Economic role of online review filtering systems in the electronic marketplaces

Abstract

The monetary effects of online reviews motivate firms to manipulate such reviews, and electronic marketplaces then adopt online review filtering systems to combat manipulation. We develop an analytical model to explore the role of a filtering system for a monopoly firm, electronic marketplaces and consumer surplus in the monopoly context and then extend it to the symmetrical competition context to explore whether competition changes the role of the filtering system. The results show that the existence of such a filtering system strengthens the possibility of manipulation by the monopoly firm when the intelligence of the filtering system is relatively low. However, in the competitive context, whether the existence of such a filtering system strengthens the possibility of manipulation by both firms relates to the difference between the total consumer base with manipulation under no filtering system and that under the filtering system. We also find that in the monopoly context, the electronic marketplace will adopt the filtering system only when the intelligence of the filtering system is relatively low and the unit misfit cost is moderate. However, in the competitive context, the intelligence of the filtering system is irrelevant to whether the electronic marketplace adopts the filtering system. Finally, the adoption of the filtering system always benefits consumers in both the monopoly and competing contexts.

Appendix

1.1 A1: Proof of Lemma 1

Proof

We derive the equilibrium results case by case.

• (a) Case NN and Case FN.

If the monopoly firm decides not to manipulate online reviews, then the case in which the electronic marketplace adopts a filtering system (i.e., Case FN) is equivalent to that in which the electronic marketplace does not adopt a filtering system (i.e., Case NN). Therefore, here, we discuss only Case NN.

In this case, the monopoly firm’s optimization problem is as follows:

$$\underset{{{\text{p}}}_{{\text{NN}}}}{{\text{max}}}{\uppi }_{{\text{NN}}}=(1-\mathrm{\alpha }){{\text{D}}}_{{\text{NN}}}{{\text{p}}}_{{\text{NN}}}$$

The monopoly firm's optimization problem is characterized by first-order conditions as follows:

$$\frac{\partial {\uppi }_{{\text{NN}}}}{\partial {{\text{p}}}_{{\text{NN}}}}=\frac{\left(1-\mathrm{\alpha }\right)({\text{q}}-2{{\text{p}}}_{{\text{NN}}})}{{\text{t}}}=0$$

Based on these equations, we can derive the equilibrium as follows:

$${p}_{NN}^{*}=\frac{q}{2}$$

Substituting the equilibrium prices into Eqs. (1) and (2), we can derive the equilibrium profits of the monopoly firm and electronic marketplace. Note that with no firm manipulation, the case with no filtering system is actually the same as that with the filtering system. Additionally, condition ${\text{t}}>\frac{{\text{q}}}{2}$ is used to ensure that the firm cannot cover the whole market, even with manipulation.

• (b) Case NM

In this case, the monopoly firm’s optimization problem is as follows:

$$\underset{{{\text{p}}}_{{\text{NM}}},{{\text{e}}}_{{\text{NM}}}}{{\text{max}}}{\uppi }_{{\text{NM}}}=(1-\mathrm{\alpha }){{\text{D}}}_{{\text{NM}}}{{\text{p}}}_{{\text{NM}}}-\upbeta {{\text{e}}}_{{\text{NM}}}^{2}$$

The monopoly firm's optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\frac{\partial {\uppi }_{{\text{NM}}}}{\partial {{\text{p}}}_{{\text{NM}}}}=\frac{\left(1-\mathrm{\alpha }\right)\uptheta }{{\text{t}}}\left({\text{q}}+{{\text{e}}}_{{\text{NM}}}-2{{\text{p}}}_{{\text{NM}}}\right)=0\\ \frac{\partial {\uppi }_{{\text{NM}}}}{\partial {{\text{e}}}_{{\text{NM}}}}=\frac{(1-\mathrm{\alpha })\uptheta {{\text{p}}}_{{\text{NM}}}}{{\text{t}}}-2\beta {{\text{e}}}_{{\text{NM}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$$\left\{\begin{array}{c}{p}_{NM}^{*}=\frac{2\beta tq}{4\beta t-\left(1-\alpha \right)\theta }\\ {{\text{e}}}_{{\text{NM}}}^{*}=\frac{(1-\mathrm{\alpha })\mathrm{\theta q}}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right)\uptheta }\end{array}\right.$$

Substituting the equilibrium prices into Eqs. (1) and (2), we can derive the equilibrium profits of the monopoly firm and electronic marketplace. Additionally, condition ${\text{t}}>\frac{{\text{q}}}{2}+\frac{\uptheta \left(1-\mathrm{\alpha }\right)}{4\upbeta }$ is used to ensure that the firm cannot cover the whole market, even with manipulation.

• (c) Case FM

In this case, the monopoly firm’s optimization problem is as follows:

$$\underset{{{\text{p}}}_{{\text{FM}}},{{\text{e}}}_{{\text{FM}}}}{{\text{max}}}{\uppi }_{{\text{FM}}}=(1-\mathrm{\alpha }){{\text{D}}}_{{\text{FM}}}{{\text{p}}}_{{\text{FM}}}-\upbeta {{\text{e}}}_{{\text{FM}}}^{2}$$

The monopoly firm's optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\frac{\partial {\uppi }_{{\text{FM}}}}{\partial {{\text{p}}}_{{\text{FM}}}}=\frac{\left(1-\mathrm{\alpha }\right){\uptheta }_{\upgamma }}{{\text{t}}}\left({\text{q}}+(1-\upgamma ){{\text{e}}}_{{\text{FM}}}-2{{\text{p}}}_{{\text{FM}}}\right)=0\\ \frac{\partial {\uppi }_{{\text{FM}}}}{\partial {{\text{e}}}_{{\text{FM}}}}=\frac{(1-\mathrm{\alpha }){\uptheta }_{\upgamma }{{\text{p}}}_{{\text{FM}}}}{{\text{t}}}-2\beta {{\text{e}}}_{{\text{FM}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$$\left\{\begin{array}{c}{{\text{p}}}_{{\text{FM}}}^{*}=\frac{2\mathrm{\beta tq}}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }}\\ {{\text{e}}}_{{\text{FM}}}^{*}=\frac{\left(1-\mathrm{\alpha }\right)(1-\upgamma ){\theta }_{\gamma }q}{4\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\theta }_{\gamma }}\end{array}\right.$$

Substituting the equilibrium prices into Eqs. (1) and (2), we can derive the equilibrium profits of the monopoly firm and electronic marketplace. Additionally, condition ${\text{t}}>\frac{{\text{q}}}{2}+\frac{{\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}}{4\upbeta }$ is used to ensure that the firm cannot cover the whole market, even with manipulation.$\square$

1.2 A2: Proof of Proposition 1

Proof

We discuss the monopoly firm’s manipulation decision without and with the filtering system.

• (a) If the electronic marketplace does not adopt the filtering system, then the monopoly firm will manipulate online reviews when ${\uppi }_{{\text{NM}}}^{*}=\frac{(1-\mathrm{\alpha })\mathrm{\theta \beta }{{\text{q}}}^{2}}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right)\uptheta }>{\pi }_{NN}^{*}=\frac{\left(1-\alpha \right){q}^{2}}{4t}$. Thus, we have that ${\text{t}}<{{\text{t}}}_{1}$.

• (b) If the electronic marketplace adopts the filtering system, then the monopoly firm will manipulate online reviews when ${\uppi }_{{\text{FM}}}^{*}=\frac{\left(1-\mathrm{\alpha }\right)\upbeta {\theta }_{\gamma }{q}^{2}}{4\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\theta }_{\gamma }}>{\pi }_{NN}^{*}=\frac{\left(1-\alpha \right){q}^{2}}{4t}$. Thus, we have that ${\text{t}}<{{\text{t}}}_{2}$.

$\square$

1.3 A3 Proof of Corollary 1

Proof

From Proposition 1, we know that when ${\text{t}}<{{\text{t}}}_{1}$, the monopoly firm will manipulate online reviews without the filtering system, and when ${\text{t}}<{{\text{t}}}_{2}$, the monopoly firm will manipulate online reviews with the filtering system. Therefore, when ${{\text{t}}}_{1}>{{\text{t}}}_{2}$ (i.e., $\upgamma >{\upgamma }_{1}$), there exists a region where the monopoly manipulates online reviews without the filtering system but does not manipulate online reviews with the filtering system. Therefore, the existence of the filtering system reduces the possibility of manipulation by the monopoly firm. When ${{\text{t}}}_{1}<{{\text{t}}}_{2}$ (i.e., $\upgamma <{\upgamma }_{1}$), there exists a region where the monopoly does not manipulate online reviews without the filtering system but manipulates online reviews with the filtering system. Therefore, the existence of the filtering system strengthens the possibility of manipulation by the monopoly firm.$\square$

1.4 A4 Proof of Proposition 2

Proof: Based on the different values of $\upgamma$, we have two subcases.

Proof

(1) When $\upgamma >{\upgamma }_{1}$, we have that ${{\text{t}}}_{2}<{{\text{t}}}_{1}$. Therefore, we have three subcases. First, when ${\text{t}}<{{\text{t}}}_{2}$, the monopoly firm will manipulate online reviews both without and with the filtering system. The electronic marketplace will adopt the filtering system when ${\uppi }_{{\text{PNM}}}^{*}=\frac{4\mathrm{t\alpha \theta }{\upbeta }^{2}{{\text{q}}}^{2}}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right)\uptheta ]}^{2}}<{\uppi }_{{\text{PFM}}}^{*}=\frac{4\mathrm{t\alpha }{\theta }_{\gamma }{\upbeta }^{2}{q}^{2}}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){\left(1-\upgamma \right)}^{2}{\theta }_{\gamma }]}^{2}}$. Thus, we have that ${\text{t}}>{{\text{t}}}_{3}$. However, in this case, ${t}_{3}$ is always larger than ${t}_{2}$. Therefore, the electronic marketplace will not adopt the filtering system. Second, when ${{\text{t}}}_{2}<t<{{\text{t}}}_{1}$, the monopoly firm will manipulate online reviews without the filtering system and will not manipulate online reviews with the filtering system. The electronic marketplace will adopt the filtering system when ${\uppi }_{{\text{PNM}}}^{*}=\frac{4\mathrm{t\alpha \theta }{\upbeta }^{2}{{\text{q}}}^{2}}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right)\uptheta ]}^{2}}<{\pi }_{PFN}^{*}=\frac{\alpha {q}^{2}}{4t}$. Thus, we have ${\text{t}}>\frac{\left(1-\mathrm{\alpha }\right)\uptheta }{4\upbeta (1-\sqrt{\uptheta })}$. Because $\frac{\left(1-\mathrm{\alpha }\right)\uptheta }{4\upbeta (1-\sqrt{\uptheta })}>\frac{(1-\mathrm{\alpha })\uptheta }{4\upbeta (1-\uptheta )}$, the electronic marketplace will not adopt the filtering system. Third, when ${\text{t}}>{{\text{t}}}_{1}$, the monopoly firm will not manipulate online reviews either without or with the filtering system. Therefore, the electronic marketplace will not adopt the filtering system. In summary, when $\upgamma >{\upgamma }_{1}$, the electronic marketplace will not adopt the filtering system.

(2) When $\upgamma <{\upgamma }_{1}$, we have that ${{\text{t}}}_{1}<{{\text{t}}}_{2}$. Therefore, we also have three subcases. First, when ${\text{t}}<{{\text{t}}}_{1}$, the monopoly firm will manipulate online reviews both without and with the filtering system. The electronic marketplace will adopt the filtering system when ${\uppi }_{{\text{PNM}}}^{*}=\frac{4\mathrm{t\alpha \theta }{\upbeta }^{2}{{\text{q}}}^{2}}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right)\uptheta ]}^{2}}<{\uppi }_{{\text{PFM}}}^{*}=\frac{4\mathrm{t\alpha }{\theta }_{\gamma }{\upbeta }^{2}{q}^{2}}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){\left(1-\upgamma \right)}^{2}{\theta }_{\gamma }]}^{2}}$. Thus, we have that ${\text{t}}>{{\text{t}}}_{3}$. Therefore, when ${\text{min}}\{{{\text{t}}}_{3},{{\text{t}}}_{1}\}<t<{{\text{t}}}_{1}$, the electronic marketplace will adopt the filtering system. Second, when ${{\text{t}}}_{1}<t<{{\text{t}}}_{2}$, the monopoly firm will not manipulate online reviews without the filtering system and will manipulate online reviews with the filtering system. The electronic marketplace will adopt the filtering system when ${\pi }_{PNN}^{*}=\frac{\alpha {q}^{2}}{4t}<{\uppi }_{{\text{PFM}}}^{*}=\frac{4\mathrm{t\alpha }{\theta }_{\gamma }{\upbeta }^{2}{q}^{2}}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){\left(1-\upgamma \right)}^{2}{\theta }_{\gamma }]}^{2}}$. Thus, we have that ${\text{t}}<\frac{{\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}}{4\upbeta (1-\sqrt{{\uptheta }_{\upgamma }})}$. Because $\frac{{\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}}{4\upbeta (1-\sqrt{{\uptheta }_{\upgamma }})}>{{\text{t}}}_{2}$, the electronic marketplace will adopt the filtering system when ${{\text{t}}}_{1}<t<{{\text{t}}}_{2}$. Third, when ${\text{t}}>{{\text{t}}}_{2}$, the monopoly firm will not manipulate online reviews either without or with the filtering system. Therefore, the electronic marketplace will not adopt the filtering system. In summary, when $\upgamma <{\upgamma }_{1}$, the electronic marketplace will adopt the filtering system when ${\text{min}}\{{{\text{t}}}_{3},{{\text{t}}}_{1}\}<t<{t}_{2}$.$\square$

1.5 A5: Proof of Lemma 2

Proof

We derive consumer surplus on a case-by-case basis.

• (a) Case NN and Case FN

If the firm decides not to manipulate online reviews, then the case in which the electronic marketplace adopts a filtering system (i.e., Case FN) is equivalent to that in which the electronic marketplace does not adopt a filtering system (i.e., Case NN). Therefore, we discuss only Case NN.

The consumer with the highest consumer surplus is ${\text{q}}-{{\text{P}}}_{{\text{NN}}}^{*}=\frac{{\text{q}}}{2}$, and that with the lowest consumer surplus is zero. Moreover, the number of consumers who buy the product is ${{\text{D}}}_{{\text{NN}}}=\frac{{\text{q}}}{2{\text{t}}}$. Therefore, consumer surplus in Case NN is $\frac{1}{2}\times \left(\frac{{\text{q}}}{2}+0\right)\times {{\text{D}}}_{{\text{NN}}}=\frac{{{\text{q}}}^{2}}{8{\text{t}}}$.

• (b) Case NM

The consumer with the highest consumer surplus is ${\text{q}}-{{\text{P}}}_{{\text{MN}}}^{*}={\text{q}}-\frac{2\beta tq}{4\beta t-\left(1-\alpha \right)\theta }$, and that with the lowest consumer surplus is zero. Moreover, the number of consumers who buy the product is ${{\text{D}}}_{{\text{MN}}}=\frac{2\mathrm{\beta q\theta }}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right)\uptheta }$. Therefore, consumer surplus in Case MN is $\frac{1}{2}\times \left({\text{q}}-\frac{2\beta tq}{4\beta t-\left(1-\alpha \right)\theta }+0\right)\times {{\text{D}}}_{{\text{MN}}}=\frac{\mathrm{\theta \beta }{{\text{q}}}^{2}[2\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{{[4\beta t-\left(1-\alpha \right)\theta ]}^{2}}$.

• (c) Case FM

The consumer with the highest consumer surplus is ${\text{q}}-{{\text{P}}}_{{\text{FM}}}^{*}={\text{q}}-\frac{2\mathrm{\beta tq}}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }}$, and that with the lowest consumer surplus is zero. Moreover, the number of consumers who buy the product is ${{\text{D}}}_{{\text{FM}}}=\frac{2\mathrm{\beta q}{\theta }_{\gamma }}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }}$. Therefore, consumer surplus in Case MN is $\frac{1}{2}\times \left({\text{q}}-\frac{2\mathrm{\beta tq}}{4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }}+0\right)\times {{\text{D}}}_{{\text{FM}}}=\frac{{\uptheta }_{\upgamma }\upbeta {{\text{q}}}^{2}[2\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }]}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }]}^{2}}$.$\square$

1.6 A6: Proof of Proposition 3

Proof

From Proposition 2, we know that the electronic marketplace will adopt the filtering system only when $\upgamma <{\upgamma }_{1}\mathrm{and min}\{{{\text{t}}}_{3},{{\text{t}}}_{1}\}<t<{{\text{t}}}_{2}$. Therefore, we need only to discuss the effect of the filtering system on consumer surplus when the electronic marketplace adopts the filtering system.

When $\upgamma <1-\sqrt{\frac{\uptheta (1-{\uptheta }_{\upgamma })}{{\uptheta }_{\upgamma }(1-\uptheta )}}$, we have two subcases. First, when ${\text{min}}\{{{\text{t}}}_{3},{{\text{t}}}_{1}\}<t<{{\text{t}}}_{1}$, the electronic marketplace will adopt the filtering system. In this case, consumer surplus in Case FM (i.e., $\frac{{\uptheta }_{\upgamma }\upbeta {{\text{q}}}^{2}[2\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }]}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }]}^{2}}$) is always larger than that in Case NM (i.e., $\frac{\mathrm{\theta \beta }{{\text{q}}}^{2}[2\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{{[4\beta t-\left(1-\alpha \right)\theta ]}^{2}}$). Second, when ${{\text{t}}}_{1}<t<{{\text{t}}}_{2}$, the electronic marketplace will adopt the filtering system. In this case, consumer surplus in Case FM (i.e., $\frac{{\uptheta }_{\upgamma }\upbeta {{\text{q}}}^{2}[2\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }]}{{[4\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\theta }_{\gamma }]}^{2}}$) is always lower than that in Case NN (i.e., $\frac{{{\text{q}}}^{2}}{8{\text{t}}}$).$\square$

1.7 A7: Proof of Lemma 3

Proof

We derive the equilibrium results on a case-by-case basis.

• (a) Case AA and Case FA

If the firm decides not to manipulate online reviews, then the case in which the electronic marketplace adopts a filtering system (i.e., Case FA) is equivalent to that in which the electronic marketplace does not adopt a filtering system (i.e., Case AA). Therefore, we discuss only Case AA.

With no firm manipulation, from Eq. (3), we derive the demand for each product as follows:

$$\left\{\begin{array}{c}{{\text{D}}}_{{\text{AAA}}}=\frac{1}{2}-\frac{\left({{\text{p}}}_{{\text{AAA}}}-{{\text{p}}}_{{\text{BAA}}}\right)}{2{\text{t}}}\\ {{\text{D}}}_{{\text{BAA}}}=\frac{1}{2}+\frac{\left({{\text{p}}}_{{\text{AAA}}}-{{\text{p}}}_{{\text{BAA}}}\right)}{2{\text{t}}}\end{array}\right.$$

In this case, firms choose the optimal prices to maximize their profits in stage 3 of the game:

$$\left\{\begin{array}{c}\underset{{p}_{\mathit{AAA}}}{\mathit{max}}{\pi }_{AAA}=\left(1-\alpha \right){D}_{AAA}{p}_{AAA}\\ \underset{{p}_{\mathit{BAA}}}{\mathit{max}}{\pi }_{BAA}=(1-\alpha ){D}_{BAA}{p}_{BAA}\end{array}\right.$$

Firms’ optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\frac{\partial {\uppi }_{{\text{AAA}}}}{\partial {{\text{p}}}_{{\text{AAA}}}}=\frac{\left(1-\mathrm{\alpha }\right)(-2{{\text{p}}}_{{\text{AAA}}}+{{\text{p}}}_{{\text{BAA}}}+{\text{t}})}{2{\text{t}}}=0\\ \frac{\partial {\uppi }_{{\text{BAA}}}}{{\partial {\text{p}}}_{{\text{BAA}}}}=\frac{\left(1-\mathrm{\alpha }\right)({{\text{p}}}_{{\text{AAA}}}-2{{\text{p}}}_{{\text{BAA}}}+{\text{t}})}{2{\text{t}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$${{\text{p}}}_{{\text{AAA}}}^{*}={{\text{p}}}_{{\text{BAA}}}^{*}={\text{t}}$$

Substituting the equilibrium prices into Eqs. (4) and (5), we can derive the equilibrium profits of firms and the electronic marketplace.

• (b) Case AO

Due to symmetry, the case in which Firm A manipulates online reviews is equivalent to that in which Firm B manipulates online reviews. Therefore, here, we consider only the case in which only Firm A manipulates online reviews.

In this case, we derive the demand for each product as follows:

$$\left\{\begin{array}{c}{{\text{D}}}_{{\text{AAO}}}=\theta [\frac{1}{2}+\frac{{{\text{e}}}_{{\text{AAO}}}-\left({{\text{p}}}_{{\text{AAO}}}-{{\text{p}}}_{{\text{BAO}}}\right)}{2{\text{t}}}]\\ {{\text{D}}}_{{\text{BAO}}}=\theta [\frac{1}{2}-\frac{{{\text{e}}}_{{\text{AAO}}}-\left({{\text{p}}}_{{\text{AAO}}}-{{\text{p}}}_{{\text{BAO}}}\right)}{2{\text{t}}}]\end{array}\right.$$

In this case, the firms’ optimization problem is as follows:

$$\left\{\begin{array}{c}\underset{{p}_{\mathit{AAO}, {e}_{\mathit{AAO}}}}{\mathit{max}}{\pi }_{AAO}=\left(1-\alpha \right){D}_{AAO}{p}_{AAO}-\beta {e}_{AAO}^{2}\\ \underset{{p}_{\mathit{BAO}}}{\mathit{max}}{\pi }_{BAO}=(1-\alpha ){D}_{BAO}{p}_{BAO}\end{array}\right.$$

Firms’ optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\frac{\partial {\uppi }_{{\text{AAO}}}}{\partial {{\text{p}}}_{{\text{AAO}}}}=\theta \left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}+\frac{{{\text{e}}}_{{\text{AAO}}}-{{\text{p}}}_{{\text{AAO}}}+{{\text{p}}}_{{\text{BAO}}}}{2{\text{t}}}\right)-\frac{\uptheta (1-\mathrm{\alpha }){{\text{p}}}_{{\text{AAO}}}}{2{\text{t}}}=0\\ \frac{\partial {\uppi }_{{\text{AAO}}}}{\partial {{\text{e}}}_{{\text{AAO}}}}=\frac{\uptheta {(1-\mathrm{\alpha }){\text{p}}}_{{\text{AAO}}}}{2{\text{t}}}-2\beta {{\text{e}}}_{{\text{AAO}}}=0\\ \frac{\partial {\uppi }_{{\text{BAO}}}}{{\partial {\text{p}}}_{{\text{BAO}}}}=\theta \left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}-\frac{{{\text{e}}}_{{\text{AAO}}}-{{\text{p}}}_{{\text{AAO}}}+{{\text{p}}}_{{\text{BAO}}}}{2{\text{t}}}\right)-\frac{\uptheta (1-\mathrm{\alpha }){{\text{p}}}_{{\text{BAO}}}}{2{\text{t}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$$\left\{\begin{array}{l}{{\text{p}}}_{{\text{AAO}}}^{*}=\frac{12\upbeta {{\text{t}}}^{2}}{12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta }\\ {{\text{p}}}_{{\text{BAO}}}^{*}=\frac{12\upbeta {{\text{t}}}^{2}-2\mathrm{t\theta }(1-\mathrm{\alpha })}{12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta }\end{array}\right.$$

$${{\text{e}}}_{{\text{AAO}}}^{*}=\frac{3{\text{t}}(1-\mathrm{\alpha })\uptheta }{12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta }$$

Substituting the equilibrium prices and equilibrium effort of the inferior firm into Eqs. (4) and (5), we can derive the equilibrium profits of firms and the electronic marketplace. To ensure that both firms stay in the market and play a role in the equilibrium, ${{\text{p}}}_{{\text{AAO}}}^{*}$ and ${{\text{p}}}_{{\text{BAO}}}^{*}$ should be larger than zero. Thus, we have that ${\text{t}}>\frac{(1-\mathrm{\alpha })\uptheta }{6\upbeta }$.

• (c) Case AB

In this case, we derive the demand for each product as follows:

$$\left\{\begin{array}{c}{{\text{D}}}_{{\text{AAB}}}=\theta \Bigg[\frac{1}{2}+\frac{({{\text{e}}}_{{\text{AAB}}}-{{\text{e}}}_{{\text{BAB}}})-\left({{\text{p}}}_{{\text{AAB}}}-{{\text{p}}}_{{\text{BAB}}}\right)}{2{\text{t}}}\Bigg]\\ {{\text{D}}}_{{\text{BAB}}}=\theta \Bigg[\frac{1}{2}-\frac{({{\text{e}}}_{{\text{AAB}}}-{{\text{e}}}_{{\text{BAB}}})-\left({{\text{p}}}_{{\text{AAB}}}-{{\text{p}}}_{{\text{BAB}}}\right)}{2{\text{t}}}\Bigg]\end{array}\right.$$

In this case, the firms’ optimization problem is as follows:

$$\left\{\begin{array}{c}\underset{{p}_{\mathit{AAB}, {e}_{\mathit{AAB}}}}{\mathit{max}}{\pi }_{AAB}=\left(1-\alpha \right){D}_{AAB}{p}_{AAB}-\beta {e}_{AAB}^{2}\\ \underset{{p}_{\mathit{BAB}}, {e}_{\mathit{BAB}}}{\mathit{max}}{\pi }_{BAB}=(1-\alpha ){D}_{BAB}{p}_{BAB}-\beta {e}_{BAB}^{2}\end{array}\right.$$

Firms’ optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\begin{array}{c}\frac{\partial {\uppi }_{{\text{AAB}}}}{\partial {{\text{p}}}_{{\text{AAB}}}}=\theta \left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}+\frac{{{\text{e}}}_{{\text{AAB}}}-{{\text{e}}}_{{\text{BAB}}}-{{\text{p}}}_{{\text{AAB}}}+{{\text{p}}}_{{\text{BAB}}}}{2{\text{t}}}\right)-\frac{\uptheta (1-\mathrm{\alpha }){{\text{p}}}_{{\text{AAB}}}}{2{\text{t}}}=0\\ \frac{\partial {\uppi }_{{\text{AAB}}}}{\partial {{\text{e}}}_{{\text{AAB}}}}=\frac{\uptheta (1-\mathrm{\alpha }){{\text{p}}}_{{\text{AAB}}}}{2{\text{t}}}-2\beta {{\text{e}}}_{{\text{AAB}}}=0\\ \frac{\partial {\uppi }_{{\text{BAB}}}}{{\partial {\text{p}}}_{{\text{BAB}}}}=\theta \left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}-\frac{{{\text{e}}}_{{\text{AAB}}}-{{\text{e}}}_{{\text{BAB}}}-{{\text{p}}}_{{\text{AAB}}}+{{\text{p}}}_{{\text{BAB}}}}{2{\text{t}}}\right)-\frac{\uptheta (1-\mathrm{\alpha }){{\text{p}}}_{{\text{BAB}}}}{2{\text{t}}}=0\end{array}\\ \frac{\partial {\uppi }_{{\text{BAB}}}}{\partial {{\text{e}}}_{{\text{BAB}}}}=\frac{\uptheta (1-\mathrm{\alpha }){{\text{p}}}_{{\text{BAB}}}}{2{\text{t}}}-2\beta {{\text{e}}}_{{\text{BAB}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$${{\text{p}}}_{{\text{AAB}}}^{*}={{\text{p}}}_{{\text{BAB}}}^{*}={\text{t}}$$

$${{\text{e}}}_{{\text{AAB}}}^{*}={{\text{e}}}_{{\text{BAB}}}^{*}=\frac{(1-\mathrm{\alpha })\uptheta }{4\upbeta }$$

Substituting the equilibrium prices and equilibrium effort of the inferior firm into Eqs. (4) and (5), we can derive the equilibrium profits of firms and the electronic marketplace.

• (d) Case FO

In this case, we derive the demand for each product as follows:

$$\left\{\begin{array}{c}{{\text{D}}}_{{\text{AFO}}}={\uptheta }_{\upgamma }\Bigg[\frac{1}{2}+\frac{(1-\upgamma ){{\text{e}}}_{{\text{AFO}}}-\left({{\text{p}}}_{{\text{AFO}}}-{{\text{p}}}_{{\text{BFO}}}\right)}{2{\text{t}}}\Bigg]\\ {{\text{D}}}_{{\text{BFO}}}={\uptheta }_{\upgamma }\Bigg[\frac{1}{2}-\frac{(1-\upgamma ){{\text{e}}}_{{\text{AFO}}}-\left({{\text{p}}}_{{\text{AFO}}}-{{\text{p}}}_{{\text{BFO}}}\right)}{2{\text{t}}}\Bigg]\end{array}\right.$$

In this case, the firms’ optimization problem is as follows:

$$\left\{\begin{array}{l}\underset{{p}_{\mathit{AFO}, {e}_{\mathit{AFO}}}}{\mathit{max}}{\pi }_{AFO}=\left(1-\alpha \right){D}_{AFO}{p}_{AFO}-\beta {e}_{AFO}^{2}\\ \underset{{p}_{\mathit{BFO}}}{\mathit{max}}{\pi }_{BFO}=(1-\alpha ){D}_{BFO}{p}_{BFO}\end{array}\right.$$

Firms’ optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\frac{\partial {\uppi }_{{\text{AFO}}}}{\partial {{\text{p}}}_{{\text{AFO}}}}={\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}+\frac{(1-\upgamma ){{\text{e}}}_{{\text{AFO}}}-{{\text{p}}}_{{\text{AFO}}}+{{\text{p}}}_{{\text{BFO}}}}{2{\text{t}}}\right)-\frac{{\uptheta }_{\upgamma }(1-\mathrm{\alpha }){{\text{p}}}_{{\text{AFO}}}}{2{\text{t}}}=0\\ \frac{\partial {\uppi }_{{\text{AFO}}}}{\partial {{\text{e}}}_{{\text{AFO}}}}=\frac{{\uptheta }_{\upgamma }(1-\upgamma ){(1-\mathrm{\alpha }){\text{p}}}_{{\text{AFO}}}}{2{\text{t}}}-2\beta {{\text{e}}}_{{\text{AFO}}}=0\\ \frac{\partial {\uppi }_{{\text{BFO}}}}{{\partial {\text{p}}}_{{\text{BFO}}}}={\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}-\frac{(1-\upgamma ){{\text{e}}}_{{\text{AFO}}}-{{\text{p}}}_{{\text{AFO}}}+{{\text{p}}}_{{\text{BFO}}}}{2{\text{t}}}\right)-\frac{{\uptheta }_{\upgamma }(1-\mathrm{\alpha }){{\text{p}}}_{{\text{BFO}}}}{2{\text{t}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$$\left\{\begin{array}{l}{{\text{p}}}_{{\text{AFO}}}^{*}=\frac{12\upbeta {{\text{t}}}^{2}}{12\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }}\\ {{\text{p}}}_{{\text{BFO}}}^{*}=\frac{2{\text{t}}(6\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){\left(1-\upgamma \right)}^{2}{\uptheta }_{\upgamma })}{12\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }}\end{array}\right.$$

$${{\text{e}}}_{{\text{AFO}}}^{*}=\frac{3{\text{t}}\left(1-\mathrm{\alpha }\right)(1-\upgamma ){\uptheta }_{\upgamma }}{12\mathrm{\beta t}-(1-\mathrm{\alpha }){{(1-\upgamma )}^{2}\uptheta }_{\upgamma }}$$

Substituting the equilibrium prices and equilibrium effort of the inferior firm into Eqs. (4) and (5), we can derive the equilibrium profits of firms and the electronic marketplace. To ensure that both firms stay in the market and play a role in the equilibrium, ${{\text{p}}}_{{\text{AFO}}}^{*}$ and ${{\text{p}}}_{{\text{BFO}}}^{*}$ should be larger than zero. Thus, we have that ${\text{t}}>\frac{\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }}{6\upbeta }$.

• (e) Case FB

In this case, we derive the demand for each product as follows:

$$\left\{\begin{array}{c}{{\text{D}}}_{{\text{AFB}}}={\uptheta }_{\upgamma }\Bigg[\frac{1}{2}+\frac{(1-\upgamma )({{\text{e}}}_{{\text{AFB}}}-{{\text{e}}}_{{\text{BFB}}})-\left({{\text{p}}}_{{\text{AFB}}}-{{\text{p}}}_{{\text{BFB}}}\right)}{2{\text{t}}}\Bigg]\\ {{\text{D}}}_{{\text{BFB}}}={\uptheta }_{\upgamma }\Bigg[\frac{1}{2}-\frac{(1-\upgamma )({{\text{e}}}_{{\text{AFB}}}-{{\text{e}}}_{{\text{BFB}}})-\left({{\text{p}}}_{{\text{AFB}}}-{{\text{p}}}_{{\text{BFB}}}\right)}{2{\text{t}}}\Bigg]\end{array}\right.$$

In this case, the firms’ optimization problem is as follows:

$$\left\{\begin{array}{c}\underset{{p}_{\mathit{AFB}, {e}_{\mathit{AFB}}}}{\mathit{max}}{\pi }_{AFB}=\left(1-\alpha \right){D}_{AFB}{p}_{AFB}-\beta {e}_{AFB}^{2}\\ \underset{{p}_{\mathit{BFB}}, {e}_{\mathit{BFB}}}{\mathit{max}}{\pi }_{BFB}=(1-\alpha ){D}_{BFB}{p}_{BFB}-\beta {e}_{BFB}^{2}\end{array}\right.$$

Firms’ optimization problem is characterized by first-order conditions as follows:

$$\left\{\begin{array}{c}\begin{array}{c}\frac{\partial {\uppi }_{{\text{AFB}}}}{\partial {{\text{p}}}_{{\text{AFB}}}}={\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}+\frac{(1-\upgamma )({{\text{e}}}_{{\text{AFB}}}-{{\text{e}}}_{{\text{BFB}}})-{{\text{p}}}_{{\text{AFB}}}+{{\text{p}}}_{{\text{BFB}}}}{2{\text{t}}}\right)-\frac{{\uptheta }_{\upgamma }(1-\mathrm{\alpha }){{\text{p}}}_{{\text{AFB}}}}{2{\text{t}}}=0\\ \frac{\partial {\uppi }_{{\text{AFB}}}}{\partial {{\text{e}}}_{{\text{AFB}}}}=\frac{{\uptheta }_{\upgamma }(1-\upgamma )(1-\mathrm{\alpha }){{\text{p}}}_{{\text{AFB}}}}{2{\text{t}}}-2\beta {{\text{e}}}_{{\text{AFB}}}=0\\ \frac{\partial {\uppi }_{{\text{BFB}}}}{{\partial {\text{p}}}_{{\text{BFB}}}}={\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right)\left(\frac{1}{2}-\frac{{(1-\upgamma )({\text{e}}}_{{\text{AFB}}}-{{\text{e}}}_{{\text{BFB}}})-{{\text{p}}}_{{\text{AFB}}}+{{\text{p}}}_{{\text{BFB}}}}{2{\text{t}}}\right)-\frac{{\uptheta }_{\upgamma }(1-\mathrm{\alpha }){{\text{p}}}_{{\text{BFB}}}}{2{\text{t}}}=0\end{array}\\ \frac{\partial {\uppi }_{{\text{BFB}}}}{\partial {{\text{e}}}_{{\text{BFB}}}}=\frac{{\uptheta }_{\upgamma }(1-\upgamma )(1-\mathrm{\alpha }){{\text{p}}}_{{\text{BFB}}}}{2{\text{t}}}-2\beta {{\text{e}}}_{{\text{BFB}}}=0\end{array}\right.$$

Based on these equations, we can derive the equilibrium as follows:

$${{\text{p}}}_{{\text{AFB}}}^{*}={{\text{p}}}_{{\text{BFB}}}^{*}={\text{t}}$$

$${{\text{e}}}_{{\text{AFB}}}^{*}={{\text{e}}}_{{\text{BFB}}}^{*}=\frac{\left(1-\mathrm{\alpha }\right)(1-\upgamma ){\uptheta }_{\upgamma }}{4\upbeta }$$

Substituting the equilibrium prices and equilibrium effort of the inferior firm into Eqs. (4) and (5), we can derive the equilibrium profits of firms and the electronic marketplace.$\square$

1.8 A8: Proof of Proposition 4

Proof

(1) If the electronic marketplace does not adopt the filtering system, then the following table summarizes the four cases under no filtering system.

		Firm A’s manipulation decision
		Firm A manipulates	Firm A does not manipulate
Firm B’s manipulation decision	Firm B manipulates	$\bigg(\frac{\left(1-\mathrm{\alpha }\right)\uptheta [8\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{16\upbeta }, \frac{\left(1-\mathrm{\alpha }\right)\uptheta [8\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{16\upbeta }\Bigg)$	$\bigg(\frac{2\mathrm{t\theta }\left(1-\mathrm{\alpha }\right){[6\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}, \frac{9\mathrm{\beta \theta }{{\text{t}}}^{2}\left(1-\mathrm{\alpha }\right)[8\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}\bigg)$
	Firm B does not manipulate	$\bigg(\frac{9\mathrm{\beta \theta }{{\text{t}}}^{2}\left(1-\mathrm{\alpha }\right)[8\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}, \frac{2\mathrm{t\theta }\left(1-\mathrm{\alpha }\right){[6\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}\bigg)$	$\bigg(\frac{\left(1-\mathrm{\alpha }\right){\text{t}}}{2}, \frac{\left(1-\mathrm{\alpha }\right){\text{t}}}{2}\bigg)$

If Firm A manipulates online reviews, then Firm B’s best response is to manipulate online reviews because $\frac{\left(1-\mathrm{\alpha }\right)\uptheta [8\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{16\upbeta }$ is always larger than $\frac{2\mathrm{t\theta }\left(1-\mathrm{\alpha }\right){[6\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}$ on the condition of ${\text{t}}>max\{\frac{(1-\mathrm{\alpha })\uptheta }{6\upbeta },\frac{(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }}{6\upbeta }\}$, which is used to ensure that both firms play a role in the equilibrium. If Firm A does not manipulate online reviews, Firm B will manipulate online reviews when $\uptheta >\frac{8}{9} and \overline{{t }_{1}}<t<\overline{{t }_{2}}$; otherwise, Firm B will not manipulate online reviews. The reason for this is that $\frac{9\mathrm{\beta \theta }{{\text{t}}}^{2}\left(1-\mathrm{\alpha }\right)[8\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha })\uptheta ]}^{2}}\mathrm{is larger than }\frac{\left(1-\mathrm{\alpha }\right){\text{t}}}{2}$ only when $\uptheta >\frac{8}{9} and \overline{{t }_{1}}<t<\overline{{t }_{2}}$. Due to symmetry, Firm A’s best response to Firm B’s manipulation decision is the same as that of Firm B. Therefore, both firms will manipulate online reviews only when $\uptheta >\frac{8}{9} and \overline{{t }_{1}}<t<\overline{{t }_{2}}$; otherwise, both firms will not manipulate online reviews.

(2) If the electronic marketplace adopts the filtering system, then the following table summarizes the four cases under no filtering system.

		Firm A’s manipulation decision
		Firm A manipulates	Firm A does not manipulate
Firm B’s manipulation decision	Firm B manipulates	$\bigg(\frac{\left(1-\mathrm{\alpha }\right){\uptheta }_{\upgamma }[8\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}{16\upbeta }, \frac{\left(1-\mathrm{\alpha }\right){\uptheta }_{\upgamma }[8\mathrm{\beta t}-\left(1-\mathrm{\alpha }\right){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}{16\upbeta }\bigg)$	$\bigg(\frac{2{\text{t}}{\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right){[6\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}^{2}}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}^{2}}, \frac{9\upbeta {{{\text{t}}}^{2}\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right)[8\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}^{2}}\bigg)$
	Firm B does not manipulate	$\bigg(\frac{9\upbeta {{{\text{t}}}^{2}\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right)[8\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}^{2}}, \frac{2{\text{t}}{\uptheta }_{\upgamma }\left(1-\mathrm{\alpha }\right){[6\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}^{2}}{{[12\mathrm{\beta t}-(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }]}^{2}}\bigg)$	$\bigg(\frac{\left(1-\mathrm{\alpha }\right){\text{t}}}{2}, \frac{\left(1-\mathrm{\alpha }\right){\text{t}}}{2}\bigg)$

Using the same logic as that for the case in which the electronic marketplace does not adopt the filtering system, we can derive that both firms will manipulate online reviews with the filtering system when ${\uptheta }_{\upgamma }>\frac{8}{9}\mathrm{ and }\overline{{t }_{3}}<t<\overline{{t }_{4}}$.$\square$

1.9 A9: Proof of Corollary 2

Proof

(1) When ${\theta }_{\gamma }>\frac{8}{9}>\theta$, both firms will not manipulate online reviews without the filtering system and will manipulate online reviews with the filtering system where $\overline{{t }_{3}}<t<\overline{{t }_{4}}$. Therefore, in this case, the existence of the filtering system strengthens the possibility of manipulation by competing firms.

(2) When ${\theta }_{\gamma }>\theta >\frac{8}{9}$, both firms will manipulate online reviews without the filtering system when $\overline{{t }_{1}}<t<\overline{{t }_{2}}$, and both firms will manipulate online reviews with the filtering system when $\overline{{t }_{3}}<t<\overline{{t }_{4}}$. Therefore, the existence of the filtering system strengthens the possibility of manipulation by competing firms when $\overline{{t }_{2}}-\overline{{t }_{1}}<\overline{{t }_{3}}-\overline{{t }_{4}}$. Thus, we have that ${\uptheta }_{\upgamma }>\overline{\uptheta }$. Otherwise, the existence of the filtering system hinders manipulation by competing firms.$\square$

1.10 A10: Proof of Proposition 5

Proof

We discuss the adoption decision of the filtering system of the electronic marketplace in three cases within the condition of ${\text{t}}>max\{\frac{(1-\mathrm{\alpha })\uptheta }{6\upbeta },\frac{(1-\mathrm{\alpha }){(1-\upgamma )}^{2}{\uptheta }_{\upgamma }}{6\upbeta }\}$.

(1) When $\uptheta <{\uptheta }_{\upgamma }<\frac{8}{9}$, both firms will not manipulate online reviews regardless of the existence of the filtering system. Therefore, the electronic marketplace will not adopt the filtering system.

(2) When $\uptheta <{\frac{8}{9}<\uptheta }_{\upgamma }$, both firms will not manipulate online reviews without the filtering system and will manipulate online reviews with the filtering system when $\overline{{t }_{3}}<t<\overline{{t }_{4}}$. Therefore, when $\uptheta <{\frac{8}{9}<\uptheta }_{\upgamma }$ and $t<\overline{{t }_{3}}$ or $\uptheta <{\frac{8}{9}<\uptheta }_{\upgamma }$ and $t>\overline{{t }_{4}}$, the electronic marketplace will not adopt the filtering system. When $\uptheta <{\frac{8}{9}<\uptheta }_{\upgamma }$ and $\overline{{t }_{3}}<t<\overline{{t }_{4}}$, the profit for the electronic marketplace without the filtering system is $\mathrm{\alpha t}$, and the profit for the electronic marketplace with the filtering system is $\mathrm{\alpha t}{\uptheta }_{\upgamma }$. Therefore, when $\uptheta <{\frac{8}{9}<\uptheta }_{\upgamma }$ and $\overline{{t }_{3}}<t<\overline{{t }_{4}}$, the electronic marketplace will not adopt the filtering system because $\mathrm{\alpha t}{\uptheta }_{\upgamma }$ is always lower than $\mathrm{\alpha t}$. In summary, when $\uptheta <{\frac{8}{9}<\uptheta }_{\upgamma }$, the electronic marketplace will not adopt the filtering system.

(3) When $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$, both firms will manipulate online reviews without the filtering system when $\overline{{t }_{1}}<t<\overline{{t }_{2}}$ and will manipulate online reviews with the filtering system when $\overline{{t }_{3}}<t<\overline{{t }_{4}}$. Because $\overline{{t }_{3}}$ is always lower than $\overline{{t }_{1}}$ and $\overline{{t }_{2}}$ could be lower or larger than $\overline{{t }_{4}}$, we should discuss the two subcases. First, when $\overline{{t }_{2}}<\overline{{t }_{4}}$, we have $\overline{{t }_{3}}<\overline{{t }_{1}}<\overline{{t }_{2}}<\overline{{t }_{4}}$. When ${\text{t}}<\overline{{t }_{3}}$ and ${\text{t}}>\overline{{t }_{4}}$, both firms will not manipulate online reviews regardless of the existence of the filtering system. Therefore, the electronic marketplace will not adopt the filtering system. When $\overline{{t }_{3}}<{\text{t}}<\overline{{t }_{1}}$ and $\overline{{t }_{2}}<{\text{t}}<\overline{{t }_{4}}$, both firms will not manipulate online reviews without the filtering system and will manipulate online reviews with the filtering system. Therefore, the electronic marketplace will not adopt the filtering system because the profit under no manipulation without the filtering system ($\mathrm{\alpha t}$) is always higher than that under manipulation with the filtering system ($\mathrm{\alpha t}{\uptheta }_{\upgamma }$). When $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{2}}$, both firms will manipulate online reviews regardless of the existence of the filtering system. Therefore, the electronic marketplace will adopt the filtering system because the profit under manipulation without the filtering system ($\mathrm{\alpha t\theta }$) is always lower than that under manipulation with the filtering system ($\mathrm{\alpha t}{\uptheta }_{\upgamma }$). Second, when $\overline{{t }_{2}}>\overline{{t }_{4}}$, we have $\overline{{t }_{3}}<\overline{{t }_{1}}<\overline{{t }_{4}}<\overline{{t }_{2}}$. When ${\text{t}}<\overline{{t }_{3}}$ and ${\text{t}}>\overline{{t }_{2}}$, both firms will not manipulate online reviews regardless of the existence of the filtering system. Therefore, the electronic marketplace will not adopt the filtering system. When $\overline{{t }_{3}}<{\text{t}}<\overline{{t }_{1}}$, both firms will not manipulate online reviews without the filtering system and will manipulate online reviews with the filtering system. Therefore, the electronic marketplace will not adopt the filtering system because the profit under no manipulation without the filtering system ($\mathrm{\alpha t}$) is always higher than that under manipulation with the filtering system ($\mathrm{\alpha t}{\uptheta }_{\upgamma }$). When $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{4}}$, both firms will manipulate online reviews regardless of the existence of the filtering system. Therefore, the electronic marketplace will adopt the filtering system because the profit under manipulation without the filtering system ($\mathrm{\alpha t\theta }$) is always lower than that under manipulation with the filtering system ($\mathrm{\alpha t}{\uptheta }_{\upgamma }$). When $\overline{{t }_{4}}<{\text{t}}<\overline{{t }_{2}}$, both firms will manipulate online reviews without the filtering system and will not manipulate online reviews with the filtering system. Therefore, the electronic marketplace will adopt the filtering system because the profit under manipulation without the filtering system ($\mathrm{\alpha t\theta }$) is always lower than that under no manipulation with the filtering system ($\mathrm{\alpha t}$). In summary, when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$, the electronic marketplace will adopt the filtering system when $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{2}}$.

In summary, the electronic marketplace will adopt the filtering system when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{2}}$.$\square$

1.11 A11: Proof of Corollary 3

Proof

We compare the electronic marketplace’s adoption of the filtering system in the monopoly context with that in the competing context in the common region (i.e., ${\text{t}}>\widehat{t}$). However, we have $\overline{{t }_{1}}<\frac{\uptheta \left(1-\mathrm{\alpha }\right)}{4\upbeta }<\frac{{\text{q}}}{2}+\frac{\uptheta \left(1-\mathrm{\alpha }\right)}{4\upbeta }$. Thus, in the common region (${\text{t}}>\widehat{t}$), the electronic marketplace will adopt the filtering system when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\widehat{t}<{\text{t}}<\overline{{t }_{2}}$.

First, when competition promotes the electronic marketplace’s adoption of the filtering system, the electronic marketplace adopts the filtering system in a competing context and does not adopt the filtering system in a monopoly context. When $\upgamma >{\upgamma }_{1}$, the electronic marketplace does not adopt the filtering system in the monopoly context and adopts the filtering system in the competing context when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\widehat{t}<{\text{t}}<\overline{{t }_{2}}$. When $\upgamma <{\upgamma }_{1}$, the electronic marketplace does not adopt the filtering system in the monopoly context when $\widehat{t}<{\text{t}}<min\{{t}_{1},{t}_{3}\}$ or $t>{t}_{2}$ and adopts the filtering system in the competing context when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\widehat{t}<{\text{t}}<\overline{{t }_{2}}$. Therefore, competition promotes the electronic marketplace’s adoption of the filtering system when ${\theta }_{\gamma }>\theta >\frac{8}{9}$ and $\upgamma >{\upgamma }_{1}$ and $\widehat{t}<t<\overline{{t }_{2}}$ or ${\theta }_{\gamma }>\theta >\frac{8}{9}$ and $\upgamma <{\upgamma }_{1}$ and $\widehat{t}<t<min\{\overline{{t }_{2}}, min\{{t}_{1},{t}_{3}\}\}$.

Second, using the same logic in the case where competition promotes the electronic marketplace’s adoption of the filtering system, we can obtain the condition where competition hinders the electronic marketplace’s adoption of the filtering system when $\theta {<\theta }_{\gamma }<\frac{8}{9}$ and $\upgamma <{\upgamma }_{1}$ and $min\{{t}_{1},{t}_{3}\}<t<{t}_{2}$ or $\theta <\frac{8}{9}<{\theta }_{\gamma }$ and $\upgamma <{\upgamma }_{1}$ and $min\{{t}_{1},{t}_{3}\}<t<{t}_{2}$ or ${\theta }_{\gamma }>\theta >\frac{8}{9}$ and $\upgamma <{\upgamma }_{1}$ and $max\{\overline{{t }_{2}}, min\{{t}_{1},{t}_{3}\}\}<t<{t}_{2}$.$\square$

1.12 A12: Proof of Lemma 4

Proof

We derive consumer surplus in the competing context on a case-by-case basis.

• (a) Case AA and Case FA

If the firm decides not to manipulate online reviews, then the case in which the electronic marketplace adopts a filtering system (i.e., Case AA) is equivalent to that in which the electronic marketplace does not adopt a filtering system (i.e., Case AA). Therefore, we discuss only Case AA.

Due to symmetry, the consumer surplus of consumers who buy from Firm A is the same as that of those who buy from Firm B. For Firm A, the consumer with the highest consumer surplus is ${\text{q}}-{{\text{P}}}_{{\text{AAA}}}^{*}={\text{q}}-{\text{t}}$, and the consumer with the lowest consumer surplus is zero. The number of consumers who buy the product is ${{\text{D}}}_{{\text{AAA}}}=\frac{1}{2}$. Therefore, the consumer surplus of consumers who buy from Firm A in Case AA is $\frac{1}{2}\times \left({\text{q}}-{\text{t}}+0\right)\times {{\text{D}}}_{{\text{AAA}}}=\frac{{\text{q}}-{\text{t}}}{4}$. Furthermore, total consumer surplus is $2\times \frac{{\text{q}}-{\text{t}}}{4}=\frac{{\text{q}}-{\text{t}}}{2}$.

• (b) Case AB

Due to symmetry, the consumer surplus of consumers who buy from Firm A is the same as that of those who buy from Firm B. For Firm A, the consumer with the highest consumer surplus is ${\text{q}}-{{\text{P}}}_{{\text{AAB}}}^{*}={\text{q}}-{\text{t}}$, and the consumer with the lowest consumer surplus is zero. The number of consumers who buy the product is ${{\text{D}}}_{{\text{AAB}}}=\frac{\uptheta }{2}$. Therefore, the consumer surplus of consumers who buy from Firm A in Case AA is $\frac{1}{2}\times \left({\text{q}}-{\text{t}}\right)\times {{\text{D}}}_{{\text{AAB}}}=\uptheta (\frac{{\text{q}}-{\text{t}}}{4})$. Furthermore, total consumer surplus is $2\times\uptheta (\frac{{\text{q}}-{\text{t}}}{4})=\frac{\uptheta ({\text{q}}-{\text{t}})}{2}$.

• (c) Case FB

Due to symmetry, the consumer surplus of consumers who buy from Firm A is the same as that of those who buy from Firm B. For Firm A, the consumer with the highest consumer surplus is ${\text{q}}-{{\text{P}}}_{{\text{AFB}}}^{*}={\text{q}}-{\text{t}}$, and the consumer with the lowest consumer surplus is zero. The number of consumers who buy the product is ${{\text{D}}}_{{\text{AAB}}}=\frac{{\uptheta }_{\upgamma }}{2}$. Therefore, the consumer surplus of consumers who buy from Firm A in Case AA is $\frac{1}{2}\times \left({\text{q}}-{\text{t}}\right)\times {{\text{D}}}_{{\text{AAB}}}={\uptheta }_{\upgamma }(\frac{{\text{q}}-{\text{t}}}{4})$. Furthermore, total consumer surplus is $2\times {\uptheta }_{\upgamma }(\frac{{\text{q}}-{\text{t}}}{4})=\frac{{\uptheta }_{\upgamma }({\text{q}}-{\text{t}})}{2}$.$\square$

1.13 A13: Proof of Proposition 6

Proof

From Proposition 5, we know that the electronic marketplace will adopt the filtering system when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{2}}$. Under the condition that $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{2}}$, both firms will manipulate online reviews without the filtering system. Therefore, in this case, consumer surplus is $\frac{\uptheta ({\text{q}}-{\text{t}})}{2}$. However, with the filtering system, both firms will either manipulate or not manipulate online reviews when $\frac{8}{9}<\uptheta {<\uptheta }_{\upgamma }$ and $\overline{{t }_{1}}<{\text{t}}<\overline{{t }_{2}}$. Consumer surplus with the filtering system under both choices of competing firms ($\frac{{\uptheta }_{\upgamma }({\text{q}}-{\text{t}})}{2}$ or $\frac{{\text{q}}-{\text{t}}}{2}$) is always higher than that without the filtering system.$\square$