Effects of risk attitudes and investment spillover on supplier encroachment

Abstract

With the development of e-commerce, a growing number of suppliers have begun to initially establish their own direct channels, competing with their retail channels. However, while this encroachment endows the suppliers with an efficient method to control downstream competition and total production output directly, it may hurt the retailer due to the loss of monopoly in the retail market. This inconsistency presents a difficulty in reaching equilibrium. In this paper, we focus on the combined effects of the risk attitudes and upstream production investment of supply chain members on supplier encroachment and verify the existence of “win–win” results for both supplier and retailer. We find that, while the two parties cannot simultaneously benefit from supplier encroachment in the absence of upstream investment, they can obtain a Pareto improvement from it in the presence of upstream investment and spillover effect. Regarding risk attitudes, we find that both the supplier and the retailer can reach agreement on the supplier encroachment in the case of a moderate confidence level. In other words, the not too risk-loving and not too risk-averse supply chain members are more likely to obtain a Pareto improvement.

Appendix: Effects of risk attitudes and investment spillover on supplier encroachment

To obtain Theorem 1, we first present the following lemma.

Lemma 1

(Liu 2011) Let the function $g(\mathbf x ,x_1,x_2,\dots ,x_n)$ be strictly increasing with $x_1,x_2, \dots , x_k$ and be strictly decreasing with $x_{k+1},x_{k+2},\dots ,x_n$. If $\xi _1,\xi _2,\dots , \xi _{n}$ are uncertain variables with uncertainty distributions $\varPhi _1,\varPhi _2,\dots ,\varPhi _n$, respectively, then the chance constraint ${\mathcal {M}}\{g(\mathbf x ,\xi _1,\xi _2,\dots ,\xi _n) \leqslant 0\}\geqslant \alpha$ holds if and only if $g\left( \mathbf x ,\varPhi _1^{-1}(\alpha ),\dots ,\varPhi _k^{-1}(\alpha ),\varPhi _{k+1}^{-1}(1-\alpha ),\dots ,\varPhi _{n}^{-1}(1-\alpha )\right) \le 0,$ where $0 \le \alpha \le 1$.

Proof

(Theorem 1) Because the supplier’s profit function $\pi _{s}(q_{r},w;x)$ is strictly decreasing with respect to the per-unit selling cost x, from Lemma 1, the chance constraint $M\{\pi _{s}(q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\xi )\geqslant \pi _{0s}\}\geqslant \alpha _{1}$ is equivalent to

$$\begin{aligned} \pi _{0s}-\pi _{s}\left( q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\varPsi ^{-1}(\alpha _{1})\right) \leqslant 0. \end{aligned}$$

Since $\pi _\mathrm{ms}(q_{r},w;\alpha _{1})=\hbox {max}\{\pi _{0s}\}$, the supplier’s maximum profit $\pi _\mathrm{ms}(\cdot ,\cdot )$ under his acceptable confidence level $\alpha _{1}$ satisfies

$$\begin{aligned}&\pi _\mathrm{ms}\left( q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\alpha _{1}\right) \nonumber \\&\quad =\max \left\{ \pi _{0s}\bigm |M\{\pi _{s}(q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\xi )\geqslant \pi _{0s}\}\geqslant \alpha _{1}\right\} \nonumber \\&\quad =\pi _{s}\left( q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\varPsi ^{-1}(\alpha _{1})\right) . \end{aligned}$$

(11)

Similarly, the retailer’s maximum profit $\pi _{mr}(\cdot ,\cdot )$ under her acceptable confidence level $\alpha _{2}$ is

$$\begin{aligned}&\pi _\mathrm{mr}\left( q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\alpha _{2}\right) \nonumber \\&\quad =\max \left\{ \pi _{0r}\bigm |M\{\pi _{r}(q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\eta )\geqslant \pi _{0r}\}\geqslant \alpha _{2}\right\} \nonumber \\&\quad =\pi _{r}\left( q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\varPhi ^{-1}(1-\alpha _{2})\right) . \end{aligned}$$

(12)

Consequently, by via Eqs. (11) and (12), model (3) can be equivalent to model (4). The proof of the theorem is completed. $\square$

Proof

(Proposition 1) For model (4), by maximizing the retailer’s profit $\pi _{r}\left( q_{r},w;\varPhi ^{-1}(1-\alpha _{2})\right)$ under her acceptable confidence level $\alpha _{2}$, the optimal quantity is $q_{r}^{{\mathrm{NN}}}(w;\alpha _{2})=\frac{\varPhi ^{-1}(1-\alpha _{2})-w}{2}$. Substituting $q_{r}^{{\mathrm{NN}}}(w;\alpha _{2})$ into the supplier’s profit function $\pi _{s}\left( q_{r}^{{\mathrm{NN}}}(w;\alpha _{2}),w;\varPsi ^{-1}(\alpha _{1})\right)$, the first-order condition yields the optimal wholesale price as follows: $w^{{\mathrm{NN}}}=\frac{\varPhi ^{-1}(1-\alpha _{2})+\varPsi ^{-1}(\alpha _{1})}{2}$. The optimal quantity and the wholesale price yield the firm’s profits under no encroachment. The proof of the proposition is completed. $\square$

Proof

(Theorem 2) The proof of this theorem is similar to that of Theorem 1. $\square$

Proof

(Proposition 2) First, solving the first-order conditions in the profit function of the retailer $\pi _{r}\left( q_{s},q_{r},w;\varPhi ^{-1}(1-\alpha _{2})\right)$ and the supplier $\pi _{s}\left( q_{s},q_{r},w;\varPhi ^{-1}(1-\alpha _{1}),\varPsi ^{-1}(\alpha _{1})\right)$, we can derive the equilibrium quantities as a function of the wholesale price as

$$\begin{aligned} \begin{array}{l} q_r^{{\mathrm{EN}}}(w;\alpha _{1},\alpha _{2})=\frac{2\varPhi ^{-1}(1-\alpha _{2})-(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-c)\theta -2w}{4-\theta ^2},\\ q_s^{{\mathrm{EN}}}(w;\alpha _{1},\alpha _{2})=\frac{2(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-c)-\theta \varPhi ^{-1}(1-\alpha _{2})+\theta w}{4-\theta ^2}. \end{array} \end{aligned}$$

Next, substituting the production decision $q_r^{{\mathrm{EN}}}(w;\alpha _{1},\alpha _{2})$ and $q_s^{{\mathrm{EN}}}(w;\alpha _{1},\alpha _{2})$ into the supplier’s profit function and maximizing the objective function yields the following optimal wholesale price

$$\begin{aligned} w^{{\mathrm{EN}}}=\frac{\left( \varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-c\right) \theta ^3-\left( 4\varPhi ^{-1}(1-\alpha _{2})+2\varPsi ^{-1}(\alpha _{1})\right) \theta ^2+8\left( \varPhi ^{-1}(1-\alpha _{2})+\varPsi ^{-1}(\alpha _{1})\right) }{2(8-3\theta ^2)}. \end{aligned}$$

So we can obtain the optimal quantities and profits of the retailer and the supplier under encroachment.

Moreover, if the game exists, the quantity must be greater than zero. $q_r^{{\mathrm{EN}}}>0$ requires $c>\widetilde{c}_1$ and $q_s^{{\mathrm{EN}}}>0$ requires $c<\widetilde{c}_2$, where $\widetilde{c}_1=\frac{(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))\theta -(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{\theta }$ and $\widetilde{c}_2=\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{2(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))\theta }{8-\theta ^2}$. Note that $\widetilde{c}_1<\widetilde{c}_2$, hence, game exits if $\widetilde{c}_1<c<\widetilde{c}_2$. The proof of the proposition is completed. $\square$

Proof

(Corollary 1) Taking the first-order condition of the equilibrium results yields

$\qquad \qquad \quad \frac{\partial w^{{\mathrm{EN}}}}{\partial c}=\frac{\theta ^3}{2(3\theta ^2-8)}<0, \frac{\partial q_r^{{\mathrm{EN}}}}{\partial c}=\frac{2\theta }{8-3\theta ^2}>0, \frac{\partial q_s^{{\mathrm{EN}}}}{\partial c}=\frac{8-\theta ^2}{2(3\theta ^2-8)}<0$.

Thus the wholesale price $w^{{\mathrm{EN}}}$ and the supplier’s optimal direct selling quantity $q_{s}^{{\mathrm{EN}}}$ always decrease with the supplier’s selling cost c, while the retailer’s optimal order quantity $q_{r}^{{\mathrm{EN}}}$ always increases with the supplier’s selling cost c.

Similarly, we have $\frac{\partial \pi _r^{{\mathrm{EN}}}}{\partial c} =-\frac{8\theta \left( (\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-c) \theta -\varPhi ^{-1}(1-\alpha _{2})+\varPsi ^{-1}(\alpha _{1})\right) }{(3\theta ^2-8)^2} >0$, thus the retailer’s optimal profit $\pi _{r}^{{\mathrm{EN}}}$ always increases with the supplier’s selling cost c. $\frac{\partial \pi _s^{{\mathrm{EN}}}}{\partial c}<0$ if $c<\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{4\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{8+\theta ^2}$ and $\frac{\partial \pi _s^{{\mathrm{EN}}}}{\partial c}>0$ if $c>\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{4\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{8+\theta ^2}$. Denote $\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{4\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{8+\theta ^2}$ as $\overline{c}$, combining with $\widetilde{c}_1<c<\widetilde{c}_2$, hence, the supplier’s optimal profit $\pi _s^{{\mathrm{EN}}}$ is decreases if $c\in (\widetilde{c}_1,\overline{c})$, and increases otherwise. The proof of the corollary is completed. $\square$

Proof

(Corollary 2) We take the first-order condition of the equilibrium results and obtain $\frac{\partial w^{{\mathrm{EN}}}}{\partial \theta }>0$ if $\widetilde{c}_1<c<\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})+\frac{16(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{3\theta (\theta ^2-8)}$ and $\frac{\partial w^{{\mathrm{EN}}}}{\partial \theta }<0$ if $\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})+\frac{16(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{3\theta (\theta ^2-8)}<c<\widetilde{c}_2$. $\frac{\partial q_r^{{\mathrm{EN}}}}{\partial \theta }<0$ and $\frac{\partial \pi _r^{{\mathrm{EN}}}}{\partial \theta }<0$ if $\widetilde{c}_1<c<\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{6(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))\theta }{8+3\theta ^2}$ while $\frac{\partial q_r^{{\mathrm{EN}}}}{\partial \theta }>0$ and $\frac{\partial \pi _r^{{\mathrm{EN}}}}{\partial \theta }>0$ if $\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{6(\theta \varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{8+3\theta ^2}<c<\widetilde{c}_2$. $\frac{\partial q_s^{{\mathrm{EN}}}}{\partial \theta }>0$ if $\widetilde{c}_1<c<\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))(8+3\theta ^2)}{16\theta }$ and $\frac{\partial q_s^{{\mathrm{EN}}}}{\partial \theta }<0$ if $\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))(8+3\theta ^2)}{16\theta }<c<\widetilde{c}_2$. $\frac{\partial \pi _s^{{\mathrm{EN}}}}{\partial \theta }<0$ if $\widetilde{c}_1<c<\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{3\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{8}$ and $\frac{\partial \pi _s^{{\mathrm{EN}}}}{\partial \theta }>0$ if $\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{3\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{8}<c<\widetilde{c}_2$. The proof of the corollary is completed. $\square$

Proof

(Proposition 3) When $\widetilde{c}_1<c<\widetilde{c}_2$, both NN and ES games exit. Comparing $\pi _{r}^{{\mathrm{EN}}}$ and $\pi _{r}^{{\mathrm{NN}}}$ yields

$$\begin{aligned} \pi _{r}^{{\mathrm{EN}}}-\pi _{r}^{{\mathrm{NN}}}=\frac{\left( -8c\theta +8\left( \varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})\right) \theta -\left( \varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})\right) (16-3\theta ^2)\right) \left( -8c+8\left( \varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})\right) -3\left( \varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})\right) \theta \right) \theta }{16(8-3\theta ^2)^2}. \end{aligned}$$

Then, the sign of $\pi _{r}^{{\mathrm{EN}}}-\pi _{r}^{{\mathrm{NN}}}$ depends on that of two roots

$$\begin{aligned} \begin{array}{ll} \widetilde{c}_{r1}=\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))(16- 3\theta ^2)}{8\theta },\\ \widetilde{c}_{r2}=\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-\frac{3\theta \left( \varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})\right) }{8}. \end{array} \end{aligned}$$

Further, we can show that $\widetilde{c}_{r1}<\widetilde{c}_{1}<\widetilde{c}_{r2}<\widetilde{c}_{2}$. Thus, $\pi _{r}^{{\mathrm{EN}}}<\pi _{r}^{{\mathrm{NN}}}$ if $\widetilde{c}_{1}<c<\widetilde{c}_{r2}$, and $\pi _{r}^{{\mathrm{EN}}}>\pi _{r}^{{\mathrm{NN}}}$ if $\widetilde{c}_{r2}<c<\widetilde{c}_{2}$. Next, comparing the supplier’s profit under two games, we have

$\pi _{s}^{{\mathrm{EN}}}-\pi _{s}^{{\mathrm{NN}}}=\frac{1}{8(8-3\theta ^2)}((2\theta ^2+16)c^2-((\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))(4\theta ^2+32)+16(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))\theta )c+(2 (\varPhi ^{-1}(1-\alpha _{1}))^2-4 \varPhi ^{-1}(1-\alpha _{1}) c+3 (\varPhi ^{-1}(1-\alpha _{2}))^2-6 \varPhi ^{-1}(1-\alpha _{2}) c+5 (\varPsi ^{-1}(\alpha _{1}))^2)\theta ^2-16(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})))\theta +16(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))^2$.

Then, the sign of $\pi _{s}^{{\mathrm{EN}}}-\pi _{s}^{{\mathrm{NN}}}$ depends on that of two roots

$$\begin{aligned} \begin{array}{ll} \widetilde{c}_{s1}=\frac{(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))(16+2\theta ^2)-8(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))-\sqrt{2(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))^2(8-3\theta ^2)\theta ^2}}{16+2\theta ^2},\\ \widetilde{c}_{s2}=\frac{(\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))(16+2\theta ^2)-8(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))+\sqrt{2(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))^2(8-3\theta ^2)\theta ^2}}{16+2\theta ^2}. \end{array} \end{aligned}$$

And we have $\widetilde{c}_{1}<\widetilde{c}_{s1}<\widetilde{c}_{2}<\widetilde{c}_{s2}$. Therefore, $\pi _{s}^{{\mathrm{EN}}}>\pi _{s}^{{\mathrm{NN}}}$ if $\widetilde{c}_{1}<c<\widetilde{c}_{s1}$, and $\pi _{s}^{{\mathrm{EN}}}<\pi _{s}^{{\mathrm{NN}}}$ if $\widetilde{c}_{s1}<c<\widetilde{c}_{2}$. The proof of the proposition is completed. $\square$

Proof

(Theorem 3) The proof of this theorem is similar to that of Theorem 1. $\square$

Proof

(Proposition 4) Here, similar to the proof of Propositions 1 and 2, we also apply the backward induction to solve this problem. First, in Case NS, a retailer’s production decision is the same as that in Proposition 1 and the optimal quantity is

$$\begin{aligned} q_r^{{\mathrm{NS}}}(w;\alpha _{2})=\frac{\varPhi ^{-1}(1-\alpha _{2})+w}{2}. \end{aligned}$$

Next, substituting the retailer’s production decision $q_r^{{\mathrm{NS}}}(w;\alpha _{2})$ into the supplier’s profit function and maximizing the objective function yield the following optimal wholesale price

$$\begin{aligned} w^{{\mathrm{NS}}}(I;\alpha _{1},\alpha _{2})=\frac{\varPhi ^{-1}(1-\alpha _{2})+\varPsi ^{-1}(\alpha _{1})(1-I)}{2}. \end{aligned}$$

Finally, substituting the above production quantity and wholesale price decision into the supplier’s profit function yields the following optimal investment level

$$\begin{aligned} I^{{\mathrm{NS}}}=\frac{\varPsi ^{-1}(\alpha _{1})\left( \varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})\right) }{4k-\left( \varPsi ^{-1}(\alpha _{1})\right) ^2}. \end{aligned}$$

So we can obtain the optimal quantities and profits of the retailer and the supplier under encroachment. The proof of the proposition is completed. $\square$

Proof

(Corollary 3) Taking the first-order condition of the equilibrium results yields

$$\begin{aligned}&\frac{\partial q_r^{{\mathrm{NS}}}}{\partial k}=-\,\frac{(\varPhi ^{-1}(1-\alpha _{2}) -\varPsi ^{-1}(\alpha _{1}))(\varPsi ^{-1}(\alpha _{1}))^2}{((\varPsi ^{-1} (\alpha _{1}))^2-4k)^2}<0,\\&\frac{\partial g^{{\mathrm{NS}}}}{\partial k}=-\,\frac{4\varPsi ^{-1}(\alpha _{1})(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)^2}<0,\\&\frac{\partial w^{{\mathrm{NS}}}}{\partial k}=\frac{(2(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))) (\varPsi ^{-1}(\alpha _{1}))^2}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)^2}>0,\\&\frac{\partial \pi _{r}^{{\mathrm{NS}}}}{\partial k}=\frac{2k(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))^2(\varPsi ^{-1}(\alpha _{1}))^2}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)^3}<0,\\&\frac{\partial \pi _{s}^{{\mathrm{NS}}}}{\partial k}=-\,\frac{(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1} (\alpha _{1}))^2(\varPsi ^{-1}(\alpha _{1}))^2}{2((\varPsi ^{-1}(\alpha _{1}))^2-4k)^2}<0. \end{aligned}$$

So we derive Corollary 3 based on above equation. $\square$

Proof

(Theorem 4) The proof of this theorem is similar to that of Theorem 1. $\square$

Proof

(Proposition 5) First, solving the first-order conditions in the profit function of the retailer $\pi _{r}\left( q_{s},q_{r},w,I;\varPhi ^{-1}(1-\alpha _{2})\right)$ and the supplier $\pi _{s}\left( q_{s},q_{r},w,I;\varPhi ^{-1}(1-\alpha _{1}),\varPsi ^{-1}(\alpha _{1})\right)$, we obtain the equilibrium quantities as a function of the wholesale price and investment level as follows

$$\begin{aligned} q_r^{{\mathrm{ES}}}(w,I;\alpha _{1},\alpha _{2})= & {} \frac{2(\varPhi ^{-1}(1-\alpha _{2})-w)-(\varPhi ^{-1}(1-\alpha _{1})-c)\theta +\theta \varPsi ^{-1}(\alpha _{1})(1-I)}{4-\theta ^2},\\ q_s^{{\mathrm{ES}}}(w,I;\alpha _{1},\alpha _{2})= & {} \frac{2(\varPhi ^{-1}(1-\alpha _{1})-c)-2\varPsi ^{-1}(\alpha _{1})(1-I)-\theta (\varPhi ^{-1}(1-\alpha _{2})- w}{4-\theta ^2}. \end{aligned}$$

Next, substituting the production decision $q_r^{{\mathrm{ES}}}(w,I;\alpha _{1},\alpha _{2})$ and $q_s^{{\mathrm{ES}}}(w,I;\alpha _{1},\alpha _{2})$ into the supplier’s profit function and maximizing the objective function yield the following optimal wholesale price

$$\begin{aligned} w^{{\mathrm{ES}}}=\frac{\left( \varPhi ^{-1}(1-\alpha _{1})-c-\varPsi ^{-1}(\alpha _{1})(1-I)\right) \theta ^{3}-\left( 4\varPhi ^{-1}(1-\alpha _{2})+2\varPsi ^{-1}(\alpha _{1})(1-I)\right) \theta ^2+8\left( \varPhi ^{-1}(1-\alpha _{2})+\varPsi ^{-1}(\alpha _{1})(1-I)\right) }{2(8-3\theta ^2)}. \end{aligned}$$

Subsequent calculation steps with Proposition 5. Substituting the above production quantity and wholesale price decision into the supplier’s profit function yields the optimal investment level. The optimal investment level yields the optimal wholesale price, production quantities and firms’ profit under encroachment with the spillover effect.

Moreover, if the game exits, the quantity must be greater than zero. $q_r^{{\mathrm{ES}}}=0$ yields

$$\begin{aligned} \widetilde{c}_3=\frac{ 2\left( (\varPhi ^{-1}(1-\alpha _{2}) - \varPsi ^{-1}(\alpha _{1}))-2\theta (\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))\right) +(\varPhi ^{-1}(1-\alpha _{1})-\varPhi ^{-1}(1-\alpha _{2}))(\varPsi ^{-1}(\alpha _{1}))^2}{(\varPsi ^{-1}(\alpha _{1}))^2-2k\theta }, \end{aligned}$$

and $q_s^{{\mathrm{ES}}}=0$ yields

$$\begin{aligned} \widetilde{c}_4=\frac{\left( (\varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1}))(8-\theta ^2)-2(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))\theta \right) k-(\varPhi ^{-1}(1-\alpha _{1})-\varPhi ^{-1}(1-\alpha _{2}))(2-\theta )}{(8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2}. \end{aligned}$$

In addition, we discuss the situation of supplier investment. From

$$\begin{aligned}I^{{\mathrm{ES}}}=\frac{\left( \left( \varPhi ^{-1}(1-\alpha _{1})-\varPsi ^{-1}(\alpha _{1})-c\right) (\theta ^2-4\theta +8)+\left( \varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})\right) (1-\theta )\right) \varPsi ^{-1}(\alpha _{1})}{2(8-3\theta ^2))k-( 12-8\theta +\theta ^2)(\varPsi ^{-1}(\alpha _{1}))^2}, \end{aligned}$$

the upper molecule is larger than zero, so the denominator $2(8-3\theta ^2))k-( 12-8\theta +\theta ^2)(\varPsi ^{-1}(\alpha _{1}))^2$ is greater than zero, i.e., $k>\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}$. Comparing $\widetilde{c}_4$ and $\widetilde{c}_3$ yields

$$\begin{aligned}\widetilde{c}_4-\widetilde{c}_3=\frac{(2(8-3\theta ^2))k-( \theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2)(\varPhi ^{-1}(1-\alpha _2)-\varPsi ^{-1}(\alpha _1))k}{((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )}. \end{aligned}$$

Upper size dependent factor $(2(8-3\theta ^2))k-(\theta ^2-8\theta +12)$, $((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)$ and $((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )$. In addition, $\frac{(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{8-\theta }<\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<$$\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$; then, we have when (i) $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$ and $c<\widetilde{c}_{4}$; (ii) $\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{\theta }$ and $\widetilde{c}_{3}<c<\widetilde{c}_{4}$, the supplier will encroach the consumer market, and both the supplier and the retailer are existing in the consumer market. The proof of the proposition is completed. $\square$

Proof

(Corollary 4) Taking the first-order condition of the equilibrium results yields

$$\begin{aligned} \begin{array}{l} \frac{\partial I^{{\mathrm{ES}}}}{\partial c}=\frac{(\theta ^2-4\theta +8)\varPsi ^{-1}(\alpha _{1})}{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2+(6\theta ^2-16)k}<0,\\ \frac{\partial q_s^{{\mathrm{ES}}}}{\partial c}=\frac{(8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2+(6\theta ^2-16)k}<0. \end{array} \end{aligned}$$

Hence, the supplier’s investment intensity $I^{{\mathrm{ES}}}$ and the direct selling quantity $q_s^{{\mathrm{ES}}}$ always decrease with the supplier’s selling cost c. And

$$\begin{aligned} \frac{\partial q_r^{{\mathrm{ES}}}}{\partial c}=\frac{2((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )}{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2+(6\theta ^2-16)k}. \end{aligned}$$

We can find that $\frac{\partial q_r^{{\mathrm{ES}}}}{\partial c}<0$ if $k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$ while $\frac{\partial q_r^{{\mathrm{ES}}}}{\partial c}>0$ if $k>\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$, combining with Proposition 5 that $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k$, so the retailer’s order quantity $q_r^{{\mathrm{ES}}}$ decrease in $k\in \left( \frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta )}, \frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }\right)$ and increase in $k\in \left( \frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }, +\infty \right)$. Because $\pi _r^{{\mathrm{ES}}}=(q_r^{{\mathrm{ES}}})^2$, $\pi _r^{{\mathrm{ES}}}$ is the same as $q_r^{{\mathrm{ES}}}$ on parameter monotonicity.

Next, we consider the trend of $w^{{\mathrm{ES}}}$, first-order condition is

$$\begin{aligned}\frac{\partial w^{{\mathrm{ES}}}}{\partial c}=\frac{k\theta ^3-(\theta ^2-2\theta +4)(\varPsi ^{-1}(\alpha _{1}))^2}{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2+(6\theta ^2-16)k}. \end{aligned}$$

Thus, the wholesale price $w^{{\mathrm{ES}}}$ increases in $k\in \left( \frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta )}, \frac{(\theta ^2-2\theta +4)(\varPsi ^{-1}(\alpha _{1}))^2}{\theta ^3}\right)$ and decreases in $k\in \left( \frac{(\theta ^2-2\theta +4)(\varPsi ^{-1}(\alpha _{1}))^2}{\theta ^3}, +\infty \right)$.

With respect to $\pi _s^{{\mathrm{ES}}}$, we have $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}=\frac{(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)c+R}{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2+(6\theta ^2-16)k}$, here

$$\begin{aligned} R= & {} (2(\varPsi ^{-1}(\alpha _{1}))^2-4k\theta )\varPhi ^{-1}(1-\alpha _{2})\\&-(2(\varPsi ^{-1}(\alpha _{1}))^2-8k-k\theta ^2)\varPhi ^{-1}(1-\alpha _{1})\\&-k(\theta ^2-4\theta +8)\varPsi ^{-1}(\alpha _{1}). \end{aligned}$$

Solving $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}=0$ yields the feasible root $\widehat{c}=\frac{-R}{2(\varPsi ^{-1}(\alpha _{1}))^2-(\theta ^2+8)k}$, and $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}<0$ if $c<\widehat{c}$ and $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}>0$ otherwise. In addition, $\widehat{c}<\widetilde{c}_4$ if $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$ and $\widetilde{c}_3<\widehat{c}<\widetilde{c}_4$ if $\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }<k$, we can obtain the monotonicity of $\pi _s^{{\mathrm{ES}}}$ about c, we can obtain

(a) When $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$, then $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}<0$ if $c<\widehat{c}$ and $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}>0$ if $\widehat{c}<c<\widetilde{c}_4$;

(b) When $\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }<k$, then $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}<0$ if $\widetilde{c}_3<c<\widehat{c}$ and $\frac{\partial \pi _s^{{\mathrm{ES}}}}{\partial c}>0$ if $\widehat{c}<c<\widetilde{c}_4$. The proof of the corollary is completed. $\square$

Proof

(Corollary 5) The proof of this corollary is similar to that of Corollary 4. $\square$

Proof

(Proposition 6) From comparison of the retailer’s profit under NS and ES games, we find that the size of $\pi _r^{{\mathrm{ES}}}-\pi _r^{{\mathrm{NS}}}$ depends on two roots

$$\begin{aligned} \begin{array}{l} \widetilde{c}_{r3}=\frac{S_1}{2(\varPsi ^{-1}(\alpha _{1}))^4-4(\theta +2)k(\varPsi ^{-1}(\alpha _{1}))^2+16k^2\theta },\\ \widetilde{c}_{r4}=\frac{S_2}{2(\varPsi ^{-1}(\alpha _{1}))^4-4(\theta +2)k(\varPsi ^{-1}(\alpha _{1}))^2+16k^2\theta }. \end{array} \end{aligned}$$

And two intermediate parts $\pi _r^{{\mathrm{ES}}}-\pi _r^{{\mathrm{NS}}}$ are less than zero, the other part $\pi _r^{{\mathrm{ES}}}-\pi _r^{{\mathrm{NS}}}$ is greater than zero, where

$$\begin{aligned} S_1= & {} (2(\varPsi ^{-1}(\alpha _{1}))^4-4(\theta -2)k(\varPsi ^{-1} (\alpha _{1}))^2+16k^2\theta )\varPhi ^{-1}(1-\alpha _{1})\\&+\,(2(\varPsi ^{-1}(\alpha _{1}))^4+(\theta -8)k\theta (\varPsi ^{-1}(\alpha _{1}))^2\\&+\,6k^2\theta ^2)\varPhi ^{-1}(1-\alpha _{2})+(\theta ^2-4\theta +8)k (\varPsi ^{-1}(\alpha _{1}))^3\\&+\,(6\theta -16)k^2\theta \varPsi ^{-1}(\alpha _{1}),\\ S_2= & {} (2(\varPsi ^{-1}(\alpha _{1}))^4-4(\theta +2)k(\varPsi ^{-1} (\alpha _{1}))^2+16k^2\theta )\varPhi ^{-1}(1-\alpha _{1})\\&-\,(2(\varPsi ^{-1}(\alpha _{1}))^4-(\theta ^2-8\theta +24)k(\varPsi ^{-1}(\alpha _{1}))^2\\&-\,(6\theta ^2-32)k^2)\varPhi ^{-1}(1-\alpha _{2})\\&-\,(\theta ^2-12\theta +16)k(\varPsi ^{-1}(\alpha _{1}))^3+(6\theta ^2+16\theta -32)k^2\varPsi ^{-1}(\alpha _{1}). \end{aligned}$$

Next, we need to compare $\widetilde{c}_{r3}$, $\widetilde{c}_{r4}$, $\widetilde{c}_{3}$ and $\widetilde{c}_{4}$, as shown in the following.

$\bullet$$\widetilde{c}_{r3}-\widetilde{c}_{4}=\frac{k\theta ((\varPsi ^{-1}(\alpha _{1}))^2-k\theta )(2(8-3\theta ^2))k-( \theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2)(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{2((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )((\varPsi ^{-1}(\alpha _{1}))^2-4k)((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)}$,

$\bullet$$\widetilde{c}_{r3}-\widetilde{c}_{3}=\frac{k(2(8-3\theta ^2))k-( 12-8\theta +\theta ^2)(\varPsi ^{-1}(\alpha _{1}))^2)(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{2((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )((\varPsi ^{-1}(\alpha _{1}))^2-4k)}$,

$\bullet$$\widetilde{c}_{r4}-\widetilde{c}_{3}=-\frac{k(2(8-3\theta ^2))k-( 12-8\theta +\theta ^2)(\varPsi ^{-1}(\alpha _{1}))^2)(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))}{2((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )((\varPsi ^{-1}(\alpha _{1}))^2-4k)}$.

Through comparison of $\widetilde{c}_{3}$ and $\widetilde{c}_{4}$ in combination with Proposition 5, we find that the relationship between the size of $\widetilde{c}_{r3}$, $\widetilde{c}_{r4}$, $\widetilde{c}_{3}$, and $\widetilde{c}_{4}$ depends on the factors of $(2(8-3\theta ^2))k-(\theta ^2 -8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2$, $((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)$, $((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )$ and $((\varPsi ^{-1}(\alpha _{1}))^2-k\theta )$. In addition, $\frac{(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{8-\theta }<\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{\theta }$; then, we have: (a) When $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$, then $\widetilde{c}_{r3}<\widetilde{c}_{4}<\widetilde{c}_{3}<\widetilde{c}_{r4}$; (b) When $\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{\theta }$, then $\widetilde{c}_{r4}<\widetilde{c}_{3}<\widetilde{c}_{4}<\widetilde{c}_{r3}$; (c) When $\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{\theta }<k$, then $\widetilde{c}_{r4}<\widetilde{c}_{3}<\widetilde{c}_{r3}<\widetilde{c}_{4}$.

In summary, we have

(i) When $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}{(\alpha _1)})^2}{2(8-3\theta ^2)}<k<\frac{(\varPsi ^{-1}{(\alpha _1)})^2}{2\theta }$, we have $\pi ^{{\mathrm{ES}}}_r>\pi ^{{\mathrm{NS}}}_r$ if $c<\widetilde{c}_{r3}$ and $\pi ^{{\mathrm{ES}}}_r<\pi ^{{\mathrm{NS}}}_r$ if $\widetilde{c}_{r3}<c<\widetilde{c}_4$;

(ii) When $\frac{(\varPsi ^{-1}{(\alpha _1)})^2}{\theta }<k$, we have $\pi ^{{\mathrm{ES}}}_r<\pi ^{{\mathrm{NS}}}_r$ if $\widetilde{c}_3<c<\widetilde{c}_{r3}$ and $\pi ^{{\mathrm{ES}}}_r>\pi ^{{\mathrm{NS}}}_r$ if $\widetilde{c}_{r3}<c<\widetilde{c}_4$;

(iii) When $\frac{(\varPsi ^{-1}{(\alpha _1)})^2}{2\theta }<k<\frac{(\varPsi ^{-1}{(\alpha _1)})^2}{\theta }$, we have $\pi ^{{\mathrm{ES}}}_r<\pi ^{{\mathrm{NS}}}_r$.

With respect to the supplier’s profit, we find that the size of $\pi _s^{{\mathrm{ES}}}-\pi _s^{{\mathrm{NS}}}$ is also dependent on two roots

$$\begin{aligned} \begin{array}{l} \widetilde{c}_{s3}=\frac{T_1}{\left( (\varPsi ^{-1}(\alpha _{1}))^2-4\,k \right) (2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)},\\ \widetilde{c}_{s4}=\frac{T_2}{\left( (\varPsi ^{-1}(\alpha _{1}))^2-4\,k \right) (2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)}. \end{array} \end{aligned}$$

And when $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k$, two intermediate parts $\pi _s^{{\mathrm{ES}}}-\pi _s^{{\mathrm{NS}}}$ are less than zero, the other part $\pi _s^{{\mathrm{ES}}}-\pi _s^{{\mathrm{NS}}}$ is greater than zero. Where

$$\begin{aligned} T_1= & {} ((4(\theta ^2+8))\varPhi ^{-1}(1-\alpha _{1}) -(4(\theta ^2-4\theta +8))\varPsi ^{-1}(\alpha _{1})-16\varPhi ^{-1}(1-\alpha _{2})\theta )k^2-((\theta ^2+16)\varPhi ^{-1}(1-\alpha _{1})\\&-(4(\theta +2))\varPhi ^{-1}(1-\alpha _{2})-(\theta ^2- 4\theta +8)\varPsi ^{-1}(\alpha _{1}))(\varPsi ^{-1}(\alpha _{1}))^2k+(2(\varPhi ^{-1}(1-\alpha _{1})-\varPhi ^{-1}(1-\alpha _{2})))(\varPsi ^{-1}(\alpha _{1}))^4\\&+k\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})) \sqrt{((\varPsi ^{-1}(\alpha _{1}))^2-4k)(\theta ^2-8\theta +12) (\varPsi ^{-1}(\alpha _{1}))^2-(2(8-3\theta ^2))k}),\\ T_2= & {} ((4(\theta ^2+8))\varPhi ^{-1}(1-\alpha _{1}) -(4(\theta ^2-4\theta +8))\varPsi ^{-1}(\alpha _{1})-16\varPhi ^{-1}(1-\alpha _{2})\theta )k^2+((\theta ^2+16)\varPhi ^{-1}(1-\alpha _{1})\\&-(4(\theta +2))\varPhi ^{-1}(1-\alpha _{2})-(\theta ^2-4\theta +8)\varPsi ^{-1}(\alpha _{1}))(\varPsi ^{-1}(\alpha _{1}))^2k+(2(\varPhi ^{-1}(1-\alpha _{1})-\varPhi ^{-1}(1-\alpha _{2}))) (\varPsi ^{-1}(\alpha _{1}))^4\\&-k\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1})) \sqrt{((\varPsi ^{-1}(\alpha _{1}))^2-4k)(\theta ^2-8\theta +12) (\varPsi ^{-1}(\alpha _{1}))^2-(2(8-3\theta ^2))k}). \end{aligned}$$

Next, we need to compare $\widetilde{c}_{s3}$, $\widetilde{c}_{s4}$, $\widetilde{c}_{3}$ and $\widetilde{c}_{4}$, as shown in the following

$\bullet$ $\widetilde{c}_{s3}-\widetilde{c}_{4}=\frac{k\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))((2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2-(8-\theta ^2)k)\sqrt{k((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2))}}{((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2))((\varPsi ^{-1}(\alpha _{1}))^2-4k)}$

$\qquad \qquad -\frac{k\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2)k)}{((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)((\varPsi ^{-1}(\alpha _{1}))^2-4k)}$,

$\bullet$ $\widetilde{c}_{s4}-\widetilde{c}_{4}=\frac{k\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)\sqrt{k((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2))}}{((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)((\varPsi ^{-1}(\alpha _{1}))^2-4k)}$

$\qquad \qquad -\frac{k\theta (\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2))}{((8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2)(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)((\varPsi ^{-1}(\alpha _{1}))^2-4k)}$,

$\bullet$ $\widetilde{c}_{s4}-\widetilde{c}_{3}=\frac{-k\theta (\varPhi ^{-1}(1-\alpha _{2})-c)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )\sqrt{((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2))}}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)}$

$\qquad \qquad \quad -\frac{k(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2)k)}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)}$,

$\bullet$ $\widetilde{c}_{s3}-\widetilde{c}_{3}=\frac{k\theta (\varPhi ^{-1}(1-\alpha _{2})-c)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )\sqrt{((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2))}}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)}$

$\qquad \qquad \quad -\frac{k(\varPhi ^{-1}(1-\alpha _{2})-\varPsi ^{-1}(\alpha _{1}))((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2-2(8-3\theta ^2)k)}{((\varPsi ^{-1}(\alpha _{1}))^2-4k)((\varPsi ^{-1}(\alpha _{1}))^2-2k\theta )(2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k)}$.

After comparing $\widetilde{c}_{3}$ and $\widetilde{c}_{4}$ in combination with Proposition 5, we find that the relationship between the size of $\widetilde{c}_{s3}$, $\widetilde{c}_{s4}$, $\widetilde{c}_{3}$ and $\widetilde{c}_{4}$ depends on the factors of $(8-\theta ^2)k-(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2$, $2(\varPsi ^{-1}(\alpha _{1}))^2-(8+\theta ^2)k$, $(\varPsi ^{-1}(\alpha _{1}))^2-4k$, $2(8-3\theta ^2))k-(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2$, $(2\theta -3)(\varPsi ^{-1}(\alpha _{1}))^2+(4-2\theta ^2)k$ and $(\varPsi ^{-1}(\alpha _{1}))^2-2k\theta$. In addition, $\frac{(2-\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{8-\theta }<\frac{2(\varPsi ^{-1}(\alpha _{1}))^2}{8+\theta ^2}<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{4}<\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<\frac{(3-2\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{2(2- \theta ^2}<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$. Then we have: (a) When $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}(\alpha _{1}))^2}{2(8-3\theta ^2)}<k<\frac{(3-2\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{2(2- \theta ^2}$, then $\widetilde{c}_{s4}<\widetilde{c}_{4}<\widetilde{c}_{3}<\widetilde{c}_{s3}$; (b) When $\frac{(3-2\theta )(\varPsi ^{-1}(\alpha _{1}))^2}{2(2- \theta ^2}<k<\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }$, then $\widetilde{c}_{s4}<\widetilde{c}_{4}<\widetilde{c}_{s3}<\widetilde{c}_{3}$; (c) When $\frac{(\varPsi ^{-1}(\alpha _{1}))^2}{2\theta }<k$, then $\widetilde{c}_{3}<\widetilde{c}_{s4}<\widetilde{c}_{4}<\widetilde{c}_{s3}$.

In summary, we have

(i) When $\frac{(\theta ^2-8\theta +12)(\varPsi ^{-1}{(\alpha _1})^2}{2(8-3\theta ^2)}<k<\frac{(\varPsi ^{-1}{(\alpha _1)})^2}{2\theta }$, we have $\pi ^{{\mathrm{ES}}}_s>\pi ^{{\mathrm{NS}}}_s$ if $c<\widetilde{c}_{s4}$ and $\pi ^{{\mathrm{ES}}}_s<\pi ^{{\mathrm{NS}}}_s$ if $\widetilde{c}_{s4}<c<\widetilde{c}_{4}$;

(ii) When $\frac{(\varPsi ^{-1}{(\alpha _1)})^2}{2\theta }<k$, we have $\pi ^{{\mathrm{ES}}}_s>\pi ^{{\mathrm{NS}}}_s$ if $\widetilde{c}_{3}<c<\widetilde{c}_{s4}$ and $\pi ^{{\mathrm{ES}}}_s<\pi ^{{\mathrm{NS}}}_s$ if $\widetilde{c}_{s4}<c<\widetilde{c}_{4}$.

The proof of the proposition is completed. $\square$