Appendix: Process of Solving Game in Each State
1.1 State R
In State R, only retailer r has signal. We first consider the retailers’ decisions in the second stage given the wholesale prices \( w_{n} {\text{and }}w_{r} \) for the two types of products. The uninformed retailer n’s decision is made according to the manufacturer’s wholesale prices. Similar to Gal-Or (2008) and Wu and Zhang (2014), we assume that retailer n uses the decision rule \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} \) to set the quantity. However, the informed retailer r sets quantity according to the wholesale prices \( w_{n} {\text{and }}w_{r} \) and signal \( \varGamma , \) i.e., \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \). The expected profit of retailer n is
$$ \pi_{n} = E\left[ {\left( {1 - q_{n} - \delta q_{r} - {\text{w}}_{\text{n}} } \right)q_{n} } \right], $$
and retailer r’s expected profit conditional on signal \( \varGamma \) is
$$ \pi_{r} = E\left[ {\left[ {\delta \left( {1 - q_{n} - q_{r} } \right) + \varepsilon - w_{r} } \right]q_{r} \left| \varGamma \right.} \right] $$
The two first-order conditions yield
$$ q_{n} = \frac{1}{2}\left( {1 - \delta E[q_{r} ] - w_{n} } \right) $$
(1)
$$ q_{r} = \frac{1}{2\delta }\left( {\delta \left( {1 - E[q_{n} \left| \varGamma \right.]} \right) + E[\varepsilon \left| \varGamma \right.] - w_{r} } \right) $$
(2)
Substituting \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} \) and \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \) into the right side of (1) and (2), we can obtain
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{{1 - \delta B_{1} }}{2}} \hfill \\ {A_{2} = - \frac{{1 + \delta B_{4} }}{2}} \hfill \\ {A_{3} = - \frac{{\delta B_{2} }}{2}} \hfill \\ {B_{1} = \frac{{1 - A_{1} }}{2}} \hfill \\ {B_{2} = - \frac{{1 + \delta A_{3} }}{2\delta }} \hfill \\ {B_{3} = \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ {B_{4} = - \frac{{A_{2} }}{2}} \hfill \\ \end{array} } \right. $$
(3)
Solving the equations in (3) leads to
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{2 - \delta }{4 - \delta }} \hfill \\ {A_{2} = - \frac{2}{4 - \delta }} \hfill \\ {A_{3} = \frac{1}{4 - \delta }} \hfill \\ {B_{1} = \frac{1}{4 - \delta }} \hfill \\ {B_{2} = - \frac{2}{{\delta \left( {4 - \delta } \right)}}} \hfill \\ {B_{3} = \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ {B_{4} = \frac{1}{4 - \delta }} \hfill \\ \end{array} } \right. $$
Thus, given the wholesale prices \( w_{n} {\text{and }}w_{r} \), retailers’ quantities are
$$ \begin{aligned} q_{n} & = \frac{2 - \delta }{4 - \delta } - \frac{2}{4 - \delta }w_{n} + \frac{1}{4 - \delta }w_{r} \\ q_{r} & = \frac{1}{4 - \delta } - \frac{2}{{\delta \left( {4 - \delta } \right)}}w_{r} + \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma + \frac{1}{4 - \delta }w_{n} \\ \end{aligned} $$
Then we proceed to analyze the wholesale price decision of manufacturer m in the first stage. Without signal, the objective function of the manufacturer m maximizing the expected profit is
$$ \pi_{m} = E\left[w_{n} \left( {\frac{{2 - \delta - 2w_{n} + w_{r} }}{4 - \delta }} \right) + w_{r} \left( {\frac{{\delta - 2w_{r} + \delta w_{n} }}{{\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma } \right)\right] $$
The wholesale prices set by manufacturer m satisfy
$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial \pi_{m} }}{{\partial w_{n} }} = \frac{{2 - \delta - 4w_{n} + 2w_{r} }}{4 - \delta } = 0} \hfill \\ {\frac{{\partial \pi_{m} }}{{dw_{r} }} = \frac{{\delta + 2\delta w_{n} - 4w_{r} }}{{\delta \left( {4 - \delta } \right)}} + E\left[ \varGamma \right] = 0 } \hfill \\ \end{array} } \right. $$
which yields equilibrium wholesale prices
$$ {\text{w}}_{\text{n}}^{R} = \frac{1}{2},w_{r}^{R} = \frac{\delta }{2} $$
1.2 State RM
In State RM, manufacturer m and retailer r have signal. Similarly, we first consider the retailers’ decisions in the second stage for given the wholesale prices \( w_{n} {\text{and}} w_{r} \) of the two products. We assume that the manufacturer doesn’t share information with retailer n, though the former has demand information. We assume retailer n doesn’t have inference ability. Because retailer n only observes wholesale price \( w_{n} {\text{and}} w_{r} \), we assume that retailer n uses the decision rule \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} \) to set the quantity. However, the informed retailer r sets quantity according to the wholesale prices \( w_{n} {\text{and}} w_{r} \) and signal Γ, i.e., \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \). The expected profit of retailer n is\( \pi_{n} = E\left[ {\left( {1 - q_{n} - \delta q_{r} - {\text{w}}_{\text{n}} } \right)q_{n} } \right] \), and retailer r’s expected profit conditional on signal \( \varGamma \) is
$$ \pi_{r} = E\left[ {\left[ {\delta \left( {1 - q_{n} - q_{r} } \right) + \varepsilon - w_{r} } \right]q_{r} \left| \varGamma \right.} \right] $$
The two first-order conditions yield
$$ q_{n} = \frac{1}{2}\left( {1 - \delta E\left[ {q_{r} } \right] - E\left[ {w_{n} } \right]} \right) $$
(4)
$$ q_{r} = \frac{1}{2\delta }\left( {\delta \left( {1 - E[q_{n} \left| \varGamma \right.]} \right) + E[\varepsilon \left| \varGamma \right.] - w_{r} } \right) $$
(5)
Substitute \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} \) and \( q_{r} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \) into (4) and (5), and we have
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{{1 - \delta B_{1} }}{2}} \hfill \\ {A_{2} = - \frac{{1 + \delta B_{4} }}{2}} \hfill \\ {A_{3} = - \frac{{\delta B_{2} }}{2}} \hfill \\ {B_{1} = \frac{{1 - A_{1} }}{2}} \hfill \\ {B_{2} = - \frac{{1 + \delta A_{3} }}{2\delta }} \hfill \\ {B_{3} = \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ {B_{4} = - \frac{{A_{2} }}{2}} \hfill \\ \end{array} } \right. $$
Solving the equations leads to
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{2 - \delta }{4 - \delta }} \hfill \\ {A_{2} = - \frac{2}{4 - \delta }} \hfill \\ {A_{3} = \frac{1}{4 - \delta }} \hfill \\ {B_{1} = \frac{1}{4 - \delta }} \hfill \\ {B_{2} = - \frac{2}{{\delta \left( {4 - \delta } \right)}}} \hfill \\ {B_{3} = \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ {B_{4} = \frac{1}{4 - \delta }} \hfill \\ \end{array} } \right. $$
Therefore, the order quantity decisions of retailers are
$$ \begin{aligned} q_{n} & = \frac{2 - \delta }{4 - \delta } - \frac{2}{4 - \delta }w_{n} + \frac{1}{4 - \delta }w_{r} \\ q_{r} & = \frac{1}{4 - \delta } - \frac{2}{{\delta \left( {4 - \delta } \right)}}w_{r} + \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma + \frac{1}{4 - \delta }w_{n} \\ \end{aligned} $$
Next, the wholesale price decision by the informed manufacturer m is in the first stage, who can accurately forecast the order quantity decision by retailer n and retailer r in the second stage, to maximize the profit. The objective function is
$$ \pi_{m} = E\left[w_{n} \left( {\frac{{2 - \delta - 2w_{n} + w_{r} }}{4 - \delta }} \right) + w_{r} \left( {\frac{{\delta - 2w_{r} + \delta w_{n} }}{{\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma } \right)\left| \varGamma \right.\right] $$
Compared to state R, in state RM the manufacturer has the signal \( \varGamma \). Thus, the wholesale prices set by manufacturer m satisfy
$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial \pi_{m} }}{{\partial w_{n} }} = \frac{2 - \delta }{4 - \delta } - \frac{4}{4 - \delta }w_{n} + \frac{2}{4 - \delta }w_{r} = 0} \hfill \\ {\frac{{\partial \pi_{m} }}{{\partial w_{r} }} = \frac{1}{4 - \delta } - \frac{4}{{\delta \left( {4 - \delta } \right)}}w_{r} + \frac{{\sigma^{2} }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma + \frac{2}{4 - \delta }w_{n} = 0} \hfill \\ \end{array} } \right. $$
Therefore, we derive the equilibrium wholesale prices
$$ w_{n}^{RM} = \frac{1}{2} + \frac{{\sigma^{2} \varGamma }}{{4\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}, w_{r}^{RM} = \frac{\delta }{2} + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} $$
1.3 State RN
In State RN, only retailer n and retailer r have signal. First we consider the retailers’ decisions in the second stage when manufacturer m has given the wholesale price \( {\text{w}}_{\text{n}} ,w_{r} \) for the two products. Since both retailers have the same signal, the expected profits of the retailers are
$$ \begin{aligned} \pi_{n} & = E[\left( {1 - q_{n} - \delta q_{r} - w_{n} } \right)q_{n} \left| \varGamma \right.] \\ \pi_{r} & = E[\left( {\delta \left( {1 - q_{n} - q_{r} } \right) + \varepsilon - w_{r} } \right)q_{r} \left| \varGamma \right.] \\ \end{aligned} $$
The two first-order conditions yield
$$ q_{n} = \frac{{1 - \delta E[q_{r} \left| \varGamma \right.] - w_{n} }}{2} $$
(6)
$$ q_{r} = \frac{1}{2\delta }\left( {\delta \left( {1 - E[q_{n} \left| \varGamma \right.]} \right) + E[\varepsilon \left| \varGamma \right.] - w_{r} } \right) $$
(7)
In this state, having signal, the retailers set their quantities according to the wholesale prices \( w_{n} {\text{and }}w_{r} \) and signal \( \varGamma , \) i.e., \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} + A_{4} \varGamma \) and \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \). Substituting \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} + A_{4} \varGamma \) and \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \) into the right sides of (6) and (7), we can obtain
$$ \begin{aligned} q_{n} & = \frac{{1 - \delta B_{1} }}{2} - \frac{{1 + \delta B_{3} }}{2}w_{n} - \frac{{\delta B_{2} }}{2}w_{r} - \frac{{\delta B_{4} \varGamma }}{2} \\ q_{r} & = \frac{{1 - A_{1} }}{2} - \frac{{1 + \delta A_{3} }}{2\delta }w_{r} - \frac{{A_{2} }}{2}w_{n} + \frac{{\left( {\sigma^{2} - \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)A_{4} } \right)}}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma \\ \end{aligned} $$
Therefore, we have
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{{1 - \delta B_{1} }}{2}} \hfill \\ {B_{1} = \frac{{1 - A_{1} }}{2}} \hfill \\ {A_{2} = - \frac{{1 + \delta B_{3} }}{2}} \hfill \\ {B_{2} = - \frac{{1 + \delta A_{3} }}{2\delta }} \hfill \\ {A_{3} = - \frac{{\delta B_{2} }}{2}} \hfill \\ {B_{3} = - \frac{{A_{2} }}{2}} \hfill \\ {A_{4} = - \frac{{\delta B_{4} }}{2}} \hfill \\ {B_{4} = \frac{{\left( {\sigma^{2} - \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)A_{4} } \right)}}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ \end{array} } \right. $$
(8)
Solving the system of equations in (8) leads to
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{2 - \delta }{4 - \delta }} \hfill \\ {B_{1} = \frac{1}{4 - \delta }} \hfill \\ {A_{2} = - \frac{2}{4 - \delta }} \hfill \\ {B_{2} = - \frac{2}{{\delta \left( {4 - \delta } \right)}}} \hfill \\ {A_{3} = \frac{1}{4 - \delta }} \hfill \\ {B_{3} = \frac{1}{4 - \delta }} \hfill \\ {A_{4} = - \frac{{\sigma^{2} }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ {B_{4} = \frac{{2\sigma^{2} }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ \end{array} } \right. $$
Thus, given the wholesale prices \( w_{n} {\text{and }}w_{r} \), retailers’ quantities are
$$ \begin{aligned} q_{n} & = \frac{2 - \delta }{4 - \delta } - \frac{2}{4 - \delta }w_{n} + \frac{1}{4 - \delta }w_{r} - \frac{{\sigma^{2} }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma \\ q_{r} & = \frac{1}{4 - \delta } - \frac{2}{{\delta \left( {4 - \delta } \right)}}w_{r} + \frac{1}{4 - \delta }w_{n} + \frac{{2\sigma^{2} }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma \\ \end{aligned} $$
Then we analyze the wholesale price decision by manufacturer m in the first stage. Without signal, manufacturer m maximizes the expected profit and the objective function is
$$ \begin{aligned} \pi_{m} & = E[w_{n} \left( {\frac{{2 - \delta - 2w_{n} + w_{r} }}{4 - \delta } - \frac{{\sigma^{2} }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma } \right) + w_{R} \left( {\frac{{\delta - 2w_{r} + \delta w_{n} }}{{\delta \left( {4 - \delta } \right)}} + \frac{{2\sigma^{2} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma } \right)] \\ & = w_{n} \left( {\frac{{2 - \delta - 2w_{n} + w_{r} }}{4 - \delta }} \right) + w_{R} \left( {\frac{{\delta - 2w_{r} + \delta w_{n} }}{{\delta \left( {4 - \delta } \right)}}} \right) \\ \end{aligned} $$
The two first-order conditions lead to the following two equations
$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial \pi_{m} }}{{\partial w_{n} }} = 0} \\ {\frac{{\partial \pi_{m} }}{{\partial w_{r} }} = 0} \\ \end{array} } \right. $$
Thus, we have
$$ {\text{w}}_{\text{n}}^{RN} = \frac{1}{2},w_{r}^{RN} = \frac{\delta }{2} $$
1.4 State RNM
In State RNM, all players have signal. First we consider the retailers’ decisions in the second stage, for given the wholesale prices \( w_{n} {\text{and}} w_{r} \) for the two products. In state RNM, both retailers have the same signal \( \varGamma \). The retailers set their quantities according to the wholesale prices \( w_{n} {\text{and }}w_{r} \) and signal \( \varGamma , \) i.e., \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} + A_{4} \varGamma \) and \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \). The expected profits of the retailers are
$$ \begin{aligned} \pi_{n} & = E[\left( {1 - q_{n} - \delta q_{r} - w_{n} } \right)q_{n} \left| \varGamma \right.] \\ \pi_{r} & = E[\left( {\delta \left( {1 - q_{n} - q_{r} } \right) + \varepsilon - w_{r} } \right)q_{r} \left| \varGamma \right.] \\ \end{aligned} $$
The two first-order conditions yield
$$ q_{n} = \frac{{1 - \delta E[q_{r} \left| \varGamma \right.] - w_{n} }}{2} $$
(9)
$$ q_{r} = \frac{1}{2\delta }\left( {\delta \left( {1 - E[q_{n} \left| \varGamma \right.]} \right) + E[\varepsilon \left| \varGamma \right.] - w_{r} } \right) $$
(10)
Substituting \( q_{n} = A_{1} + A_{2} w_{n} + A_{3} w_{r} + A_{4} \varGamma \) and \( q_{R} = B_{1} + B_{2} w_{r} + B_{3} \varGamma + B_{4} w_{n} \) into the right side of (9) and (10), we can obtain
$$ \begin{aligned} q_{n} & = \frac{{1 - \delta B_{1} }}{2} - \frac{{1 + \delta B_{3} }}{2}w_{n} - \frac{{\delta B_{2} }}{2}w_{r} - \frac{{\delta B_{4} \varGamma }}{2} \\ q_{r} & = \frac{{1 - A_{1} }}{2} - \frac{{1 + \delta A_{3} }}{2\delta }w_{r} - \frac{{A_{2} }}{2}w_{n} + \frac{{\left( {\sigma^{2} - \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)A_{4} } \right)}}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma \\ \end{aligned} $$
Therefore, we have
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{{1 - \delta B_{1} }}{2}} \hfill \\ {B_{1} = \frac{{1 - A_{1} }}{2}} \hfill \\ {A_{2} = - \frac{{1 + \delta B_{3} }}{2}} \hfill \\ {B_{2} = - \frac{{1 + \delta A_{3} }}{2\delta }} \hfill \\ {A_{3} = - \frac{{\delta B_{2} }}{2}} \hfill \\ {B_{3} = - \frac{{A_{2} }}{2}} \hfill \\ {A_{4} = - \frac{{\delta B_{4} }}{2}} \hfill \\ {B_{4} = \frac{{\left( {\sigma^{2} - \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)A_{4} } \right)}}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ \end{array} } \right. $$
Solving the equations leads to
$$ \left\{ {\begin{array}{*{20}l} {A_{1} = \frac{2 - \delta }{4 - \delta }} \hfill \\ {B_{1} = \frac{1}{4 - \delta }} \hfill \\ {A_{2} = - \frac{2}{4 - \delta }} \hfill \\ {B_{2} = - \frac{2}{{\delta \left( {4 - \delta } \right)}}} \hfill \\ {A_{3} = \frac{1}{4 - \delta }} \hfill \\ {B_{3} = \frac{1}{4 - \delta }} \hfill \\ {A_{4} = - \frac{{\sigma^{2} }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ {B_{4} = \frac{{2\sigma^{2} }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \hfill \\ \end{array} } \right. $$
So, the retailers’ quantity decisions are as follows
$$ \begin{aligned} q_{n} & = \frac{2 - \delta }{4 - \delta } - \frac{2}{4 - \delta }w_{n} + \frac{1}{4 - \delta }w_{r} - \frac{{\sigma^{2} }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma \\ q_{r} & = \frac{1}{4 - \delta } - \frac{2}{{\delta \left( {4 - \delta } \right)}}w_{r} + \frac{1}{4 - \delta }w_{n} + \frac{{2\sigma^{2} }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma \\ \end{aligned} $$
Next we consider the wholesale price decision by manufacturer M, with the signal, m can accurately forecast the order quantity decision by retailer n and retailer r so as to maximize the profit. Therefore manufacturer m’s objective function is
$$ \pi_{m} = E\left[w_{n} \left( {\frac{{2 - \delta - 2w_{n} + w_{r} }}{4 - \delta } - \frac{{\sigma^{2} }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma } \right) + w_{R} \left( {\frac{{\delta - 2w_{r} + \delta w_{n} }}{{\delta \left( {4 - \delta } \right)}} + \frac{{2\sigma^{2} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\varGamma } \right)\left| \varGamma \right.\right] $$
The wholesale prices given by manufacturer m satisfy
$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial \pi_{\text{m}} }}{{\partial w_{n} }} = \frac{{2 - \delta - 4w_{n} + 2w_{r} }}{4 - \delta } = 0} \hfill \\ {\frac{{\partial \pi_{m} }}{{dw_{r} }} = \frac{{\delta + 2\delta w_{n} - 4w_{r} }}{{\delta \left( {4 - \delta } \right)}} + = 0 } \hfill \\ \end{array} } \right. $$
by the first-order conditions, we derive the equilibrium wholesale prices
$$ {\text{w}}_{\text{n}}^{RNM} = \frac{1}{2},w_{r}^{RNM} = \frac{\delta }{2} + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} $$
Proof for Proposition
4
Cost-Equilibrium Analysis
We first summarize the equilibrium outcomes as follows.
Equilibrium Outcomes in State R
$$ \begin{aligned} w_{n}^{R - O} & = \frac{1}{2}\left( {1 + c_{n} } \right) \\ w_{r}^{R - O} & = \frac{1}{2}\left( {\delta + c_{r} } \right) \\ q_{n}^{R - O} & = \frac{{2 - \delta - 2c_{n} + c_{r} }}{8 - 2\delta } \\ q_{r}^{R - O} & = \frac{{\delta + \delta c_{n} - 2c_{r} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{n}^{R - O} & = \frac{{\left( {2 - \delta - 2c_{n} + c_{r} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} \\ \widehat{\Pi }_{r}^{R - O} & = \frac{{\left( {\delta + \delta c_{n} - 2c_{r} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{4\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{m}^{R - O} & = \frac{{\delta - 2\delta c_{n} + \delta^{2} c_{n} + \delta c_{n}^{2} - \delta c_{r} - \delta c_{n} c_{r} + c_{r}^{2} }}{{2\delta \left( {4 - \delta } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RM
$$ \begin{aligned} w_{n}^{RM - O} & = \frac{1}{2}\left( {1 + c_{n} } \right) + \frac{{\sigma^{2} \varGamma }}{{4\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ w_{r}^{RM - O} & = \frac{1}{2}\left( {\delta + c_{r} } \right) + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RM - O} & = \frac{{2 - \delta - 2c_{n} + c_{r} }}{{2\left( {4 - \delta } \right)}} \\ q_{r}^{RM - O} & = \frac{{\delta + \delta c_{n} - 2c_{r} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{4\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{n}^{RM - O} & = \frac{{\left( {2 - \delta - 2c_{n} + c_{r} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} \\ \widehat{\Pi }_{r}^{RM - O} & = \frac{{\left( {\delta + \delta c_{n} - 2c_{r} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{m}^{RM - O} & = \frac{{\delta - 2\delta c_{n} + \delta^{2} c_{n} + \delta c_{n}^{2} - \delta c_{r} - \delta c_{n} c_{r} + c_{r}^{2} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RN
$$ \begin{aligned} w_{n}^{RN - O} & = \frac{1}{2}\left( {1 + c_{n} } \right) \\ w_{r}^{RN - O} & = \frac{1}{2}\left( {\delta + c_{r} } \right) \\ q_{n}^{RN - O} & = \frac{{2 - \delta - 2c_{n} + c_{r} }}{{2\left( {4 - \delta } \right)}} - \frac{{\sigma^{2} \varGamma }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RN - O} & = \frac{{\delta + \delta c_{n} - 2c_{r} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{2\sigma^{2} \varGamma }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RN - O} & = \frac{{\left( {2 - \delta - 2c_{n} + c_{r} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{r}^{RN - O} & = \frac{{\left( {\delta + \delta c_{n} - 2c_{r} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{4\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{m}^{RN - O} & = \frac{{\delta - 2\delta c_{n} + \delta^{2} c_{n} + \delta c_{n}^{2} - \delta c_{r} - \delta c_{n} c_{r} + c_{r}^{2} }}{{2\delta \left( {4 - \delta } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RNM
$$ \begin{aligned} w_{n}^{RNM - O} & = \frac{1}{2}\left( {1 + c_{n} } \right) \\ w_{r}^{RNM - O} & = \frac{1}{2}\left( {\delta + c_{r} } \right) + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RNM - O} & = \frac{{2 - \delta - 2c_{n} + c_{r} }}{{2\left( {4 - \delta } \right)}} - \frac{{\sigma^{2} \varGamma }}{{2\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RNM - O} & = \frac{{\delta + \delta c_{n} - 2c_{r} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RNM - O} & = \frac{{\left( {2 - \delta - 2c_{n} + c_{r} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{r}^{RNM - O} & = \frac{{\left( {\delta + \delta c_{n} - 2c_{r} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{m}^{RNM - O} & = \frac{{\delta - 2\delta c_{n} + \delta^{2} c_{n} + \delta c_{n}^{2} - \delta c_{r} - \delta c_{n} c_{r} + c_{r}^{2} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
Then, we conduct the comparison and have the results as follows:
Comparison: State R versus State RM
$$ \begin{aligned} & \widehat{\Pi }_{n}^{R - O} - \widehat{\Pi }_{n}^{RM - O} = 0 \\ & \widehat{\Pi }_{r}^{R - O} - \widehat{\Pi }_{r}^{RM - O} = \frac{{3\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\Pi }_{m}^{R - O} - \widehat{\Pi }_{m}^{RM - O} = - \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{n}^{R - O} + \widehat{\Pi }_{r}^{R - O} + \widehat{\Pi }_{m}^{R - O} - \left( {\widehat{\Pi }_{n}^{RM - O} + \widehat{\Pi }_{r}^{RM - O} + \widehat{\Pi }_{m}^{RM - O} } \right) = \frac{{\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ \end{aligned} $$
Comparison: State R versus State RN
$$ \begin{aligned} & \widehat{\Pi }_{n}^{R - O} - \widehat{\Pi }_{n}^{RN - O} = - \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{r}^{R - O} - \widehat{\Pi }_{r}^{RN - O} = - \frac{{\left( {8 - \delta } \right)\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{m}^{R - O} - \widehat{\Pi }_{m}^{RN - O} = 0 \\ & \widehat{\Pi }_{n}^{R - O} + \widehat{\Pi }_{r}^{R - O} + \widehat{\Pi }_{m}^{R - O} - \left( {\widehat{\Pi }_{n}^{RN - O} + \widehat{\Pi }_{r}^{RN - O} + \widehat{\Pi }_{m}^{RN - O} } \right) = - \frac{{\left( {12 - \delta } \right)\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State R versus State RNM
$$ \begin{aligned} & \widehat{\Pi }_{n}^{R - O} - \widehat{\Pi }_{n}^{RNM - O} = - \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{r}^{R - O} - \widehat{\Pi }_{r}^{RNM - O} = \frac{{\left( {12 - 8\delta + \delta^{2} } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\Pi }_{m}^{R - O} - \widehat{\Pi }_{m}^{RNM - O} = - \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{n}^{R - O} + \widehat{\Pi }_{r}^{R - O} + \widehat{\Pi }_{m}^{R - O} - \left( {\widehat{\Pi }_{n}^{RNM - O} + \widehat{\Pi }_{r}^{RNM - O} + \widehat{\Pi }_{m}^{RNM - O} } \right) = \frac{{\left( {4 - 7\delta + \delta^{2} } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\left\{ {\begin{array}{*{20}c} { \ge 0, \;\;if \delta \in \left[ {0,\frac{{7 - \sqrt {33} }}{2}} \right]} \\ { < 0,\;\; if \delta \in \left( {\frac{{7 - \sqrt {33} }}{2},1} \right)} \\ \end{array} } \right. \\ \end{aligned} $$
Comparison: State RM versus State RN
$$ \begin{aligned} & \widehat{\Pi }_{n}^{RM - O} - \widehat{\Pi }_{n}^{RN - O} = - \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{r}^{RM - O} - \widehat{\Pi }_{r}^{RN - O} = - \frac{{\left( {48 + 8\delta - \delta^{2} } \right)\sigma^{4} }}{{16\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{m}^{RM - O} - \widehat{\Pi }_{m}^{RN - O} = \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\Pi }_{n}^{RM - O} + \widehat{\Pi }_{r}^{RM - O} + \widehat{\Pi }_{m}^{RM - O} - \left( {\widehat{\Pi }_{n}^{RN - O} + \widehat{\Pi }_{r}^{RN - O} + \widehat{\Pi }_{m}^{RN - O} } \right) = - \frac{{\left( {16 + 40\delta - 3\delta^{2} } \right)\sigma^{4} }}{{16\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State RM versus State RNM
$$ \begin{aligned} & \widehat{\Pi }_{n}^{RM - O} - \widehat{\Pi }_{n}^{RNM - O} = - \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{r}^{RM - O} - \widehat{\Pi }_{r}^{RNM - O} = - \frac{{\left( {8 - \delta } \right)\sigma^{4} }}{{16\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{m}^{RM - O} - \widehat{\Pi }_{m}^{RNM - O} = - \frac{{\sigma^{4} }}{{8\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{n}^{RM - O} + \widehat{\Pi }_{r}^{RM - O} + \widehat{\Pi }_{m}^{RM - O} - \left( {\widehat{\Pi }_{n}^{RNM - O} + \widehat{\Pi }_{r}^{RNM - O} + \widehat{\Pi }_{m}^{RNM - O} } \right) = - \frac{{\left( {20 - 3\delta } \right)\sigma^{4} }}{{16\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State RN versus State RNM
$$ \begin{aligned} & \widehat{\Pi }_{n}^{RN - O} - \widehat{\Pi }_{n}^{RNM - O} = \frac{{3\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\Pi }_{r}^{RN - O} - \widehat{\Pi }_{r}^{RNM - O} = \frac{{3\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\Pi }_{m}^{RN - O} - \widehat{\Pi }_{m}^{RNM - O} = - \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{n}^{RN - O} + \widehat{\Pi }_{r}^{RN - O} + \widehat{\Pi }_{m}^{RN - O} - \left( {\widehat{\Pi }_{n}^{RNM - O} + \widehat{\Pi }_{r}^{RNM - O} + \widehat{\Pi }_{m}^{RNM - O} } \right) = \frac{{\left( {4 + 5\delta } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ \end{aligned} $$
Based on the foregoing equilibria and the comparison results, we have
$$ \begin{aligned} & \widehat{\Pi }_{n}^{RN - O} > \widehat{\Pi }_{n}^{RNM - O} > \widehat{\Pi }_{n}^{R - O} = \widehat{\Pi }_{n}^{RM - O} \\ & \widehat{\Pi }_{r}^{RN - O} > \widehat{\Pi }_{r}^{R - O} > \widehat{\Pi }_{r}^{RNM - O} > \widehat{\Pi }_{r}^{RM - O} \\ & \widehat{\Pi }_{m}^{RNM - O} > \widehat{\Pi }_{m}^{RM - O} > \widehat{\Pi }_{m}^{R - O} = \widehat{\Pi }_{m}^{RN - O} \\ & \widehat{\Pi }_{n}^{RN - O} + \widehat{\Pi }_{r}^{RN - O} + \widehat{\Pi }_{m}^{RN - O} > \widehat{\Pi }_{n}^{RNM - O} + \widehat{\Pi }_{r}^{RNM - O} \\ & \quad + \widehat{\Pi }_{m}^{RNM - O} \left( {\widehat{\Pi }_{n}^{R - O} + \widehat{\Pi }_{r}^{R - O} + \widehat{\Pi }_{m}^{R - O} } \right) > \widehat{\Pi }_{n}^{RM - O} + \widehat{\Pi }_{r}^{RM - O} + \widehat{\Pi }_{m}^{RM - O} \\ \end{aligned} $$
It is clearly, the equilibrium state is state RN, i.e., the retailer r will transmit demand signal to the retailer n. Neither retailer r or the retailer n voluntarily transmits the signal to manufacturer.
Proof for Proposition
5
Upstream Subsidy-Equilibrium Analysis
Similar to that for Proposition 1(in Sect. 3.2), we derive the outcomes in state R, RM, RN, and RNM as follows:
Equilibrium Outcomes in State R
$$ \begin{aligned} w_{n}^{R - U} & = \frac{1}{2} \\ w_{r}^{R - U} & = \frac{1}{2}\left( {\delta + \eta_{u} } \right) \\ q_{n}^{R - U} & = \frac{{2 - \delta - \eta_{u} }}{{2\left( {4 - \delta } \right)}} \\ q_{r}^{R - U} & = \frac{{\delta + 2\eta_{u} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{r}^{R - U} & = \frac{{\left( {\delta + 2\eta_{u} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{4\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{m}^{R - U} & = \frac{{\delta + \delta \eta_{u} + \eta_{u}^{2} }}{{2\delta \left( {4 - \delta } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RM
$$ \begin{aligned} w_{n}^{RM - U} & = \frac{1}{2} + \frac{{\sigma^{2} \varGamma }}{{4\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ w_{r}^{RM - U} & = \frac{1}{2}\left( {\delta + \eta_{u} } \right) + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RM - U} & = \frac{{2 - \delta - \eta_{u} }}{{2\left( {4 - \delta } \right)}} \\ q_{r}^{MU} & = \frac{{\delta + 2\eta_{u} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{4\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{n}^{RM - U} & = \frac{{\left( {2 - \delta - \eta_{u} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} \\ \widehat{\Pi }_{r}^{RM - U} & = \frac{{\left( {\delta + 2\eta_{u} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{m}^{RM - U} & = \frac{{\delta + \eta_{u} \delta + \eta_{u}^{2} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RN
$$ \begin{aligned} w_{n}^{RN - U} & = \frac{1}{2} \\ w_{r}^{RN - U} & = \frac{1}{2}\left( {\delta + \eta_{u} } \right) \\ q_{n}^{RN - U} & = \frac{{2 - \delta - \eta_{u} }}{{2\left( {4 - \delta } \right)}} - \frac{{\sigma^{2} \varGamma }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RN - U} & = \frac{{\delta + 2\eta_{u} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{2\sigma^{2} \varGamma }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{n}^{RN - U} & = \frac{{\left( {2 - \delta - \eta_{u} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{r}^{RN - U} & = \frac{{\left( {\delta + 2\eta_{u} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{4\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{m}^{RN - U} & = \frac{{\delta + \delta \eta_{u} + \eta_{u}^{2} }}{{2\delta \left( {4 - \delta } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RNM
$$ \begin{aligned} w_{n}^{RNM - U} & = \frac{1}{2} \\ w_{r}^{RNM - U} & = \frac{1}{2}\left( {\delta + \eta_{u} } \right) + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RNM - U} & = \frac{{2 - \delta - \eta_{u} }}{{2\left( {4 - \delta } \right)}} - \frac{{\sigma^{2} \varGamma }}{{2\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RNM - U} & = \frac{{\delta + 2\eta_{u} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{n}^{RNM - U} & = \frac{{\left( {2 - \delta - \eta_{u} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\Pi }_{r}^{RNM - U} & = \frac{{\left( {\delta + 2\eta_{u} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
We then conduct the comparison and derive the results as follows:
Comparison: State R versus State RM
$$ \begin{aligned} & \widehat{\Pi }_{n}^{R - U} - \widehat{\Pi }_{n}^{RM - U} = 0 \\ & \widehat{\Pi }_{r}^{R - U} - \widehat{\Pi }_{r}^{RM - U} = \frac{{3\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\Pi }_{m}^{R - U} - \widehat{\Pi }_{m}^{RM - U} = - \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{n}^{R - U} + \widehat{\Pi }_{r}^{R - U} + \widehat{\Pi }_{m}^{R - U} - \left( {\widehat{\Pi }_{n}^{RM - U} + \widehat{\Pi }_{r}^{RM - U} + \widehat{\Pi }_{m}^{RM - U} } \right) = \frac{{\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ \end{aligned} $$
Comparison: State R versus State RN
$$ \begin{aligned} & \widehat{\Pi }_{n}^{R - U} - \widehat{\Pi }_{n}^{RN - U} = - \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{r}^{R - U} - \widehat{\Pi }_{r}^{RN - U} = - \frac{{\left( {8 - \delta } \right)\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\Pi }_{m}^{R - U} - \widehat{\Pi }_{m}^{RN - U} = 0 \\ & \widehat{\Pi }_{n}^{R - U} + \widehat{\Pi }_{r}^{R - U} + \widehat{\Pi }_{m}^{R - U} - \left( {\widehat{\Pi }_{n}^{RN - U} + \widehat{\Pi }_{r}^{RN - U} + \widehat{\Pi }_{m}^{RN - U} } \right) = - \frac{{\left( {12 - \delta } \right)\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State R versus State RNM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{R - U} - \widehat{\varPi }_{n}^{RNM - U} = - \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{r}^{R - U} - \widehat{\varPi }_{r}^{RNM - U} = \frac{{\left( {12 - 8\delta + \delta^{2} } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{m}^{R - U} - \widehat{\varPi }_{m}^{RNM - U} = - \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{R - U} + \widehat{\varPi }_{r}^{R - U} + \widehat{\varPi }_{m}^{R - U} - \left( {\widehat{\varPi }_{n}^{RNM - U} + \widehat{\varPi }_{r}^{RNM - U} + \widehat{\varPi }_{m}^{RNM - U} } \right) = \frac{{\left( {4 - 7\delta + \delta^{2} } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\left\{ {\begin{array}{*{20}c} { \ge 0,\;\; if \delta \in \left[ {0,\frac{{7 - \sqrt {33} }}{2}} \right]} \\ { < 0, \;\;if \delta \in \left( {\frac{{7 - \sqrt {33} }}{2},1} \right)} \\ \end{array} } \right. \\ \end{aligned} $$
Comparison: State RM versus State RN
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RM - U} - \widehat{\varPi }_{n}^{RN - U} = - \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{r}^{RM - U} - \widehat{\varPi }_{r}^{RN - U} = - \frac{{\left( {48 + 8\delta - \delta^{2} } \right)\sigma^{4} }}{{16\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{m}^{RM - U} - \widehat{\varPi }_{m}^{RN - U} = \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{n}^{RM - U} + \widehat{\varPi }_{r}^{RM - U} + \widehat{\varPi }_{m}^{RM - U} - \left( {\widehat{\varPi }_{n}^{RN - U} + \widehat{\varPi }_{r}^{RN - U} + \widehat{\varPi }_{m}^{RN - U} } \right) = - \frac{{\left( {16 + 40\delta - 3\delta^{2} } \right)\sigma^{4} }}{{16\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State RM versus State RNM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RM - U} - \widehat{\varPi }_{n}^{RNM - U} = - \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{r}^{RM - U} - \widehat{\varPi }_{r}^{RNM - U} = - \frac{{\left( {8 - \delta } \right)\sigma^{4} }}{{16\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{m}^{RM - U} - \widehat{\varPi }_{m}^{RNM - U} = - \frac{{\sigma^{4} }}{{8\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{RM - U} + \widehat{\varPi }_{r}^{RM - U} + \widehat{\varPi }_{m}^{RM - U} - \left( {\widehat{\varPi }_{n}^{RNM - U} + \widehat{\varPi }_{r}^{RNM - U} + \widehat{\varPi }_{m}^{RNM - U} } \right) = - \frac{{\left( {20 - 3\delta } \right)\sigma^{4} }}{{16\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State RM versus State RNM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RN - U} - \widehat{\varPi }_{n}^{RNM - U} = \frac{{3\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{r}^{RN - U} - \widehat{\varPi }_{r}^{RNM - U} = \frac{{3\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{m}^{RN - U} - \widehat{\varPi }_{m}^{RNM - U} = - \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{RN - U} + \widehat{\varPi }_{r}^{RN - U} + \widehat{\varPi }_{m}^{RN - U} - \left( {\widehat{\varPi }_{n}^{RNM - U} + \widehat{\varPi }_{r}^{RNM - U} + \widehat{\varPi }_{m}^{RNM - U} } \right) = \frac{{\left( {4 + 5\delta } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ \end{aligned} $$
Based on the foregoing equilibria and the comparison results, we have:
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RN - U} > \widehat{\varPi }_{n}^{RNM - U} > \widehat{\varPi }_{n}^{R - U} = \widehat{\varPi }_{n}^{RM - U} \\ & \widehat{\varPi }_{r}^{RN - U} > \widehat{\varPi }_{r}^{R - U} > \widehat{\varPi }_{r}^{RNM - U} > \widehat{\varPi }_{r}^{RM - U} \\ & \widehat{\varPi }_{m}^{RNM - U} > \widehat{\varPi }_{m}^{RM - U} > \widehat{\varPi }_{m}^{R - U} = \widehat{\varPi }_{m}^{RN - U} \\ & \widehat{\varPi }_{n}^{RN - U} + \widehat{\varPi }_{r}^{RN - U} + \widehat{\varPi }_{m}^{RN - U} > \widehat{\varPi }_{n}^{RNM - U} + \widehat{\varPi }_{r}^{RNM - U} + \widehat{\varPi }_{m}^{RNM - U} \left( {\widehat{\varPi }_{n}^{R - U} + \widehat{\varPi }_{r}^{R - U} + \widehat{\varPi }_{m}^{R - U} } \right) \\ & \quad > \widehat{\varPi }_{n}^{RM - U} + \widehat{\varPi }_{r}^{RM - U} + \widehat{\varPi }_{m}^{RM - U} \\ \end{aligned} $$
We find that, when the government subsidy is given to the retailer, the five rules in Proposition 1 still hold.
Downstream Subsidy-Equilibrium Analysis
We derive the outcomes in state R, RM, RN, and RNM as follows:
Equilibrium Outcomes in State R
$$ \begin{aligned} w_{n}^{R - D} & = \frac{1}{2} \\ w_{r}^{R - D} & = \frac{1}{2}\left( {\delta - \eta_{d} } \right) \\ q_{n}^{R - D} & = \frac{{2 - \delta - \eta_{d} }}{{2\left( {4 - \delta } \right)}} \\ q_{r}^{R - D} & = \frac{{\delta + 2\eta_{d} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{2\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{R - D} & = \frac{{\left( {2 - \delta - \eta_{d} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} \\ \widehat{\varPi }_{r}^{R - D} & = \frac{{\left( {\delta + 2\eta_{d} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{4\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{m}^{R - D} & = \frac{{\delta + \delta \eta_{d} + \eta_{d}^{2} }}{{2\delta \left( {4 - \delta } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RM
$$ \begin{aligned} w_{n}^{RM - D} & = \frac{1}{2} + \frac{{\sigma^{2} \varGamma }}{{4\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ w_{r}^{RM - D} & = \frac{1}{2}\left( {\delta - \eta_{d} } \right) + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RM - D} & = \frac{{2 - \delta - \eta_{d} }}{{2\left( {4 - \delta } \right)}} \\ q_{r}^{RM - D} & = \frac{{\delta + 2\eta_{d} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{4\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RM - D} & = \frac{{\left( {2 - \delta - \eta_{d} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} \\ \widehat{\varPi }_{r}^{RM - D} & = \frac{{\left( {\delta + 2\eta_{d} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{m}^{RM - D} & = \frac{{\delta + \eta_{d} \delta + \eta_{d}^{2} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RN
$$ \begin{aligned} w_{n}^{RN - D} & = \frac{1}{2} \\ w_{r}^{RN - D} & = \frac{1}{2}\left( {\delta - \eta_{d} } \right) \\ q_{n}^{RN - D} & = \frac{{2 - \delta - \eta_{d} }}{{2\left( {4 - \delta } \right)}} - \frac{{\sigma^{2} \varGamma }}{{\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RN - D} & = \frac{{\delta + 2\eta_{d} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{2\sigma^{2} \varGamma }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RN - D} & = \frac{{\left( {2 - \delta - \eta_{d} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{r}^{RN - D} & = \frac{{\left( {\delta + 2\eta_{d} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{4\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{m}^{RN - D} & = \frac{{\delta + \delta \eta_{d} + \eta_{d}^{2} }}{{2\delta \left( {4 - \delta } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RNM
$$ \begin{aligned} w_{n}^{RNM - D} & = \frac{1}{2} \\ w_{r}^{RNM - D} & = \frac{1}{2}\left( {\delta - \eta_{d} } \right) + \frac{{\sigma^{2} \varGamma }}{{2\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RNM - D} & = \frac{{2 - \delta - \eta_{d} }}{{2\left( {4 - \delta } \right)}} - \frac{{\sigma^{2} \varGamma }}{{2\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RNM - D} & = \frac{{\delta + 2\eta_{d} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RNM - D} & = \frac{{\left( {2 - \delta - \eta_{d} } \right)^{2} }}{{4\left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{r}^{RNM - D} & = \frac{{\left( {\delta + 2\eta_{d} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{m}^{RNM - D} & = \frac{{\delta + \delta \eta_{d} + \eta_{d}^{2} }}{{2\delta \left( {4 - \delta } \right)}} + \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
We then conduct the comparison and derive the results as follows:
Comparison: State R versus State RM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{R - D} - \widehat{\varPi }_{n}^{RM - D} = 0 \\ & \widehat{\varPi }_{r}^{R - D} - \widehat{\varPi }_{r}^{RM - D} = \frac{{3\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{m}^{R - D} - \widehat{\varPi }_{m}^{RM - D} = - \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{R - D} + \widehat{\varPi }_{r}^{R - D} + \widehat{\varPi }_{m}^{R - D} - \left( {\widehat{\varPi }_{n}^{RM - D} + \widehat{\varPi }_{r}^{RM - D} + \widehat{\varPi }_{m}^{RM - D} } \right) = \frac{{\sigma^{4} }}{{16\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ \end{aligned} $$
Comparison: State R versus State RN
$$ \begin{aligned} & \widehat{\varPi }_{n}^{R - D} - \widehat{\varPi }_{n}^{RN - D} = - \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{r}^{R - D} - \widehat{\varPi }_{r}^{RN - D} = - \frac{{\left( {8 - \delta } \right)\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{m}^{R - D} - \widehat{\varPi }_{m}^{RN - D} = 0 \\ & \widehat{\varPi }_{n}^{R - D} + \widehat{\varPi }_{r}^{R - D} + \widehat{\varPi }_{m}^{R - D} - \left( {\widehat{\varPi }_{n}^{RN - D} + \widehat{\varPi }_{r}^{RN - D} + \widehat{\varPi }_{m}^{RN - D} } \right) = - \frac{{\left( {12 - \delta } \right)\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison: State R versus State RNM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{R - D} - \widehat{\varPi }_{n}^{RNM - D} = - \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{r}^{R - D} - \widehat{\varPi }_{r}^{RNM - D} = \frac{{\left( {12 - 8\delta + \delta^{2} } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{m}^{R - D} - \widehat{\varPi }_{m}^{RNM - D} = - \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{R - D} + \widehat{\varPi }_{r}^{R - D} + \widehat{\varPi }_{m}^{R - D} - \left( {\widehat{\varPi }_{n}^{RNM - D} + \widehat{\varPi }_{r}^{RNM - D} + \widehat{\varPi }_{m}^{RNM - D} } \right) = \frac{{\left( {4 - 7\delta + \delta^{2} } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\left\{ {\begin{array}{*{20}c} { \ge 0,\;\; if \delta \in \left[ {0,\frac{{7 - \sqrt {33} }}{2}} \right]} \\ { < 0, \;\;if \delta \in \left( {\frac{{7 - \sqrt {33} }}{2},1} \right)} \\ \end{array} } \right. \\ \end{aligned} $$
Comparison: State RM versus State RN
$$ \begin{aligned} &\widehat{\varPi }_{n}^{RM - D} - \widehat{\varPi }_{n}^{RN - D} = - \frac{{\sigma^{4} }}{{\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \hfill \\ &\widehat{\varPi }_{r}^{RM - D} - \widehat{\varPi }_{r}^{RN - D} = - \frac{{\left( {48 + 8\delta - \delta^{2} } \right)\sigma^{4} }}{{16\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \hfill \\ &\widehat{\varPi }_{m}^{RM - D} - \widehat{\varPi }_{m}^{RN - D} = \frac{{\sigma^{4} }}{{8\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \hfill \\ &\widehat{\varPi }_{n}^{RM - D} + \widehat{\varPi }_{r}^{RM - D} + \widehat{\varPi }_{m}^{RM - D} - \left( {\widehat{\varPi }_{n}^{RN - D} + \widehat{\varPi }_{r}^{RN - D} + \widehat{\varPi }_{m}^{RN - D} } \right) = - \frac{{\left( {16 + 40\delta - 3\delta^{2} } \right)\sigma^{4} }}{{16\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \hfill \\ \end{aligned} $$
Comparison: State RM versus State RNM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RM - D} - \widehat{\varPi }_{n}^{RNM - D} = - \frac{{\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{r}^{RM - D} - \widehat{\varPi }_{r}^{RNM - D} = - \frac{{\left( {8 - \delta } \right)\sigma^{4} }}{{16\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{m}^{RM - D} - \widehat{\varPi }_{m}^{RNM - D} = - \frac{{\sigma^{4} }}{{8\left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{RM - D} + \widehat{\varPi }_{r}^{RM - D} + \widehat{\varPi }_{m}^{RM - D} - \left( {\widehat{\varPi }_{n}^{RNM - D} + \widehat{\varPi }_{r}^{RNM - D} + \widehat{\varPi }_{m}^{RNM - D} } \right) = - \frac{{\left( {20 - 3\delta } \right)\sigma^{4} }}{{16\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ \end{aligned} $$
Comparison between State RN versus State RNM
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RN - D} - \widehat{\varPi }_{n}^{RNM - D} = \frac{{3\sigma^{4} }}{{4\left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{r}^{RN - D} - \widehat{\varPi }_{r}^{RNM - D} = \frac{{3\sigma^{4} }}{{\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ & \widehat{\varPi }_{m}^{RN - D} - \widehat{\varPi }_{m}^{RNM - D} = - \frac{{\sigma^{4} }}{{2\delta \left( {4 - \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} < 0 \\ & \widehat{\varPi }_{n}^{RN - D} + \widehat{\varPi }_{r}^{RN - D} + \widehat{\varPi }_{m}^{RN - D} - \left( {\widehat{\varPi }_{n}^{RNM - D} + \widehat{\varPi }_{r}^{RNM - D} + \widehat{\varPi }_{m}^{RNM - D} } \right) = \frac{{\left( {4 + 5\delta } \right)\sigma^{4} }}{{4\delta \left( {4 - \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 \\ \end{aligned} $$
Based on the foregoing equilibria and the comparison results, we have:
$$ \begin{aligned} & \widehat{\varPi }_{n}^{RN - D} > \widehat{\varPi }_{n}^{RNM - D} > \widehat{\varPi }_{n}^{R - D} = \widehat{\varPi }_{n}^{RM - D} \\ & \widehat{\varPi }_{r}^{RN - D} > \widehat{\varPi }_{r}^{R - D} > \widehat{\varPi }_{r}^{RNM - D} > \widehat{\varPi }_{r}^{RM - D} \\ & \widehat{\varPi }_{m}^{RNM - D} > \widehat{\varPi }_{m}^{RM - D} > \widehat{\varPi }_{m}^{R - D} = \widehat{\varPi }_{m}^{RN - D} \\ & \widehat{\varPi }_{n}^{RN - D} + \widehat{\varPi }_{r}^{RN - D} + \widehat{\varPi }_{m}^{RN - D} > \widehat{\varPi }_{n}^{RNM - D} + \widehat{\varPi }_{r}^{RNM - D} + \widehat{\varPi }_{m}^{RNM - D} \left( {\widehat{\varPi }_{n}^{R - D} + \widehat{\varPi }_{r}^{R - D} + \widehat{\varPi }_{m}^{R - D} } \right) \\ & \quad > \widehat{\varPi }_{n}^{RM - D} + \widehat{\varPi }_{r}^{RM - D} + \widehat{\varPi }_{m}^{RM - D} \\ \end{aligned} $$
Thus, similar to upstream subsidy model, we find the three rules in Proposition 1 still hold.
Proof for Proposition
6
We take the derivate of the equilibrium outcomes with respect to \( \eta_{u} {\text{and}} \eta_{d} \) respectively, and derive:
In upstream subsidy model (\( {\text{i}} \in \left\{ {{\text{R}},{\text{RM}},{\text{RN}},{\text{RNM}}} \right\} \))
$$ \begin{aligned} \frac{{dw_{n}^{i - U} }}{{d\eta_{u} }} & = 0 \\ \frac{{dw_{r}^{i - U} }}{{d\eta_{u} }} & = \frac{1}{2} > 0 \\ \frac{{dq_{n}^{i - U} }}{{d\eta_{u} }} & = \frac{ - 1}{{2\left( {4 - \delta } \right)}} < 0 \\ \frac{{dq_{r}^{i - U} }}{{d\eta_{u} }} & = \frac{1}{{\delta \left( {4 - \delta } \right)}} > 0 \\ \frac{{d\widehat{\varPi }_{n}^{i - U} }}{{d\eta_{u} }} & = - \frac{{2 - \delta - \eta_{u} }}{{2\left( {4 - \delta } \right)^{2} }} < 0 \\ \frac{{d\widehat{\varPi }_{r}^{i - U} }}{{d\eta_{u} }} & = \frac{{\delta + 2\eta_{u} }}{{\delta \left( {4 - \delta } \right)^{2} }} > 0 \\ \frac{{d\widehat{\varPi }_{m}^{i - U} }}{{d\eta_{u} }} & = \frac{{\delta + 2\eta_{u} }}{{2\delta \left( {4 - \delta } \right)}} > 0 \\ \end{aligned} $$
In downstream subsidy model (\( {\text{i}} \in \left\{ {{\text{R}},{\text{RM}},{\text{RN}},{\text{RNM}}} \right\} \))
$$ \begin{aligned} \frac{{dw_{n}^{i - D} }}{{d\eta_{d} }} & = 0 \\ \frac{{dw_{r}^{i - D} }}{{d\eta_{d} }} & = - \frac{1}{2} > 0 \\ \frac{{dq_{n}^{i - D} }}{{d\eta_{d} }} & = \frac{ - 1}{{2\left( {4 - \delta } \right)}} < 0 \\ \frac{{dq_{r}^{i - D} }}{{d\eta_{d} }} & = \frac{1}{{\delta \left( {4 - \delta } \right)}} > 0 \\ \frac{{d\widehat{\varPi }_{n}^{i - D} }}{{d\eta_{d} }} & = - \frac{{2 - \delta - \eta_{d} }}{{2\left( {4 - \delta } \right)^{2} }} < 0 \\ \frac{{d\widehat{\varPi }_{r}^{i - D} }}{{d\eta_{d} }} & = \frac{{\delta + 2\eta_{d} }}{{\delta \left( {4 - \delta } \right)^{2} }} > 0 \\ \frac{{d\widehat{\varPi }_{m}^{i - D} }}{{d\eta_{d} }} & = \frac{{\delta + 2\eta_{d} }}{{2\delta \left( {4 - \delta } \right)}} > 0 \\ \end{aligned} $$
According to the sensitivity analysis, it is easy to obtain the results in Proposition 6.
Proof for Proposition
7
The first-order derivation of \( \widehat{\varPi }_{n}^{RN} \) with respect to \( \delta \) is as follows:
$$ \frac{{\partial \widehat{\varPi }_{n}^{RN} }}{\partial \delta } = \frac{2A - 2 + \delta }{{\left( {4 - \delta } \right)^{3} }} $$
Obviously, If \( A \in \left( {0,\frac{1}{2}} \right] \), \( \frac{{\partial \widehat{\varPi }_{n}^{RN} }}{\partial \delta } < 0 \); If \( A \in \left( {\frac{1}{2},1} \right) \), \( \frac{{\partial \widehat{\varPi }_{n}^{RN} }}{\partial \delta } \le 0 \) when \( \delta \in \left( {0,2 - 2A} \right] \), but \( \frac{{\partial \widehat{\varPi }_{n}^{RN} }}{\partial \delta } > 0 \) when \( \delta \in \left( {2 - 2{\text{A}},1} \right) \); If \( {\text{A}} \in \left[ {1, + \infty } \right) \), \( \frac{{\partial \widehat{\varPi }_{n}^{RN} }}{\partial \delta } > 0 \).
Thus, we obtain the result of Proposition 7.
Proof for Proposition
8
The first-order derivation of \( \widehat{\varPi }_{r}^{RN} \) with respect to \( \delta \) is as follows:
$$ \frac{{\partial \widehat{\varPi }_{r}^{RN} }}{\partial \delta } = \frac{{\delta^{3} + 4\delta^{2} + 48A\delta - 64A}}{{4\delta^{2} \left( {4 - \delta } \right)^{3} }} $$
We define a function \( f\left( \delta \right) = \delta^{3} + 4\delta^{2} + 48A\delta - 64A \). It is clear that, function \( f\left( \delta \right) \) is increasing in \( \delta \), and \( f\left( \delta \right)\left| {_{{\delta \to 0^{ + } }} } \right. < 0 \), \( f\left( \delta \right)\left| {_{{\delta \to 1^{ - } }} } \right. \approx 5 - 16A \). Therefore, we find that, if \( A \ge \frac{5}{16} \), then \( f\left( \delta \right) < 0 \) for \( \delta \in \left( {0,1} \right) \); if \( A < \frac{5}{16} \), there is a solution \( \hat{\delta } \in \left( {0,1} \right) \) to let \( f\left( {\hat{\delta }} \right) = 0.\) According to the monotonically increasing property, we have
$$ f\left( {\hat{\delta }} \right) = \left\{ {\begin{array}{*{20}l} { < 0,} \hfill & {0 < \delta < \hat{\delta }} \hfill \\ { = 0,} \hfill & {\delta = \hat{\delta }} \hfill \\ { > 0,} \hfill & {\hat{\delta } < \delta < 1} \hfill \\ \end{array} } \right. $$
.
Proof for Proposition
9
Revenue Sharing Contract-Equilibrium Analysis
Similar to that for Proposition 1 (in Sect. 3.2), we derive the outcomes in state R, RM, RN, and RNM as follows:
Equilibrium Outcomes in State R
$$ \begin{aligned} w_{n}^{R - R} & = \frac{1}{2} \\ w_{r}^{R - R} & = \frac{{\delta \left( { - 4 + 3r + \delta } \right)}}{{2\left( { - 4 + 2r + \delta } \right)}} \\ q_{n}^{R - R} & = \frac{ - 2 + r + \delta }{{2\left( { - 4 + 2r + \delta } \right)}} \\ q_{r}^{R - R} & = \frac{1}{2}\left( {\frac{1}{4 - 2r - \delta } + \frac{{\sigma^{2} \varGamma }}{{\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \right) \\ \widehat{\varPi }_{n}^{R - R} & = \frac{{\left( { - 2 + r + \delta } \right)^{2} }}{{4\left( { - 4 + 2r + \delta } \right)^{2} }} \\ \widehat{\varPi }_{r}^{R - R} & = \frac{{\left( { - 1 + r} \right)\left( { - \frac{{\delta^{2} }}{{\left( { - 4 + 2r + \delta } \right)^{2} }} - \frac{{\sigma^{4} }}{{\sigma^{2} + \sigma_{1}^{2} }}} \right)}}{4\delta } \\ \widehat{\varPi }_{m}^{R - R} & = \frac{1}{4}\left( {\frac{ - 2 + r}{ - 4 + 2r + \delta } - \frac{{r\sigma^{4} }}{{\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}} \right) \\ \end{aligned} $$
Equilibrium Outcomes in State RM
$$ \begin{aligned} w_{n}^{RM - R} & = \frac{1}{2} + \frac{{\sigma^{2} \varGamma }}{{4\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ w_{r}^{RM - R} & = \frac{{2\delta \left( { - 4 + 3r + \delta } \right)}}{{4\left( { - 4 + 2r + \delta } \right)}} + \frac{{\sigma^{2} \varGamma \left( { - 8 + 8r + 2\delta - r\delta } \right)}}{{4\left( { - 4 + 2r + \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RM - R} & = \frac{{2\left( { - 2 + \delta + r} \right)}}{{4\left( { - 4 + 2r + \delta } \right)}} + \frac{{r\sigma^{2} \varGamma }}{{4\left( { - 4 + 2r + \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RM - R} & = \frac{ - 2\delta }{{4\delta \left( { - 4 + 2r + \delta } \right)}} + \frac{{\varGamma \sigma^{2} \left( { - 4 + \delta } \right)}}{{4\delta \left( { - 4 + 2r + \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RM - R} & = \frac{{4\left( { - 2 + r + \delta } \right)^{2} + \frac{{r\left( {8 - 3r - 2\delta } \right)\sigma^{4} }}{{\sigma^{2} + \sigma_{1}^{2} }}}}{{16\left( { - 4 + 2r + \delta } \right)^{2} }} \\ \widehat{\varPi }_{r}^{RM - R} & = \left( {1 - r} \right)\frac{{4\delta^{2} + \frac{{\left( { - 4 + \delta } \right)^{2} \sigma^{4} }}{{\sigma^{2} + \sigma_{1}^{2} }}}}{{16\delta \left( { - 4 + 2r + \delta } \right)^{2} }} \\ \widehat{\varPi }_{m}^{RM - R} & = \frac{{4\left( { - 2 + r} \right)^{2} - 4\delta + \frac{{\left( {4r\left( { - 4 + \delta } \right) + \left( { - 4 + \delta } \right)^{2} + r^{2} \delta } \right)\sigma^{4} }}{{\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}}}{{8\left( { - 4 + 2r + \delta } \right)^{2} }} \\ \end{aligned} $$
Equilibrium Outcomes in State RN
$$ \begin{aligned} w_{n}^{RN - R} & = \frac{1}{2} \\ w_{r}^{RN - R} & = \frac{{\delta \left( { - 4 + 3r + \delta } \right)}}{{2\left( { - 4 + 2r + \delta } \right)}} \\ q_{n}^{RN - R} & = \frac{ - 2 + r + \delta }{{2\left( { - 4 + 2r + \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{\left( { - 4 + \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RN - R} & = \frac{1}{8 - 4r - 2\delta } - \frac{{2\sigma^{2} \varGamma }}{{\left( { - 4 + \delta } \right)\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RN - R} & = \frac{{\left( { - 2 + r + \delta } \right)^{2} }}{{4\left( { - 4 + 2r + \delta } \right)^{2} }} + \frac{{\sigma^{4} }}{{\left( { - 4 + \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{r}^{RN - R} & = \frac{{\left( {1 - r} \right)\delta }}{{4\left( { - 4 + 2r + \delta } \right)^{2} }} + \frac{{4\left( {1 - r} \right)\sigma^{4} }}{{\left( { - 4 + \delta } \right)^{2} \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{m}^{RN - R} & = \frac{ - 2 + r}{{4\left( { - 4 + 2r + \delta } \right)}} + \frac{{4r\sigma^{4} }}{{\left( { - 4 + \delta } \right)^{2} \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \end{aligned} $$
Equilibrium Outcomes in State RNM
$$ \begin{aligned} w_{n}^{RNM - R} & = \frac{1}{2} \\ w_{r}^{RNM - R} & = \frac{{\delta \left( { - 4 + 3r + \delta } \right)}}{{2\left( { - 4 + 2r + \delta } \right)}} + \frac{{\sigma^{2} \varGamma \left( { - 4 + 4r + \delta } \right)}}{{2\left( { - 4 + 2r + \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{n}^{RNM - R} & = \frac{ - 2 + r + \delta }{{2\left( { - 4 + 2r + \delta } \right)}} + \frac{{\sigma^{2} \varGamma }}{{2\left( { - 4 + 2r + \delta } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ q_{r}^{RNM - R} & = \frac{\delta }{{8\delta - 4r\delta - 2\delta^{2} }} + \frac{{\sigma^{2} \varGamma }}{{\left( {4\delta - 2r\delta - \delta^{2} } \right)\left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} \\ \widehat{\varPi }_{n}^{RNM - R} & = \frac{{\left( { - 2 + r + \delta } \right)^{2} + \frac{{\sigma^{4} }}{{\sigma^{2} + \sigma_{1}^{2} }}}}{{4\left( { - 4 + 2r + \delta } \right)^{2} }} \\ \widehat{\varPi }_{r}^{RNM - R} & = \left( {1 - r} \right)\frac{{\delta^{2} + \frac{{4\sigma^{4} }}{{\sigma^{2} + \sigma_{1}^{2} }}}}{{4\delta \left( { - 4 + 2r + \delta } \right)^{2} }} \\ \widehat{\varPi }_{m}^{RNM - R} & = \frac{{ - 2 + r - \frac{{2\sigma^{4} }}{{\delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}}}{{4\left( { - 4 + 2r + \delta } \right)}} \\ \end{aligned} $$
We then conduct the comparison and derive the results as follows:
$$ \begin{aligned} \widehat{\varPi }_{r}^{R - R} - \widehat{\varPi }_{r}^{RM - R} & = \frac{{\left( {1 - r} \right)\left( {4r + 3\left( { - 4 + \delta } \right)} \right)\left( { - 4 + 4r + \delta } \right)\sigma^{4} }}{{16\delta \left( { - 4 + 2r + \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\left\{ {\begin{array}{*{20}c} { > 0,\;\;r < \frac{4 - \delta }{4}} \\ { \le 0,\;\;r \ge \frac{4 - \delta }{4}} \\ \end{array} } \right. \\ \widehat{\varPi }_{r}^{RN - R} - \widehat{\varPi }_{r}^{RNM - R} & = \frac{{\left( {1 - r} \right)\left( {4r + 3\left( { - 4 + \delta } \right)} \right)\left( { - 4 + 4r + \delta } \right)\sigma^{4} }}{{\left( { - 4 + \delta } \right)^{2} \delta \left( { - 4 + 2r + \delta } \right)^{2} \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}}\left\{ {\begin{array}{*{20}c} { > 0,\;\;r < \frac{4 - \delta }{4}} \\ { \le 0,\;\;r \ge \frac{4 - \delta }{4}} \\ \end{array} } \right. \\ \end{aligned} $$
Proof for Proposition
10
Based on equilibrium outcomes, we conduct the comparison and derive the results as follows:
$$ \widehat{\varPi }_{r}^{R} - \widehat{\varPi }_{r}^{RM} = \frac{{\left( { - 6 + \delta } \right)\left( { - 2 + \delta } \right)\sigma^{4} }}{{4\left( { - 4 + \delta } \right)^{2} \delta \left( {\sigma^{2} + \sigma_{1}^{2} } \right)}} > 0 $$
According to the comparison result, it is easy to obtain the results in Proposition 10.
