Optimal sequential releasing strategy for software products in the presence of word-of-mouth and requirements uncertainty

Abstract

This study proposes a two-period analytical model to explore the relative optimality of three sequential releasing strategies for a software vendor: (1) Skipping Strategy—skip the limited-functionality version and introduce a full-functionality version in the next period, (2) Replacement Strategy—release a limited-functionality version first and replace it with a full-functionality version later, and (3) Line-extension Strategy—release a limited-functionality version first and extend the product line by releasing a full-functionality version later. Word-of-mouth (WOM) effect and uncertainty in consumers’ requirements are taken into consideration. Our analysis shows that Skipping Strategy is optimal when the net WOM is negative and sufficiently small, while the Replacement Strategy becomes the optimal choice when the net WOM is negative but not sufficiently small. However, the Line-extension Strategy dominates other two strategies when the net WOM is positive. Different pricing patterns may be chosen when adopting Line-extension Strategy, i.e. when the net WOM is positive and not very large, Low–High pricing pattern (a relatively low price for the limited-functionality version and a relatively high price for the full-functionality version) is optimal, while High–Low pricing pattern (a relatively high price for the limited-functionality version and a relatively low price for the full-functionality version) becomes optimal when the net WOM is positive and sufficiently large. Numerical experiments show that the software vendor could improve its profit by choosing optimal quality design but the overall dominance pattern of the three release strategies is similar no matter whether the quality design is exogenously given or optimally chosen.

Appendix

1.1 A1: Proof of Proposition 1

• (I) Firstly, we give the proof of the local optimum RS-I, corresponding to type-I market segmentation in Fig. 4

• (II) In this case, $0 < \theta_{f}^{RS {\text{-}} I} = \theta_{l}^{RS {\text{-}} I} < \theta_{u}^{RS {\text{-}} I} < 1$, $\theta_{f}^{RS {\text{-}} I} = \theta_{l}^{RS {\text{-}} I} = \frac{{p_{l}^{RS {\text{-}} I} }}{{q_{l} }}$, $\theta_{u}^{RS {\text{-}} I} = \frac{{p_{f}^{RS {\text{-}} I} - W_{N} }}{{q_{f} }}$, $\pi^{RS {\text{-}} I} = \left( {1 - \theta_{u}^{RS {\text{-}} I} } \right)p_{f}^{RS {\text{-}} I} + \left( {\theta_{u}^{RS {\text{-}} I} - \theta_{l}^{RS {\text{-}} I} } \right)p_{l}^{RS {\text{-}} I} = \left( {1 - \frac{{p_{f}^{RS {\text{-}} I} - W_{N} }}{{q_{f} }}} \right)p_{f}^{RS {\text{-}} I} + \left( {\frac{{p_{f}^{RS {\text{-}} I} - W_{N} }}{{q_{f} }} - \frac{{p_{l}^{RS {\text{-}} I} }}{{q_{l} }}} \right)p_{l}^{RS {\text{-}} I}$. Let $\varvec{p} = \left( {p_{f}^{RS {\text{-}} I} ,p_{l}^{RS {\text{-}} I} } \right)$, then $H\left( \varvec{p} \right) = \left[ {\begin{array}{*{20}c} { - \frac{2}{{q_{f} }}} & {\frac{1}{{q_{f} }}} \\ {\frac{1}{{q_{f} }}} & { - \frac{2}{{q_{l} }}} \\ \end{array} } \right]$, $D_{1} \left( H \right) = - \frac{2}{{q_{f} }} < 0$, $D_{2} \left( H \right) = \frac{{4q_{f} - q_{l} }}{{q_{f}^{2} q_{l} }} > 0$, thus the profit function of RS-I is concave. The first-order conditions of $\pi^{RS {\text{-}} I}$ with respect to $p_{f}^{RS {\text{-}} I}$ and $p_{l}^{RS {\text{-}} I}$ are $\left\{ {\begin{array}{*{20}l} {\frac{{\partial \pi^{RS {\text{-}} I} }}{{\partial p_{f}^{RS {\text{-}} I} }} = \frac{{q_{f} + W_{N} - 2p_{f}^{RS {\text{-}} I} + p_{l}^{RS {\text{-}} I} }}{{q_{f} }} = 0} \hfill \\ {\frac{{\partial \pi^{RS {\text{-}} I} }}{{\partial p_{l}^{RS {\text{-}} I} }} = \frac{{ - q_{l} W_{N} + q_{l} p_{f}^{RS {\text{-}} I} - 2q_{f} p_{l}^{RS {\text{-}} I} }}{{q_{f} q_{l} }} = 0} \hfill \\ \end{array} } \right.$, and then $\left\{ {\begin{array}{*{20}c} {p_{f}^{RS {\text{-}} I*} = \frac{{\left( {2q_{f} - q_{l} } \right)W_{N} + 2q_{f}^{2} }}{{4q_{f} - q_{l} }}} \\ {p_{l}^{RS {\text{-}} I*} = \frac{{q_{l} \left( {q_{f} - W_{N} } \right)}}{{4q_{f} - q_{l} }}} \\ \end{array} } \right.$. The optimal segmenting point of different versions are $\theta_{u}^{RS {\text{-}} I*} = \frac{{2\left( {q_{f} - W_{N} } \right)}}{{4q_{f} - q_{l} }}$ and $\theta_{l}^{RS {\text{-}} I*} = \frac{{q_{f} - W_{N} }}{{4q_{f} - q_{l} }}$, satisfying the constraint $0 < \theta_{l}^{RS {\text{-}} I} < \theta_{u}^{RS {\text{-}} I} < 1$. Therefore, The number of all types of consumers are $D_{LU}^{RS {\text{-}} I*} = 1 - \frac{{2\left( {q_{f} - W_{N} } \right)}}{{4q_{f} - q_{l} }}$, $D_{LN}^{RS {\text{-}} I*} = \theta_{u}^{RS {\text{-}} I*} - \theta_{l}^{RS {\text{-}} I*} = \frac{{q_{f} - W_{N} }}{{4q_{f} - q_{l} }}$, $D_{NF}^{RS {\text{-}} I*} = 0$ and $D_{Total}^{RS {\text{-}} I*} = D_{LU}^{RS {\text{-}} I*} + D_{LN}^{RS {\text{-}} I*} + D_{NF}^{RS {\text{-}} I*} = 1 - \frac{{q_{f} - W_{N} }}{{4q_{f} - q_{l} }}$. Finally, the optimal total profit of Scheme RS-I is $\pi^{RS {\text{-}} I*} = \frac{{W_{N}^{2} + \left( {2q_{f} - q_{l} } \right)W_{N} + q_{f}^{2} }}{{4q_{f} - q_{l} }}$.

  According to the constraint $p_{f} \ge p_{l} > 0$, $p_{f}^{RS {\text{-}} I*} - p_{l}^{RS {\text{-}} I*} = \frac{{q_{f} \left( {2q_{f} - q_{l} + 2W_{N} } \right)}}{{4q_{f} - q_{l} }} > 0$, then we have $W_{N} > \frac{1}{2}q_{l} - q_{f}$, and $p_{l}^{RS {\text{-}} I*} = \frac{{q_{l} \left( {q_{f} - W_{N} } \right)}}{{4q_{f} - q_{l} }} > 0$ is obvious. Because $- q_{l}^{2} < W_{N} < q_{l}$, we need to examine which is larger between $\frac{1}{2}q_{l} - q_{f}$ and $- q_{l}^{2}$. If $\frac{1}{2}q_{l} - q_{f} < - q_{l}^{2}$ (namely $q_{l} < A = \frac{{ - 1 + \sqrt {1 + 16q_{f} } }}{4}$), then $p_{f}^{RS {\text{-}} I*} > p_{l}^{RS {\text{-}} I*}$. But if $\frac{1}{2}q_{l} - q_{f} \ge - q_{l}^{2}$ (namely $q_{l} \ge A$), then $p_{f}^{RS {\text{-}} I*} > p_{l}^{RS {\text{-}} I*}$ when $W_{N} > \frac{1}{2}q_{l} - q_{f}$, but $p_{f}^{RS {\text{-}} I*} = p_{l}^{RS {\text{-}} I*}$ when $W_{N} \le \frac{1}{2}q_{l} - q_{f}$. If $p_{f}^{RS {\text{-}} I*} = p_{l}^{RS {\text{-}} I*}$, the software vendor will not earn any profit in the second period but just earn profit from the first period, namely $\pi^{RS {\text{-}} I} = \left( {1 - \theta_{l}^{RS {\text{-}} I} } \right)p_{l}^{RS {\text{-}} I}$. In this case, the optimal prices for both versions are $p_{l}^{RS {\text{-}} I*} = p_{f}^{RS {\text{-}} I*} = \frac{{q_{l} }}{2}$. $\theta_{u} = \frac{{\frac{{q_{l} }}{2} - W_{N} }}{{q_{f} }} \ge 1$. Namely, no consumers upgrade in the second period. All types of demand are $D_{LN}^{RS {\text{-}} I*} = \frac{1}{2}$, $D_{LU}^{RS {\text{-}} I*} = D_{NF}^{RS {\text{-}} I*} = 0$ and $D_{Total}^{RS {\text{-}} I*} = \frac{1}{2}$. The optimal profit of the software vendor is $\pi^{RS {\text{-}} I*} = \frac{{q_{l} }}{4}$.

• (III) Now we give the proof of the local optimum RS-II, corresponding to type-II market segmentation in Fig. 4

• (IV) In this case, $0 < \theta_{f}^{RS {\text{-}} II} < \theta_{l}^{RS {\text{-}} II} = \theta_{u}^{RS {\text{-}} II} < 1$, $\theta_{f}^{RS {\text{-}} II} = \frac{{p_{f}^{RS {\text{-}} II} - W_{N} }}{{q_{f} }},$$\pi^{RS {\text{-}} II} = \left( {1 - \theta_{u}^{RS {\text{-}} II} } \right)p_{f}^{RS {\text{-}} II} + \left( {\theta_{l}^{RS {\text{-}} II} - \theta_{f}^{RS {\text{-}} II} } \right)p_{f}^{RS {\text{-}} II} = \left( {1 - \frac{{p_{f}^{RS {\text{-}} II} - W_{N} }}{{q_{f} }}} \right)p_{f}^{RS {\text{-}} II}$. The first- order and second-order conditions of $\pi^{RS {\text{-}} II}$ with respect to $p_{f}^{RS {\text{-}} II}$ are $\frac{{\partial \pi^{RS {\text{-}} II} }}{{\partial p_{f}^{RS {\text{-}} II} }} = \frac{{q_{f} + W_{N} - 2p_{f}^{RS {\text{-}} II} }}{{q_{f} }}$ and $\frac{{\partial^{2} \pi^{RS {\text{-}} II} }}{{\partial \left( {p_{f}^{RS {\text{-}} II} } \right)^{2} }} = - \frac{2}{{q_{f} }} < 0$, meaning that $\pi^{RS {\text{-}} II}$ is concave. Let $\frac{{\partial \pi^{RS {\text{-}} II} }}{{\partial p_{f}^{RS {\text{-}} II} }} = 0$, we get the optimal price of the full-functionality version $p_{f}^{RS {\text{-}} II*} = \frac{{q_{f} + W_{N} }}{2}$. The indifferent point of the consumer between buying the full-functionality version and nothing is $\theta_{f}^{RS {\text{-}} II*} = \frac{{q_{f} - W_{N} }}{{2q_{f} }}$, which is between 0 and 1. So the resulting demand of the full-functionality version is $D_{F}^{RS {\text{-}} II*} = D_{LU}^{RS {\text{-}} II*} + D_{NF}^{RS {\text{-}} II*} = 1 - \theta_{f}^{RS {\text{-}} II*} = \frac{{q_{f} + W_{N} }}{{2q_{f} }}$. As for the price for the limited-functionality version, the software vendor just sets it in $\left[ {\frac{{q_{l} \left( {q_{f} - W_{N} } \right)}}{{2q_{f} }},min\left\{ {q_{l} ,\frac{{q_{f} + W_{N} }}{2}} \right\}} \right]$ so that $\theta_{f}^{RS {\text{-}} II} < \theta_{l}^{RS {\text{-}} II}$. As a result, the number of type LU consumer is between ${ \hbox{max} }\left\{ {0,1 - \frac{{q_{f} + W_{N} }}{{2q_{l} }}} \right\}$ and $\frac{{q_{f} + W_{N} }}{{2q_{f} }}$. Finally, the optimal total demand and profit of the software vendor is $D_{Total}^{RS {\text{-}} II*} = D_{F}^{RS {\text{-}} II*} = \frac{{q_{f} + W_{N} }}{{2q_{f} }}$ and $\pi^{RS {\text{-}} II*} = \frac{{\left( {q_{f} + W_{N} } \right)^{2} }}{{4q_{f} }}$.

• (V) Finally, we give the globally optimal solution of Replacement Strategy by comparing two local optima RS-I and RS-II. When $q_{l} > A$ and $W_{N} > \frac{1}{2}q_{l} - q_{f}$, or $q_{l} \le A$, $\pi^{RS {\text{-}} I*} - \pi^{RS {\text{-}} II*} = \frac{{q_{l} \left( {q_{f} + W_{N} } \right)^{2} }}{{4q_{f} \left( {4q_{f} - q_{l} } \right)}} > 0$. When $q_{l} > A$ and $W_{N} \le \frac{1}{2}q_{l} - q_{f}$, $\pi^{RS {\text{-}} I*} - \pi^{RS {\text{-}} II*} = \frac{{q_{l} q_{f} - \left( {q_{f} + W_{N} } \right)^{2} }}{{4q_{f} }} \ge \frac{{q_{l} q_{f} - \left( {q_{f} + \frac{1}{2}q_{l} - q_{f} } \right)^{2} }}{{4q_{f} }} = \frac{{4q_{l} q_{f} - q_{l}^{2} }}{{16q_{f} }} > 0$. Therefore, RS-I is always the optimal choice if the software vendor adopts the Replacement Strategy.

1.2 A2: Proof of Corollary 1

If $q_{l} > A = \frac{{\sqrt {1 + 16q_{f} } - 1}}{4}$ and $W_{N} > \frac{1}{2}q_{l} - q_{f},$$\frac{{dp_{l}^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{{ - q_{l} }}{{4q_{f} - q_{l} }} < 0,$$\frac{{dp_{f}^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{{2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} > 0$, $\frac{{dD_{Total}^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{1}{{4q_{f} - q_{l} }} > 0$, $\frac{{d\pi^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{{2W_{N} + 2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} > \frac{{2\left( {\frac{1}{2}q_{l} - q_{f} } \right) + 2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} = 0$.

If $q_{l} \le A$, then $\frac{1}{2}q_{l} - q_{f} \le - q_{l}^{2} \le W_{N}$, $\frac{{dp_{l}^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{{ - q_{l} }}{{4q_{f} - q_{l} }} < 0$, $\frac{{dp_{f}^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{{2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} > 0$, $\frac{{dD_{Total}^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{1}{{4q_{f} - q_{l} }} > 0$, $\frac{{d\pi^{RS {\text{-}} I*} }}{{dW_{N} }} = \frac{{2W_{N} + 2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} > 0$.

1.3 A3: Proof of Proposition 2

• (I) Firstly, we give the proof of the local optimum LES-I, corresponding to the type-I market segmentation in Fig. 6. In this case, $0 < \theta_{2l}^{LES {\text{-}} I} < \theta_{f}^{LES {\text{-}} I} = \theta_{1l}^{LES {\text{-}} I} < \theta_{u}^{LES {\text{-}} I} < 1$, $\theta_{f}^{LES {\text{-}} I} = \theta_{1l}^{LES {\text{-}} I} = \frac{{p_{l} }}{{q_{l} }}$, $\theta_{u}^{LES {\text{-}} I} = \frac{{p_{f} - W_{N} }}{{q_{f} }}$ and $\theta_{2l}^{LES {\text{-}} I} = \frac{{p_{l} - W_{N} }}{{q_{l} }}$. $\uppi^{LES {\text{-}} I} = \left( {1 - \frac{{p_{f}^{LES {\text{-}} I} - W_{N} }}{{q_{f} }}} \right)p_{f}^{LES {\text{-}} I} + \left( {\frac{{p_{f}^{LES {\text{-}} I} - W_{N} }}{{q_{f} }} - \frac{{p_{l}^{LES {\text{-}} I} - W_{N} }}{{q_{l} }}} \right)p_{l}^{LES {\text{-}} I}$. Let $\varvec{p} = \left( {p_{f}^{LES {\text{-}} I} ,p_{l}^{LES {\text{-}} I} } \right)$, then $H\left( \varvec{p} \right) = \left[ {\begin{array}{*{20}c} { - \frac{2}{{q_{f} }}} & {\frac{1}{{q_{f} }}} \\ {\frac{1}{{q_{f} }}} & { - \frac{2}{{q_{l} }}} \\ \end{array} } \right]$, $D_{1} \left( H \right) = - \frac{2}{{q_{f} }} < 0$, $D_{2} \left( H \right) = \frac{{4q_{f} - q_{l} }}{{q_{f}^{2} q_{l} }} > 0$, thus the profit function of LES-I is concave. The first-order conditions of $\pi^{LES {\text{-}} I}$ with respect to $p_{f}^{LES {\text{-}} I}$ and $p_{l}^{LES {\text{-}} I}$ are $\left\{ {\begin{array}{*{20}l} {\frac{{\partial \pi^{LES {\text{-}} I} }}{{\partial p_{f}^{LES {\text{-}} I} }} = \frac{{q_{f} + W_{N} - 2p_{f}^{LES {\text{-}} I} + p_{l}^{LES {\text{-}} I} }}{{q_{f} }} = 0} \hfill \\ {\frac{{\partial \pi^{LES {\text{-}} I} }}{{\partial p_{l}^{LES {\text{-}} I} }} = \frac{{\left( {q_{f} - q_{l} } \right)W_{N} + q_{l} p_{f}^{LES {\text{-}} I} - 2q_{f} p_{l}^{LES {\text{-}} I} }}{{q_{f} q_{l} }} = 0} \hfill \\ \end{array} } \right.$, and then $\left\{ {\begin{array}{*{20}l} {p_{f}^{LES {\text{-}} I*} = \frac{{\left( {3q_{f} - q_{l} } \right)W_{N} + 2q_{f}^{2} }}{{4q_{f} - q_{l} }}} \hfill \\ {p_{l}^{LES {\text{-}} I*} = \frac{{\left( {2q_{f} - q_{l} } \right)W_{N} + q_{l} q_{f} }}{{4q_{f} - q_{l} }}} \hfill \\ \end{array} } \right.$. $p_{f}^{LES {\text{-}} I*} - p_{l}^{LES {\text{-}} I*} = \frac{{q_{f} \left( {W_{N} + 2q_{f} - q_{l} } \right)}}{{4q_{f} - q_{l} }} > 0$, which is obvious because $\left| {W_{N} } \right| < q_{l} < q_{f}$.

  The optimal segmenting point of different versions are $\theta_{u}^{LES {\text{-}} I *} = \frac{{2q_{f} - W_{N} }}{{4q_{f} - q_{l} }},$$\theta_{f}^{LES {\text{-}} I *} = \theta_{1l}^{LES {\text{-}} I *} = \frac{{\left( {2q_{f} - q_{l} } \right)W_{N} + q_{l} q_{f} }}{{q_{l} \left( {4q_{f} - q_{l} } \right)}},$ and $\theta_{2l}^{LES {\text{-}} I *} = \frac{{q_{f} \left( { - 2W_{N} + q_{l} } \right)}}{{q_{l} \left( {4q_{f} - q_{l} } \right)}}$, according to the constraint $0 < \theta_{2l}^{LES {\text{-}} I} < \theta_{f}^{LES {\text{-}} I} = \theta_{1l}^{LES {\text{-}} I} < \theta_{u}^{LES {\text{-}} I} < 1$, then we get: $\theta_{2l}^{LES {\text{-}} I *} = \frac{{q_{f} \left( { - 2W_{N} + q_{l} } \right)}}{{q_{l} \left( {4q_{f} - q_{l} } \right)}} > 0$, then $W_{N} < \frac{1}{2}q_{l}$; $\theta_{1l}^{LES {\text{-}} I *} - \theta_{2l}^{LES {\text{-}} I *} = \frac{{W_{N} }}{{q_{l} }} > 0$, then $W_{N} > 0$; $\theta_{u}^{LES {\text{-}} I *} - \theta_{1l}^{LES {\text{-}} I *} = \frac{{q_{f} \left( { - 2W_{N} + q_{l} } \right)}}{{q_{l} \left( {4q_{f} - q_{l} } \right)}} > 0$, then $W_{N} < \frac{1}{2}q_{l}$; $\theta_{u}^{LES {\text{-}} I *} = \frac{{2q_{f} - W_{N} }}{{4q_{f} - q_{l} }} < 1$, then $W_{N} > - 2q_{f} + q_{l}$. Therefore, when $0 < W_{N} < \frac{1}{2}q_{l}$, not all old consumers upgrade and not all new consumers purchase the limited-functionality version in the second period. The number of all types of consumers are $D_{LU}^{LES {\text{-}} I *} = 1 - \theta_{u}^{LES {\text{-}} I *} = \frac{{2q_{f} - q_{l} + W_{N} }}{{4q_{f} - q_{l} }}$, $D_{LN}^{LES {\text{-}} I *} = \theta_{u}^{LES {\text{-}} I *} - \theta_{1l}^{LES {\text{-}} I *} = \frac{{q_{f} \left( { - 2W_{N} + q_{l} } \right)}}{{q_{l} \left( {4q_{f} - q_{l} } \right)}}$, $D_{NF}^{LES {\text{-}} I *} = 0$, $D_{NL}^{LES {\text{-}} I *} = \theta_{1l}^{LES {\text{-}} I *} - \theta_{2l}^{LES {\text{-}} I *} = \frac{{W_{N} }}{{q_{l} }}$, $D_{Total}^{LES {\text{-}} I *} = D_{LU}^{LES {\text{-}} I *} + D_{LN}^{LES {\text{-}} I *} + D_{NF}^{LES {\text{-}} I *} + D_{NL}^{LES {\text{-}} I *} = \frac{{3q_{l} q_{f} - q_{l}^{2} + 2q_{f} W_{N} }}{{q_{l} \left( {4q_{f} - q_{l} } \right)}}.$ Finally, the optimal total profit of LES-I is $\pi^{LES {\text{-}} I *} = \frac{{q_{f} W_{N}^{2} + \left( {3q_{l} q_{f} - q_{f}^{2} } \right)W_{N} + q_{l} q_{f}^{2} }}{{q_{l} \left( {4q_{f} - q_{l} } \right)}}$.

  When $W_{N} > \frac{1}{2}q_{l}$, all old consumers upgrade and all new consumers will purchase the limited-functionality version. In this case, $\theta_{1l}^{LES {\text{-}} I} = \theta_{u}^{LES {\text{-}} I} = \theta_{f}^{LES {\text{-}} I} = \frac{{p_{l}^{LES {\text{-}} I} }}{{q_{l} }}$, and $\theta_{2l} = 0$. $\uppi^{LES {\text{-}} I} = p_{l}^{LES {\text{-}} I} + \left( {1 - \frac{{p_{l}^{LES {\text{-}} I} }}{{q_{l} }}} \right)\left( {p_{f}^{LES {\text{-}} I} - p_{l}^{LES {\text{-}} I} } \right)$. Let $\varvec{p} = \left( {p_{f}^{LES {\text{-}} I} ,p_{l}^{LES {\text{-}} I} } \right)$, then $H\left( \varvec{p} \right) = \left[ {\begin{array}{*{20}c} 0 & { - \frac{1}{{q_{l} }}} \\ { - \frac{1}{{q_{l} }}} & {\frac{1}{{q_{l} }}} \\ \end{array} } \right]$, $D_{1} \left( H \right) = 0$, $D_{2} \left( H \right) = - \frac{1}{{q_{l}^{2} }} < 0$, thus the profit function in LES-I is not concave, then the optimal price should be chosen at boundary value. To ensure the consumer at 0 to buy the limited-functionality, $p_{l}^{LES {\text{-}} I}$ should be not larger than $W_{N}$, namely $0 \le p_{l}^{LES {\text{-}} I} \le W_{N}$. To ensure that all old consumers choose to upgrade and no new consumers choose to buy the full-functionality version, then $\theta_{1l} q_{f} + W_{N} - p_{f}^{LES {\text{-}} I} \ge 0$ and $\theta_{1l} q_{f} + W_{N} - p_{f}^{LES {\text{-}} I} \le \theta_{1l} q_{l} + W_{N} - p_{l}^{LES {\text{-}} I}$, namely $\frac{{q_{f} }}{{q_{l} }}p_{l}^{LES {\text{-}} I} \le p_{f}^{LES {\text{-}} I} \le \frac{{q_{f} }}{{q_{l} }}p_{l}^{LES {\text{-}} I} + W_{N}$. We examine the optimality of four groups of $\varvec{p}$, namely $\varvec{p}_{1} = \left( {0,0} \right)$, $\varvec{p}_{2} = \left( {W_{N} ,0} \right)$,$\varvec{ p}_{3} = \left( {\frac{{q_{f} }}{{q_{l} }}W_{N} ,W_{N} } \right)$,$\varvec{ p}_{4} = \left( {\frac{{q_{f} + q_{l} }}{{q_{l} }}W_{N} ,W_{N} } \right)$, then the corresponding profits under different $\varvec{p}$ are $\pi^{LES {\text{-}} I} \left( {\varvec{p}_{1} } \right) = 0$, $\pi^{LES {\text{-}} I} \left( {\varvec{p}_{2} } \right) = W_{N}$, $\pi^{LES {\text{-}} I} \left( {\varvec{p}_{3} } \right) = \frac{{W_{N} \left[ {q_{l} q_{f} - \left( {q_{f} - q_{l} } \right)W_{N} } \right]}}{{q_{l}^{2} }}$, $\pi^{LES {\text{-}} I} \left( {\varvec{p}_{4} } \right) = \frac{{W_{N} \left[ {q_{l}^{2} + \left( {q_{l} - W_{N} } \right)q_{f} } \right]}}{{q_{l}^{2} }}$. By comparing four groups of profits, we find $\pi^{LES {\text{-}} I} \left( {\varvec{p}_{4} } \right)$ is the largest, thus we set the optimal prices for both versions as $p_{l}^{LES {\text{-}} I*} = W_{N}$ and $p_{f}^{LES {\text{-}} I*} = \frac{{q_{f} + q_{l} }}{{q_{l} }}W_{N}$. The optimal segmentation points of different versions are $\theta_{1l}^{LES {\text{-}} I*} = \theta_{f}^{LES {\text{-}} I*} = \theta_{u}^{LES {\text{-}} I*} = \frac{{W_{N} }}{{q_{l} }}$ and $\theta_{2l}^{LES {\text{-}} I*} = 0$. The number of all types of consumers are $D_{LU}^{LES {\text{-}} I *} = 1 - \theta_{u}^{LES {\text{-}} I *} = \frac{{q_{l} - W_{N} }}{{q_{l} }}$, $D_{LN}^{LES {\text{-}} I *} = \theta_{u}^{LES {\text{-}} I *} - \theta_{1l}^{LES {\text{-}} I *} = 0$, $D_{NF}^{LES {\text{-}} I*} = 0$, $D_{NL}^{LES {\text{-}} I *} = \theta_{1l}^{LES {\text{-}} I *} - \theta_{2l}^{LES {\text{-}} I *} = \frac{{W_{N} }}{{q_{l} }}$, $D_{Total}^{LES {\text{-}} I *} = D_{LU}^{LES {\text{-}} I *} + D_{LN}^{LES {\text{-}} I *} + D_{NF}^{LES {\text{-}} I *} + D_{NL}^{LES {\text{-}} I *} = 1$. The resulting profit is $\pi^{LES {\text{-}} I*} = \frac{{W_{N} \left[ {q_{l}^{2} + \left( {q_{l} - W_{N} } \right)q_{f} } \right]}}{{q_{l}^{2} }}$.

• (II) Secondly, we give the proof of the local optimum LES-II, corresponding to the type-II market segmentation in Fig. 6. In this case, $0 < \theta_{2l}^{LES {\text{-}} II} < \theta_{f}^{LES {\text{-}} II} < \theta_{1l}^{LES {\text{-}} II} = \theta_{u}^{LES {\text{-}} II} < 1$$\theta_{u}^{LES {\text{-}} II} = \theta_{1l}^{LES {\text{-}} II} = \frac{{p_{l}^{LES {\text{-}} II} }}{{q_{l} }}$, $\theta_{f}^{LES {\text{-}} II} = \frac{{p_{f}^{LES {\text{-}} II} - p_{l}^{LES {\text{-}} II} }}{{q_{f} - q_{l} }}$ and $\theta_{2l}^{LES {\text{-}} II} = \frac{{p_{l}^{LES {\text{-}} II} - W_{N} }}{{q_{l} }}$. $\uppi^{LES {\text{-}} II} = \left( {1 - \frac{{p_{f}^{LES {\text{-}} II} - p_{l}^{LES {\text{-}} II} }}{{q_{f} - q_{l} }}} \right)p_{f}^{LES {\text{-}} II} + \left( {\frac{{p_{f}^{LES {\text{-}} II} - p_{l}^{LES {\text{-}} II} }}{{q_{f} - q_{l} }} - \frac{{p_{l}^{LES {\text{-}} II} - W_{N} }}{{q_{l} }}} \right)p_{l}^{LES {\text{-}} II}$. Let $\varvec{p} = \left( {p_{f}^{LES {\text{-}} II} ,p_{l}^{LES {\text{-}} II} } \right)$, then $H\left( \varvec{p} \right) = \left[ {\begin{array}{*{20}c} { - \frac{2}{{q_{f} - q_{l} }}} & {\frac{2}{{q_{f} - q_{l} }}} \\ {\frac{2}{{q_{f} - q_{l} }}} & { - \frac{{2q_{f} }}{{q_{l} \left( {q_{f} - q_{l} } \right)}}} \\ \end{array} } \right]$, $D_{1} \left( H \right) = - \frac{2}{{q_{f} - q_{l} }} < 0$, $D_{2} \left( H \right) = \frac{4}{{q_{l} \left( {q_{f} - q_{l} } \right)}} > 0$, H is negative definite, so $\pi^{LES {\text{-}} II}$ is concave and there exists maximal value. The first-order conditions of $\pi^{LES {\text{-}} II}$ with respect to $p_{l}^{LES {\text{-}} II}$ and $p_{f}^{LES {\text{-}} II}$ are $\left\{ {\begin{array}{*{20}c} {\frac{{\partial \pi^{LES {\text{-}} II} }}{{\partial p_{l}^{LES {\text{-}} II} }} = \frac{{\left( {q_{f} - q_{l} } \right)W_{N} + 2q_{l} p_{f}^{LES {\text{-}} II} - 2q_{f} p_{l}^{LES {\text{-}} II} }}{{q_{l} \left( {q_{f} - q_{l} } \right)}} = 0} \\ {\frac{{\partial \pi^{LES {\text{-}} II} }}{{\partial p_{f}^{LES {\text{-}} II} }} = \frac{{q_{f} - q_{l} - 2p_{f}^{LES {\text{-}} II} + 2p_{l}^{LES {\text{-}} II} }}{{q_{f} - q_{l} }} = 0} \\ \end{array} } \right.$, then $\left\{ {\begin{array}{*{20}c} {p_{l}^{LES {\text{-}} II*} = \frac{{W_{N} + q_{l} }}{2}} \\ {p_{f}^{LES {\text{-}} II*} = \frac{{W_{N} + q_{f} }}{2}} \\ \end{array} } \right.$. $p_{f}^{LES {\text{-}} II*} > p_{l}^{LES {\text{-}} II*}$ is obvious.

  The optimal segmenting point of all kinds of versions are $\theta_{f}^{LES {\text{-}} II*} = \frac{1}{2}$, $\theta_{1l}^{LES {\text{-}} II*} = \theta_{u}^{LES {\text{-}} II*} = \frac{{W_{N} + q_{l} }}{{2q_{l} }}$, and $\theta_{2l}^{LES {\text{-}} II*} = \frac{{ - W_{N} + q_{l} }}{{2q_{l} }}$. According to the constraint $0 < \theta_{2l}^{LES {\text{-}} II} < \theta_{f}^{LES {\text{-}} II} < \theta_{1l}^{LES {\text{-}} II} = \theta_{u}^{LES {\text{-}} II} < 1$, then $\theta_{2l}^{LES {\text{-}} II*} = \frac{{ - W_{N} + q_{l} }}{2} > 0$, which is obvious; $\theta_{f}^{LES {\text{-}} II*} - \theta_{2l}^{LES {\text{-}} II*} = \frac{{W_{N} }}{{2q_{l} }} > 0$, then $W_{N} > 0$; $\theta_{1l}^{LES {\text{-}} II*} - \theta_{f}^{LES {\text{-}} II*} = \frac{{W_{N} }}{{2q_{l} }} > 0$, then $W_{N} > 0$; $\theta_{1l}^{LES {\text{-}} II*} = \frac{{W_{N} + q_{l} }}{{2q_{l} }} < 1$, which is obvious.

  All types of demand are $D_{LU}^{LES {\text{-}} II *} = 1 - \theta_{u}^{LES {\text{-}} II *} = \frac{{ - W_{N} + q_{l} }}{{2q_{l} }}$, $D_{LN}^{LES {\text{-}} II *} = \theta_{u}^{LES {\text{-}} II *} - \theta_{1l}^{LES {\text{-}} II *} = 0$, $D_{NF}^{LES {\text{-}} II *} = \theta_{1l}^{LES {\text{-}} II *} - \theta_{f}^{LES {\text{-}} II*} = \frac{{W_{N} }}{{2q_{l} }}$, $D_{NL}^{LES {\text{-}} II*} = \theta_{f}^{LES {\text{-}} II *} - \theta_{2l}^{LES {\text{-}} II *} = \frac{{W_{N} }}{{2q_{l} }}$, $D_{Total}^{LES {\text{-}} II *} = D_{LU}^{LES {\text{-}} II *} + D_{LN}^{LES {\text{-}} II *} + D_{NF}^{LES {\text{-}} II *} + D_{NL}^{LES {\text{-}} II *} = \frac{{W_{N} + q_{l} }}{{2q_{l} }}$. The optimal total profit of LES-II is $\pi^{LES {\text{-}} II*} = \frac{{W_{N}^{2} + 2q_{l} W_{N} + q_{l} q_{f} }}{{4q_{l} }}$.

• (III) Finally, we give the globally optimal solution of Line-extension Strategy by comparing two local optimum LES-I and LES-II. When $0 < W_{N} < \frac{1}{2}q_{l}$, $\pi^{LES {\text{-}} I*} - \pi^{LES {\text{-}} II*} = \frac{{W_{N}^{2} + \left( {4q_{f} - 2q_{l} } \right)W_{N} + q_{l} q_{f} }}{{4\left( {4q_{f} - q_{l} } \right)}} > 0$. When $\frac{1}{2}q_{l} \le W_{N} \le q_{l}$, $\pi^{LES {\text{-}} I*} - \pi^{LES {\text{-}} II*} = \frac{{ - \left( {4q_{f} + q_{l} } \right)W_{N}^{2} + \left( {4q_{f} q_{l} + 2q_{l}^{2} } \right)W_{N} - q_{f} q_{l}^{2} }}{{4q_{l}^{2} }}$. Let $\pi^{LES {\text{-}} I*} - \pi^{LES {\text{-}} II*} > 0$ we could get $F_{1} < W_{N} < F_{2}$, where $F_{1} = \frac{{q_{l} \left( {2q_{f} + q_{l} - \sqrt {q_{l} \left( {3q_{f} + q_{l} } \right)} } \right)}}{{4q_{f} + q_{l} }}$ and $F_{2} = \frac{{q_{l} \left( {2q_{f} + q_{l} + \sqrt {q_{l} \left( {3q_{f} + q_{l} } \right)} } \right)}}{{4q_{f} + q_{l} }}$. In particular, $F_{1} - \frac{1}{2}q_{l} = \frac{{q_{l} \left( {q_{l} - 2\sqrt {q_{l} \left( {3q_{f} + q_{l} } \right)} } \right)}}{{8q_{f} + 2q_{l} }} < 0$, $F_{2} - \frac{1}{2}q_{l} = \frac{{q_{l} \left( {q_{l} + 2\sqrt {q_{l} \left( {3q_{f} + q_{l} } \right)} } \right)}}{{8q_{f} + 2q_{l} }} > 0$ and $F_{2} - q_{l} = \frac{{q_{l} \left( { - 2q_{f} + \sqrt {q_{l} \left( {3q_{f} + q_{l} } \right)} } \right)}}{{4q_{f} + q_{l} }} < 0$ because $q_{l} \left( {3q_{f} + q_{l} } \right) - \left( {2q_{f} } \right)^{2} = \left( {q_{l} - q_{f} } \right)\left( {4q_{f} + q_{l} } \right) < 0$. Therefore, $\pi^{LES {\text{-}} I*} > \pi^{LES {\text{-}} II*}$ when $W_{N} \in \left[ {\frac{1}{2}q_{l} ,F_{2} } \right)$ but $\pi^{LES {\text{-}} I*} \le \pi^{LES {\text{-}} II*}$ when $W_{N} \in \left[ {F_{2} ,q_{l} } \right]$.

  In sum, $\pi^{LES {\text{-}} I*} > \pi^{LES {\text{-}} II*}$ when $W_{N} \in \left( {0,F_{2} } \right)$ but $\pi^{LES {\text{-}} I*} \le \pi^{LES {\text{-}} II*}$ when $W_{N} \in \left[ {F_{2} ,q_{l} } \right]$.

1.4 A4: Proof of Lemma 1

When $0 < W_{N} < \frac{1}{2}q_{l}$, $p_{l}^{LES {\text{-}} I*} - p_{l}^{LES {\text{-}} II*} = \frac{{q_{l} }}{4}\left[ {\frac{{q_{l} - 2W_{N} }}{{4q_{f} - q_{l} }} - 1} \right] < 0$; $p_{f}^{LES {\text{-}} I*} - p_{f}^{LES {\text{-}} II*} = \frac{{q_{l} q_{f} + \left( {2q_{f} - q_{l} } \right)W_{N} }}{{2\left( {4q_{f} - q_{l} } \right)}} > 0$.

When $\frac{1}{2}q_{l} \le W_{N} \le q_{l}$, $p_{l}^{LES {\text{-}} I*} - p_{l}^{LES {\text{-}} II*} = \frac{{W_{N} - q_{l} }}{2} \le 0$; $p_{f}^{LES {\text{-}} I*} - p_{f}^{LES {\text{-}} II*} = \frac{{\left( {2q_{f} + q_{l} } \right)W_{N} - q_{l} q_{f} }}{{2q_{l} }} \ge 0$.

1.5 A5: Proof of Corollary 2

When $0 < W_{N} < \frac{1}{2}q_{l}$, $\frac{{dp_{l}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{{2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} > 0$; $\frac{{dp_{f}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{{3q_{f} - q_{l} }}{{4q_{f} - q_{l} }} > 0$; $\frac{{dD_{LU}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{1}{{4q_{f} - q_{l} }} > 0$; $\frac{{dD_{LN}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{{ - 2q_{f} }}{{q_{l} \left( {4q_{f} - q_{l} } \right)}} < 0$; $\frac{{dD_{NF}^{LES {\text{-}} I*} }}{{dW_{N} }} = 0$; $\frac{{dD_{NL}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{1}{{q_{l} }} > 0$; $\frac{{dD_{Total}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{{2q_{f} }}{{q_{l} \left( {4q_{f} - q_{l} } \right)}} > 0$; $\frac{{d\pi^{LES {\text{-}} I *} }}{{dW_{N} }} = q_{f } \left[ {2W_{N} + 3q_{l} - q_{f } } \right]$, then $\frac{{d\pi^{LES {\text{-}} I *} }}{{dW_{N} }} > 0$ when $W_{N} > \frac{{q_{f} - 3q_{l} }}{2}$, and $\frac{{d\pi^{LES {\text{-}} I *} }}{{dW_{N} }} < 0$ when $0 < W_{N} < \frac{{q_{f} - 3q_{l} }}{2}$ if $q_{l} < \frac{{q_{f} }}{3}$.

When $\frac{1}{2}q_{l} \le W_{N} \le F_{2}$, $\frac{{dp_{l}^{LES {\text{-}} I*} }}{{dW_{N} }} = 1 > 0$; $\frac{{dp_{f}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{{q_{f} + q_{l} }}{{q_{l} }} > 0$; $\frac{{dD_{LU}^{LES {\text{-}} I*} }}{{dW_{N} }} = - \frac{1}{{q_{l} }} > 0$; $\frac{{dD_{LN}^{LES {\text{-}} I*} }}{{dW_{N} }} = 0$; $\frac{{dD_{NF}^{LES {\text{-}} I*} }}{{dW_{N} }} = 0$; $\frac{{dD_{NL}^{LES {\text{-}} I*} }}{{dW_{N} }} = \frac{1}{{q_{l} }} > 0$; $\frac{{dD_{Total}^{LES {\text{-}} I*} }}{{dW_{N} }} = 0$; $\frac{{d\pi^{LES {\text{-}} I *} }}{{dW_{N} }} = \frac{{q_{l}^{2} + q_{f} q_{l} - 2q_{f } W_{N} }}{{q_{l}^{2} }}$, then $\frac{{d\pi^{LES {\text{-}} I *} }}{{dW_{N} }} > 0$ when $W_{N} < \frac{{q_{l}^{2} + q_{f} q_{l} }}{{2q_{f} }}$, and $\frac{{d\pi^{LES {\text{-}} I *} }}{{dW_{N} }} < 0$ when $W_{N} > \frac{{q_{l}^{2} + q_{f} q_{l} }}{{2q_{f} }}$.

When $F_{2} \le W_{N} \le q_{l}$, $\frac{{dp_{l}^{LES {\text{-}} II*} }}{{dW_{N} }} = \frac{1}{2} > 0$, $\frac{{dp_{f}^{LES {\text{-}} II*} }}{{dW_{N} }} = \frac{1}{2} > 0$, $\frac{{dD_{LU}^{LES {\text{-}} II*} }}{{dW_{N} }} = - \frac{1}{{2q_{l} }} < 0$, $\frac{{dD_{LN}^{LES {\text{-}} II*} }}{{dW_{N} }} = 0$, $\frac{{dD_{NF}^{LES {\text{-}} II*} }}{{dW_{N} }} = \frac{1}{{2q_{l} }} > 0$, $\frac{{dD_{NL}^{LES {\text{-}} II*} }}{{dW_{N} }} = \frac{1}{{2q_{l} }} > 0$, $\frac{{dD_{Total}^{LES {\text{-}} II*} }}{{dW_{N} }} = \frac{1}{{2q_{l} }} > 0$. $\frac{{d\pi^{LES {\text{-}} II *} }}{{dW_{N} }} = \frac{{W_{N} + q_{l} }}{{2q_{l} }} > 0 .$

1.6 A6: Proof of Proposition 3

According to $0 \le\upalpha \le 1$, $0 \le k \le 1$ and $W_{N} = \alpha q_{l} - kq_{l}^{2}$, we could get $- q_{l}^{2} \le W_{N} \le q_{l}$. We examine the optimality of the three strategies (SS, RS, LES-I and LES-II) under different WOM level.

• 1. When $0 < W_{N} < \frac{1}{2}q_{l}$, according to Proposition 2, we know that $\pi^{LES {\text{-}} I*} > \pi^{LES {\text{-}} II*}$. $\pi^{LES {\text{-}} I*} - \pi^{RS*} = \frac{{W_{N} \left[ {q_{f} q_{l} + \left( {q_{f} - q_{l} } \right)W_{N} } \right]}}{{q_{l} \left( {4q_{f} - q_{l} } \right)}} > 0$; $\pi^{RS*} - \pi^{SS*} = \frac{{4W_{N}^{2} + \left( {8q_{f} - 4q_{l} } \right)W_{N} + q_{f} q_{l} }}{{4\left( {4q_{f} - q_{l} } \right)}} > 0.$ Then $\pi^{LES {\text{-}} I*} > \pi^{RS*} > \pi^{SS*}$, thus LES-I is optimal when $0 < W_{N} < \frac{1}{2}q_{l}$.

• 2. When $\frac{1}{2}q_{l} \le W_{N} \le F_{2} = \frac{{q_{l} \left( {2q_{f} + q_{l} + \sqrt {q_{l} \left( {3q_{f} + q_{l} } \right)} } \right)}}{{4q_{f} + q_{l} }}$, according to Proposition 2, we know that $\pi^{LES {\text{-}} I*} > \pi^{LES {\text{-}} II*}$. Then $\pi^{LES {\text{-}} I*} - \pi^{RS*} = - \left( {\frac{1}{{4q_{f} - q_{l} }} + \frac{{q_{f} }}{{q_{l}^{2} }}} \right)W_{N}^{2} + \left( {1 - \frac{{2q_{f} - q_{l} }}{{4q_{f} - q_{l} }} + \frac{{q_{f} }}{{q_{l} }}} \right) - \frac{{q_{f}^{2} }}{{4q_{f} - q_{l} }}$, When $W_{N} = \frac{1}{2}q_{l}$, $\pi^{LES {\text{-}} I *} - \pi^{RS *} = \frac{{q_{l} \left( {3q_{f} - q_{l} } \right)}}{{4\left( {4q_{f} - q_{l} } \right)}} > 0$, when $W_{N} = F_{2}$, $\pi^{LES {\text{-}} I *} - \pi^{RS *} = \frac{{4q_{f}^{2} q_{l}^{2} - 4q_{f} q_{l}^{3} - 2q_{l}^{4} + \left( {4q_{f}^{2} - q_{f} q_{l} - 2q_{l}^{2} } \right)\sqrt {q_{l}^{3} \left( {3q_{f} + q_{l} } \right)} }}{{\left( {4q_{f} - q_{l} } \right)\left( {4q_{f} + q_{l} } \right)^{2} }} > \frac{{4q_{f}^{2} q_{l}^{2} - 4q_{f} q_{l}^{3} - 2q_{l}^{4} + \left( {4q_{f}^{2} - q_{f} q_{l} - 2q_{l}^{2} } \right)2q_{l}^{2} }}{{\left( {4q_{f} - q_{l} } \right)\left( {4q_{f} + q_{l} } \right)^{2} }} = \frac{{q_{l}^{2} \left[ {\left( {q_{f} - q_{l} } \right)\left( {11q_{f} + q_{l} } \right) + q_{f}^{2} } \right]}}{{\left( {4q_{f} - q_{l} } \right)\left( {4q_{f} + q_{l} } \right)^{2} }} > 0$. Then, $\pi^{LES {\text{-}} I *} > \pi^{RS *}$ for $W_{N} \in \left[ {\frac{1}{2}q_{l} ,F_{2} } \right]$. $\pi^{RS *} - \pi^{SS *} = \frac{{4W_{N}^{2} + \left( {8q_{f} - 4q_{l} } \right)W_{N} + q_{f} q_{l} }}{{4\left( {4q_{f} - q_{l} } \right)}} > 0$. Therefore, LES-I is the optimal strategy when $W_{N} \in \left[ {\frac{1}{2}q_{l} ,F_{2} } \right]$.

• 3. When $F_{2} < W_{N} \le q_{l}$, all strategies are feasible. According to Proposition 2, $\pi^{LES {\text{-}} I*} < \pi^{LES {\text{-}} II*}$. $\pi^{RS*} - \pi^{SS*} = \frac{{4W_{N}^{2} + \left( {8q_{f} - 4q_{l} } \right)W_{N} + q_{f} q_{l} }}{{4\left( {4q_{f} - q_{l} } \right)}} > 0$. So we just examine the relative optimality of RS and LES-II. Then $\pi^{RS*} - \pi^{LES {\text{-}} II*} = \frac{{\left( {5q_{l} - 4q_{f} } \right)W_{N}^{2} - 2q_{l}^{2} W_{N} + q_{f} q_{l}^{2} }}{{4q_{l} \left( {4q_{f} - q_{l} } \right)}}$. When $5q_{l} - 4q_{f} < 0$, namely $q_{l} < \frac{4}{5}q_{f}$, to make $\pi^{RS *} - \pi^{LES {\text{-}} II *} > 0$, $W_{N}$ should satisfy $G_{1} = \frac{{q_{l} \left( {q_{l} + \sqrt {q_{l}^{2} - q_{f} \left( {5q_{l} - 4q_{f} } \right)} } \right)}}{{5q_{l} - 4q_{f} }} < W_{N} < \frac{{q_{l} \left( {q_{l} - \sqrt {q_{l}^{2} - q_{f} \left( {5q_{l} - 4q_{f} } \right)} } \right)}}{{5q_{l} - 4q_{f} }} = G_{2}$. $G_{1} < 0$ is obvious. By comparing $G_{2}$ and $F_{2}$, we find that $G_{2}$ is always less than $F_{2}$ for all $q_{l} \in \left[ {0,q_{f} } \right]$. Then $G_{2} < F_{2}$ for all $q_{l} \in \left[ {0,\frac{4}{5}q_{f} } \right]$, thus $\pi^{RS *} < \pi^{LES {\text{-}} II *}$ for all $W_{N} \in \left( {F_{2} ,q_{l} } \right]$.

  When $5q_{l} - 4q_{f} > 0$, namely $q_{l} > \frac{4}{5}q_{f}$, to make $\pi^{RS*} - \pi^{LES {\text{-}} II*} > 0$, $W_{N}$ should satisfy $0 < W_{N} < G_{2}$ or $W_{N} > G_{1}$. Because $G_{1} > q_{l}$ for $q_{l} > 0$, which is also for $q_{l} > \frac{4}{5}q_{f}$, so $W_{N} > G_{1}$ is impossible. When $q_{l} > \frac{4}{5}q_{f}$, $G_{2}$ is less than $\frac{1}{2}q_{l} < F_{2}$, then $\pi^{RS*} - \pi^{LES {\text{-}} II*} > 0$ is impossible for $W_{N} > F_{2}$. Therefore, when $q_{l} > \frac{4}{5}q_{f}$, $\pi^{RS*} < \pi^{LES {\text{-}} II*}$ for $W_{N} \in \left( {F_{2} ,q_{l} } \right]$. Therefore, when $F_{2} < W_{N} \le q_{l}$, the optimal strategy is LES-II.

• 4. We examine the optimality of three strategies when $- q_{l}^{2} \le W_{N} < 0$. If $\frac{1}{2}q_{l} - q_{f} > - q_{l}^{2}$, then for $\frac{1}{2}q_{l} - q_{f} \le W_{N} < 0$, $\pi^{RS {\text{-}} I*} - \pi^{SS*} = \frac{{4W_{N}^{2} + \left( {8q_{f} - 4q_{l} } \right)W_{N} + q_{f} q_{l} }}{{4\left( {4q_{f} - q_{l} } \right)}}$. To make $\pi^{RS {\text{-}} I*} > \pi^{SS*}$,$W_{N}$ should satisfy $W_{N} < \frac{{ - \sqrt {\left( {q_{f} - q_{l} } \right)\left( {4q_{f} - q_{l} } \right)} + q_{l} - 2q_{f} }}{2} = X_{1}$ or $\frac{{\sqrt {\left( {q_{f} - q_{l} } \right)\left( {4q_{f} - q_{l} } \right)} + q_{l} - 2q_{f} }}{2} = X_{2} < W_{N} < 0$. Because $X_{1} < \frac{1}{2}q_{l} - q_{f}$ and $X_{2} > \frac{1}{2}q_{l} - q_{f}$, then $\pi^{RS {\text{-}} I*} > \pi^{SS*}$ when $W_{N} > X_{2}$, but $\pi^{RS*} < \pi^{SS*}$ when $W_{N} < X_{2}$. For $- q_{l}^{2} < W_{N} < \frac{1}{2}q_{l} - q_{f}$, it is easy to obtain that $\pi^{RS*} - \pi^{SS*} = \frac{1}{4}\left( {q_{l} - q_{f} } \right) < 0$. Therefore, If $\frac{1}{2}q_{l} - q_{f} > - q_{l}^{2}$, then SS is optimal when $W_{N} \in \left[ { - q_{l}^{2} ,X_{2} } \right]$, while RS is the dominant strategy when $W_{N} \in \left[ {X_{2} ,0} \right]$. If $\frac{1}{2}q_{l} - q_{f} < - q_{l}^{2}$, then for $- q_{l}^{2} < W_{N} < 0$, if $X_{2} < - q_{l}^{2}$, then $\pi^{RS*} > \pi^{SS*}$ when $- q_{l}^{2} < W_{N} < 0$; if $X_{2} > - q_{l}^{2}$, then $\pi^{RS*} > \pi^{SS*}$ when $X_{2} < W_{N} < 0$ and $\pi^{RS*} < \pi^{SS*}$ for $- q_{l}^{2} < W_{N} < X_{2}$. In sum, $\pi^{RS*} > \pi^{SS*}$ when $W_{N} > X_{2}$, but $\pi^{RS*} < \pi^{SS*}$ when $W_{N} < X_{2}$.