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doi: 10.3934/jimo.2021200
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## Information sharing in two-tier supply chains considering cost reduction effort and information leakage

 1 School of Management, Xi'an Jiaotong University, Xi'an, 710049, China 2 School of Economics and Management, Xidian University, Xi'an, 710126, China 3 Business School, Central South University, Changsha, 410083, China

* Corresponding author: Xiaomei Li

Received  January 2021 Revised  September 2021 Early access November 2021

Fund Project: This resarch is supported by the Major Program of National Fund of Philosophy and Social Science of China (18ZDA104), National Natural Science Foundation of China (72102174), Ministry of Education of Humanities and Social Science Project (21XJC630004, 19YJA630068), Natural Science Basic Research Program of Shaanxi(2021JM-144), Fundamental Research Funds for the Central Universities (XJS200601)

This study investigates information sharing in two-tier supply chai-ns considering cost reduction effort and information leakage, with either upstream competition (system SC) or downstream competition (system RC). Results show that in system SC without information leakage, the retailer shares information with one supplier when suppliers are efficient in cost reduction, shares information with neither supplier when suppliers are inefficient in cost reduction, and shares information with two suppliers when suppliers are intermediate in cost reduction efficiency. nformation leakage won't affect the information sharing decisions of the retailer. In system RC with or without information leakage, both retailers share information with the supplier when the supplier is efficient in cost reduction and neither retailer shares information with the supplier when the supplier is inefficient in cost reduction. However, the threshold of cost reduction efficiency without information leakage is always lower than that with information leakage, which demonstrates that it is less likely for retailers to share information with information leakage. What's more, the two retailers choose the same information sharing strategies without information leakage but the opposite information sharing strategies with information leakage when the cost reduction efficiency is intermediate.

Citation: Xiaomei Li, Renjing Liu, Zhongquan Hu, Jiamin Dong. Information sharing in two-tier supply chains considering cost reduction effort and information leakage. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021200
##### References:

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##### References:
Supply chain members' expected profit in system SC without information leakage
The retailer's information sharing strategies in system SC without information leakage
Supply chain members' expected profits in system RC without information leakage
Retailers' information sharing strategies in system RC without information leakage
Supply chain members' expected profits in system RC with information leakage
Retailers' information sharing strategies in system RC without information leakage
Summary table of literature review
 Papers Influence factors Supply chain structures Information leakage Cost reduction System SC System RC Fang and Ren[6] $\checkmark$ $\times$ $\times$ $\times$ Wang et al.[29] $\checkmark$ $\times$ $\times$ $\checkmark$ Chen and Özer[5] $\checkmark$ $\times$ $\times$ $\checkmark$ Ha et al.[8] $\times$ $\checkmark$ $\times$ $\times$ Sun et al.[27] $\times$ $\checkmark$ $\times$ $\times$ Cao and Chen[4] $\checkmark$ $\checkmark$ $\times$ $\checkmark$ Our paper $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
 Papers Influence factors Supply chain structures Information leakage Cost reduction System SC System RC Fang and Ren[6] $\checkmark$ $\times$ $\times$ $\times$ Wang et al.[29] $\checkmark$ $\times$ $\times$ $\checkmark$ Chen and Özer[5] $\checkmark$ $\times$ $\times$ $\checkmark$ Ha et al.[8] $\times$ $\checkmark$ $\times$ $\times$ Sun et al.[27] $\times$ $\checkmark$ $\times$ $\times$ Cao and Chen[4] $\checkmark$ $\checkmark$ $\times$ $\checkmark$ Our paper $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Parameters and notations
 Parameters and notations Description $p_{i}$ Selling price of product $i$ $q_{i}$ Order quantity of product $i$ $\alpha$ Potential market size $\theta$ Demand uncertainty with the mean of $0$ and variance of $\theta^2$ $\gamma$ Competition intensity and a larger $\gamma$ implies more intense competition $c_i$ Production cost of product $i$ $x_i$ Production cost reduction level of product $i$ $k_i$ Cost reduction efficiency of product $i$ and a lower $k_i$ indicates a higher efficiency $Y_{(i)}$ Demand signal of retailer $(i)$ $s$ Accuracy of demand signal and a larger $s$ indicates a less accurate information $n=(n_1,n_2)$ Information sharing decisions $\omega_i$ Wholesale price of product $i$ $\pi_{R_{(i)}}^{(n_1,n_2)}$ Profit of the retailer $(i)$ under information sharing arrangement $n=(n_1,n_2)$ $\pi_{S_{(i)}}^{(n_1,n_2)}$ Profit of the supplier $(i)$ under information sharing arrangement $n=(n_1,n_2)$
 Parameters and notations Description $p_{i}$ Selling price of product $i$ $q_{i}$ Order quantity of product $i$ $\alpha$ Potential market size $\theta$ Demand uncertainty with the mean of $0$ and variance of $\theta^2$ $\gamma$ Competition intensity and a larger $\gamma$ implies more intense competition $c_i$ Production cost of product $i$ $x_i$ Production cost reduction level of product $i$ $k_i$ Cost reduction efficiency of product $i$ and a lower $k_i$ indicates a higher efficiency $Y_{(i)}$ Demand signal of retailer $(i)$ $s$ Accuracy of demand signal and a larger $s$ indicates a less accurate information $n=(n_1,n_2)$ Information sharing decisions $\omega_i$ Wholesale price of product $i$ $\pi_{R_{(i)}}^{(n_1,n_2)}$ Profit of the retailer $(i)$ under information sharing arrangement $n=(n_1,n_2)$ $\pi_{S_{(i)}}^{(n_1,n_2)}$ Profit of the supplier $(i)$ under information sharing arrangement $n=(n_1,n_2)$
Decisions and ex-ante profits in system SC without information leakage
 $n$ Decisions Ex-ante profits $(0,0)$ $\omega_i^{(0,0)}=\omega^0$ $\pi_{R}^{(0,0)}=\pi_{R}^{0}+\frac{\sigma ^2}{2(1+s)(1+\gamma )}$ $x_i^{(0,0)}=x^0$ $q_i^{(0,0)}=q^0+\frac{Y}{2(1+s)(1+\gamma )}$ $\pi_{S_i}^{(0,0)}=\pi_{S}^{0}$ $(1,1)$ $\omega_i^{(1,1)}=\omega^0+\frac{(2k(1-\gamma^2)-1)Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)}$ $\pi_{R}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2(1+\gamma)\sigma ^2}{(1+s)(2k(2-\gamma)(1+\gamma)-1)^2}$ $x_i^{(1,1)}=x^0+\frac{Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)}$ $q_i^{(1,1)}=q^0+\frac{kY}{(1+s)(2k(2-\gamma)(1+\gamma)-1)}$ $\pi_{S_i}^{(1,1)}=\pi_{S}^{0}+\frac{k(4k(1-\gamma^2)-1)\sigma ^2}{2(1+s)(2k(2-\gamma)(1+\gamma)-1)^2}$ $(1,0)$ $\omega_1^{(1,0)}=\omega^0+\frac{(1-\gamma)(2k(1-\gamma^2)-1)Y}{(1+s)(4k(1+\gamma^2)-1)}$ $\pi_{R}^{(1,0)}=\pi_{R}^{0}$ $x_1^{(1,0)}=x^0+\frac{(1-\gamma)Y}{(1+s)(4k(1+\gamma^2)-1)}$ $+\frac{(4k^2(1-\gamma)^2(1+\gamma)(5+3\gamma)-8k(1-\gamma^2)+1)\sigma ^2}{4(1+s)(4k(1-\gamma^2)-1)^2}$ $\omega_2^{(1,0)}=\omega^0$ $\pi_{S_1}^{(1,0)}=\pi_{S}^{0}+\frac{(k(1-\gamma)^2\sigma ^2}{2(1+s)(4k(1-\gamma^2)-1)^2}$ $x_2^{(1,0)}=x^0$ $q_1^{(1,0)}=q^0+\frac{(1-\gamma)kY}{(1+s)(4k(1+\gamma^2)-1)}$ $\pi_{S_2}^{(1,0)}=\pi_{S}^{0}$ $q_2^{(1,0)}=q^0+\frac{2k(2+\gamma)(1-\gamma)Y}{2(1+s)(4k(1+\gamma^2)-1)}$ Notes. $\omega^0=\frac{(2k(1-\gamma^2)-1)\alpha+2k(1+\gamma)c}{(2k(2-\gamma)(1+\gamma)-1}, x^0=\frac{\alpha-c}{(2k(2-\gamma)(1+\gamma)-1)}, q^0=\frac{k(\alpha-c)}{(2k(2-\gamma)(1+\gamma)-1},$ $\pi_{R}^{0}=\frac{2k^2(1+\gamma)(\alpha-c)^2}{(2k(2-\gamma)(1+\gamma)-1)^2},\pi_{S}^{0}=\frac{k(4k(1-\gamma^2)-1)(\alpha-c)^2}{2(2k(2-\gamma)(1+\gamma)-1)^2}.$
 $n$ Decisions Ex-ante profits $(0,0)$ $\omega_i^{(0,0)}=\omega^0$ $\pi_{R}^{(0,0)}=\pi_{R}^{0}+\frac{\sigma ^2}{2(1+s)(1+\gamma )}$ $x_i^{(0,0)}=x^0$ $q_i^{(0,0)}=q^0+\frac{Y}{2(1+s)(1+\gamma )}$ $\pi_{S_i}^{(0,0)}=\pi_{S}^{0}$ $(1,1)$ $\omega_i^{(1,1)}=\omega^0+\frac{(2k(1-\gamma^2)-1)Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)}$ $\pi_{R}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2(1+\gamma)\sigma ^2}{(1+s)(2k(2-\gamma)(1+\gamma)-1)^2}$ $x_i^{(1,1)}=x^0+\frac{Y}{(1+s)(2k(2-\gamma)(1+\gamma)-1)}$ $q_i^{(1,1)}=q^0+\frac{kY}{(1+s)(2k(2-\gamma)(1+\gamma)-1)}$ $\pi_{S_i}^{(1,1)}=\pi_{S}^{0}+\frac{k(4k(1-\gamma^2)-1)\sigma ^2}{2(1+s)(2k(2-\gamma)(1+\gamma)-1)^2}$ $(1,0)$ $\omega_1^{(1,0)}=\omega^0+\frac{(1-\gamma)(2k(1-\gamma^2)-1)Y}{(1+s)(4k(1+\gamma^2)-1)}$ $\pi_{R}^{(1,0)}=\pi_{R}^{0}$ $x_1^{(1,0)}=x^0+\frac{(1-\gamma)Y}{(1+s)(4k(1+\gamma^2)-1)}$ $+\frac{(4k^2(1-\gamma)^2(1+\gamma)(5+3\gamma)-8k(1-\gamma^2)+1)\sigma ^2}{4(1+s)(4k(1-\gamma^2)-1)^2}$ $\omega_2^{(1,0)}=\omega^0$ $\pi_{S_1}^{(1,0)}=\pi_{S}^{0}+\frac{(k(1-\gamma)^2\sigma ^2}{2(1+s)(4k(1-\gamma^2)-1)^2}$ $x_2^{(1,0)}=x^0$ $q_1^{(1,0)}=q^0+\frac{(1-\gamma)kY}{(1+s)(4k(1+\gamma^2)-1)}$ $\pi_{S_2}^{(1,0)}=\pi_{S}^{0}$ $q_2^{(1,0)}=q^0+\frac{2k(2+\gamma)(1-\gamma)Y}{2(1+s)(4k(1+\gamma^2)-1)}$ Notes. $\omega^0=\frac{(2k(1-\gamma^2)-1)\alpha+2k(1+\gamma)c}{(2k(2-\gamma)(1+\gamma)-1}, x^0=\frac{\alpha-c}{(2k(2-\gamma)(1+\gamma)-1)}, q^0=\frac{k(\alpha-c)}{(2k(2-\gamma)(1+\gamma)-1},$ $\pi_{R}^{0}=\frac{2k^2(1+\gamma)(\alpha-c)^2}{(2k(2-\gamma)(1+\gamma)-1)^2},\pi_{S}^{0}=\frac{k(4k(1-\gamma^2)-1)(\alpha-c)^2}{2(2k(2-\gamma)(1+\gamma)-1)^2}.$
Decisions and ex-ante profits in system SC without information leakage
 $n$ Decisions Ex-ante profits $(0,0)$ $\omega_i^{(0,0)}=\omega^0$ $\pi_{R_i}^{(0,0)}=\pi_{R_i}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2}$ $x_i^{(0,0)}=x^0$ $q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma}$ $\pi_{S}^{(0,0)}=\pi_{S}^{0}$ $(1,1)$ $\omega_i^{(1,1)}=\omega^0+\alpha_i^{(1,1)}$ $\pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\zeta_i^{(1,1)}$ $x_i^{(1,1)}=x^0+\beta_i^{(1,1)}$ $q_i^{(1,1)}=q^0+\delta_i^{(1,1)}$ $\pi_{S}^{(1,1)}=\pi_{S}^{0}+\eta_i^{(1,1)}$ $(1,0)$ $\omega_1^{(1,0)}=\omega^0+\alpha_1^{(1,0)}$ $\pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\zeta_1^{(1,0)}$ $x_1^{(1,0)}=x^0+\beta_1^{(1,0)}$ $\omega_2^{(1,0)}=\omega^0+\alpha_2^{(1,0)}$ $\pi_{R_2}^{(1,0)}=\pi_{R}^{0}+\zeta_2^{(1,0)}$ $x_2^{(1,0)}=x^0+\beta_2^{(1,0)}$ $q_1^{(1,0)}=q^0+\delta_1^{(1,0)}$ $\pi_{S}^{(1,0)}=\pi_{S}^{0}+\eta^{(1,0)}$ $q_2^{(1,0)}=q^0+\delta_2^{(1,0)}$ Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1},x^0=\frac{\alpha-c}{2k(2+\gamma)-1},q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1},\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2},$, $\pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2} \qquad \left\{\begin{array}{l} \alpha _1^{(1,0)}=\frac{(2k^2(2+2s+\gamma)(4-\gamma ^2)+k((1+s)\gamma ^2-4(3+3s+\gamma ))+2+2s+\gamma )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \alpha _2^{(1,0)}=\frac{(2k^2(2+\gamma+s\gamma)(4-\gamma ^2)-k(12+(4+4s-\gamma )\gamma )+2+\gamma+s\gamma)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \alpha _i^{(1,1)}=\frac{(4k-1)(k(4-\gamma^2)-2)Y_i+(2k^2(4-\gamma^2)-4k+1)\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \beta _1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _2^{(1,0)}=\frac{(2k(4-\gamma ^2)-(2+\gamma+s\gamma))Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _i^{(1,1)}=\frac{2(k(4-\gamma^2)-1)Y_i-\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \delta _1^{(1,0)}=\frac{k(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \delta _2^{(1,0)}=\frac{(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}+\frac{Y_2}{2+2s+\gamma},\\ \delta _i^{(1,1)}=\frac{2k(k(4-\gamma^2)-1)Y_i-k\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \eta^{(1,0)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \eta^{(0,1)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \eta^{(1,1)}=\frac{(2k(2+2s+\gamma)(4-\gamma^2)-s(4+\gamma^2)-(2+\gamma)^2)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \zeta_1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )^2k^2\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}, \\ \zeta_2^{(1,0)}=\frac{(2k(2+\gamma )-1)(2k(2-\gamma )-1)(2k^2(1+2s(2+s))(4-\gamma^2)-k(2+\gamma+s(16+8s+\gamma))+s(2+s))\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}+\\ \frac{(2k^2(1+2s(2-\gamma))(4-\gamma^2)-k(2+\gamma+s(17-7\gamma))+s(2-\gamma)(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2},\\ \zeta_i^{(1,1)}=\frac{(4k^3(1+s)(4-\gamma^2)^2+4k^2(4-\gamma^2)(s\gamma^2-(1+2s)\gamma-2(1+s)))k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}\\+\frac{(k((1-7s)\gamma^2+4(1+4s)\gamma+4(1+s))-s(2-\gamma)\gamma)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}. \end{array}\right.$
 $n$ Decisions Ex-ante profits $(0,0)$ $\omega_i^{(0,0)}=\omega^0$ $\pi_{R_i}^{(0,0)}=\pi_{R_i}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2}$ $x_i^{(0,0)}=x^0$ $q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma}$ $\pi_{S}^{(0,0)}=\pi_{S}^{0}$ $(1,1)$ $\omega_i^{(1,1)}=\omega^0+\alpha_i^{(1,1)}$ $\pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\zeta_i^{(1,1)}$ $x_i^{(1,1)}=x^0+\beta_i^{(1,1)}$ $q_i^{(1,1)}=q^0+\delta_i^{(1,1)}$ $\pi_{S}^{(1,1)}=\pi_{S}^{0}+\eta_i^{(1,1)}$ $(1,0)$ $\omega_1^{(1,0)}=\omega^0+\alpha_1^{(1,0)}$ $\pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\zeta_1^{(1,0)}$ $x_1^{(1,0)}=x^0+\beta_1^{(1,0)}$ $\omega_2^{(1,0)}=\omega^0+\alpha_2^{(1,0)}$ $\pi_{R_2}^{(1,0)}=\pi_{R}^{0}+\zeta_2^{(1,0)}$ $x_2^{(1,0)}=x^0+\beta_2^{(1,0)}$ $q_1^{(1,0)}=q^0+\delta_1^{(1,0)}$ $\pi_{S}^{(1,0)}=\pi_{S}^{0}+\eta^{(1,0)}$ $q_2^{(1,0)}=q^0+\delta_2^{(1,0)}$ Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1},x^0=\frac{\alpha-c}{2k(2+\gamma)-1},q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1},\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2},$, $\pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2} \qquad \left\{\begin{array}{l} \alpha _1^{(1,0)}=\frac{(2k^2(2+2s+\gamma)(4-\gamma ^2)+k((1+s)\gamma ^2-4(3+3s+\gamma ))+2+2s+\gamma )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \alpha _2^{(1,0)}=\frac{(2k^2(2+\gamma+s\gamma)(4-\gamma ^2)-k(12+(4+4s-\gamma )\gamma )+2+\gamma+s\gamma)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \alpha _i^{(1,1)}=\frac{(4k-1)(k(4-\gamma^2)-2)Y_i+(2k^2(4-\gamma^2)-4k+1)\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \beta _1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _2^{(1,0)}=\frac{(2k(4-\gamma ^2)-(2+\gamma+s\gamma))Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \beta _i^{(1,1)}=\frac{2(k(4-\gamma^2)-1)Y_i-\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \delta _1^{(1,0)}=\frac{k(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \delta _2^{(1,0)}=\frac{(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)Y_1}{(1+s)(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}+\frac{Y_2}{2+2s+\gamma},\\ \delta _i^{(1,1)}=\frac{2k(k(4-\gamma^2)-1)Y_i-k\gamma Y_j}{(2+2s+\gamma)(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \eta^{(1,0)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \\ \eta^{(0,1)}=\frac{(4k(2+\gamma+s(2+s+\gamma))(4-\gamma^2)-2(1+s)(2+\gamma)^2-s^2(4+\gamma^2))k\sigma^2}{2(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)},\\ \eta^{(1,1)}=\frac{(2k(2+2s+\gamma)(4-\gamma^2)-s(4+\gamma^2)-(2+\gamma)^2)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)(2k(2-\gamma )-1)}, \end{array}\right.$ $\left\{\begin{array}{l} \zeta_1^{(1,0)}=\frac{(2k(1+s)(4-\gamma ^2)-(2+2s+\gamma) )^2k^2\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}, \\ \zeta_2^{(1,0)}=\frac{(2k(2+\gamma )-1)(2k(2-\gamma )-1)(2k^2(1+2s(2+s))(4-\gamma^2)-k(2+\gamma+s(16+8s+\gamma))+s(2+s))\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}+\\ \frac{(2k^2(1+2s(2-\gamma))(4-\gamma^2)-k(2+\gamma+s(17-7\gamma))+s(2-\gamma)(k(6-\gamma-s\gamma)-2k^2(4-\gamma^2)-1)\sigma^2}{(1+s)(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2},\\ \zeta_i^{(1,1)}=\frac{(4k^3(1+s)(4-\gamma^2)^2+4k^2(4-\gamma^2)(s\gamma^2-(1+2s)\gamma-2(1+s)))k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}\\+\frac{(k((1-7s)\gamma^2+4(1+4s)\gamma+4(1+s))-s(2-\gamma)\gamma)k\sigma^2}{(2+2s+\gamma)^2(2k(2+\gamma )-1)^2(2k(2-\gamma )-1)^2}. \end{array}\right.$
Decisions and ex-ante profits in system SC without information leakage
 $n$ Decisions Ex-ante profits $(0,0)$ $\omega^{(0,0)}=\omega^0$ $\pi_{R_i}^{(0,0)}=\pi_{R}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2}$ $x^{(0,0)}=x^0$ $q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma}$ $\pi_{S}^{(0,0)}=\pi_{S}^{0}$ $(1,1)$ $\omega^{(1,1)}=\omega^0+\frac{(2k+k\gamma-1)(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)}$ $\pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2\sigma^2}{(2+s)(2k(2+\gamma)-1)^2}$ $x^{(1,1)}=x^0+\frac{Y_1+Y_2}{(2+s)(2k(2+\gamma)-1)}$ $q_i^{(1,1)}=q^0+\frac{k(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)}$ $\pi_{S}^{(1,1)}=\pi_{S}^{0}+\frac{2k\sigma^2}{(2+s)(2k(2+\gamma)-1)}$ $(1,0)$ $\omega^{(1,0)}=\omega^0+\frac{(2k+k\gamma-1)Y_1}{(1+s)(2k(2+\gamma)-1)}$ $\pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\frac{k^2\sigma^2}{(1+s)(2k(2+\gamma)-1)^2}$ $x^{(1,0)}=x^0+\frac{Y_1}{(1+s)(2k(2+\gamma)-1)}$ $\pi_{R_2}^{(1,0)}=\pi_{R}^{0}$ $q_1^{(1,0)}=q^0+\frac{kY_1}{(1+s)(2k(2+\gamma)-1)}$ $+\frac{(4k^2(2+5s)+4ks\gamma(4k+k\gamma-1)-(8k-1)s)\sigma^2}{4(1+s)(2+s)(2k(2+\gamma)-1)^2}$ $q_2^{(1,0)}=q^0+\frac{(1+2ks-2k\gamma)Y_1}{2(1+s)(2+s)(2k(2+\gamma)-1)}+\frac{Y_2}{2(2+s)}$ $\pi_{S}^{(1,0)}=\pi_{S}^{0}+\frac{k\sigma^2}{(1+s)(2k(2+\gamma)-1)}$ Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1}, \quad x^0=\frac{\alpha-c}{2k(2+\gamma)-1}, \quad q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1},$ $\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2}, \qquad \pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2}$
 $n$ Decisions Ex-ante profits $(0,0)$ $\omega^{(0,0)}=\omega^0$ $\pi_{R_i}^{(0,0)}=\pi_{R}^{0}+\frac{(1+s)\sigma ^2}{(2+2s+\gamma)^2}$ $x^{(0,0)}=x^0$ $q_i^{(0,0)}=q^0+\frac{Y_i}{2+2s+\gamma}$ $\pi_{S}^{(0,0)}=\pi_{S}^{0}$ $(1,1)$ $\omega^{(1,1)}=\omega^0+\frac{(2k+k\gamma-1)(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)}$ $\pi_{R_i}^{(1,1)}=\pi_{R}^{0}+\frac{2k^2\sigma^2}{(2+s)(2k(2+\gamma)-1)^2}$ $x^{(1,1)}=x^0+\frac{Y_1+Y_2}{(2+s)(2k(2+\gamma)-1)}$ $q_i^{(1,1)}=q^0+\frac{k(Y_1+Y_2)}{(2+s)(2k(2+\gamma)-1)}$ $\pi_{S}^{(1,1)}=\pi_{S}^{0}+\frac{2k\sigma^2}{(2+s)(2k(2+\gamma)-1)}$ $(1,0)$ $\omega^{(1,0)}=\omega^0+\frac{(2k+k\gamma-1)Y_1}{(1+s)(2k(2+\gamma)-1)}$ $\pi_{R_1}^{(1,0)}=\pi_{R}^{0}+\frac{k^2\sigma^2}{(1+s)(2k(2+\gamma)-1)^2}$ $x^{(1,0)}=x^0+\frac{Y_1}{(1+s)(2k(2+\gamma)-1)}$ $\pi_{R_2}^{(1,0)}=\pi_{R}^{0}$ $q_1^{(1,0)}=q^0+\frac{kY_1}{(1+s)(2k(2+\gamma)-1)}$ $+\frac{(4k^2(2+5s)+4ks\gamma(4k+k\gamma-1)-(8k-1)s)\sigma^2}{4(1+s)(2+s)(2k(2+\gamma)-1)^2}$ $q_2^{(1,0)}=q^0+\frac{(1+2ks-2k\gamma)Y_1}{2(1+s)(2+s)(2k(2+\gamma)-1)}+\frac{Y_2}{2(2+s)}$ $\pi_{S}^{(1,0)}=\pi_{S}^{0}+\frac{k\sigma^2}{(1+s)(2k(2+\gamma)-1)}$ Notes. $\omega^0=\frac{(k(2+\gamma)-1)\alpha+k(2+\gamma)c}{2k(2+\gamma)-1}, \quad x^0=\frac{\alpha-c}{2k(2+\gamma)-1}, \quad q^0=\frac{k(\alpha-c)}{2k(2+\gamma)-1},$ $\pi_{R}^{0}=\frac{k^2(\alpha-c)^2}{(2k(2+\gamma)-1)^2}, \qquad \pi_{S}^{0}=\frac{k(\alpha-c)^2}{(2k(2+\gamma)-1)^2}$
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