Competitive crowdfunding under asymmetric quality information

Abstract

We proposed a framework on how to design the optimal crowdfunding strategy for the competitive creators under quality information asymmetry. And we also considered the platform uses different funding mechanisms (fixed or flexible) and the creators’ different orders of initiation (lead or follow). We find that when high-quality creator initiates first and the degree of quality information asymmetry is low, the optimal price of low-quality creator is negatively correlated with the quality utility and success rate of his own product, but positively correlated with the quality utility and success rate of his competitor’s product. More interestingly, we also show that it is more conducive to maximize the profits of creators to retain quality information asymmetry when high-quality creator initiates crowdfunding first. We also find that when the degree of quality information asymmetry between creators and contributors is high, it is always better that the high-quality creator initiates crowdfunding first, whereas it is better for the low-quality creator to initiate first if the of degree of quality information asymmetry between creators and contributors is low. Finally, when the degree of quality information asymmetry between creators and contributors is very high or very low, the two creators and the platform can obtain higher profits by a fixed mechanism; otherwise, the flexible mechanism is more beneficial to them.

Appendices

Appendix

The proof of Proposition 1

The conclusion of Proposition 1 comes from the equilibrium solution of Stackelberg-game between high-Quality creator and low-quality creator when the high-quality creator first publishes his crowdfunding program. We use the backward induction method to solve the game.

First, for the following low-quality creator, the aim is to maximize his profit:

$$ \max \pi_{L} \left( {p_{L} } \right) = \left\{ {\begin{array}{*{20}l} {Np_{L} \left( {1 - \gamma } \right) - S_{L} p_{H} > \overline{{p\left( {p_{L} } \right)}}_{1} } \\ {\left( {1 - \alpha } \right) Np_{L} \left( {1 - \gamma } \right) - S_{L} { }p_{H} \le \overline{{p\left( {p_{L} } \right)}}_{1} } \\ \end{array} } \right. $$

Note ${ }p_{LH} > p_{LL}$,which are all the prices can be set by the following low-quality creator, and $p_{H} > \overline{{p\left( {p_{LL} } \right)}}_{1}$, $p_{H} \le \overline{{p\left( {p_{LH} } \right)}}_{1}$. So, the low-quality creator has two choose: sell to uninformed contributors at a higher price $p_{LH} { }$ or to all contributors at a lower price ${p}_{LL}$. Obviously, if an equilibrium solution exists, the low-quality creator can only sell to a part of contributors but not all of them, and his price can only be $p_{LH}$, and his profit function is:

$$ \begin{aligned} & \max \pi_{L} \left( {p_{LH} } \right) = \left( {1 - \alpha } \right){ }Np_{LH} \left( {1 - \gamma } \right) - S_{L} \\ & s.t. p_{LH} \le p_{H} \\ & p_{LH} \le V + u_{E} \\ & p_{H} \le \overline{{p\left( {p_{LH} } \right)}}_{1} \\ \end{aligned} $$

(4)

Obviously, the higher the price $p_{LH}$, the greater the low-quality creator’s profit. The optimal price of the low-quality creator is $p_{L}^{*} = \max \left( {p_{LH} } \right) = \max \left( {p_{H} , V + u_{E} } \right)$.

Then, for the high-quality creator who first publishes crowdfunding, his purpose is to maximize his profit without being squeezed out of the market. In order not to be squeezed out of the market, his price need to meet the constraints as below:

• (1) The profit of the low-quality creator with the high price $p_{LH}$ is greater than that with the low price $p_{LL}$, i.e.

  $$ p_{LL} N\left( {1 - \gamma } \right) - S_{L} \le \left( {1 - \alpha } \right){ }Np_{LH} \left( {1 - \gamma } \right) - S_{L} $$

  After simplification, the following equation can be obtained:

  $$ p_{LL} \le \left( {1 - \alpha } \right){ }p_{LH} $$

• (2) The high-quality creator can get the support of informed contributors, while the low-quality creator will not get all the support from contributors by lowering $p_{LH}$, as shown in Eq. (3):

  $$ p_{H} \le \overline{{p\left( {p_{LL} } \right)}}_{1} $$

Constraint (2) is actually a supplement to constraint (1). Combining these two constraints can ensure that the profits of the low-quality creator get from the whole market by lowering the price are not as good as selling to some contributors at a higher price. This is because, when the $p_{H}$ satisfies the constraint (2), the low-quality creator must set a price lower than $p_{LL}$ in order to obtain all the markets. Then, even if he obtains all the markets, according to constraint (1), his profit is lower than that of selling to uninformed contributors at the price of $p_{LH}$. Under such constraints, the low-quality creator's maximum profit can only come from maximizing $p_{LH}$, i.e. $p_{L}^{*} = \max \left( {p_{LH} } \right) = \max \left( {p_{H} , V + u_{E} } \right)$. And the high-quality creator can take $p_{L}^{*}$ as the higher price $p_{LH}$ when making decision. So, we can rewrite constraint (1) as

$$ p_{LL} \le \left( {1 - \alpha } \right)p_{L}^{*} $$

In addition, to gain contributors’ support, the creators' prices need to meet the following conditions:

$$ p_{H} \le V + u_{H} ; $$

So, the subjective function of the high-quality creator is:

$$ \begin{aligned} & max \pi_{H} { }\left( {p_{H} } \right) = \alpha N p_{H} \left( {1 - \gamma } \right) - S_{H} \\ & s.t. p_{H} \le V + u_{H} \\ & p_{LL} \le \left( {1 - \alpha } \right)p_{L}^{*} \\ & p_{H} \le \overline{{p\left( {p_{LL} } \right)}}_{1} \\ \end{aligned} $$

(5)

As shown in Eqs. (4) and (5), we transform the complex Stackelberg-game problem into two relatively simple linear programming problem by backward induction. By introducing $p_{L}^{*} = \max \left( {p_{LH} } \right) = \max \left( {p_{H} , V + u_{E} } \right)$ Into Eq. (5), the linear programming problem can be solved, we can get the optimal price of the high-quality creator:

$$ p_{H}^{*} = \left\{ {\begin{array}{*{20}l} {V + u_{H} } \hfill & {{\upalpha } \le \alpha_{2} } \hfill \\ {V + u_{H} - \frac{{x_{L} }}{{x_{H} }}\left[ {V + u_{L} - \left( {1 - \alpha } \right)\left( {V + u_{E} } \right)} \right]} \hfill & {\alpha_{2} < {\upalpha } \le \alpha_{1} } \hfill \\ {\frac{1}{{x_{H} - \left( {1 - \alpha } \right)x_{L} }}\left[ {x_{H} \left( {V + u_{H} } \right) - x_{L} \left( {V + u_{L} } \right)} \right]} \hfill & { {\upalpha } > \alpha_{1} } \hfill \\ \end{array} } \right. $$

Then, from $p_{L}^{*} = \max \left( {p_{LH} } \right) = \max \left( {p_{H} , V + u_{E} } \right)$, we can obtain the optimal price of the low-quality creator:

$$ p_{L}^{*} = \left\{ {\begin{array}{*{20}l} {V + u_{E} } \hfill & {{\upalpha } \le \alpha_{1} } \hfill \\ {\frac{1}{{x_{H} - \left( {1 - \alpha } \right)x_{L} }}\left[ {x_{H} \left( {V + u_{H} } \right) - x_{L} \left( {V + u_{L} } \right)} \right] - \varepsilon } \hfill & {{\upalpha } > \alpha_{1} } \hfill \\ \end{array} } \right. $$

where $u_{E} = \beta_{0} u_{H} + \left( {1 - \beta_{0} } \right)u_{L}$, $\alpha_{1} = 1 - \frac{{x_{L} \left( {V + u_{L} } \right) - x_{H} \left( {u_{H} - u_{E} } \right)}}{{x_{L} \left( {V + u_{E} } \right)}}$, $\alpha_{2} = 1 - \frac{{V + u_{L} }}{{V + u_{E} }}$, and $\varepsilon$ is a constant that makes $p_{L} < p_{H}$.

Therefore, the Proposition 1 can be proved.

The proof of Proposition 3

The conclusion of Proposition 3 comes from the equilibrium solution of Stackelberg-game between high-quality creator and low-quality creator when the low-quality creator first publishes his crowdfunding program. We also use the backward induction method to solve the game.

First, for the following high-quality creator, his aim is to maximize his profit:

$$ {\text{max}}\pi_{H} \left( {p_{H} } \right) = \left\{ {\begin{array}{*{20}l} {0{ }} \hfill & {p_{H} > \overline{{p\left( {p_{L} } \right)}}_{1} } \hfill \\ {\alpha N{ }p_{H} \left( {1 - \gamma } \right) - S_{H} } \hfill & {p_{H} \le \overline{{p\left( {p_{L} } \right)}}_{1} } \hfill \\ {N{ }p_{H} \left( {1 - \gamma } \right) - S_{H} } \hfill & {p_{H} \le p_{L} { } \le \overline{{p\left( {p_{L} } \right)}}_{1} } \hfill \\ \end{array} } \right. $$

Note ${ }p_{HH} > p_{HL}$, which are all the prices can be set by the following high-quality creator, and $p_{L} < { }p_{HH} \le \overline{{p\left( {p_{L} } \right)}}_{1}$, $p_{HL} \le p_{L} { } \le \overline{{p\left( {p_{L} } \right)}}_{1}$. So, the high-quality creator has two strategies: sell to informed contributors at a higher price $p_{HH} { }$ or to all contributors at a lower price ${ }p_{HL}$. Obviously, if an equilibrium solution exists, the high-quality creator can only sell to a part of contributors but not all of them, his price can only be $p_{HH}$, and his subjective function is:

$$ \begin{aligned} & {\text{max}}\pi_{H} \left( {p_{HH} } \right) = \alpha N{ }p_{H} \left( {1 - \gamma } \right) - S_{H} \\ & p_{HH} \le V + u_{H} \\ & p_{HH} \le \overline{{p\left( {p_{L} } \right)}}_{1} \\ \end{aligned} $$

(7)

Obviously, the higher the price $p_{HH}$, the greater the high-quality creator’s profits. The optimal price of the high-quality creator is $p_{H}^{*} = \max \left( {p_{HH} } \right) = \max \left( {\overline{{p\left( {p_{L} } \right)}}_{1} , V + u_{H} } \right)$.

Then, for the low-quality creator who first publishes crowdfunding, his purpose also is to maximize his profit without being squeezed out of the market. In order not to be squeezed out of the market, his price need to meet the constraints as below:

• (1) The profit of high-quality creator with the high price $p_{HH}$ is greater than that with the low price $p_{HL}$, i.e.

  $$ p_{HL} N\left( {1 - \gamma } \right) - S_{H} \le \alpha Np_{HH} \left( {1 - \gamma } \right) - S_{H} $$

  After simplification, the following equation can be obtained:

  $$ p_{HL} \le \alpha { }p_{HH} $$

• (2) The high-quality creator can get the support of informed contributors with the higher price $p_{HH}$, as shown in Eq. (1):

  $$ p_{HH} \le \overline{{p\left( {p_{L} } \right)}}_{1} $$

• (3) The high-quality creator will not get all the support from contributors by lowering $p_{HH}$:

  $$ p_{L} \le p_{HL} $$

Under such constraints, the high-quality creator's maximum profit can only come from maximizing $p_{HH}$, i.e. $p_{H}^{*} = \max \left( {p_{HH} } \right) = \max \left( {\overline{{p\left( {p_{L} } \right)}}_{1} , V + u_{H} } \right)$. And the low-quality creator can take $p_{L}^{*}$ as the higher price $p_{LH}$ when making decision. So, we can rewrite constraint (1) and constraint (2) as

$$ \begin{aligned} & p_{HL} \le \alpha { }p_{H}^{*} \\ & p_{H}^{*} \le \overline{{p\left( {p_{L} } \right)}}_{1} \\ \end{aligned} $$

In addition, to gain contributors’ support, the creators' prices need to meet the following conditions:

$$ p_{L} \le V + u_{E} $$

So, the subjective function of the low-quality creator is:

$$ \begin{aligned} & {\text{max}}\pi_{L} \left( {p_{L} } \right) = \left( {1 - \alpha } \right) Np_{L} \left( {1 - \gamma } \right) - S_{L} \\ & s.t.{ }p_{L} \le p_{HL} \\ & p_{L} \le V + u_{E} \\ & p_{HL} \le \alpha p_{H}^{*} \\ & p_{H}^{*} \le \overline{{p\left( {p_{L} } \right)}}_{1} \\ \end{aligned} $$

(9)

As shown in Eqs. (7) and (9), we transform the complex Stackelberg-game problem into two relatively simple linear programming problem by backward induction. By introducing $p_{H}^{*} = \max \left( {p_{HH} } \right) = \max \left( {\overline{{p\left( {p_{L} } \right)}}_{1} , V + u_{H} } \right)$ into Eq. (9), the linear programming problem can be solved, we can get the optimal price of the low-quality creator:

$$ p_{L}^{*} = \left\{ {\begin{array}{*{20}l} {\frac{\alpha }{{x_{H} - \alpha x_{L} }}\left[ {x_{H} \left( {V + u_{H} } \right) - x_{L} \left( {V + u_{L} } \right)} \right] } \hfill & {\alpha < \alpha_{3} } \hfill \\ {\alpha \left( {V + u_{H} } \right)} \hfill & {\alpha_{3} \le \alpha < \alpha_{4} } \hfill \\ {V + u_{E} } \hfill & { \alpha \ge \alpha_{4} } \hfill \\ \end{array} } \right. $$

Then, from $p_{H}^{*} = \max \left( {p_{HH} } \right) = \max \left( {\overline{{p\left( {p_{L} } \right)}}_{1} , V + u_{H} } \right)$, we can obtain the optimal price of the high-quality creator:

$$ p_{H}^{*} = \left\{ {\begin{array}{*{20}l} {\frac{1}{{x_{H} - \alpha x_{L} }}\left[ {x_{H} \left( {V + u_{H} } \right) - x_{L} \left( {V + u_{L} } \right)} \right] } \hfill & {\alpha < \alpha_{3} } \hfill \\ {V + u_{H} } \hfill & {\alpha_{3} \le \alpha < \alpha_{4} } \hfill \\ {V + u_{H} } \hfill & {\alpha \ge \alpha_{4} } \hfill \\ \end{array} } \right. $$

where $u_{E} = \beta_{0} u_{H} + \left( {1 - \beta_{0} } \right)u_{L}$, $\alpha_{3} = \frac{{V + u_{L} }}{{V + u_{H} }}$, and $\alpha_{4} = \frac{{x_{H} \left( {V + u_{E} } \right)}}{{x_{H} \left( {V + u_{H} } \right) + x_{L} \left( {u_{E} - u_{L} } \right)}}$.

Therefore, the Proposition 3 can be proved.

The proof of Proposition 5

AS same as the Proposition 1, the conclusion of Proposition 5 comes from the equilibrium solution of Stackelberg-game between high-quality creator and low-quality creator when the high-quality creator first publishes his crowdfunding program.

The only difference is that Proposition 1 get the conclusions in a platform which use a fixed funding mechanism, and the Proposition 5 get the conclusions in a platform which adopted the flexible mechanism, the difference is that the equations in solving reflected in (3) into (10) in the form of the rest of the proof process are the same, so we omit this part of the duplicate certificate content.

The proof of Proposition 7

AS same as the Proposition 3, the conclusion of Proposition 7 comes from the equilibrium solution of Stackelberg-game between high-quality creator and low-quality creator when the high-quality creator first publishes his crowdfunding program.

The only difference is that Proposition 3 get the conclusions in a platform which use a fixed funding mechanism, and the Proposition 7 get the conclusions in a platform which adopted the flexible mechanism, the difference is that the equations in solving reflected in (3) into (10) in the form of the rest of the proof process are the same, so we omit this part of the duplicate certificate content.