Risk-aversion information in a supply chain with price and warranty competition

Abstract

We consider a supply chain with one supplier providing basic products to two risk-averse retailers. Each retailer provides a warranty policy for the products to differentiate their products along with pricing strategy. Considering that the risk-aversion level of each retailer is private in most applications, we assume that the retailers’ risk-aversion levels are their respective private information in this paper. Each retailer may choose to disclose her private risk-aversion information to the supplier. We explore the risk-aversion information revelation mechanism of the retailers and propose the supplier’s decisions on whether to accept the revealed information. Our results show that the warranty period added to the product by each retailer is only related to each retailer’s warranty cost efficiency and the consumers’ sensitivity toward warranty. Each retailer is willing to share her risk-aversion level with the supplier only when the real value of her risk-aversion level is lower than the mean value of other participants’ estimation under both the no information leakage scenario and the information leakage scenario. Furthermore, we find that the information leakage has a negative effect on the supplier’s profit if he accepts the shared risk-aversion information.

Appendix A. Proofs

Proof of Proposition 1

For the retailer 1:

$$ \begin{aligned} \mathop {Max}\limits_{{p_{ 1} , t_{ 1} }} E\left[ {U_{{R_{ 1} }} } \right] & = \mathop \int \limits_{{\mu_{ 2} - \varepsilon_{ 2} }}^{{\mu_{ 2} + \varepsilon_{ 2} }} \left( {\left( {p_{ 1} - w - k_{ 1} t_{ 1}^{ 2} } \right)D_{ 1} - \eta_{ 1} \left( {p_{ 1} - w - k_{ 1} t_{ 1}^{ 2} } \right){\upsigma }} \right)f\left( {\eta_{ 2} } \right)d\eta_{ 2} \\ & = \frac{ 1}{{ 2\varepsilon_{ 2} }}\mathop \int \limits_{{\mu_{ 2} - \varepsilon_{ 2} }}^{{\mu_{ 2} + \varepsilon_{ 2} }} \left( {\left( {p_{ 1} - w - k_{ 1} t_{ 1}^{ 2} } \right)D_{ 1} - \eta_{ 1} \left( {p_{ 1} - w - k_{ 1} t_{ 1}^{ 2} } \right){\upsigma }} \right)d\eta_{ 2} .\\ \end{aligned} $$

(A.1)

For the retailer 2:

$$ \begin{aligned} \mathop {Max}\limits_{{p_{ 2} , t_{ 2} }} E\left[ {U_{{R_{ 2} }} } \right] & = \mathop \int \limits_{{\mu_{ 1} - \varepsilon_{ 1} }}^{{\mu_{ 1} + \varepsilon_{ 1} }} \left( {\left( {p_{ 2} - w - k_{ 2} t_{ 2}^{ 2} } \right)D_{ 2} - \eta_{ 2} \left( {p_{ 2} - w - k_{ 2} t_{ 2}^{ 2} } \right){\upsigma }} \right)f\left( {\eta_{ 1} } \right)d\eta_{ 1} \\ & = \frac{ 1}{{ 2\varepsilon_{ 1} }}\mathop \int \limits_{{\mu_{ 1} - \varepsilon_{ 1} }}^{{\mu_{ 1} + \varepsilon_{ 1} }} \left( {\left( {p_{ 2} - w - k_{ 2} t_{ 2}^{ 2} } \right)D_{ 2} - \eta_{ 2} \left( {p_{ 2} - w - k_{ 2} t_{ 2}^{ 2} } \right){\upsigma }} \right)d\eta_{ 1} .\\ \end{aligned} $$

(A.2)

By simultaneously solving Eqs: $\partial E\left[ {U_{{R_{ 1} }} } \right]/\partial p_{ 1} = 0 , \partial E\left[ {U_{{R_{ 1} }} } \right]/\partial t_{ 1} = 0$, we have three different equilibrium solutions that satisfy the necessary condition for the optimal solution. However, considering that we must have $t_{ 1} > 0$, only the following solution is feasible:

$$ p_{ 1}^{*} = \frac{{\alpha - \eta_{ 1} {\upsigma } - \tau \theta t_{ 2} + \left( { 1+ \theta } \right)w + \theta p_{ 2} }}{{ 2\left( { 1+ \theta } \right)}} + \frac{{ 3\tau^{ 2} }}{{ 8k_{ 1} }} , $$

(A.3)

$$ t_{ 1}^{*} = \frac{\tau }{{ 2k_{ 1} }} . $$

(A.4)

The Hessian matrix of $E\left[ {U_{{R_{ 1} }} } \right]$ is as follows:

$$ \left[ {\begin{array}{*{20}c} {\frac{{\partial E\left[ {U_{{R_{ 1} }} } \right]}}{{\partial p_{ 1}^{ 2} }}} & {\frac{{\partial E\left[ {U_{{R_{ 1} }} } \right]}}{{\partial p_{ 1} \partial t_{ 1} }}} \\ {\frac{{\partial E\left[ {U_{{R_{ 1} }} } \right]}}{{\partial t_{ 1} \partial p_{ 1} }}} & {\frac{{\partial E\left[ {U_{{R_{ 1} }} } \right]}}{{\partial t_{ 1}^{ 2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2- 2\theta } & {\left( { 1 { + }\theta } \right)\left( { 2k_{ 1} t_{ 1} + \tau } \right)} \\ {} & { - 2k_{ 1} (\alpha + \left( { 3t_{ 1} \tau - p_{ 1} } \right)\left( { 1 { + }\theta } \right)} \\ {\left( { 1 { + }\theta } \right)\left( { 2k_{ 1} t_{ 1} + \tau } \right)} & { + \theta p_{ 2} - \eta_{ 1} {\upsigma } - \tau \theta t_{ 2} )} \\ \end{array} } \right] . $$

Substituting $p_{ 1}^{*}$ and $t_{ 1}^{*}$ into the Hessian matrix of $E\left[ {U_{{R_{ 1} }} } \right]$, we obtain the Hessian matrix on the equilibrium point as follows:

$$ H_{0} = \left[ {\begin{array}{*{20}c} { - 2- 2\theta } & { 2\tau \left( { 1 { + }\theta } \right)} \\ { 2\tau \left( { 1 { + }\theta } \right)} & { - \left( {\alpha - w} \right)k_{ 1} - \frac{ 9}{ 4}\tau^{ 2} \left( { 1 { + }\theta } \right) - \left( {p_{ 2} - w} \right)\theta + \eta_{ 1} {\upsigma } + \tau \theta t_{ 2} } \\ \end{array} } \right] . $$

Hence, we have

$$ \left| {H_{0} } \right| = \frac{ 1}{ 2}\left( { 1 { + }\theta } \right)\left( { 4\left( {\alpha - w} \right)k_{ 1} + 4\left( {p_{ 2} - w} \right)\theta + \tau^{ 2} \left( { 1 { + }\theta } \right) - 4\eta_{ 1} {\upsigma } - 4\tau \theta t_{ 2} } \right) . $$

It is obvious that $- 2- 2\theta < 0$, and $- \left( {\alpha - w} \right)k_{ 1} - \frac{ 9}{ 4}\tau^{ 2} \left( { 1 { + }\theta } \right) - \left( {p_{ 2} - w} \right)\theta + \eta_{ 1}\upsigma + \tau \theta t_{ 2} < 0$ and $\left| {H_{0} } \right| > 0$ under the assumption that market base $\alpha$ is large enough compared to other parameters, which is a very realistic assumption (Taleizadeh et al. 2017b; Yao et al. 2008; Dai and Chao 2016; Ha et al. 2017). Thus the expected utility function $E\left[ {U_{{R_{ 1} }} } \right]$ of retailer 1 is jointly concave in $p_{ 1}$ and $t_{ 1}$ and we get that $p_{ 1}^{*}$ and $t_{ 1}^{*}$ are the optimal response function of $E\left[ {U_{{R_{ 1} }} } \right]$.

Due to the symmetry, similarly, from the first-order conditions (FOC): $\partial E\left[ {U_{{R_{ 2} }} } \right]/\partial p_{ 2} = 0 , \partial E\left[ {U_{{R_{ 2} }} } \right]/\partial t_{ 2} = 0 ,$ the optimal response function for retailer 2 is:

$$ p_{ 2}^{*} = \frac{{\alpha - \eta_{ 2} {\upsigma } - \tau \theta t_{ 1} + \left( { 1+ \theta } \right)w + \theta p_{ 1} }}{{ 2\left( { 1+ \theta } \right)}} + \frac{{ 3\tau^{ 2} }}{{ 8k_{ 2} }} , $$

(A.5)

$$ t_{ 2}^{*} = \frac{\tau }{{ 2k_{ 2} }} . $$

(A.6)

Eq. (A.3) is the best response function of retailer 1. Because retailer 1 has incomplete information of retailer 2, she has no complete knowledge about retailer 2’s retail price $p_{ 2}$. Thereby retailer 1 finds the expected retail price for retailer 2, i.e.,

$$ p_{ 2} = \mathop \int \limits_{{\mu_{ 2} - \varepsilon_{ 2} }}^{{\mu_{ 2} + \varepsilon_{ 2} }} \left( {\frac{{\alpha - \eta_{ 2} {\upsigma } - \tau \theta t_{ 1} + \left( { 1+ \theta } \right)w + \theta p_{ 1} }}{{ 2\left( { 1+ \theta } \right)}} + \frac{{ 3\tau^{ 2} }}{{ 8k_{ 2} }}} \right)f\left( {\eta_{ 2} } \right)d\eta_{ 2} . $$

(A.7)

Similarly, retailer 2 finds the expected retail price for retailer 1.

$$ p_{ 1} = \mathop \int \limits_{{\mu_{ 1} - \varepsilon_{ 1} }}^{{\mu_{ 1} + \varepsilon_{ 1} }} \left( {\frac{{\alpha - \eta_{ 1} {\upsigma } - \tau \theta t_{ 2} + \left( { 1+ \theta } \right)w + \theta p_{ 2} }}{{ 2\left( { 1+ \theta } \right)}} + \frac{{ 3\tau^{ 2} }}{{ 8k_{ 1} }}} \right)f\left( {\eta_{ 1} } \right)d\eta_{ 1} . $$

(A.8)

Substituting (A.7) into (A.3), (A.8) into (A.5), and solving (A.3), (A.5) simultaneously, we obtain the equilibriums for the retailers:

$$ \begin{aligned} p_{i}^{*} & = \frac{{\left( { 3\theta + 2} \right)\alpha + \left( { 3\theta^{ 2} + 5\theta + 2} \right)w - \left( { 2\theta + 2} \right)\sigma \eta_{i} - \theta \sigma \mu_{ 3- i} }}{{ 3\theta^{ 2} + 8\theta + 4}} \\ & \quad + \frac{{ 2\left( { 2\theta^{ 2} + 6\theta + 3} \right)\tau^{ 2} k_{ 3- i} - \left( {\theta^{ 2} + \theta } \right)\tau^{ 2} k_{i} }}{{ 4k_{i} k_{ 3- i} \left( { 3\theta^{ 2} + 8\theta + 4} \right)}} ,\\ \end{aligned} $$

(A.9)

$$ t_{i}^{*} = \frac{\tau }{{ 2k_{i} }} , \quad i = 1 , 2. $$

(A.10)

Proof of Proposition 2

Substituting Eq. (A.9) and Eq. (A.10) into Eq. (11), Eq. (12) and Eq. (13) respectively, we can obtain the supplier’s expected profit functions under the three different information sharing cases as follows:

$$ \begin{aligned} E\left( {\pi_{\text{M}}^{\text{BI}} } \right) & = \frac{{ 2\left( { 1+ \theta } \right)w\alpha }}{ 2+ \theta } + \frac{{ 2w\sigma \left( { 1+ \theta } \right)\left( {\eta_{ 1} + \eta_{ 2} } \right) + w\sigma \theta \left( {\mu_{ 1} + \mu_{ 2} } \right)}}{{\left( { 2+ \theta } \right)\left( { 3\theta + 2} \right)}} \\ & \quad + \frac{{\left( { 1+ \theta } \right)\left( {k_{ 1} \tau^{ 2} + k_{ 2} \tau^{ 2} - 8k_{ 1} k_{ 2} w} \right)w}}{{ 4\left( { 2+ \theta } \right)k_{ 1} k_{ 2} }} ; \\ \end{aligned} $$

(A.11)

$$ \begin{aligned} E\left( {\pi_{\text{M}}^{\text{OI}} } \right) & = \frac{{ 2\left( { 1+ \theta } \right)w\alpha }}{ 2+ \theta } + \frac{{ 2w\sigma \left( { 1+ \theta } \right)\eta_{i} + w\sigma \left( {\mu_{i} \theta + 2\mu_{ 3- i} + 3\theta \mu_{ 3- i} } \right)}}{{\left( { 2+ \theta } \right)\left( { 3\theta + 2} \right)}} \\ & \quad + \frac{{\left( { 1+ \theta } \right)\left( { - 8k_{i} k_{ 3- i} w + k_{i} \tau^{ 2} + k_{ 3- i} \tau^{ 2} } \right)w}}{{ 4k_{i} k_{ 3- i} \left( { 2+ \theta } \right)}} ;\\ \end{aligned} $$

(A.12)

$$ E\left( {\pi_{\text{M}}^{\text{NI}} } \right) = \frac{{ 2\left( { 1+ \theta } \right)w\alpha }}{ 2+ \theta } + \frac{{w\sigma \left( {\mu_{ 1} + \mu_{ 2} } \right)}}{ 2+ \theta } + \frac{{\left( { 1+ \theta } \right)\left( {k_{ 1} \tau^{ 2} + k_{ 2} \tau^{ 2} - 8k_{ 1} k_{ 2} w} \right)w}}{{ 4\left( { 2+ \theta } \right)k_{ 1} k_{ 2} }} ; $$

(A.13)

where the superscript $BI , OI$ and $NI$ represent the case 1, 2 and 3, respectively. Since $\frac{{d^{ 2} E\left( {\pi_{\text{M}}^{\text{BI}} } \right)}}{{dw^{ 2} }} = \frac{{d^{ 2} E\left( {\pi_{\text{M}}^{\text{OI}} } \right)}}{{dw^{ 2} }} = \frac{{d^{ 2} E\left( {\pi_{\text{M}}^{\text{NI}} } \right)}}{{dw^{ 2} }} = - \frac{{ 4\left( { 1+ \theta } \right)}}{ 2+ \theta } < 0$, $E\left( {\pi_{\text{M}}^{\text{BI}} } \right)$, $E\left( {\pi_{\text{M}}^{\text{OI}} } \right)$ and $E\left( {\pi_{\text{M}}^{\text{NI}} } \right)$ are all concave in $w$. Let $\frac{{dE\left( {\pi_{\text{M}}^{\text{BI}} } \right)}}{dw} = 0$, $\frac{{dE\left( {\pi_{\text{M}}^{\text{OI}} } \right)}}{dw} = 0$, $\frac{{dE\left( {\pi_{\text{M}}^{\text{NI}} } \right)}}{dw} = 0$, then we obtain the optimal wholesale prices of the supplier under the three different information sharing cases as shown in Proposition 2.

Then we discuss the positiveness of the equilibrium results obtained in this paper. From Proposition 1 and 2, we can easily see that the optimal warranty periods of the retailers $t_{i}^{*} > 0$ and supplier’s wholesale prices $w_{j}^{*} > 0$ in the three cases under the no information leakage scenario, where $i = 1 , 2$ and $j = {\text{BI, OI, NI}} .$ In case 1, substituting $w_{BI}^{*}$ into $p_{i}^{*}$ in Proposition 1, we get the retailers’ optimal retail price in the case of both retailers sharing information as follows:

$$ \begin{aligned} p_{i}^{*} & = \frac{{\left( {\theta + 3} \right)\alpha }}{{ 2\left( {\theta + 2} \right)}} + \frac{{ 2\sigma \eta_{ 3- i} \left( {\theta + 1} \right) + \mu_{i} \sigma \theta - 3\sigma \mu_{ 3- i} \theta - 3 \left( { 2\theta + 2} \right)\sigma \eta_{i} }}{{ 4\left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} \\ & \quad + \frac{{\left( {\theta + 2- \theta^{ 2} } \right)\tau^{ 2} k_{i} + \left( { 1 9\theta^{ 2} + 5 3\theta + 2 6} \right)\tau^{ 2} k_{ 3- i} }}{{ 1 6k_{i} k_{ 3- i} \left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} .\\ \end{aligned} $$

(A.14)

Then, substituting $t_{i}^{*}$, $w_{BI}^{*}$ and Eq. (A.14) into Eq. (A.11) and Eq. (10), we can obtain the supplier’s optimal expected profit $\pi_{\text{M}}^{ *} \left( 2\right)$ and retailer $i$’s utility $U_{{R_{i} }} \left( { 1_{s} , 2_{s} } \right)$, $i = 1 , 2.$ We can easily see that Eq. (A.14), $\pi_{\text{M}}^{ *} \left( 2\right)$ and $U_{{R_{i} }} \left( { 1_{s} , 2_{s} } \right)$ are positive when the market base $\alpha$ is large enough compared to other parameters, which is a realistic assumption (Taleizadeh et al. 2017b; Yao et al. 2008; Dai and Chao 2016; Ha et al. 2017). In case 2, substituting ${\text{w}}_{\text{OI}}^{ *}$ into ${\text{p}}_{\text{i}}^{ *}$ in Proposition 1, we get the retailers’ optimal retail price in the case of only retailer ${\text{i}}$ sharing information as follows:

$$ \begin{aligned} p_{i}^{*} & = \frac{{\left( {\theta + 3} \right)\alpha }}{{ 2\left( {\theta + 2} \right)}} + \frac{{\mu_{i} \sigma \theta + \sigma \mu_{ 3- i} \left( { 2- \theta } \right) - 3 \left( { 2\theta + 2} \right)\sigma \eta_{i} }}{{ 4\left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} \\ & + \frac{{\left( {\theta + 2- \theta^{ 2} } \right)\tau^{ 2} k_{i} + \left( { 1 9\theta^{ 2} + 5 3\theta + 2 6} \right)\tau^{ 2} k_{ 3- i} }}{{ 1 6k_{i} k_{ 3- i} \left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} . \\ \end{aligned} $$

(A.15)

$$ \begin{aligned} p_{ 3- i}^{*} & = \frac{{\left( {\theta + 3} \right)\alpha }}{{ 2\left( {\theta + 2} \right)}} + \frac{{ 2\sigma \eta_{i} \left( {\theta + 1} \right) + \sigma \mu_{ 3- i} \left( { 3\theta + 2} \right) - 4 \left( { 2\theta + 2} \right)\sigma \eta_{ 3- i} - 3\mu_{i} \sigma \theta }}{{ 4\left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} \\ & \quad + \frac{{\left( {\theta + 2- \theta^{ 2} } \right)\tau^{ 2} k_{ 3- i} + \left( { 1 9\theta^{ 2} + 5 3\theta + 2 6} \right)\tau^{ 2} k_{i} }}{{ 1 6k_{i} k_{ 3- i} \left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} . \\ \end{aligned} $$

(A.16)

Substituting $t_{i}^{*}$, $w_{OI}^{*}$, Eqs. (A.15) and (A.16) into Eq. (A.12) and Eq. (10), we can obtain the supplier’s optimal expected profit $\pi_{{{\text{M}}i}}^{ *} \left( 1\right)$ and retailer $i$’s utility $U_{{R_{i} }} \left( { 1_{n} , 2_{s} } \right)$ or $U_{{R_{i} }} \left( { 1_{s} , 2_{n} } \right)$, $i = 1 , 2.$ Similarly, we can easily see that Eqs. (A.15) and (A.16), $\pi_{{{\text{M}}i}}^{ *} \left( 1\right)$ and $U_{{R_{i} }} \left( { 1_{n} , 2_{s} } \right)$ or $U_{{R_{i} }} \left( { 1_{s} , 2_{n} } \right)$ are positive under the assumption that the market base $\alpha$ is large enough compared to other parameters. In case 3, substituting $w_{NI}^{*}$ into $p_{i}^{*}$ in Proposition 1, we get the retailers’ optimal retail price in the case of no retailers sharing information as follows:

$$ \begin{aligned} p_{i}^{*} & = \frac{{\left( {\theta + 3} \right)\alpha }}{{ 2\left( {\theta + 2} \right)}} + \frac{{\left( { 3\theta + 2} \right)\mu_{i} \sigma + \sigma \mu_{ 3- i} \left( { 2- \theta } \right) - 4\left( { 2\theta + 2} \right)\sigma \eta_{i} }}{{ 4\left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} \\ & \quad + \frac{{\left( {\theta + 2- \theta^{ 2} } \right)\tau^{ 2} k_{i} + \left( { 1 9\theta^{ 2} + 5 3\theta + 2 6} \right)\tau^{ 2} k_{ 3- i} }}{{ 1 6k_{i} k_{ 3- i} \left( {\theta + 2} \right)\left( { 3\theta + 2} \right)}} .\\ \end{aligned} $$

(A.17)

Substituting $t_{i}^{*}$, $w_{NI}^{*}$ and Eq. (A.17) into Eq. (A.13) and Eq. (10), we can obtain the supplier’s optimal expected profit $\pi_{\text{M}}^{ *} \left( 0\right)$ and retailer $i$’s utility $U_{{R_{i} }} \left( { 1_{n} , 2_{n} } \right)$, $i = 1 , 2.$ We can easily see that Eq. (A.17), $\pi_{\text{M}}^{ *} \left( 0\right)$ and $U_{{R_{i} }} \left( { 1_{n} , 2_{n} } \right)$ are positive when the market base $\alpha$ is large enough compared to other parameters.

Proof of Proposition 3

$$ \pi_{\text{M}}^{ *} \left( 2\right) - \pi_{\text{M}}^{ *} \left( 0\right) = \frac{{\left( { 1+ \theta } \right)\left( {\eta_{ 1} + \eta_{ 2} - \mu_{ 1} - \mu_{ 2} } \right)^{ 2} {\upsigma }^{ 2} }}{{ 2\left( { 2+ \theta } \right)\left( { 3\theta + 2} \right)^{ 2} }} \ge 0 ; $$

$$ \pi_{\text{M}}^{ *} \left( 2\right) - \pi_{{{\text{M}}i}}^{ *} \left( 1\right) = \frac{{\left( { 1+ \theta } \right)\left( {\eta_{ 3- i} - \mu_{ 3- i} } \right)^{ 2} {\upsigma }^{ 2} }}{{ 2\left( { 2+ \theta } \right)\left( { 3\theta + 2} \right)^{ 2} }} \ge 0 ; $$

$$ \pi_{{{\text{M}}i}}^{ *} \left( 1\right) - \pi_{\text{M}}^{ *} \left( 0\right) = \frac{{\left( { 1+ \theta } \right)\left( {\eta_{i} - \mu_{i} } \right)\left( {\eta_{i} + 2\eta_{ 3- i} - \mu_{i} - 2\mu_{ 3- i} } \right){\upsigma }^{ 2} }}{{ 2\left( { 2+ \theta } \right)\left( { 3\theta + 2} \right)^{ 2} }} . $$

When $\eta_{i} > \mu_{i}$, we can obtain $\pi_{{{\text{M}}i}}^{ *} \left( 1\right) > \pi_{\text{M}}^{ *} \left( 0\right)$ if $\mu_{ 3- i} < \frac{{\eta_{i} - \mu_{i} }}{ 2} ,$ or if $\mu_{ 3- i} > \frac{{\eta_{i} - \mu_{i} }}{ 2}$ and $\eta_{ 3- i} > \frac{{\left( { 2\mu_{ 3- i} + \mu_{i} - \eta_{i} } \right)}}{ 2}$; when $\eta_{i} < \mu_{i}$, we can obtain $\pi_{{{\text{M}}i}}^{ *} \left( 1\right) > \pi_{\text{M}}^{ *} \left( 0\right)$ if $\eta_{ 3- i} < \frac{{\left( { 2\mu_{ 3- i} + \mu_{i} - \eta_{i} } \right)}}{ 2}$.

Proof of Proposition 4

$$ U_{{R_{ 1} }} \left( { 1_{n} , 2_{n} } \right) - U_{{R_{ 1} }} \left( { 1_{s} , 2_{n} } \right) = {\upsigma }\left( {\eta_{ 1} - \mu_{ 1} } \right)\left[ { \frac{{ 2\left( { 1 { + }\theta } \right)\left( { 3\theta { + 2}} \right){{\upsigma \upalpha }} - {\text{A}}}}{{ 4\left( { 2 { + }\theta } \right)^{ 2} \left( { 3\theta { + 2}} \right)^{ 2} }}{ + }\frac{{\left( { 1 { + }\theta } \right)\left( { 4\theta^{ 2} { + 13}\theta { + 6}} \right)k_{ 2} \tau^{ 2} - \left( { 1 { + }\theta } \right)\left( { 4\theta^{ 2} { + 7}\theta { + 2}} \right)k_{ 1} \tau^{ 2} }}{{ 1 6k_{ 1} k_{ 2} \left( { 2 { + }\theta } \right)^{ 2} \left( { 3\theta { + 2}} \right)^{ 2} }}} \right] , $$

where $A = \left( { 6\theta^{ 2} { + 17}\theta + 9} \right)\upsigma\eta_{ 1} + 4\theta \left( { 1+ \theta } \right)\upsigma\eta_{ 2} + \left( { 2\theta^{ 2} { + 2}\theta + 1} \right)\upsigma\mu_{ 1} + \left( { 3\theta + 2} \right)\upsigma\mu_{ 2}$.

We assume that the market base $\alpha$ is large enough, which is a very realistic assumption (Taleizadeh et al. 2017b; Yao et al. 2008; Dai and Chao 2016; Ha et al. 2017). If $\eta_{ 1} > \mu_{ 1}$, then $U_{{R_{ 1} }} \left( { 1_{n} , 2_{n} } \right) > U_{{R_{ 1} }} \left( { 1_{s} , 2_{n} } \right)$. Similarly for $U_{{R_{ 1} }} \left( { 1_{n} , 2_{s} } \right) > U_{{R_{ 1} }} \left( { 1_{s} , 2_{s} } \right)$, $\eta_{ 1} > \mu_{ 1}$ must hold. For retailer 2, because of the symmetry, we obtain the condition is $\eta_{ 2} > \mu_{ 2}$.

Proof of Proposition 5

From Proposition 4, we have $U_{{R_{ 1} }} \left( { 1_{n} , 2_{n} } \right) < U_{{R_{ 1} }} \left( { 1_{s} , 2_{n} } \right)$ and $U_{{R_{ 1} }} \left( { 1_{n} , 2_{s} } \right) < U_{{R_{ 1} }} \left( { 1_{s} , 2_{s} } \right)$ if $\eta_{ 1} < \mu_{ 1}$. For retailer 2, the condition is $\eta_{ 2} < \mu_{ 2}$. Moreover, from Proposition 3, we get that the supplier is always better off by acquiring the risk-aversion information from two retailers. Therefore, this Proposition is proved.

Proof of Proposition 6

Similar to the Proof of Proposition 5.

Proof of Proposition 7

Without loss of generality, suppose retailer 1 shares her risk-aversion information with the supplier while retailer 2 does not. The following inequality should be satisfied:

$$ \pi_{\text{M}}^{ *} \left( 2\right) - \pi_{\text{M1}}^{ *} \left( 1\right) > U_{{R_{ 2} }} \left( { 1_{s} , 2_{n} } \right) - U_{{R_{ 2} }} \left( { 1_{s} , 2_{s} } \right) . $$

By some algebra,

$$ \begin{aligned} & \left[ {\pi_{\text{M}}^{ *} \left( 2\right) - \pi_{\text{M1}}^{ *} \left( 1\right)} \right] - \left[ {U_{{R_{ 2} }} \left( { 1_{s} , 2_{n} } \right) - U_{{R_{ 2} }} \left( { 1_{s} , 2_{s} } \right)} \right] \\ & = {\upsigma }\left( {\mu_{ 2} - \eta_{ 2} } \right)\left[ { \frac{{ 2\left( { 1 { + }\theta } \right)\left( { 3\theta { + 2}} \right){{\upalpha }} - B}}{{ 4\left( { 2 { + }\theta } \right)^{ 2} \left( { 3\theta { + 2}} \right)^{ 2} }}{ + }\frac{{\left( { 1 { + }\theta } \right)\left( { 4\theta^{ 2} { + 13}\theta { + 6}} \right)k_{ 1} \tau^{ 2} - \left( { 1 { + }\theta } \right)\left( { 4\theta^{ 2} { + 7}\theta { + 2}} \right)k_{ 2} \tau^{ 2} }}{{ 1 6k_{ 1} k_{ 2} \left( { 2 { + }\theta } \right)^{ 2} \left( { 3\theta { + 2}} \right)^{ 2} }}} \right] ,\\ \end{aligned} $$

where $B = 2\left( { 1+ \theta } \right)\left( { 1+ 2\theta } \right){\upsigma }\eta_{ 1} + \left( { 8\theta^{ 2} { + 23}\theta + 1 3} \right){\upsigma }\eta_{ 2} + \theta {\upsigma }\mu_{ 1} - \left( { 4\theta + 3} \right){\upsigma }\mu_{ 2}$.

Since we assume that the market base $\alpha$ is large enough, then for$\left[ {\pi_{\text{M}}^{ *} \left( 2\right) - \pi_{\text{M1}}^{ *} \left( 1\right)} \right] - \left[ {U_{{R_{ 2} }} \left( { 1_{s} , 2_{n} } \right) - U_{{R_{ 2} }} \left( { 1_{s} , 2_{s} } \right)} \right] > 0$, we have the condition that $\eta_{ 2} < \mu_{ 2}$. This condition is contradictory to our assumption that retailer 2 has no incentive to share her risk-aversion information ($\eta_{ 2} > \mu_{ 2}$). Thus it is proved that$\pi_{\text{M}}^{ *} \left( 2\right) - \pi_{\text{M1}}^{ *} \left( 1\right) < U_{{R_{ 2} }} \left( { 1_{s} , 2_{n} } \right) - U_{{R_{ 2} }} \left( { 1_{s} , 2_{s} } \right)$. Because of the symmetry, this Proposition is proved.

Proof of Proposition 8

In this case, the retailers have complete knowledge with respect to each other’s risk-aversion level because of the information leakage. Retailer $i$’s utility function in this case is:

$$ U_{{R_{i} }} = \left( {p_{i} - w - k_{i} t_{i}^{ 2} } \right)D_{i} - \eta_{i} \left( {p_{i} - w - k_{i} t_{i}^{ 2} } \right){\upsigma } . $$

(A.18)

By simultaneously solving Eqs: $\partial U_{{R_{i} }} /\partial p_{i} = 0 , \partial U_{{R_{i} }} /\partial t_{i} = 0$, we have the following response functions:

$$ p_{i}^{*} = \frac{{\alpha - \eta_{i} {\upsigma } - \tau \theta t_{ 3- i} + \left( { 1+ \theta } \right)w + \theta p_{ 3- i} }}{{ 2\left( { 1+ \theta } \right)}} + \frac{{ 3\tau^{ 2} }}{{ 8k_{i} }} , $$

(A.19)

$$ t_{i}^{*} = \frac{\tau }{{ 2k_{i} }} . $$

(A.20)

The Hessian matrix of $U_{{R_{i} }}$ is as follows:

$$ \left[ {\begin{array}{*{20}c} {\frac{{\partial U_{{R_{i} }} }}{{\partial p_{i}^{ 2} }}} & {\frac{{\partial U_{{R_{i} }} }}{{\partial p_{i} \partial t_{i} }}} \\ {\frac{{\partial U_{{R_{i} }} }}{{\partial t_{i} \partial p_{i} }}} & {\frac{{\partial U_{{R_{i} }} }}{{\partial t_{i}^{ 2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2- 2\theta } & {\left( { 1 { + }\theta } \right)\left( { 2k_{i} t_{i} + \tau } \right)} \\ {} & { - 2k_{i} (\alpha + \left( { 3t_{i} \tau - p_{i} } \right)\left( { 1 { + }\theta } \right)} \\ {\left( { 1 { + }\theta } \right)\left( { 2k_{i} t_{i} + \tau } \right)} & { + \theta p_{ 3- i} - \eta_{i} {\upsigma } - \tau \theta t_{ 3- i} )} \\ \end{array} } \right] . $$

Substituting $p_{i}^{*}$ and $t_{i}^{*}$ into the Hessian matrix of $U_{{R_{i} }}$, we obtain the Hessian Matrix on the equilibrium point as follows:

$$ H_{ 1} = \left[ {\begin{array}{*{20}c} { - 2- 2\theta } & { 2\tau \left( { 1 { + }\theta } \right)} \\ { 2\tau \left( { 1 { + }\theta } \right)} & { - \left( {\alpha - w} \right)k_{i} - \frac{ 9}{ 4}\tau^{ 2} \left( { 1 { + }\theta } \right) - \left( {p_{ 3- i} - w} \right)\theta + \eta_{i} {\upsigma } + \tau \theta t_{ 3- i} } \\ \end{array} } \right] . $$

Hence, we have

$$ \left| {H_{ 1} } \right| = \frac{ 1}{ 2}\left( { 1 { + }\theta } \right)\left( { 4\left( {\alpha - w} \right)k_{i} + 4\left( {p_{ 3- i} - w} \right)\theta + \tau^{ 2} \left( { 1 { + }\theta } \right) - 4\eta_{i} {\upsigma } - 4\tau \theta t_{ 3- i} } \right) . $$

Similarly, it shows that $- 2- 2\theta < 0$, and $- \left( {\alpha - w} \right)k_{i} - \frac{ 9}{ 4}\tau^{ 2} \left( { 1 { + }\theta } \right) - \left( {p_{ 3- i} - w} \right)\theta + \eta_{i} {\upsigma } + \tau \theta t_{ 3- i} < 0$ and $\left| {H_{ 1} } \right| > 0$ when the market base $\alpha$ is large enough compared to other parameters. So we get that $p_{i}^{*}$ and $t_{i}^{*}$ are the optimal response functions of $U_{{R_{i} }}$. Then solving $p_{ 1}^{*}$ and $p_{ 2}^{*}$ in Eq. (A. 19), we obtain the equilibriums for the retailers:

$$ \begin{aligned} p_{i}^{*} & = \frac{{\left( { 3\theta + 2} \right)\alpha + \left( { 3\theta + 2} \right)\left( {\theta + 1} \right)w - \left( { 2\theta + 2} \right)\sigma \eta_{i} - \theta \sigma \eta_{ 3- i} }}{{ 3\theta^{ 2} + 8\theta + 4}} \\ & \quad + \frac{{ 2\left( { 2\theta^{ 2} + 6\theta + 3} \right)\tau^{ 2} k_{ 3- i} - \left( {\theta^{ 2} + \theta } \right)\tau^{ 2} k_{i} }}{{ 4k_{i} k_{ 3- i} \left( { 3\theta^{ 2} + 8\theta + 4} \right)}} , \\ \end{aligned} $$

(A.21)

$$ t_{i}^{*} = \frac{\tau }{{ 2k_{i} }} , \quad i = 1 , 2. $$

(A.22)

Substituting Eq. (A.21) and Eq. (A.22) into Eq. (11), we can obtain the supplier’s expected profit function in this cases as follows:

$$ E\left( {\pi_{\text{M}} } \right) = \frac{{ 2\left( { 1+ \theta } \right)w\alpha + w\sigma \left( {\eta_{ 1} + \eta_{ 2} } \right) - 2w^{ 2} \left( { 1+ \theta } \right)}}{ 2+ \theta } + \frac{{\tau^{ 2} w\left( { 1+ \theta } \right)\left( {k_{ 1} + k_{ 2} } \right)}}{{ 4\left( { 2+ \theta } \right)k_{ 1} k_{ 2} }} $$

(A.23)

Since $\frac{{d^{ 2} E\left( {\pi_{\text{M}} } \right)}}{{dw^{ 2} }} = - \frac{{ 4\left( { 1+ \theta } \right)}}{ 2+ \theta } < 0$, $E\left( {\pi_{\text{M}} } \right)$ is concave in $w$. Let $\frac{{dE\left( {\pi_{\text{M}} } \right)}}{dw} = 0$, then we obtain the optimal wholesale prices $w_{BI}^{IL*}$ of the supplier under this case as shown in Proposition 8. Then substituting $w_{BI}^{IL*}$ into Eq. (A. 21), we can obtain the retailer $i$’s optimal retail price and warranty period as shown in Proposition 8. Given the optimal equilibrium results above, then we can obtain the supplier’s expected profit and retailers’ utility. Similarly, it can show that the equilibrium results are positive under the assumption that the market base $\alpha$ is large enough.

Proof of Proposition 9

Similar to the Proofs of Proposition 1 and Proposition 2.

Proof of Proposition 10

Similar to the Proof of Proposition 3

Proof of Proposition 11

Similar to the Proof of Proposition 5.