ^R₁�2V⁰Sð ÞIIð Þ þK VS0ð ÞIð ÞK � , (43) and

ω₂¼

α₂ð2�3θÞ þϑ^F₁h2V⁰_S^{ð ÞII}ð Þ �K V_S^{0ð ÞI}ð ÞK i

α₂ð2�θÞ þϑ^F₁�2V⁰Sð ÞIIð Þ þK VS0ð ÞIð ÞK � : (44) Hence, the solution of the HJB equation is an unary function with K (K as the independent variable), we define V^{ð Þ}_S^I ¼a1Kþb1and V^{ð Þ}_S^II ¼a2Kþb2,

where a1, b1, a2, and b2are constants that need to be solved. Simplifying (Eq. (39)), obtain:

ρ₁V^{ð Þ}_S^Ið Þ ¼K θ α₁ θα₁þa1ϑ^R₁ β^Rð1�ω₁Þ

� �

þα₂ θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �

þβ₁ ð1�θÞβ₁þa2ϑ^R₂

~β^R

! (45)

þβ₂ ð1�θÞβ₂þa₂ϑ^F₂

~β^F

!

þðΓþδÞKÞ

!

�1

2β^Rð1�ω₁Þ θα₁þa₁ϑ^R₁ β^Rð1�ω₁Þ

� �2

�1

2β^Fð1�ω₂Þ θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �2

�ξKa1

þ ϑ^R₁�θα₁þa₁ϑ^R₁�

β^Rð1�ω₁Þ þϑ^F₁�θα₂þa₁ϑ^F₁� β^Fð1�ω₂Þ

" #

a₁

þ ϑ^R₂�ð1�θÞβ₁þa2ϑ^R₂�

~β^R þϑ^F₂�ð1�θÞβ₂þa2ϑ^F₂�

~β^F

" #

a1,

(46)

and simplifying (Eq. (40)), obtain:

ρ₂V^{ð Þ}_S^IIð Þ ¼K ð1�θÞ α₁�θα₁þa1ϑ^R₁�

β^Rð1�ω₁Þ þα₂�θα₂þa1ϑ^F₁�

β^Fð1�ω₂Þ þβ₁ ð1�θÞβ₁þa₂ϑ^R₂

~β^R

!

þβ₂ ð1�θÞβ₂þa2ϑ^F₂ β~^F

!

þðΓþδÞK

!

�1

2~β^R ð1�θÞβ₁þa2ϑ^R₂

~β^R

!2

�1

2~β^F ð1�θÞβ₂þa2ϑ^F₂

~β^F

!2

�1

2ω₁β^R θα₁þa1ϑ^R₁ β^Rð1�ω₁Þ

� �²

�1

2ω₂β^F θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �²

þ ϑ^R₁�θα₁þa₁ϑ^R₁�

β^Rð1�ω₁Þ þϑ^F₁�θα₂þa₁ϑ^F₁� β^Fð1�ω₂Þ

" #

a2�ξKa2

þ ϑ^R₂�ð1�θÞβ₁þa2ϑ^R₂�

~β^R þϑ^F₂�ð1�θÞβ₂þa2ϑ^F₂�

~β^F

" #

a2:

(47) This implies that,

a₁¼θðΓþδÞ ρ₁þξ

ð Þ, b₁¼Φ1

ρ₁, a₂¼ð1�θÞðΓþδÞ ρ₂þξ

ð Þ , b₂¼Φ2

ρ₂ , (48) where

Φ1¼ α₁θ��θα₁þa₁ϑ^R₁� 2 þϑ^R₁a₁

! θα₁þa₁ϑ^R₁ β^Rð1�ω₁Þ

� �

þ α₂θ��θα₂þa1ϑ^F₁� 2 þϑ^F₁a1

! θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �

þ�β₁θþϑ^R₂a₁� ð1�θÞβ₁þa₂ϑ^R₂

~β^R

!

þ�β₂θþϑ^F₂a2� ð1�θÞβ₂þa2ϑ^F₂

~β^F

!

>0,

(49)

Substituting the results of Eqs. (37) and (38) into Eq. (39), obtain

ρ₂V^{ð Þ}_S^IIð Þ ¼K max

~L^R_S,~L^F_S

� �

≥0

1�θ

ð Þ

α₁�θα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁� β^Rð1�ω₁Þ þ

α₂�θα₂þV^{0ð Þ}_S^Ið ÞK ϑ^F₁� β^Fð1�ω₂Þ 0

@ 2

4 8<

:

þβ₁L~^R_Sþβ₂L~^F_SþðΓþδÞK�

�1

2~β^R�L~^R_Sð Þt �2

�1

2β~^F� �~L^F_S 2

�1

2ω₁β^R θα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁ β^Rð1�ω₁Þ

!2

�1

2ω₂β^F θα₂þV^{0ð Þ}_S^Ið ÞKϑ^F₁ β^Fð1�ω₂Þ

!23 5

þ∂V^{ð Þ}_S^IIð ÞK

∂K

ϑ^R₁hθα₁þV^{0ð Þ}_S^Ið ÞK ϑ^R₁i

β^Rð1�ω₁Þ þϑ^F₁hθα₂þV^{0ð Þ}_S^Ið ÞK ϑ^F₁i

β^Fð1�ω₂Þ þϑ₂�L~^R_S,L~^F_S�

�ξK 2

4

3 5

þ1 2

∂²V^{ð Þ}_S^IIð ÞK

∂K² φ²ð ÞK )

:

(40) Via the first order conditions of (Eq. (40)), we obtain the optimal values L~^R,L~^F

� �

for Player II as:

L~^R_S ¼ð1�θÞβ₁þV^{0ð Þ}_S^IIð ÞK ϑ^R₂

~β^R , (41)

L~^F_S ¼ð1�θÞβ₂þV^{0ð Þ}_S^IIð ÞK ϑ^F₂

~β^F : (42)

And the optimal value forðω₁,ω₂Þ

ω₁¼

α₁ð2�3θÞ þϑ^R₁h2V⁰Sð ÞIIð Þ �K VS0ð ÞIð ÞK i

α₁ð2�θÞ þϑ^R₁�2V⁰Sð ÞIIð Þ þK VS0ð ÞIð ÞK � , (43) and

ω₂¼

α₂ð2�3θÞ þϑ^F₁h2V⁰_S^{ð ÞII}ð Þ �K V_S^{0ð ÞI}ð ÞK i

α₂ð2�θÞ þϑ^F₁�2V⁰Sð ÞIIð Þ þK VS0ð ÞIð ÞK � : (44) Hence, the solution of the HJB equation is an unary function with K (K as the independent variable), we define V^{ð Þ}_S^I ¼a1Kþb1and V^{ð Þ}_S^II ¼a2Kþb2,

where a1, b1, a2, and b2are constants that need to be solved. Simplifying (Eq. (39)), obtain:

ρ₁V^{ð Þ}_S^Ið Þ ¼K θ α₁ θα₁þa1ϑ^R₁ β^Rð1�ω₁Þ

� �

þα₂ θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �

þβ₁ ð1�θÞβ₁þa2ϑ^R₂

~β^R

! (45)

þβ₂ ð1�θÞβ₂þa₂ϑ^F₂

~β^F

!

þðΓþδÞKÞ

!

�1

2β^Rð1�ω₁Þ θα₁þa₁ϑ^R₁ β^Rð1�ω₁Þ

� �2

�1

2β^Fð1�ω₂Þ θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �2

�ξKa1

þ ϑ^R₁�θα₁þa₁ϑ^R₁�

β^Rð1�ω₁Þ þϑ^F₁�θα₂þa₁ϑ^F₁� β^Fð1�ω₂Þ

" #

a₁

þ ϑ^R₂�ð1�θÞβ₁þa2ϑ^R₂�

~β^R þϑ^F₂�ð1�θÞβ₂þa2ϑ^F₂�

~β^F

" #

a1,

(46)

and simplifying (Eq. (40)), obtain:

ρ₂V^{ð Þ}_S^IIð Þ ¼K ð1�θÞ α₁�θα₁þa1ϑ^R₁�

β^Rð1�ω₁Þ þα₂�θα₂þa1ϑ^F₁�

β^Fð1�ω₂Þ þβ₁ ð1�θÞβ₁þa₂ϑ^R₂

~β^R

!

þβ₂ ð1�θÞβ₂þa2ϑ^F₂

~β^F

!

þðΓþδÞK

!

�1

2~β^R ð1�θÞβ₁þa2ϑ^R₂

~β^R

!2

�1

2~β^F ð1�θÞβ₂þa2ϑ^F₂

~β^F

!2

�1

2ω₁β^R θα₁þa1ϑ^R₁ β^Rð1�ω₁Þ

� �²

�1

2ω₂β^F θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �²

þ ϑ^R₁�θα₁þa₁ϑ^R₁�

β^Rð1�ω₁Þ þϑ^F₁�θα₂þa₁ϑ^F₁� β^Fð1�ω₂Þ

" #

a2�ξKa2

þ ϑ^R₂�ð1�θÞβ₁þa2ϑ^R₂�

~β^R þϑ^F₂�ð1�θÞβ₂þa2ϑ^F₂� β~^F

" #

a2:

(47) This implies that,

a₁¼θðΓþδÞ ρ₁þξ

ð Þ, b₁¼Φ1

ρ₁, a₂¼ð1�θÞðΓþδÞ ρ₂þξ

ð Þ , b₂¼Φ2

ρ₂, (48) where

Φ1¼ α₁θ��θα₁þa₁ϑ^R₁� 2 þϑ^R₁a₁

! θα₁þa₁ϑ^R₁ β^Rð1�ω₁Þ

� �

þ α₂θ��θα₂þa1ϑ^F₁� 2 þϑ^F₁a1

! θα₂þa1ϑ^F₁ β^Fð1�ω₂Þ

� �

þ�β₁θþϑ^R₂a₁� ð1�θÞβ₁þa₂ϑ^R₂ β~^R

!

þ�β₂θþϑ^F₂a2� ð1�θÞβ₂þa2ϑ^F₂

~β^F

!

>0,

(49)

and

Φ2¼ ð1�θÞα₁�ω₁�θα₁þa1ϑ^R₁� 2 1ð �ω₁Þ þϑ^R₁a2

!�θα₁þa1ϑ^R₁� β^Rð1�ω₁Þ

þ ð1�θÞα₂�ω₂�θα₂þa₁ϑ^F₁� 2 1ð �ω₂Þ þϑ^F₁a₂

!�θα₂þa₁ϑ^F₁� β^Fð1�ω₂Þ

þ ð1�θÞβ₁��ð1�θÞβ₁þa₂ϑ^R₂� 2 þϑ^R₂a₂

! ð1�θÞβ₁þa2ϑ^R₂

~β^R

!

þ ð1�θÞβ₂��ð1�θÞβ₂þa2ϑ^F₂� 2 þϑ^F₂a2

! ð1�θÞβ₂þa₂ϑ^F₂

~β^F

!

>0:

(50)

Substituting the results of a1and a2into Eqs. (37), (38), (41) and (42), and simplifying, we obtain the optimal effort level of hybrid-enabling technological improvements. By substituting optimal values given in Eqs. (48)–(50) into Eqs. (46) and (47) obtain the optimal sharing payoff functions under hybrid- enabling technology on renewable sources and fossil fuel for Player I and Player II.

3.1 The limit of expectation and variance

The payoff of Player I and Player II, under the Stackelberg game paradigm is related to the improvement degree of hybrid-enabling technology via Proposition 4.

To analyze the limit of expectations and variance under Stackelberg game equilib- rium rewrite (Eq. (19)) as follows.

dK tð Þ ¼½μ₁þμ₂�ξK tð Þ�dtþφ ffiffiffiffi pK

dW tð Þ K 0ð Þ ¼K₀>0,

(

(51) where

μ₁¼ϑ₁ α₁ð2�θÞðρ₂þξÞðρ₁þξÞ þϑ^R₁ðΓþδÞðð2�2θÞðρ₁þξÞ þθ ρð ₂þξÞÞ 2β^Rðρ₂þξÞðρ₁þξÞ

�

þα₂ð2�θÞðρ₂þξÞðρ₁þξÞ þϑ^F₁ðΓþδÞðð2�2θÞðρ₁þξÞ þθ ρð ₂þξÞÞ 2β^Fðρ₂þξÞðρ₁þξÞ

�

, (52) and

μ₂¼ϑ₂ ð1�θÞðβ₁ðρ₂þξÞ þðΓþδÞÞϑ^R₂

~β^Rðρ₂þξÞ þð1�θÞðβ₂ðρ₂þξÞ þðΓþδÞÞϑ^F₂

~β^Fðρ₂þξÞ

# :

"

(53) Proposition 6. The limit of expectation E K tð ð ÞÞ, and variance D K tð ð ÞÞin the Stackelberg game feedback equilibrium must satisfy

E K tð ð ÞÞ ¼μ₁þμ₂

ξ þe^�ξt K~0�μ₁þμ₂ ξ

� �

, lim_t!∞E K tð ð ÞÞ ¼μ₁þμ₂

ξ : (54)

D K tð ð ÞÞ ¼φ² ðμ₁þμ₂Þ �2�μ₁þμ₂�ξK~0�

e^�ξtþ�μ₁þμ₂�2ξK~0� e^�2ξt

� �

2ξ² (55)

t!∞limD E K tð ð ð ÞÞÞ ¼φ²ðμ₁þμ₂Þ

2ξ² , (56)

Proof. Applying Itô’s lemma to (Eq. (51)), obtain:

d K tð ð ÞÞ²¼�2ðμ₁þμ₂þφ²ÞK�2ξK²�

dtþ2φK ffiffiffiffi pK

dW tð Þ K 0ð Þ

ð Þ²¼K~²₀>0: (

(57)

Then E K tð ð ÞÞand E K tð ð ÞÞ²can be defined as:

dE K tð ð ÞÞ ¼½μ₁þμ₂�ξK tð Þ�dt K 0ð Þ ¼K0>0:

�

(58)

dE K tð ð ÞÞ²¼�½2ðμ₁þμ₂þφ²ÞK�E Kð Þ �2ξE K� �² � dt K 0ð Þ

ð Þ²¼K²₀>0, (

(59) Solving the above non-homogeneous linear differential equation, will obtain the results.