Appendix A
Proof of Theorem 1. In scenario
, we need to establish the existence of continuously differentiable value functions
and
to ensure that there is a unique solution
to Equation (3) and the
equations. The
equations for each player are as follows:
Since the game is a Stackelberg game and
is the leader, we first determine the first-order necessary conditions for scenario
as
,
, and
. Substituting these into
’s
Equation (A1), the following equation can be obtained:
Thus, the necessary condition for is . And we obtain the optimal decisions of the players to the game as Equation (A4). ☐
Replacing the strategies of each player in Equations (A1) and (A2), we obtain Equations (A5) and (A6):
Based on the structure of (A5) and (A6), we conjecture the linear-valued functions,
and
, where
,
,
, and
are the constant parameters to be identified. Substituting
and
and their derivatives into Equations (A5) and (A6), we obtain Equations (A7) and (A8):
then we can identify the parameter values as follows:
which are all strictly positive; therefore, the results show the concave profit functions with respect to the players’ decision variables and the existence of a unique equilibrium that maximizes the objective function. The optimal value functions then become
By solving the differential equation of Equation (3), we obtain the optimal brand goodwill trajectory as , where , is the steady-state goodwill.
Substituting into Equations (A10) and (A11), E-CLSC members profits are solved; replacing and Equation (A9) into Equation (A4), E-CLSC members’ equilibrium strategies are solved. Theorem 1 is proved.
Proof of Theorem 2. In scenario
, we need to establish the existence of continuously differentiable value functions
and
to ensure that there is a unique solution
to Equation (3) and the
equations. The
equations for each player are as follows:
Since
is the leader, we first determine the first-order necessary conditions for scenario
as
and
. Substituting these into
’s
Equation (A12), the following equation can be obtained:
Thus, the necessary condition for is . And we obtain the optimal decisions of the players to the game as Equation (A15). ☐
Replacing the strategies of each player in Equations (A12) and (A13), we obtain the following:
Based on the structure of (A16) and (A17), we conjecture the linear-valued functions,
and
, where
,
,
, and
are the constant parameters to be identified. Substituting
and
and their derivatives into Equations (A16) and (A17) we obtain the following:
then we can identify the parameter values as follows:
which are all strictly positive; therefore, the results show the concave profit functions with respect to the players’ decision variables and the existence of a unique equilibrium that maximizes the objective function. The optimal value functions then become
By solving the differential equation of Equation (3), we obtain the optimal brand goodwill trajectory as , where , is the steady-state goodwill.
Substituting into Equations (A21) and (A22), E-CLSC members profits are solved; replacing and Equation (A20) into (A15), E-CLSC members’ equilibrium strategies are solved. Theorem 2 is proved.
Proof of Corollary 1. In scenarios and , the platform profits are and , respectively. Then, we have . Obviously, when , ; when , .
In scenarios and , similarly, the manufacturer profits are and , respectively. Then, we have . Obviously, when or , ; when , . ☐
Proof of Theorem 3. In scenario
, we need to establish the existence of continuously differentiable value functions
and
to ensure that there is a unique solution
to Equation (3) and the
equations. The
equations for each player are as follows:
Since
is the leader, we first determine the first-order necessary conditions for scenario
as
,
, and
. Substituting these into
’s
Equation (A23), the following equation can be obtained:
Thus, the necessary condition for is . And we obtain the optimal decisions of the players to the game as Equation (A26). ☐
Replacing the strategies of each player in Equations (A23) and (A24), we obtain the following:
Based on the structure of (A27) and (A28), we conjecture the linear-valued functions,
and
, where
,
,
, and
are the constant parameters to be identified. Substituting
and
and their derivatives into Equations (A27) and (A28) we obtain the following:
then we can identify the parameter values as follows:
which are all strictly positive; therefore, the results show the concave profit functions with respect to the players’ decision variables and the existence of a unique equilibrium that maximizes the objective function. The optimal value functions then become
By solving the differential equation of Equation (3), we obtain the optimal brand goodwill trajectory as , where , is the steady-state goodwill.
Following the proof procedure of Theorems 1 and 2, the equilibrium strategies and E-CLSC members’ profits under scenario can be obtained, which will not be repeated here.
Proof of Theorem 4. In scenario
, we need to establish the existence of continuously differentiable value functions
and
to ensure that there is a unique solution
to Equation (3) and the
equations. The
equations for each player are the following:
Since
is the leader, we first determine the first-order necessary conditions for scenario
as
and
. Substituting these into
’s
Equation (A34), the following equation can be obtained:
Thus, the necessary condition for is . And we obtain the optimal decisions of the players to the game as Equation (A37). ☐
Replacing the strategies of each player in Equations (A34) and (A35), we obtain the following:
Based on the structure of (A38) and (A39), we conjecture the linear-valued functions,
and
, where
,
,
, and
are the constant parameters to be identified. Substituting
and
and their derivatives into Equations (A38) and (A39), we obtain the following:
then we can identify the parameter values as follows:
which are all strictly positive; therefore, the results show the concave profit functions with respect to the players’ decision variables and the existence of a unique equilibrium that maximizes the objective function. The optimal value functions then become
By solving the differential equation of Equation (3), we obtain the optimal brand goodwill trajectory as , where , is the steady-state goodwill. Similarly, the equilibrium strategies and optimal value functions for each enterprise under scenario can be further obtained. The proof procedure is no different from the rest of Theorems 1, 2, and 3, and will not be repeated here.
Proof of Corollary 2. In scenarios and , the platform profits are and , respectively. Then, we have . Obviously, when , ; when , .
In scenarios and , similarly, the manufacturer profits are and , respectively. Then, we have . Obviously, when or , ; when , . ☐
Proof of Theorem 5. In the reselling model, the comparative relationships between the quantity of used products recycling, steady-state goodwill, demand, equilibrium strategies, and manufacturer profits with and without blockchain technology are as follows:
- (1)
, because , we have
- (2)
, because , we have .
- (3)
, because , we have .
- (4)
, because , we have .
- (5)
, because , we have .
- (6)
, because , we have .
- (7)
, because , we have .
- (8)
, because , we have .
We omit the proof in the marketplace model since the proof procedure is similar. ☐
Proof of Theorem 6. In the reselling model, the blockchain deployment conditions need to be satisfied: , where and , respectively. Thus, we have , i.e., . ☐
In the marketplace model, the blockchain deployment conditions need to be satisfied: , where and , respectively. Thus, we have , i.e., .
Proof of Theorem 7. According to the proof process of Corollaries 1 and 2, we obtain the sales model selection for the platform and manufacturer’s intention to collaborate in Theorem 7. ☐
Proof of Theorem 8. When the online platform collaborates with a blockchain provider, there is a per-unit usage cost . The proof process is consistent with the process of Theorems 1, 2, 3, and 4 and will not be expanded here. We can obtain the following:
In scenario , the optimal brand trajectory is , and its steady-state . The E-CLSC members’ equilibrium strategies are , , , . ☐
The optimal profits for the manufacturer and the platform are and , respectively.
In scenario , the optimal brand trajectory is , and its steady-state . The E-CLSC members’ equilibrium strategies are , , .
The optimal profits for the manufacturer and the platform are and , respectively.