## Appendix
The profit gets the maximum value when \(B\) gets the minimum value \(\beta(p-\
u)\), and \(p\) gets the maximum value \(L\). Therefore, candidate 2 is: \(p=L\), \(B=\beta(L-\
u)\), \(E(\pi)=L-\beta(L-\
u)\). Third, when \(L<p\leq H\), \(Pr(p\leq c)=q\). When \(I\geq p-\
u\), \(\chi=1\). Thus, when \(I\geq p-\
u\), \(Pr(\delta=1)=q\), which meets constraint (3). \(E(u_{S})\geq 0\) can be expressed as \(qB-\beta I\geq 0\), and \(u_{S}=qB-\beta I\). Therefore, problem (4) becomes \(\max\limits_{I}qB-\beta I\), s.t. \(qB-\beta I\geq 0\), \(I\geq p-\
u\). To assure \(u_{S}\) max, \(I\) gets the minimum value \(p-\
u\), and \(qB-\beta I\geq 0\)can be expressed as \(B\geq\frac{\beta(p-\
u)}{q}\). Therefore, when \(B\geq\frac{\beta(p-\
u)}{q}\), the optimal \(I\) is \(p-\
u\). Then, discuss problem (5). The firm's expected profit is \(E(\pi)=q(p-B)\). Substituting the optimal \(I\)into problem (5), the problem becomes: \(\max\limits_{p,B}q(p-B)\), s.t. \(p\geq 0\), \(B\geq 0\), \(B\geq\frac{\beta(p-\
u)}{q}\). The profit gets the maximum value when \(B\) gets the minimum value \(\frac{\beta(p-\
u)}{q}\). Substituting optimal \(B\) into problem (5), the problem becomes: \(\max\limits_{p,B}(q-\beta)p+\beta\
u\), s.t. \(p\geq 0\). If \(q>\beta\), the profit gets the maximum value when \(p\) gets the maximum value \(H\). Therefore, candidate 3 is: \(p=H\), \(B=\frac{\beta(H-\
u)}{q}\), \(E(\pi)=qH-\beta(H-\
u)\). If \(q\leq\beta\), the profit gets the maximum value when \(p\) gets the minimum value \(L\). Therefore, candidate 4 is: \(p=L\), \(B=\frac{\beta(L-\
u)}{q}\), \(E(\pi)=qL-\beta(L-\
u)\). Compare the expected profits \(E(\pi)\), and we get the optimal solution as Proposition 2 when \(\
u<L\). When \(\
u\geq H\), there are two probable situations for the price: (1), \(p\leq L\); (2), \(L<p\leq H\). First, when \(p\leq L\), \(Pr(p\leq c)=1\), \(\chi=1\). Therefore, \(Pr(\delta=1)=1\), which meets constraint (3). \(E(u_{S})\geq 0\) can be expressed as \(B-\beta I\geq 0\). Therefore, problem (4) becomes \(\max\limits_{I}B-\beta I\), s.t. \(B-\beta I\geq 0\). \(u_{S}\) gets the maximum value when \(I=0\). The firm's problem (5) can be expressed as \(\max\limits_{p,B}\)\(\max\limits_{p,B}-B\), s.t. \(p\geq 0\), \(B\geq 0\). The profit gets the maximum value when \(B\) gets the minimum value \(0\), and \(p\) gets the maximum value \(L\). Therefore, candidate 1 is: \(p=L\), \(B=0\), \(E(\pi)=L\). Second, when \(L<p\leq H\), \(Pr(p\leq c)=q\), \(\chi=1\). Thus, \(Pr(\delta=1)=q\), which meets constraint (3). \(E(u_{S})\geq 0\) can be expressed as \(qB-\beta I\geq 0\), and \(u_{S}=qB-\beta I\). Therefore, problem (4) becomes \(\max\limits_{I}qB-\beta I\), s.t. \(qB-\beta I\geq 0\) and \(I\geq 0\). \(u_{S}\) gets the max value when \(I\) gets the minimum value \(I=0\) and \(B\geq 0\). The firm's problem (5) can be expressed as \(\max\limits_{p,B}q(p-B)\), s.t. \(p\geq 0\), \(B\geq 0\). The profit gets the maximum value when \(B\) gets the minimum value \(0\), and \(p\) gets the maximum value \(H\). Therefore, candidate 2 is: \(p=H\), \(B=0\), \(E(\pi)=qH\). Compare the expected profits \(E(\pi)\), and we get the optimal solution as Proposition 2 when \(\
u\geq H\). **Result of Problem (6). ** The optimal RRP adoption conditions and solutions are given in the following table (Table A1). (The proof is the same as the proofs of Propositions 1 and 2.) \begin{tabular}{c c c} \hline \hline
**Case** & **Conditions** & **Optimal Priceand Reward** \\ \hline Case 1 & \(\
u<L<H\),\(q\leq\beta+(1-\beta)\frac{L}{H}\) & \(p^{*}=L\), \(B^{*}=\beta(L-\
u)\),\(m^{*}=0\) \\ Case 2 & \(\
u<L<H\), \(q>\beta+(1-\beta)\frac{L}{H}\) & \(p^{*}=H\),\(B^{*}=\frac{\beta}{q}(H-\
u)\),\(m^{*}=0\) \\ Case 3 & \(L\leq\
u<H\), \(q>\beta