## Appendix  
**Proof of Lemma 1. ** When \(L\leq v\leq H\), there are three probable situations for the price: (1), \(p\leq L\); (2), \(L<p\leq v\); (3), \(v<p\leq H\). First, when \(p\leq L\), \(Pr(p\leq c)=1\), \(p\leq v+I\), and thus \(\chi=1\). Therefore, \(Pr(\delta=1)=1\), which meets the constraint (3). \(E(u_{S})\geq 0\) can be expressed as \(B-\beta I\geq 0\). Therefore, problem (4) becomes \(\max\limits_{l}B-\beta I\), 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 \(\underset{p,B}{max}p-B\), 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 \(L\). Therefore, candidate 1 is: \(p=L\), \(B=0\), \(E(\pi)=L\). Second, when \(L<p\leq v,Pr(p\leq c)=q,\chi=1\). 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. \(qB-\beta I\geq 0\). \(u_{S}\) gets the max value when \(I=0\). 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 \(v\). Therefore, candidate 2 is: \(p=v\), \(B=0\), \(E(\pi)=qv\). Third, when \(v<p\leq H\), \(Pr(p\leq c)=q\). When \(I\geq p-v\), \(\chi=1\). Thus, when \(I\geq p-v\), \(Pr(\delta=1)=q\), 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. \(qB-\beta I\geq 0\), \(I\geq p-v\). To assure \(u_{S}\) max, \(I\) gets the minimum value \(p-v\), and \(qB-\beta I\geq 0\) can be expressed as \(B\geq\frac{\beta(p-v)}{q}\). Therefore, when \(B\geq\frac{\beta(p-v)}{q}\), the optimal \(I\) is \(p-v\). Then, discuss problem (5). The firm's expected profit \(E(\pi)=q(p-B)\). Substituting optimal \(I\)into problem (5), the problem becomes: \(\max\limits_{p,B}q(p-B)\), s.t. \(p\geq 0\), \(B\geq 0\), \(B\geq\frac{\beta(p-v)}{q}\). The profit gets the maximum value when \(B\) gets the minimum value \(\frac{\beta(p-v)}{q}\). Substituting optimal \(B\) into problem (5), the problem becomes: \(\max\limits_{p,B}(q-\beta)p+\beta v\), 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: \(p=H\), \(B=\frac{\beta(H-v)}{q}\), \(E(\pi)=qH-\beta(H-v)\). If \(q\leq\beta\), The profit gets the maximum value when \(p\) gets the minimum value \(v\). Therefore, candidate 2 is: \(p=v\), \(B=0\), \(E(\pi)=qv\). Summarize the candidates, we get Lemma 1. **Proof of Proposition 1. ** Proposition 1 is the result of the comparison between the three candidates. First, compare the profits of candidate 1 and candidate 2. Solving \(\pi_{1}\geq\pi_{2}\), we get \(q\leq\frac{L}{v}\). When \(q\leq\frac{L}{v}\), compare candidate 1 and candidate 3. Then \(\pi_{3}\geq\pi_{1}\) can be expressed as \(qH-\beta(H-v)>L\), from which we can get \(q>\frac{L+\beta(H-v)}{H}\). When \(q>\frac{L}{v}\), compare candidate 2 and candidate 3. Then \(\pi_{3}\geq\pi_{2}\)can be expressed as \(qH-\beta(H-v)>qv\), from which we can get \(q>\beta\). Therefore, if candidate 3 is optimal among the three candidates, the two conditions should be met: \(q>\frac{L+\beta(H-v)}{H}\)and \(q>\beta\). In a similar way, we get the conditions when one dominates the other among the three candidates. **Proof of Proposition 2. ** When \(v<L\), there are three probable situations of the price: (1), \(p\leq v\); (2), \(v<p\leq L\); (3), \(L<p\leq H\). First, when \(p\leq v\), \(Pr(p\leq c)=1\), \(p\leq v+I\), and thus \(\chi=1\). 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\s.t. \(p\geq 0\), \(B\geq\beta(p-\
u)\).