State- and Control-Dependent Incentives in a Closed-Loop Supply Chain with Dynamic Returns

Abstract

This paper analyzes two incentive schemes available for a closed-loop supply chain (CLSC) in which a manufacturer and a retailer contribute to the return rate dynamics through their investments in green activity programs. Both firms have economic motivations to perform the return rate because customers who return end-of-use goods also repurchase new ones. In addition, the manufacturer exploits the returns’ residual value in operations to increase profits. Because the manufacturer has both operational and marketing motivations to close the loop, he can provide an incentive to the retailer to boost her investments in green activity programs. The incentive can be either state dependent or control dependent. The former assumes that the incentive depends on the fraction of customers who are willing to return end-of-use products; the latter is proportional to the retailer’s green activity programs efforts. Our results show that a state-dependent incentive is profit-Pareto-improving only when the retailer’s environmental effectiveness is large. In contrast, a control-dependent incentive mechanism is profit-Pareto-improving for low incentive values, high retailer’s environmental effectiveness, and customers’ repurchasing intention. In all other cases, players have divergent preferences and neither mechanism coordinates the CLSC.

Appendices

Appendix 1

Proof of Proposition 1

In the non-coordinated scenario, we search for a pair of bounded and continuously differentiable value functions $V_{M}^{B}\left( r^{B}\right) ,V_{R}^{B}\left( r^{B}\right)$ for which a unique solution for $r^{B}\left( t\right)$ does exist, and the Hamilton–Jacobi–Bellman (HJB) equations:

$$\begin{aligned} \rho V_{M}^{B}\left( r^{B}\right)&= \left( \alpha +r^{B}\theta -\beta p^{B}\right) \left( \omega ^{B}+r^{B}\varDelta \right) -\frac{A_{M}^{B^{2}}}{2} +V_{M}^{B^{\prime }}\left( aA_{M}^{B}+bA_{R}^{B}-\delta r^{B}\right) \end{aligned}$$

(36)

$$\begin{aligned} \rho V_{R}^{B}\left( r^{B}\right)&= \left( \alpha +r^{B}\theta -\beta p^{B}\right) \left( p^{B}-\omega ^{B}\right) -\frac{A_{R}^{B^{2}}}{2} +V_{R}^{B^{\prime }}\left( aA_{M}^{B}+bA_{R}^{B}-\delta r^{B}\right) \nonumber \\ \end{aligned}$$

(37)

are satisfied for any value of $r^{B}\in (0,1].$ Maximization of the $R$’s HJB gives pricing and $R$’s GAP strategies.

$$\begin{aligned} p^{B}\left( r^{B}\right)&= \frac{\alpha +r^{B}\theta +\beta \omega ^{B}}{2\beta } \end{aligned}$$

(38)

$$\begin{aligned} A_{R}^{B}&= bV_{R}^{B^{\prime }} \end{aligned}$$

(39)

Substituting Eqs. (38) and (39) inside $M$’s HJB provides:

$$\begin{aligned} \rho V_{M}^{B}\left( r^{B}\right) =\left( \frac{\alpha +r^{B}\theta -\beta \omega ^{B}}{2}\right) \left( \omega ^{B}+r^{B}\varDelta \right) -\frac{ A_{M}^{B^{2}}}{2}+V_{M}^{B^{\prime }}\left( aA_{M}^{B}+b^{2}V_{R}^{B^{\prime }}-\delta r^{B}\right) \end{aligned}$$

(40)

Maximization of Eq. (40) with respect to $M\prime s$ GAP strategies and wholesale price gives

$$\begin{aligned} A_{M}^{B}&= aV_{M}^{B^{\prime }} \end{aligned}$$

(41)

$$\begin{aligned} \omega ^{B}\left( r^{B}\right)&= \frac{\alpha +r^{B}\left( \theta -\varDelta \beta \right) }{2\beta } \end{aligned}$$

(42)

Substituting Eq. (42) in (38), pricing results:

$$\begin{aligned} p^{B}\left( r^{B}\right) =\frac{3\alpha +r^{B}\left( 3\theta -\varDelta \beta \right) }{4\beta } \end{aligned}$$

(43)

Plagging Eqs. (43), (42), (39), and (41) in Eqs. (40) and (37), it provides

$$\begin{aligned} \rho V_{M}^{B}\left( r^{B}\right)&= \frac{1}{2\beta }\left( \frac{\alpha +r^{B}\left( \theta +\varDelta \beta \right) }{2}\right) ^{2}+V_{M}^{B^{\prime }}\left( \frac{a^{2}V_{M}^{B^{\prime }}}{2}+b^{2}V_{R}^{B^{\prime }}-\delta r^{B}\right) \end{aligned}$$

(44)

$$\begin{aligned} \rho V_{R}^{B}\left( r^{B}\right)&= \frac{1}{\beta }\left( \frac{\alpha +r^{B}\left( \theta +\varDelta \beta \right) }{4}\right) ^{2}+V_{R}^{B^{\prime }}\left( a^{2}V_{M}^{B^{\prime }}+\frac{b^{2}V_{R}^{B^{\prime }}}{2}-\delta r^{B}\right) \end{aligned}$$

(45)

We conjecture quadratic value functions $V_{M}^{B}\left( r^{B}\right) =\frac{ d_{1}}{2}r^{B^{2}}+d_{2}r^{B}+d_{3}$ and $V_{R}^{B}\left( r^{B}\right) = \frac{f_{1}}{2}r^{B^{2}}+f_{2}r^{B}+f_{3},$ where the pairs $\left( d_{j},f_{j}\right) ,j=1\ldots 3$ are the constant parameters to be identified. Substituting our conjectures and their derivatives in Eqs. (44) and (45) gives

$$\begin{aligned}&8\beta \rho \left( \frac{d_{1}}{2}r^{B^{2}}+d_{2}r^{B}+d_{3}\right) =\left( \alpha +r^{B}\left( \theta +\varDelta \beta \right) \right) ^{2} \nonumber \\&\quad +\,4\beta \left( d_{1}r^{B}+d_{2}\right) \left( a^{2}\left( d_{1}r^{B}+d_{2}\right) +2b^{2}\left( f_{1}r^{B}+f_{2}\right) -2\delta r^{B}\right) \end{aligned}$$

(46)

$$\begin{aligned}&16\beta \rho \left( \frac{f_{1}}{2}r^{B^{2}}+f_{2}r^{B}+f_{3}\right) =\left( \alpha +r^{B}\left( \theta +\varDelta \beta \right) \right) ^{2} \nonumber \\&\quad +\,8\beta \left( f_{1}r^{B}+f_{2}\right) \left( 2a^{2}\left( d_{1}r^{B}+d_{2}\right) +b^{2}\left( f_{1}r^{B}+f_{2}\right) -2\delta r^{B}\right) \end{aligned}$$

(47)

By identification, the constant parameters can be derived by solving the following set of coupled algebraic Riccati equations:

$$\begin{aligned} \varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}+4\beta \left( 2b^{2}f_{1}-2\delta -\rho \right) d_{1}+4a^{2}\beta d_{1}^{2}&= 0 \end{aligned}$$

(48)

$$\begin{aligned} 2\left( \alpha \left( \theta +\varDelta \beta \right) +4b^{2}\beta d_{1}f_{2}+4\beta \left( a^{2}d_{1}+b^{2}f_{1}-\delta -\rho \right) d_{2}\right)&= 0 \end{aligned}$$

(49)

$$\begin{aligned} \alpha ^{2}+4\beta \left( 2b^{2}f_{2}+a^{2}d_{2}\right) d_{2}-8\beta \rho d_{3}&= 0 \end{aligned}$$

(50)

$$\begin{aligned} \left( \varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}+8\beta \left( 2a^{2}d_{1}-2\delta -\rho \right) f_{1}+8b^{2}\beta f_{1}^{2}\right)&= 0 \end{aligned}$$

(51)

$$\begin{aligned} 2\left( \alpha \left( \theta +\varDelta \beta \right) +8a^{2}\beta d_{2}f_{1}+8\beta \left( a^{2}d_{1}+b^{2}f_{1}-\delta -\rho \right) f_{2}\right)&= 0 \end{aligned}$$

(52)

$$\begin{aligned} \alpha ^{2}+8\beta \left( 2a^{2}d_{2}+b^{2}f_{2}\right) f_{2}-16\rho \beta f_{3}&= 0 \end{aligned}$$

(53)

To derive the coefficients, we can start from Eq. (48) and obtain $f_{1}$ as a function of $d_{1}:f_{1}=f\left( d_{1}\right)$ where

$$\begin{aligned} f\left( d_{1}\right) =\frac{4\beta d_{1}\left( 2\delta +\rho -a^{2}d_{1}^{{}}\right) -B_{3}}{8\beta b^{2}f_{1}d_{1}}=\varOmega _{1} \end{aligned}$$

(54)

with $B_{3}=\varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}$. Substituting Eq. (54) for Eqs. (49) and (52), we can derive both $d_{2}$ and $f_{2}$ as a function of $d_{1}$

$$\begin{aligned} d_{2}\left( d_{1}\right)&= \frac{b^{2}d_{1}-2B_{4}}{8\beta \left( B_{1}^{2}-a^{2}b^{2}d_{1}\varOmega _{1}\right) }B_{2}=\varOmega _{2} \end{aligned}$$

(55)

$$\begin{aligned} f_{2}\left( d_{1}\right)&= \frac{2a_{1}^{2}\varOmega -B_{4}}{8\beta \left( B_{1}^{2}-a^{2}b^{2}d_{1}\varOmega _{1}\right) }B_{2}=\varOmega _{3} \end{aligned}$$

(56)

with $B_{1}=a^{2}m_{1}^{*}+b^{2}n_{1}^{*}-\delta -\rho <0,\, B_{2}=\alpha \left( \theta +\varDelta \beta \right) >0,\, B_{4}=a^{2}d_{1}+b^{2}\varOmega _{1}-\delta -\rho$. We then substitute Eqs. (55) and (56) in Eqs. (50) and (53) to derive $d_{3}$ and $f_{3}$ as a function of $d_{1}$:

$$\begin{aligned} d_{3}\left( d_{1}\right)&= \frac{\alpha ^{2}+4\beta \left( 2b^{2}\varOmega _{3}+a^{2}\varOmega _{2}\right) \varOmega _{2}}{8\beta \rho }=\varOmega _{4} \end{aligned}$$

(57)

$$\begin{aligned} f_{3}\left( d_{1}\right)&= \frac{\alpha ^{2}+8\beta \left( 2a^{2}\varOmega _{2}+b^{2}\varOmega _{3}\right) \varOmega _{3}}{16\rho \beta }=\varOmega _{5} \end{aligned}$$

(58)

Finally, replacing Eq. (54) into (51) gives a nonlinear equation that we have solved numerically in Mathematica 6.0.$\square$

Proof of Proposition 2

To show the inefficiency of a per-return incentive mechanism, we need to search for a pair of bounded and continuously differentiable value functions $V_{M}^{P}\left( r^{P}\right) ,V_{R}^{P}\left( r^{P}\right)$ for which a unique solution for $r^{P}\left( t\right)$ exists, and the HJBs are as follows:

$$\begin{aligned} \rho V_{M}^{P}\left( r^{P}\right)&= \left( \alpha +r^{P}\theta -\beta p^{P}\right) \left( \omega ^{P}+r^{P}\varDelta -\mu r^{P}\right) \nonumber \\&-\,\frac{ A_{M}^{P^{2}}}{2}+V_{M}^{P^{\prime }}\left( aA_{M}^{P}+bA_{R}^{P}-\delta r^{P}\right) \end{aligned}$$

(59)

$$\begin{aligned} \rho V_{R}^{P}\left( r^{P}\right)&= \left( \alpha +r^{P}\theta -\beta p^{P}\right) \left( p^{P}-\omega ^{P}+\mu r^{P}\right) \nonumber \\&-\,\frac{A_{R}^{P^{2}}}{ 2}+V_{R}^{P^{\prime }}\left( aA_{M}^{P}+bA_{R}^{P}-\delta r^{P}\right) \end{aligned}$$

(60)

Because the coordination game also has a leader–follower structure where $M$ is the leader, we start from the maximization of $R$’s HJB with respect to price and GAP strategies:

$$\begin{aligned} p^{P}\left( r^{P}\right)&= \frac{\alpha +\beta \omega ^{P}+\left( \theta -\beta \mu \right) r^{P}}{2\beta } \end{aligned}$$

(61)

$$\begin{aligned} A_{R}^{P}&= bV_{R}^{P^{\prime }} \end{aligned}$$

(62)

Substituting Eqs. (61) and (62) inside $M$’s HJB gives

$$\begin{aligned} \rho V_{M}^{P}\left( r^{P}\right)&= \left( \frac{\alpha -\beta \omega ^{P}+\left( \theta +\beta \mu \right) r^{P}}{2}\right) \left( \omega ^{P}+r^{P}\varDelta -\mu r^{P}\right) -\frac{A_{M}^{P^{2}}}{2}\nonumber \\&\quad +V_{M}^{P^{\prime }}\left( aA_{M}^{P}+bA_{R}^{P}-\delta r^{P}\right) \end{aligned}$$

(63)

whose maximization with respect to wholesale price and GAP strategies yields:

$$\begin{aligned} \omega ^{P}\left( r^{P}\right)&= \frac{\alpha +\left( \theta +\left( 2\mu -\varDelta \right) \beta \right) r^{P}}{2\beta } \end{aligned}$$

(64)

$$\begin{aligned} A_{M}^{P}&= aV_{M}^{P^{\prime }} \end{aligned}$$

(65)

Plugging Eq. (64) in Eq. (61) leads to

$$\begin{aligned} p^{P}\left( r^{P}\right) =\frac{3\alpha +r^{P}\left( 3\theta -\varDelta \beta \right) }{4\beta } \end{aligned}$$

(66)

Subsituiting Eqs. (64), (65), (66) and (62) in (63) and (60) gives

$$\begin{aligned} \rho V_{M}^{P}\left( r^{P}\right)&= \frac{\left( \alpha +\left( \theta +\varDelta \beta \right) r^{P}\right) ^{2}}{8\beta }+V_{M}^{P^{\prime }}\left( \frac{a^{2}V_{M}^{P^{\prime }}}{2}+b^{2}V_{R}^{P^{\prime }}-\delta r^{P}\right) \end{aligned}$$

(67)

$$\begin{aligned} \rho V_{R}^{P}\left( r^{P}\right)&= \frac{\left( \alpha +\left( \theta +\varDelta \beta \right) r^{P}\right) ^{2}}{16\beta }+V_{R}^{P^{\prime }}\left( a^{2}V_{M}^{P^{\prime }}+\frac{b^{2}V_{R}^{P^{\prime }}}{2}-\delta r^{P}\right) \end{aligned}$$

(68)

from which it turns out that $V_{M}^{B}\left( r^{B}\right) =V_{M}^{P}\left( r^{P}\right)$ and $V_{R}^{B}\left( r^{B}\right) =V_{R}^{P}\left( r^{P}\right) ,$ and thus, the implementation of a per-return incentive does not lead to any form of coordination.$\square$

Proof of Proposition 3

Here we follow the same steps as in the proof of Proposition 1 to derive the equilibrium strategies under the assumption that the CLSC is coordinated through a state-dependent incentive mechanism. The HJBs for this game are given by

$$\begin{aligned} \rho V_{M}^{S}\left( r^{S}\right)&= \left( \alpha +r^{S}\theta -\beta p^{S}\right) \left( \omega ^{S}+r^{S}\varDelta \right) -\mu r^{S}-\frac{ A_{M}^{S^{2}}}{2}+V_{M}^{S^{\prime }}\left( aA_{M}^{S}+bA_{R}^{S}-\delta r^{S}\right) \end{aligned}$$

(69)

$$\begin{aligned} \rho V_{R}^{S}\left( r^{S}\right)&= \left( \alpha +r^{S}\theta -\beta p^{S}\right) \left( p^{S}-\omega ^{S}\right) +\mu r^{S}-\frac{A_{R}^{S^{2}}}{ 2}+V_{R}^{S^{\prime }}\left( aA_{M}^{S}+bA_{R}^{S}-\delta r^{S}\right) \nonumber \\ \end{aligned}$$

(70)

Maximization of $R$’s HJB with respect to pricing and GAP strategies gives

$$\begin{aligned} p^{S}\left( r^{S}\right)&= \frac{\alpha +r^{S}\theta +\beta \omega ^{S}}{ 2\beta } \end{aligned}$$

(71)

$$\begin{aligned} A_{R}^{S}&= bV_{R}^{S^{\prime }} \end{aligned}$$

(72)

These expressions must be satisfied by the pricing and $R$’s GAP strategies. Replacing Eqs. (71) and (72) inside Eq. (69), it gives the following expression:

$$\begin{aligned} \rho V_{M}^{S}\left( r^{S}\right)&= \left( \frac{\alpha +r^{S}\theta -\beta \omega ^{S}}{2}\right) \left( \omega ^{S}+r^{S}\varDelta \right) -\mu r^{S}\nonumber \\&-\,\frac{\left( A_{M}^{S}\right) ^{2}}{2}+V_{M}^{S^{\prime }}\left( aA_{M}^{S}+b^{2}V_{R}^{S^{\prime }}-\delta r^{S}\right) \end{aligned}$$

(73)

$M$’s GAP equilibrium strategy is characterized by

$$\begin{aligned} \omega ^{S}\left( r^{S}\right)&= \frac{\alpha +r^{S}\left( \theta -\varDelta \beta \right) }{2\beta } \end{aligned}$$

(74)

$$\begin{aligned} A_{M}^{S}&= aV_{M}^{S^{\prime }} \end{aligned}$$

(75)

Plugging Eq. (74) inside (71), it gives

$$\begin{aligned} p^{S}\left( r^{S}\right) =\frac{3\alpha +r^{S}\left( 3\theta -\varDelta \beta \right) }{4\beta } \end{aligned}$$

(76)

Substituting, (72), (74) and (75), (76) inside Eqs. (70), and (73), the HBJ become:

$$\begin{aligned} \rho V_{M}^{S}\left( r^{S}\right)&= \frac{1}{2\beta }\left( \frac{\alpha +r^{S}\left( \theta +\varDelta \beta \right) }{2}\right) ^{2}-\mu r^{S}+V_{M}^{S^{\prime }}\left( \frac{a^{2}V_{M}^{S^{\prime }}}{2} +b^{2}V_{R}^{S^{\prime }}-\delta r^{S}\right) \end{aligned}$$

(77)

$$\begin{aligned} \rho V_{R}^{S}\left( r^{S}\right)&= \frac{1}{\beta }\left( \frac{\alpha +r^{S}\left( \theta +\varDelta \beta \right) }{4}\right) ^{2}+\mu r^{S}+V_{R}^{S^{\prime }}\left( a^{2}V_{M}^{S^{\prime }}+\frac{ b^{2}V_{R}^{S^{\prime }}}{2}-\delta r^{S}\right) \end{aligned}$$

(78)

We can conjecture quadratic value functions also in this scenario, specifically: $V_{M}^{S}\left( r^{S}\right) =\frac{m_{1}}{2} r^{S^{2}}+m_{2}r^{S}+m_{3}$ and $V_{R}^{S}\left( r^{S}\right) =\frac{n_{1}}{2 }r^{S^{2}}+n_{2}r^{S}+n_{3},$ where the pairs $\left( m_{j},n_{j}\right) ,j=1\ldots 3$ are the constant parameters to be identified. Substituting the value functions and their derivatives inside Eqs. (77) and (78), the constant parameters can be identified solving the following set of coupled Riccati equations:

$$\begin{aligned} \varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}+4\beta \left( 2b^{2}n_{1}-2\delta -\rho \right) m_{1}+4a^{2}\beta m_{1}^{2}&= 0 \end{aligned}$$

(79)

$$\begin{aligned} 2\left( \alpha \left( \theta +\varDelta \beta \right) +4b^{2}\beta m_{1}n_{2}+4\beta \left( a^{2}m_{1}+b^{2}n_{1}-\delta -\rho -\mu \right) m_{2}\right)&= 0 \end{aligned}$$

(80)

$$\begin{aligned} \alpha ^{2}+4\beta \left( 2b^{2}n_{2}+a^{2}m_{2}\right) m_{2}-8\beta \rho m_{3}&= 0 \end{aligned}$$

(81)

$$\begin{aligned} \left( \varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}+8\beta \left( 2a^{2}m_{1}-2\delta -\rho \right) n_{1}+8b^{2}\beta n_{1}^{2}\right)&= 0 \end{aligned}$$

(82)

$$\begin{aligned} 2\left( \alpha \left( \theta +\varDelta \beta \right) +8a^{2}\beta m_{2}n_{1}+8\beta \left( a^{2}m_{1}+b^{2}n_{1}-\delta -\rho +\mu \right) n_{2}\right)&= 0 \end{aligned}$$

(83)

$$\begin{aligned} \alpha ^{2}+8\beta \left( 2a^{2}m_{2}+b^{2}n_{2}\right) n_{2}-16\rho \beta n_{3}&= 0 \end{aligned}$$

(84)

The coefficients can be simply derived as $m_{1}=d_{1}$ and $n_{1}=f_{1}.$ Thus, we can obtain $n_{1}$ as a function of $d_{1}:n_{1}=f_{1}=f\left( d1\right) =\varOmega _{1}$ as it is displayed in Eq. (54). Substituting Eq. (54) for Eqs. (80) and (83), we can derive both $m_{2}$ and $n_{2}$ as a function of $d_{1}$

$$\begin{aligned} m_{2}\left( d_{1}\right)&= \frac{b^{2}d_{1}-2\mu -2B_{4}}{8\beta \left( B_{4}^{2}-\mu ^{2}-a^{2}b^{2}d_{1}\varOmega _{1}\right) }B_{2}=\varOmega _{6} \end{aligned}$$

(85)

$$\begin{aligned} n_{2}\left( d_{1}\right)&= \frac{2a_{1}^{2}\varOmega +\mu -B_{4}}{8\beta \left( B_{4}^{2}-\mu ^{2}-a^{2}b^{2}d_{1}\varOmega _{1}\right) }B_{2}=\varOmega _{7} \end{aligned}$$

(86)

We then substitute Eqs. (85) and (86) in Eqs. (81) and (84) to derive $m_{3}$ and $n_{3}$ as a function of $d_{1}$:

$$\begin{aligned} m_{3}\left( d_{1}\right)&= \frac{\alpha ^{2}+4\beta \left( 2b^{2}\varOmega _{7}+a^{2}\varOmega _{6}\right) \varOmega _{6}}{8\beta \rho }=\varOmega _{8} \end{aligned}$$

(87)

$$\begin{aligned} n_{3}\left( d_{1}\right)&= \frac{\alpha ^{2}+8\beta \left( 2a^{2}\varOmega _{6}+b^{2}\varOmega _{7}\right) \varOmega _{7}}{16\rho \beta }=\varOmega _{9} \end{aligned}$$

(88)

See Proof of Proposition 1 to check the solution for $d_{1}$.$\square$

Proof of Proposition 6

This proof follows the proof for Proposition 2, with the difference that the incentive depends on the control $A_{R}^{C}\left( r^{C}\right)$. The HJB functions should be written as follows:

$$\begin{aligned} \rho V_{M}^{C}\left( r^{C}\right)&= \left( \alpha +r^{C}\theta -\beta p^{C}\right) \left( \omega ^{C}+r^{C}\varDelta \right) -\mu A_{R}^{C}\nonumber \\&\quad -\,\frac{ A_{M}^{C^{2}}}{2}+V_{M}^{C^{\prime }}\left( aA_{M}^{C}+bA_{R}^{C}-\delta r^{C}\right) \end{aligned}$$

(89)

$$\begin{aligned} \rho V_{R}^{C}\left( r^{C}\right)&= \left( \alpha +r^{C}\theta -\beta p^{C}\right) \left( p^{C}-\omega ^{C}\right) +\mu A_{R}^{C}\nonumber \\&\quad -\,\frac{ A_{R}^{C^{2}}}{2}+V_{R}^{C^{\prime }}\left( aA_{M}^{C}+bA_{R}^{C}-\delta r^{C}\right) \end{aligned}$$

(90)

Maximization of $R$’s HJB gives pricing and $R$’s GAP strategies:

$$\begin{aligned} p^{C}\left( r^{C}\right)&= \frac{\alpha +r^{C}\theta +\beta \omega ^{C}}{ 2\beta } \end{aligned}$$

(91)

$$\begin{aligned} A_{R}&= bV_{R}^{C^{\prime }}+\mu \end{aligned}$$

(92)

Substituting these strategies inside Eq. (89) to get

$$\begin{aligned} V_{M}^{C}\left( r^{C}\right)&= \left( \frac{\alpha +r^{C}\theta -\beta \omega ^{C}}{2}\right) \left( \omega ^{C}+r^{C}\varDelta \right) -\mu \left( bV_{R}^{C^{\prime }}+\mu \right) -\frac{A_{M}^{C^{2}}}{2}\nonumber \\&\quad +\,V_{M}^{C^{\prime }}\left( aA_{M}^{C}+b\left( bV_{R}^{C^{\prime }}+\mu \right) -\delta r^{C}\right) \end{aligned}$$

(93)

First-order condition for $M$’s GAP strategy gives

$$\begin{aligned} \omega ^{C}\left( r^{C}\right)&= \frac{\alpha +r^{C}\left( \theta -\varDelta \beta \right) }{2\beta } \end{aligned}$$

(94)

$$\begin{aligned} A_{M}^{C}&= aV_{M}^{C^{\prime }} \end{aligned}$$

(95)

Plugging Eq. (94) inside (91) gives

$$\begin{aligned} p^{C}\left( r^{C}\right) =\frac{3\alpha +r^{C}\left( 3\theta -\varDelta \beta \right) }{4\beta } \end{aligned}$$

(96)

Substitute Eqs. (92), (94), (95), (96) in (90) and (93) to get

$$\begin{aligned} \rho V_{M}^{C}\left( r^{C}\right)&= \frac{1}{2\beta }\left( \frac{\alpha +r^{C}\left( \theta +\varDelta \beta \right) }{2}\right) ^{2}\nonumber \\&\quad +\,\left( bV_{M}^{C^{\prime }}-\mu \right) \left( bV_{R}^{C^{\prime }}+\mu \right) +V_{M}^{C^{\prime }}\left( \frac{a^{2}V_{M}^{C^{\prime }}}{2}-\delta r^{C}\right) \end{aligned}$$

(97)

$$\begin{aligned} \rho V_{R}^{C}\left( r^{C}\right)&= \frac{1}{\beta }\left( \frac{\alpha +r^{C}\left( \theta +\varDelta \beta \right) }{4}\right) ^{2}+\frac{\left( bV_{R}^{C^{\prime }}+\mu \right) ^{2}}{2}+V_{R}^{C^{\prime }}\left( a^{2}V_{M}^{C^{\prime }}-\delta r^{C}\right) \end{aligned}$$

(98)

To obtain a solution for this game, we conjectured quadratic value functions, $V_{M}^{C}\left( r^{C}\right) =\frac{l_{1}}{2} r^{C^{2}}+l_{2}r^{C}+l_{3}$ and $V_{R}^{C}\left( r^{C}\right) =\frac{k_{1}}{2 }r^{C^{2}}+k_{2}r^{C}+k_{3},$ where $\left( l_{j},k_{j}\right) ,j=1\ldots 3,$ are the constant parameters to be identified. Replacing our conjectures and their derivatives into Eqs. (97) and (98), it gives

$$\begin{aligned} 8\beta \rho \left( \frac{l_{1}}{2}r^{C^{2}}+l_{2}r^{C}+l_{3}\right)&= \left( \alpha +r^{C}\left( \theta +\varDelta \beta \right) \right) ^{2}+8\beta \left( bV_{M}^{C^{\prime }}-\mu \right) \left( bV_{R}^{C^{\prime }}+\mu \right) \nonumber \\&\quad +\,4\beta V_{M}^{C^{\prime }}\left( a^{2}V_{M}^{C^{\prime }}-2\delta r^{C}\right) \end{aligned}$$

(99)

$$\begin{aligned} 16\beta \rho \left( \frac{k_{1}}{2}r^{C^{2}}+k_{2}r^{C}+k_{3}\right)&= \left( \alpha +r^{C}\left( \theta +\varDelta \beta \right) \right) ^{2}+8\beta \left( bV_{R}^{C^{\prime }}+\mu \right) ^{2}\nonumber \\&\quad +\,16\beta V_{R}^{C^{\prime }}\left( a^{2}V_{M}^{C^{\prime }}-\delta r^{C}\right) \end{aligned}$$

(100)

We identified the constant parameters from the following set of coupled algebraic Riccati equations:

$$\begin{aligned} \varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}+4\beta \left( 2b^{2}k_{1}-2\delta -\rho \right) l_{1}+4\beta a^{2}l_{1}^{2}&= 0 \end{aligned}$$

(101)

$$\begin{aligned} 2\left( \alpha \left( \theta +\varDelta \beta \right) +4\beta \left( b^{2}k_{2}l_{1}+\left( a^{2}l_{1}+b^{2}k_{1}-\delta -\rho \right) l_{2}\right) +b\mu \left( l_{1}-k_{1}\right) \right)&= 0 \end{aligned}$$

(102)

$$\begin{aligned} \left( \alpha ^{2}+4\beta \left( \left( 2b\mu +2b^{2}k_{2}+a^{2}l_{2}^{{}}\right) l_{2}-2\mu \left( \mu +bk_{2}\right) \right) \right) -8\beta \rho l_{3}&= 0 \end{aligned}$$

(103)

$$\begin{aligned} \varDelta \beta \left( 2\theta +\varDelta \beta \right) +\theta ^{2}+8\beta \left( 2a^{2}l_{1}-2\delta -\rho \right) k_{1}+8\beta b^{2}k_{1}^{2}&= 0 \end{aligned}$$

(104)

$$\begin{aligned} 2\left( \alpha \left( \theta +\varDelta \beta \right) +8\beta \left( \left( \left( a^{2}l_{1}+b^{2}k_{1}-\delta -\rho \right) k_{2}\right) +a^{2}k_{1}l_{2}\right) +b\mu k_{1}\right)&= 0 \end{aligned}$$

(105)

$$\begin{aligned} \left( \alpha ^{2}+8\beta \mu ^{2}+8\beta \left( 2a^{2}l_{2}+2b\mu +b^{2}k_{2}^{{}}\right) k_{2}\right) -16\beta \rho k_{3}&= 0 \end{aligned}$$

(106)

As for the state-dependent case, the coefficients $l_{i},k_{i},i=1\ldots 3$ can be simply derived as $l_{1}=d_{1}$ and $k_{1}=f_{1}$. Thus, we can obtain $k_{1}$ as a function of $d_{1}:k_{1}=f_{1}=f\left( d1\right) =\varOmega _{1}$ as it is reported in Eq. (54). Substituting Eq. (54) for Eqs. (102) and (105), we can derive both $l_{2}$ and $k_{2}$ as a function of $d_{1}$

$$\begin{aligned} l_{2}\left( d_{1}\right)&= \frac{8b^{3}\beta \mu \varOmega _{1}d_{1}+8\mu \beta b\left( B_{4}+a^{2}d_{1}\right) \left( \varOmega _{1}-d_{1}\right) -\left( 2B_{4}+\left( 2a^{2}-b^{2}\right) \varOmega _{1}\right) B_{2}}{8\beta \left( B_{4}^{2}-a^{2}d_{1}\left( b_{{}}^{2}\varOmega _{1}-B_{4}\right) \right) }\nonumber \\&= \varOmega _{10} \end{aligned}$$

(107)

$$\begin{aligned} k_{2}\left( d_{1}\right)&= \frac{-B_{2}\left( B_{4}-2a_{{}}^{2}\varOmega _{1}\right) +8b\beta \mu k_{1}\left( a^{2}\left( d_{1}-\varOmega _{1}\right) -B_{4}\right) }{8\beta \left( B_{4}^{2}-a^{2}d_{1}\left( b^{2}\varOmega _{1}-B_{4}\right) \right) }=\varOmega _{11} \end{aligned}$$

(108)

We then substitute Eqs. (107) and (108) in Eqs. (103) and (106) to derive $l_{3}$ and $k_{3}$ as a function of $d_{1}$:

$$\begin{aligned} l_{3}\left( d_{1}\right)&= \frac{\alpha ^{2}+4\beta \left( \left( 2b\mu +2b^{2}\varOmega _{11}+a^{2}\varOmega _{10}\right) \varOmega _{10}-2\mu \left( \mu +b\varOmega _{11}\right) \right) }{8\beta \rho }=\varOmega _{12} \end{aligned}$$

(109)

$$\begin{aligned} k_{3}\left( d_{1}\right)&= \frac{\alpha ^{2}+8\beta \mu ^{2}+8\beta \left( 2a^{2}\varOmega _{10}+2b\mu +b^{2}\varOmega _{11}\right) \varOmega _{11}}{16\rho \beta }=\varOmega _{13} \end{aligned}$$

(110)

See Proof of Proposition 1 to check the solution for $d_{1}$. $\square$

Appendix 2

	Solution ${\mathcal {I}}$	Solution ${\mathcal {II}}$	Solution ${\mathcal {III}}$	Solution ${\mathcal {IV}}$
$A_{M}^{B}\left( r_{SS}^{B}\right)$	.2096	.4397	.1098	$-$.4544
$A_{R}^{B}\left( r_{SS}^{B}\right)$	.1028	.2027	$-$.1582	.0687
$r_{SS}^{B}$	.3881	.7816	$-$.3268	$-$ .1121
$V_{M}^{B}\left( r_{SS}^{B}\right)$	.192	.5309	1.123	.0284
$V_{R}^{B}\left( r_{SS}^{B}\right)$	.0953	.2008	$-$.0509	.0873

Steady-state $(SS)$ value of GAP efforts, return rates, and profits in scenario $B.$ Bold values highlight the positivity assumptions that solutions ${\mathcal {III}}$ and ${\mathcal {IV}}$ violate

Appendix 3

Parameter values	$A_{M}^{B}\left( r_{SS}^{B}\right) \ge 0$	$A_{R}^{B}\left( r_{SS}^{B}\right) \ge 0$	$r_{SS}^{B}\in (0,1]$	$V_{M}^{B}\left( r_{SS}^{B}\right) \ge 0$	$V_{R}^{B}\left( r_{SS}^{B}\right) \ge 0$	$\delta -a^{2}d_{1}-b^{2}f_{1}>0$
$\alpha$(1.1;1.2;1.3)	.23;.252;.272	.11;.12;.134	.426;.46;.505	.235;.284;.337	.115;.138;.134	.324;.324;.324
$\beta$(1.1;1.2;1.3)	.204;.2;.197	.1;.098;.097	.378;.371;.365	.176;.163;.152	.085;.079;.073	.322;.320;.317
$\varDelta$(.6;.7;.8)	.261;.329;.426	.127;.159;.204	.4812;.604;.777	.226;.287;.407	.109;.138;.194	.3025;.276;.246
$\theta$(.4;.5;.6)	.261;.329;.426	.127;.159;.204	.4812;.604;.777	.226;.287;.407	.109;.138;.194	.3025;.276;.246
a(.6;.7;.8)	.22;.234;.253	.109;.117;.128	.471;.58;.726	.211;.24;.289	.104;.12;.146	.312;.297;.28
b(1.1;.1.2;1.3)	.221;.235;.252	.107;.113;.12	.464;.555;.667	.211;.238;.277	.101;.113;.13	.313;.3;.287
$\rho$(.95;.97;.99)	.199;.195;.191	.097;.096;.094	.368;.361;.354	.177;.172;.167	.086;.084;.081	.327;.328;.329
$\delta$(.5;.6;.7)	.179;.158;.143	.088;.078;.071	.265;.196;.152	.169;.16;.154	.083;.078;.076	.433;.54;.646

1. Sensitivity analysis on Solution $\mathcal {I}$ in scenario $B$. Note that $m_{1}$ and $n_{1}$ are not influenced by $\alpha$ (see “Appendix 1”)

Parameter values	$A_{M}^{B}\left( r_{SS}^{B}\right) \ge 0$	$A_{R}^{B}\left( r_{SS}^{B}\right) \ge 0$	$r^{B}\in (0,1]$	$V_{M}^{B}\left( r_{SS}^{B}\right) \ge 0$	$V_{R}^{B}\left( r_{SS}^{B}\right) \ge 0$	$\delta -a^{2}d_{1}-b^{2}f_{1}<0$
$\alpha$(1.1;1.2;1.3)	.484;.528;.572	.223;.243;.263	.86;.937;1.02	.559;.568;.551	.225;.247;.266	$-$1.32;$-$1.32;$-$1.32
$\beta$(1.1;1.2;1.3)	.43;.425;.422	.198;.195;.193	.76;.754;.747	56;.499;.486	.19;.184;.178	$-$1.322;$-$1.323;$-$1.324
$\varDelta$(.6;.7;.8)	.581;.798;1.17	.363;.354;.514	1.022;1.34;2.02	.433;$-$ .242;$-$ 3.6	.21;.105; $-$ .626	$-$1.327;$-$1.333;$-$1.339
$\theta$(.4;.5;.6)	.581;.798;1.17	.363;.354;.514	1.022;1.34;2.02	.433;$-$ .242;$-$ 3.6	.21;.105; $-$ .626	$-$1.327;$-$1.333;$-$1.339
a(.6;.7;.8)	.471;.515;.579	.225;.257;.302	.987; 1.27;1.68	.412;.207;$-$ .268	.198;.106;$-$ .27	$-$1.312;$-$1.30;$-$1.287
b(1.1;.1.2;1.3)	.481;.533;.599	.215;.23;.25	.95; 1.16;1.43	.461;.173;$-$ .6	.195;.172;.11	$-$1.33;$-$1.34;$-$1.35
$\rho$(.95;.97;.99)	.411;.4;.39	.19;.186;.181	.733;.715;.697	.525;.521;.518	.19;.188;.18	$-$1.371;$-$1.391;$-$1.411
$\delta$(.5;.6;.7)	.369;.327;.298	.172;.153;.141	.528;.392;.308	.471;.388;.327	.164;.135;.117	$-$1.42;$-$1.518;$-$1.617

1. Sensitivity analysis on Solution $\mathcal {II}$ in the $B$-scenario. Bold values indicate that some positivity assumptions as well as assumptions on $r^{B}\in (0,1]$ are not met. Note that $m_{1}$ and $n_{1}$ are not influenced by $\alpha$ (see “Appendix 1”) while stability condition for Solution $\mathcal {II}$ requires $\delta -a^{2}d_{1}-b^{2}f_{1}<0$ as $d_{2}<0$ and $f_{2}<0$

Parameter values	$A_{M}^{B}\left( r_{SS}^{B}\right) \ge 0$	$A_{R}^{B}\left( r_{SS}^{B}\right) \ge 0$	$r^{B}\in (0,1]$	$V_{M}^{B}\left( r_{SS}^{B}\right) \ge 0$	$V_{R}^{B}\left( r_{SS}^{B}\right) \ge 0$	$\delta -a^{2}d_{1}-b^{2}f_{1}<0$
$\alpha$(1.1;1.2;1.3)	.484;.528;.572	.223;.243;.263	.86;.937;1.02	.559;.568;.551	.225;.247;.266	$-$1.32;$-$1.32;$-$1.32
$\beta$(1.1;1.2;1.3)	.43;.425;.422	.198;.195;.193	.76;.754;.747	56;.499;.486	.19;.184;.178	$-$1.322;$-$1.323;$-$1.324
$\varDelta$(.6;.7;.8)	.581;.798;1.17	.363;.354;.514	1.022;1.34;2.02	.433;$-$ .242;$-$ 3.6	.21;.105; $-$ .626	$-$1.327;$-$1.333;$-$1.339
$\theta$(.4;.5;.6)	.581;.798;1.17	.363;.354;.514	1.022;1.34;2.02	.433;$-$ .242;$-$ 3.6	.21;.105; $-$ .626	$-$1.327;$-$1.333;$-$1.339
a(.6;.7;.8)	.471;.515;.579	.225;.257;.302	.987; 1.27;1.68	.412;.207;$-$ .268	.198;.106;$-$ .27	$-$1.312;$-$1.30;$-$1.287
b(1.1;.1.2;1.3)	.481;.533;.599	.215;.23;.25	.95; 1.16;1.43	.461;.173;$-$ .6	.195;.172;.11	$-$1.33;$-$1.34;$-$1.35
$\rho$(.95;.97;.99)	.411;.4;.39	.19;.186;.181	.733;.715;.697	.525;.521;.518	.19;.188;.18	$-$1.371;$-$1.391;$-$1.411
$\delta$(.5;.6;.7)	.369;.327;.298	.172;.153;.141	.528;.392;.308	.471;.388;.327	.164;.135;.117	$-$1.42;$-$1.518;$-$1.617

1. Sensitivity analysis on Solution $\mathcal {II}$ in the $B$-scenario. Bold values indicate that some positivity assumptions as well as assumptions on $r^{B}\in (0,1]$ are not met. Note that $m_{1}$ and $n_{1}$ are not influenced by $\alpha$ (see “Appendix 1”) while stability condition for Solution $\mathcal {II}$ requires $\delta -a^{2}d_{1}-b^{2}f_{1}<0$ as $d_{2}<0$ and $f_{2}<0$

Parameter values	$A_{M}^{C}\left( r_{SS}^{C}\right) \ge 0$	$A_{R}^{C}\left( r_{SS}^{C}\right) \ge 0$	$r^{C}\in (0,1]$	$V_{M}^{C}\left( r_{SS}^{C}\right) \ge 0$	$V_{R}^{C}\left( r_{SS}^{C}\right) \ge 0$	$\delta -a^{2}l_{1}-b^{2}k_{1}>0$
$\alpha$(1.1;1.2;1.3)	.23;.25;.275	.114;.124;.135	.43;.47;.51	.214;.265;.321	.158;.184;.21	.324;.324;.324
$\beta$(1.1;1.2;1.3)	.223;.202;.199	.101;.099;.097	.383;.375;.369	.152;.138;.127	.126;.119;.114	.322;.320;.317
$\varDelta$(.6;.7;.8)	.265;.334;.434	.129;.162;.208	.488;.614;.791	.208;.276;.407	.155;.188;.25	.3025;.276;.246
$\theta$(.4;.5;.6)	.265;.334;.434	.129;.162;.208	.488;.614;.791	.208;.276;.407	.155;.188;.25	.3025;.276;.246
a(.6;.7;.8)	.223;.238;.257	.11;.12;.129	.477;.588;.736	.188;.219;.269	.146;.163;.191	.312;.297;.28
b(1.1;.1.2;1.3)	.224;.239;.257	.109;.115;.122	.47;.564;.678	.191;.223;.267	.146;.16;.18	.313;.3;.287
$\rho$(.95;.97;.99)	.201;.197;.193	.099;.097;.095	.373;.366;.358	.154;.149;.145	.125;.122;.118	.327;.328;.329
$\delta$(.5;.6;.7)	.18;.16;.144	.089;.079;.0716	.268;.198;.154	.143;.132;.126	.123;.116;.113	.433;.54;.646
$\mu$(.025;.05;.2)	.210;.210;.212	.1029;.103;.104	.388;.390;.393	.194;.195;.168	.096;.099;.135	.324;.324;.324

1. Sensitivity analysis in the $C$-scenario. Note that all values for the stability condition are the same as in the benchmark as $l_{1}=d_{1}$ and $k_{1}=f_{1}$, which are also $\mu$-independent