Subscription strategy choices of network video platforms in the presence of social influence

Abstract

In this paper, we investigate the optimal subscription strategy for network video platforms and show how it is affected by social influence. The strategy decision is made among paid strategies, free strategies, and trial strategies, and revenue models are presented in two cases: positive social influence and negative social influence. We show that regardless of which strategy a platform adopts, positive social influence always makes a platform better. Results run counter to the conventional wisdom that positive social influence has an adverse effect on subscription demand without a free trial, which is benefited under negative social influence. A platform can always benefit from offering trial clips in the presence of positive social influence. A paid strategy is optimal if a video generates less social influence and advertising becomes more of a nuisance for consumers. A free strategy, otherwise, is dominant. In the presence of negative social influence, however, a free strategy is always the worst choice for a platform. Moreover, we found that positive social influence expands a consumer’s tolerance of advertising when compared to a video with no social influence.

Appendix

1.1 Proof of Proposition 1

The derivatives of revenues under the three strategies with respect to $\lambda$ in the presence of positive social influence are: $\frac{{\partial \Pi^{{SF^{ + *} }} }}{\partial \lambda } = \frac{1}{{4\delta (\lambda - 1)^{2} }} > 0$, $\frac{{\partial \Pi^{{SP^{ + *} }} }}{\partial \lambda } = \frac{1}{{4(\lambda - 1)^{2} }} > 0$, $\frac{{\partial \Pi^{{SM^{ + *} }} }}{\partial \lambda } = \frac{4\chi (1 - \lambda )}{{(\delta \chi^{2} + 2(\lambda - 1)^{2} )^{2} }} > 0$.

Similarly, the derivatives of revenues under the three strategies with respect to $\lambda$ in the presence of negative social influence are: $\frac{{\partial \Pi^{{SF^{ - *} }} }}{\partial \lambda } = - \frac{1}{{4\delta (\lambda + 1)^{2} }} < 0$, $\frac{{\partial \Pi^{{SP^{ - } *}} }}{\partial \lambda } = - \frac{1}{{4\delta (\lambda + 1)^{2} }} < 0$, $\frac{{\partial \Pi^{{SM^{ - *} }} }}{\partial \lambda } = - \frac{4\chi (1 + \lambda )}{{(\delta \chi^{2} + 2(\lambda + 1)^{2} )^{2} }} < 0$.

To conclude, we obtain Proposition 1.

1.2 Proof of Proposition 2

1. The derivatives of demand under a free strategy or a paid strategy with respect to $\lambda$ in the case of positive or negative social influence are $\frac{{\partial D^{{SF^{ + *} }} }}{\partial \lambda } = \frac{1}{{2(\lambda - 1)^{2} }} > 0$, $\frac{{\partial D^{{SP^{ + *} }} }}{\partial \lambda } = \frac{1}{{2(\lambda - 1)^{2} }} > 0$, $\frac{{\partial D^{{SF^{ - *} }} }}{\partial \lambda } = - \frac{1}{{2(\lambda + 1)^{2} }} < 0$ and $\frac{{\partial D^{{SP^{ - *} }} }}{\partial \lambda } = - \frac{1}{{2(\lambda + 1)^{2} }} < 0$.

2. The derivatives of the demand under a trial strategy with respect to $\lambda$ in the case of negative or positive social influence are:

$$\begin{gathered} \frac{{\partial D_{1}^{{^{{SM^{ - *} }} }} }}{\partial \lambda } = \frac{{2\chi \left( {2\lambda + \chi + 2} \right)\delta - 4\left( {\lambda + 1} \right)^{2} }}{{\left( {\delta \chi^{2} + 2\left( {\lambda + 1} \right)^{2} } \right)^{2} }} > 0,\quad \frac{{\partial D_{2}^{{^{{SM^{ - *} }} }} }}{\partial \lambda } = \frac{{2\left( {\lambda + 1} \right)^{2} - \chi \left( {4\lambda + \chi + 4} \right)\delta }}{{\left( {\delta \chi^{2} + 2\left( {\lambda + 1} \right)^{2} } \right)^{2} }} < 0, \hfill \\ \frac{{\partial D_{3}^{{^{{SM^{ - *} }} }} }}{\partial \lambda } = \frac{{\delta \chi^{2} - 2\left( {\lambda + 1} \right)^{2} }}{{\left( {\delta \chi^{2} + 2\left( {\lambda + 1} \right)^{2} } \right)^{2} }} < 0. \hfill \\ \end{gathered}$$

$$\begin{gathered} \frac{{\partial D_{1}^{{^{{SM^{ + *} }} }} }}{\partial \lambda } = \frac{{\chi \left( {2\lambda - \chi - 2} \right)\delta + 4\left( {\lambda - 1} \right)^{2} }}{{\left( {\delta \chi^{2} + 2\left( {\lambda - 1} \right)^{2} } \right)^{2} }} < 0,\quad \frac{{\partial D_{2}^{{^{{SM^{ + *} }} }} }}{\partial \lambda } = \frac{{\chi \left( {\chi + 4 - 4\lambda } \right)\delta - 2\left( {\lambda - 1} \right)^{2} }}{{\left( {\delta \chi^{2} + 2\left( {\lambda - 1} \right)^{2} } \right)^{2} }} > 0, \hfill \\ \frac{{\partial D_{2}^{{^{{SM^{ + *} }} }} }}{\partial \lambda } = \frac{{2\left( {\lambda - 1} \right)^{2} - \delta \chi^{2} }}{{\left( {\delta \chi^{2} + 2\left( {\lambda - 1} \right)^{2} } \right)^{2} }}. \hfill \\ \end{gathered}$$

For $\frac{{\partial D_{3}^{{^{{SM^{{ + *}} }} }} }}{\partial \lambda }$, we identify three cases with the constraint $\delta \in \left( {\frac{1 - \lambda }{\chi },\frac{2(1 - \lambda )}{\chi }} \right)$: (i). If $0 < \lambda < 1 - \chi$ we have $\frac{{\partial D_{3}^{{^{{SM^{{ + *}} }} }} }}{\partial \lambda } > 0$. (ii) If $1 - \chi < \lambda < 1 - \frac{\chi }{2}$, we obtain the threshold $\delta_{1} = \frac{{2(\lambda - 1)^{2} }}{{\chi^{2} }}$. When $\delta < \delta_{1}$, we have $\frac{{\partial D_{3}^{{SM^{ + *} }} }}{\partial \lambda } > 0$. When $\delta > \delta_{1}$, we have $\frac{{\partial D_{3}^{{SM^{ + *} }} }}{\partial \lambda } < 0$. (iii) If $1 - \frac{\chi }{2} < \lambda < 1$, we have $\frac{{\partial D_{3}^{{SM^{ + *} }} }}{\partial \lambda } < 0$. To conclude, we obtain Proposition 2.

1.3 Proof of Proposition 3

The derivatives of a subscription fee under a trial strategy with respect to $\lambda$ in the presence of social influences are $\frac{{\partial P^{{SM^{ + *} }} }}{\partial \lambda } = \frac{{2\chi (2(\lambda - 1)^{2} - \delta \chi^{2} )}}{{(\delta \chi^{2} + 2(\lambda - 1)^{2} )^{2} }}$ and $\frac{{\partial P^{{SM^{ - *} }} }}{\partial \lambda } = \frac{{2\chi (\delta \chi^{2} - 2(\lambda + 1)^{2} )}}{{(\delta \chi^{2} + 2(\lambda - 1)^{2} )^{2} }} < 0$. For $\frac{{\partial P^{{SM^{ + *} }} }}{\partial \lambda }$, we identify three cases with the constraint $\delta \in \left( {\frac{1 - \lambda }{\chi },\frac{2(1 - \lambda )}{\chi }} \right)$: (i) If $0 < \lambda < 1 - \chi$ we have $\frac{{\partial P^{{SM^{ + *} }} }}{\partial \lambda } > 0$. (ii) If $1 - \chi < \lambda < 1 - \frac{\chi }{2}$, we obtain the threshold $\delta_{1} = \frac{{2(\lambda - 1)^{2} }}{{\chi^{2} }}$. When $\delta < \delta_{1}$, we have $\frac{{\partial P^{{SM^{ + *} }} }}{\partial \lambda } > 0$. When $\delta > \delta_{1}$, we have $\frac{{\partial P^{{SM^{ + *} }} }}{\partial \lambda } < 0$. (iii) If $1 - \frac{\chi }{2} < \lambda < 1$ we have $\frac{{\partial P^{{SM^{ + *} }} }}{\partial \lambda } < 0$. From the optimal outcomes in Sect. 3.1 and Sect. 3.2, we obtain $P^{NP*} = P^{{SP^{{ + *}} }} = P^{{SP^{ - *} }}$.

To conclude, we obtain Proposition 3.

1.4 Proof of Proposition 4

The derivatives of the quantity of advertising under a trial strategy with respect to $\lambda$ in the case where there is positive or negative social influence are $\frac{{\partial a^{{SM^{ + *} }} }}{\partial \lambda } = \frac{{4\chi^{2} (1 - \lambda )}}{{(\delta \chi^{2} + 2(\lambda - 1)^{2} )^{2} }} > 0$ and $\frac{{\partial a^{{SM^{ - *} }} }}{\partial \lambda } = - \frac{{4\chi^{2} (1 + \lambda )}}{{(\delta \chi^{2} + 2(\lambda + 1)^{2} )^{2} }} < 0$. From the optimal outcomes in Sects. 3.1 and 3.2, we can obtain $a^{NF*} = a^{{SF^{ + *} }} = a^{{SF^{ - *} }}$.

1.5 Proof of Corollary 1

Comparing the revenue under a free strategy with a paid strategy in the presence of social influence, we have:$\Pi^{{SF^{ + *} }} - \Pi^{{SP^{ + *} }} = \frac{\delta - 1}{{4\delta \left( {\lambda - 1} \right)}}$ 和 $\Pi^{{SF^{ - *} }} - \Pi^{{SP^{ - *} }} = \frac{1 - \delta }{{4\delta \left( {\lambda + 1} \right)}}$. Based on the constraint of $\delta \in \left( {\frac{1}{\chi },\frac{2}{\chi }} \right)$ in the presence of no social influence and $\delta \in \left( {\frac{1 + \lambda }{\chi },\frac{2(1 + \lambda )}{\chi }} \right)$ in the presence of negative social influence, we can derive $\delta > 1$. Futhermore, the degree of consumers’ attitudes towards advertising can be derived as $\delta < 1$ and $\delta > 1$ under the condition of $\delta \in \left( {\frac{1 - \lambda }{\chi },\frac{2(1 - \lambda )}{\chi }} \right)$ in the presence of positive social influence. To conclude, we obtain Corollary 1.

1.6 Proof of Corollary 2

Comparing the revenue under a free strategy with a trial strategy in the presence of social influence, we have $\Pi^{{SF^{ + *} }} - \Pi^{{SM^{ + *} }} = \frac{{H\left( \delta \right)_{1} }}{{4\delta \left( {1 - \lambda } \right)\left( {\delta \chi^{2} { + }2\left( {\lambda - 1} \right)^{2} } \right)}}$ and $\Pi^{{SF^{ - *} }} - \Pi^{{SM^{ - *} }} = \frac{{H\left( \delta \right)_{2} }}{{4\delta \left( {1 + \lambda } \right)\left( {\delta \chi^{2} + 2\left( {\lambda + 1} \right)^{2} } \right)}}$. i. There are four cases to consider in function $H\left( \delta \right)_{1} = \left( {4\lambda \chi + \chi^{2} - 4\chi } \right)\delta + 2\left( {\lambda - 1} \right)^{2}$ with the constraint $\delta \in \left( {\frac{1 - \lambda }{\chi },\frac{2(1 - \lambda )}{\chi }} \right)$:

Case 1: For $0 < \lambda < 1 - \frac{\chi }{2}$, $H\left( \delta \right)_{1}$, a decreasing function, we have $H\left( \delta \right)_{1} < 0$.

Case 2: For $1 - \frac{\chi }{2} < \lambda < 1 - \frac{\chi }{3}$, we obtain the threshold $\delta_{2} = \frac{{2\left( {\lambda - 1} \right)^{2} }}{{\chi \left( {4\lambda + \chi - 4} \right)}}$. We have $H\left( \delta \right)_{1} > 0$ when $\delta < \delta_{2}$, and we have $H\left( \delta \right)_{1} < 0$ when $\delta > \delta_{2}$.

Case 3: For $1 - \frac{\chi }{3} < \lambda < 1 - \frac{\chi }{4}$, we can derive $\delta_{2} > \frac{{2\left( {1 - \lambda } \right)}}{\chi }$ and $H\left( \delta \right)_{1}$ is a decreasing function, thus $H\left( \delta \right)_{1} > 0$.

Case 4: For $1 - \frac{\chi }{4} < \lambda < 1$, $H\left( \delta \right)_{1}$ is an increasing function and we have $H\left( \delta \right)_{1} > 0$.

ii. We know $H\left( \delta \right)_{2} = \left( {\chi^{2} - 4\chi - 4\lambda \chi } \right)\delta + 2\left( {\lambda + 1} \right)^{2}$ is a decreasing function and $H\left( \delta \right)_{2} < 0$ in the constraint of $\delta \in \left( {\frac{1 + \lambda }{\chi },\frac{2(1 + \lambda )}{\chi }} \right)$, and, thus, we have $\Pi^{{SF^{ - *} }} < \Pi^{{SM^{ - *} }}$. To conclude, we obtain Corollary 2.

1.7 Proof of Corollary 3

i. Comparing the revenue under a paid strategy with a trial strategy in the presence of positive social influence, we have $\Pi^{{SP^{ + *} }} - \Pi^{{SM^{ + *} }} = \frac{{H(\delta )_{3} }}{{4\delta \left( {1 - \lambda } \right)(\delta \chi^{2} { + }2\left( {\lambda - 1} \right)^{2} )}}$, where $H\left( \delta \right)_{3} = \delta \chi^{2} + 2\left( {\lambda - 1} \right)^{2} + 4\chi \left( {\lambda { + }1} \right)$.

$H\left( \delta \right)_{3}$ is an increasing function. We now identify three cases in the constraint of $\delta \in \left( {\frac{1 - \lambda }{\chi },\frac{2(1 - \lambda )}{\chi }} \right)$:

Case 1: $0 < \lambda < 1 - \frac{3\chi }{2}$, we can easily derive $H\left( \delta \right)_{3} > 0$, and thus we obtain $\Pi^{{SP^{ + *} }} > \Pi^{{SM^{ + *} }}$.

Case 2: $1 - \frac{3\chi }{2} < \lambda < 1 - \chi$, we obtain the threshold $\delta_{3} = \frac{{2\left( {\lambda + 2\chi - 1} \right)\left( {1 - \lambda } \right)}}{{\chi^{2} }}$, if $\delta < \delta_{3}$,$H\left( \delta \right)_{3} < 0$. If $\delta > \delta_{3}$,$H\left( \delta \right)_{3} > 0$.

Case 3: $1 - \chi < \lambda < 1$, we can derive $H\left( \delta \right)_{3} < 0$, and, thus, we obtain $\Pi^{{SP^{ + *} }} < \Pi^{{SM^{ + *} }}$.

ii. Comparing the revenue under a paid strategy with a trial strategy in the presence of negative social influence, we have $\Pi^{{SP^{ - *} }} - \Pi^{{SM^{ - *} }} = \frac{{H(\delta )_{4} }}{{4\delta \left( {1 + \lambda } \right)(\delta \chi^{2} + 2\left( {\lambda + 1} \right)^{2} )}}$, where $H\left( \delta \right)_{4} = \delta \chi^{2} + 2\left( {\lambda + 1} \right)\left( {\lambda + 1 - 2\chi } \right)$.

$H\left( \delta \right)_{4}$ is a monotonic increasing function. We identify two cases with the constraint $\delta \in \left( {\frac{1 + \lambda }{\chi },\frac{2(1 + \lambda )}{\chi }} \right)$:

Case 1: $0 < \lambda < \frac{3\chi }{2} - 1$, we obtain the threshold $\delta_{4} = \frac{{2\left( {1 - \lambda + 2\chi } \right)\left( {1 + \lambda } \right)}}{{\chi^{2} }}$. If $\delta < \delta_{4}$, we can derive $H\left( \delta \right)_{4} < 0$. If $\delta > \delta_{4}$, we can derive $H\left( \delta \right)_{4} > 0$.

Case 2: $\frac{3\chi }{2} - 1 < \lambda < 1$, we can derive $H\left( \delta \right)_{4} > 0$, and, thus, we obtain $\Pi^{{SP^{ - *} }} > \Pi^{{SM^{ - *} }}$.

To conclude, we obtain Corollary 3.