A distinctive early bird price in reward-based crowdfunding

Abstract

Many crowdfunding platforms allow creators maximum flexibility in terms of the prices and rewards offered in a project to gain sufficient capital. Early bird prices, with the original purpose of attracting more early consumers by providing a discounted price for the same product sold in traditional e-commerce retail, are widely used in crowdfunding worldwide and unexpectedly result in “overpay” behaviour. This research aims to explore whether and how creators can use this “overpay” behaviour through dynamic theory with incomplete information, verifying our point of view through empirical analysis. The results show that providing multiple prices for same product generally increases funding performance because backers will balance the surplus and the success rate of their payment decisions to maximize their expected surplus; however, when projects face the low heterogeneity of backer groups, providing a uniform price may be an optimal pricing decision. These findings have direct implications for launching crowdfunding projects that will be more effective in funding more capital by offering reasonable prices.

Appendix

1.1 Uniform pricing strategy

Uniform pricing strategy means that the creator provides one uniform price $p$ for three backers. Given the two-point distribution of product valuations, price $p$ should be $p \le L$ or $L < p \le H$, while funding goal $T$ should be $T \le 3p$.

1.1.1 $p \le L$

As a result of $p \le L$, and $T \le 3p$, the campaign must be successful because backers’ valuations are higher than price $p$. Although funding goal $T \le 3p$, the actual funding must be $3p$. Hence, when $p = L$, the maximum expected campaign revenue under this situation is $EP_{\hbox{max} } = 3L$.

1.1.2 $L < p \le H$

Following the same logic, $L < p \le H$, and $T \le 3p$. In this scenario, the campaign will be successful as long as three backers are high-valuation; hence, the success rate is $\alpha^{3}$. Although the funding goal is $T \le 3p$, the actual funding must be $3p$. Hence, when $p = H$, the maximum expected revenue of a campaign under this situation is $EP_{\hbox{max} } = 3\alpha^{3} H$.

1.2 Early bird pricing strategy

The early bird pricing strategy means that the creator provides two price options ($P_{H} ,P_{L}$) for the same product, and backers make their own decision about which price to pay. The effective prices should satisfy the following conditions: $L < P_{H} \le H$, $P_{L} \le L P_{S} < P_{H} \le LL < P_{S} < P_{H} \le H$. Moreover, funding goal $T$ will be $T \le 3P_{L}$, $T \le 2P_{L} + P_{H}$, $T \le P_{L} + 2P_{H}$ and $T \le 3P_{H}$. As follows.

1.2.1 $T \le 3P_{L}$

Because of $P_{L} \le L < P_{H} \le H$ and $T \le 3P_{L}$, the same as the uniform pricing strategy with a low price, this approach must be successful with success rate 1. Although the funding goal is $T \le 3P_{L}$, the actual funding must be $3P_{L}$. Hence, when $P_{L} = L$ and $L < P_{H} \le H$ (condition ② of Fig. 4), the maximum expected revenue of the campaign under this situation is $EP_{max} = 3L$.

Fig. 4

The logic of backers’ decisions

1.2.2 $T \le 2P_{L} + P_{H}$

Because of $T \le 2P_{L} + P_{H}$, the campaign will be successful as long as one backer pays high price $P_{H}$; the figure above shows the behaviour logic under $3P_{L} < T \le 2P_{L} + P_{H}$.

(1) Condition ③

Under $\left\{ {\begin{array}{*{20}c} {\left( {H - P_{H} } \right) \ge \left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$, high-valuation backer $B_{1}$ with probability $\alpha$ will think that if $B_{1}$ pays $P_{H}$, the campaign must succeed with conditional success rate 1 and surplus $H - P_{H}$, and therefore, the conditional expected surplus of paying high price $P_{H}$ is $H - P_{H}$; however, if $B_{1}$ pay the low price $P_{L}$, because of $\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)$, as long as one of the remaining two backers is a high-valuation backer, the campaign will succeed, hence, the conditional success rate of paying low price is $2\alpha - \alpha^{2}$, with surplus $H - P_{L}$. Therefore, the conditional expected surplus of paying low price $P_{L}$ is $\left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)$; thus, under $\left\{ {\begin{array}{*{20}c} {\left( {H - P_{H} } \right) \ge \left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$ (feasible region DEF of Fig. 5), high-valuation backer $B_{1}$ would pay the high price. In addition, if $B_{1}$ pay the low price, because of $\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)$, high-valuation backer $B_{2}$ will also pay the high price. Moreover, if both $B_{1}$ and $B_{2}$ pay the low price, high-valuation backer $B_{3}$ will pay the high price because $H - P_{H} > 0$.

Fig. 5

Analysis of optimal prices under $T \le 2P_{L} + P_{H}$

Therefore, for the creator, the conditional success rate of the creator’s price decision is $\left( {3\upalpha - 3\alpha^{2} + \alpha^{3} } \right)$, and the conditional expected revenue of $T \le 2P_{L} + P_{H}$ and $\left\{ {\begin{array}{*{20}c} {\left( {H - P_{H} } \right) \ge \left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$ is $\left( {3\upalpha - 3\alpha^{2} + \alpha^{3} } \right)\left( {2P_{L} + P_{H} } \right)$. When the prices are $P_{L} = L - \varepsilon ,P_{H} = \left( {1 -\upalpha} \right)^{2} H + \left( {2\upalpha - \alpha^{2} } \right){\text{L}} - \epsilon$, (see where $\varepsilon$ and $\epsilon$ are bigger than (but close to) 0, the expected revenue can be simplified as $EP_{\hbox{max} } = \left( {3\alpha - 3\alpha^{2} + \alpha^{3} } \right)[\left( {1 - \alpha )^{2} H + \left( {2 + 2\alpha - \alpha^{2} } \right)L} \right]$).

(2) Condition ⑤

Under $\left\{ {\begin{array}{*{20}c} {\left( {H - P_{H} } \right) < \left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$, high-valuation backer $B_{1}$ with probability $\alpha$ would think that if $B_{1}$ pays $P_{H}$, the campaign must succeed with conditional success rate 1 and surplus $H - P_{H}$, therefore, the conditional expected surplus of paying high price $P_{H}$ is $H - P_{H}$; if $B_{1}$ pays the low price $P_{L}$, because of $\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)$, as long as one of the remaining two backers is high valuation, the campaign will succeed, hence, the conditional success rate of paying the low price is $2\alpha - \alpha^{2}$ with surplus $H - P_{L}$; therefore, the conditional expected surplus of paying low price $P_{L}$ is $\left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)$. Accordingly, because of $\left( {H - P_{H} } \right) < \left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)$, high-valuation backer $B_{1}$ will pay the low price, but for $\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)$ and $H - P_{H} > 0$, the campaign will succeed as long as one of $B_{2}$ and $B_{3}$ is a high-valuation backer, hence; for the creator, the conditional success rate of the creator’s price decision is $\left( {2\alpha - \alpha^{2} } \right)$ and the conditional expected revenue of $T \le 2P_{L} + P_{H}$ and $\left\{ {\begin{array}{*{20}c} {\left( {H - P_{H} } \right) < \left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$ (feasible region CDFGI of Fig. 5) is $\left( {2\alpha - \alpha^{2} } \right)\left( {2P_{L} + P_{H} } \right)$. When the prices are $P_{L} = L - \varepsilon$, $P_{H} = \left( {1 - \alpha } \right)H + \alpha {\text{L}} - \epsilon$, where $\varepsilon$ and $\epsilon$ are bigger than 0, but close to 0, the expected revenue can be simplified as $EP_{\hbox{max} } = \left( {2\alpha - \alpha^{2} } \right)\left[ {\left( {1 - \alpha } \right)H + \left( {2 + \alpha } \right)L} \right]$.

(3) Condition ⑦

Under $\left\{ {\begin{array}{*{20}c} {H - P_{H} > 0} \\ {\left( {H - P_{H} } \right) < \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$, both $B_{1}$ and $B_{2}$ will not pay the high price and the campaign will succeed where only backer $B_{3}$ is a high-valuation backer with success rate $\alpha$. Hence, the conditional expected revenue of $T \le 2P_{L} + P_{H}$ and $\left\{ {\begin{array}{*{20}c} {H - P_{H} > 0} \\ {\left( {H - P_{H} } \right) < \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$ (feasible region ABCI) is $\alpha$. When the prices are $P_{L} = L - \varepsilon$, $P_{H} = H - \epsilon$, where $\varepsilon$ and $\epsilon$ are bigger than 0, but close to 0, the expected revenue can be simplified as $EP_{\hbox{max} } = \alpha \left( {H + 2L} \right)$.

1.2.3 $T \le P_{L} + 2P_{H}$

Because of $T \le 2P_{L} + P_{H}$, the campaign will be successful at least two backers pay high price $P_{H}$; the figure above shows the behaviour logic under $P_{H} + 2P_{L} < T \le 2P_{H} + P_{L}$.

(1) Condition ④

Under $\left\{ {\begin{array}{*{20}c} {\left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{H} } \right) \ge \alpha^{2} \left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$, high-valuation backer $B_{1}$ with probability $\alpha$ will think that if $B_{1}$ pays the $P_{H}$, because of $\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)$, as long as one of the remaining two backers is high valuation, the campaign will succeed; hence, the conditional success rate of paying the low price is $2\alpha - \alpha^{2}$, with surplus $H - P_{H}$. However, if $B_{1}$ pays the $P_{L}$, the campaign will succeed only if both of the remaining backers are high valuation; hence, the conditional success rate of paying the low price is $\alpha^{2}$, with surplus $H - P_{L}$. Because of $\left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{H} } \right) \ge \alpha^{2} \left( {H - P_{L} } \right)$, high-valuation backer $B_{1}$ will pay the high price $P_{H}$. In addition, if $B_{1}$ pays the $P_{H}$, because of $\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)$, high-valuation backer $B_{2}$ will also pay the high price $P_{H}$. For the creator, the conditional success rate of $\left\{ {\begin{array}{*{20}c} {\left( {2\alpha - \alpha^{2} } \right)\left( {H - P_{H} } \right) \ge \alpha^{2} \left( {H - P_{L} } \right)} \\ {\left( {H - P_{H} } \right) \ge \alpha \left( {H - P_{L} } \right)} \\ \end{array} } \right.$ and $P_{H} + 2P_{L} < T \le 2P_{H} + P_{L}$ (feasible region CDEF of Fig. 6 below) is $3\alpha^{2} - 2\alpha^{3}$; Table 5 details our success rate analysis. When the prices are $P_{L} = L - \varepsilon$, $P_{H} = \left( {1 - \alpha } \right)H + \alpha {\text{L}} - \epsilon$, where $\varepsilon$ and $\epsilon$ are bigger than (but close to) 0, the expected revenue can be simplified as $EP_{max} = \left( {3\alpha^{2} - 2\alpha^{3} } \right)\left[ {2\left( {1 -\upalpha} \right)H + \left( {2\upalpha + 1} \right){\text{L}}} \right]$.

Fig. 6

Analysis of optimal price under $T \le P_{L} + 2P_{H}$

Table 5 Success rate

(2) Condition ⑥

If high-valuation backer $B_{1}$ is willing to pay the high price (the feasible region ABCF of Fig. 6), for the creator, the success rate of the pricing decision is $\alpha^{2}$, and the campaign will succeed only if both $B_{2}$ and $B_{3}$ are high valuation. Hence, when the prices are $P_{L} = L - \varepsilon$, $P_{H} = H - \epsilon$, where $\varepsilon$ and $\epsilon$ are bigger than (but close to) 0, the expected revenue can be simplified as $EP_{\hbox{max} } =\upalpha\left[ {2H + L} \right]$.

1.2.4 $T \le 3P_{H}$

Because of $P_{L} \le L < P_{H} \le H$ and $P_{L} + 2P_{H} < T \le 3P_{H}$, the same as the uniform pricing strategy with the high price, the campaign will succeed when three backers are high-valuation with success rate $\alpha^{3}$. Although the funding goal is $T \le 3P_{L}$, the actual funding must be $3P_{H}$. Hence, when $P_{L} \le L$ and $P_{H} = H$ (Condition ① of Fig. 4), the conditional expected revenue of the campaign under this situation is $EP_{\hbox{max} } = 3\alpha^{3}$ (Table 6).

Table 6 The expected revenues and price(s) of different pricing strategies