# American Institute of Mathematical Sciences

January  2018, 14(1): 105-134. doi: 10.3934/jimo.2017039

## Advertising games on national brand and store brand in a dual-channel supply chain

 1 School of Economics and Commerce, South China University of Technology, Guangzhou, Guangdong 510006, China 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China

* Corresponding author: Jingna Ji

Received  May 2015 Revised  February 2017 Published  April 2017

Fund Project: This research is partly supported by (ⅰ) the National Natural Science Foundation of China (71101054,71201083,71572058,71571100); (ⅱ) Jiangsu Province Science Foundation for Youths (BK2012379); and (ⅲ) the Fundamental Research Funds for the Central Universities (NS2015076).

This paper investigates a dual-channel supply chain, where one national brand manufacturer has both online and retail channels. The retailer is assumed to sell the national brand as well as his store brand to customers. The following five scenarios are considered: Centralized case, Stackelberg-manufacturer (SM) game, Stackelberg-retailer (SR) game, Nash-manufacturer (NM) game and Nash-retailer (NR) game. The paper derives the conditions under which the supply chain members would like to participate in cooperative advertising. The results show that in the Stackelberg games, the leader in Stackelberg game will reduce its investment in cooperative advertising when it gets a lower marginal profit from the cooperative advertising; In addition, the dual-channel supply chain can get a higher profit if it is dominated by the member whose marginal profit from cooperative advertising is higher. In the Nash games, in order to increase the whole supply chain's profit, the member who has a higher marginal profit in the cooperative advertising should give up the decision power on cost-sharing rate voluntarily. In addition, if there exists a leader in the supply chain, the cooperative advertising will be higher. Furthermore, the introduction of store brand will trigger the manufacturer's antipathy for the low profit.

Citation: Lei Yang, Jingna Ji, Kebing Chen. Advertising games on national brand and store brand in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2018, 14 (1) : 105-134. doi: 10.3934/jimo.2017039
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##### References:
Dual-channel supply chain framework
The profits of the manufacturer versus $\lambda$
The profits of the retailer versus $\lambda$
The profits of whole supply chain versus $\lambda$
Equilibrium advertising strategies under four games considering advertising competition in retail channel
 Games ${A_1^a}^*$ ${A_2^a}^*$ SM $\big[ \frac{2B_1+B_2}{4-4(\rho_1+\rho_2-\rho_3)\gamma} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$ SR $\big[ \frac{B_1+2B_2}{4-4(\rho_1+\rho_2-\rho_3)\gamma} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$ NM $\big[ \frac{B_2}{2-2(\rho_2-\rho_3)\gamma} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$ NR $\big[ \frac{B_1}{2(1-\rho_1\gamma)} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$
 Games ${A_1^a}^*$ ${A_2^a}^*$ SM $\big[ \frac{2B_1+B_2}{4-4(\rho_1+\rho_2-\rho_3)\gamma} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$ SR $\big[ \frac{B_1+2B_2}{4-4(\rho_1+\rho_2-\rho_3)\gamma} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$ NM $\big[ \frac{B_2}{2-2(\rho_2-\rho_3)\gamma} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$ NR $\big[ \frac{B_1}{2(1-\rho_1\gamma)} \big]^2$ $\big[ \frac{\rho_3\varphi_2-\rho_2\varphi_1}{2-2(\rho_3-\rho_2)\gamma} \big]^2$
Equilibrium solutions under four games without store brand
 Equilibria SM game SR game NM game NR game ${A_1^b}^*$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ ${A_2^b}^*$ $\frac{(2B_1+B_3)^2}{16}$ $\frac{(2B_3+B_1)^2}{16}$ $\frac{B_3^2}{4}$ $\frac{B_1^2}{4}$ ${t^b}^*$ \left\{\begin{aligned}&\frac{2B_1-B_3}{2B_1+B_3}, &q_1>1/2\\&0, &q_1<1/2\end{aligned}\right. $\frac{2B_1}{2B_3+B_1}$ 0 1
 Equilibria SM game SR game NM game NR game ${A_1^b}^*$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ $\frac{(\rho_0\theta_0+\rho_1\theta_1)^2}{4}$ ${A_2^b}^*$ $\frac{(2B_1+B_3)^2}{16}$ $\frac{(2B_3+B_1)^2}{16}$ $\frac{B_3^2}{4}$ $\frac{B_1^2}{4}$ ${t^b}^*$ \left\{\begin{aligned}&\frac{2B_1-B_3}{2B_1+B_3}, &q_1>1/2\\&0, &q_1<1/2\end{aligned}\right. $\frac{2B_1}{2B_3+B_1}$ 0 1
Effects of store brand's introduction on advertising decisions
 Games $A_1$ $t$ ${A_1}_m$ ${A_1}_r$ SM ${A_1^{S_M}}^* \leq {A_1^{bS_M}}^*$ ${t^{S_M}}^* \geq {t^{bS_M}}^*$ ${A_{1m}^{S_M}}^* \geq {A_{1m}^{bS_M}}^*$ ${A_{1r}^{S_M}}^* \leq {A_{1r}^{bS_M}}^*$ SR ${A_1^{S_R}}^* \leq {A_1^{bS_R}}^*$ ${t^{S_R}}^* \geq {t^{bS_R}}^*$ ${A_{1m}^{S_R}}^* \leq {A_{1m}^{bS_R}}^*$ ${A_{1r}^{S_R}}^* \leq {A_{1r}^{bS_R}}^*$ NM ${A_1^{N_M}}^* \leq {A_1^{bN_M}}^*$ ${t^{N_M}}^* = {t^{bN_M}}^*$ ${A_{1m}^{N_M}}^* = {A_{1m}^{bN_M}}^*$ ${A_{1r}^{N_M}}^* \leq {A_{1r}^{bN_M}}^*$ NM ${A_1^{N_R}}^* = {A_1^{bN_R}}^*$ ${t^{N_R}}^* = {t^{bN_R}}^*$ ${A_{1m}^{N_R}}^* = {A_{1m}^{bN_R}}^*$ ${A_{1r}^{N_R}}^* \leq {A_{1r}^{bN_R}}^*$
 Games $A_1$ $t$ ${A_1}_m$ ${A_1}_r$ SM ${A_1^{S_M}}^* \leq {A_1^{bS_M}}^*$ ${t^{S_M}}^* \geq {t^{bS_M}}^*$ ${A_{1m}^{S_M}}^* \geq {A_{1m}^{bS_M}}^*$ ${A_{1r}^{S_M}}^* \leq {A_{1r}^{bS_M}}^*$ SR ${A_1^{S_R}}^* \leq {A_1^{bS_R}}^*$ ${t^{S_R}}^* \geq {t^{bS_R}}^*$ ${A_{1m}^{S_R}}^* \leq {A_{1m}^{bS_R}}^*$ ${A_{1r}^{S_R}}^* \leq {A_{1r}^{bS_R}}^*$ NM ${A_1^{N_M}}^* \leq {A_1^{bN_M}}^*$ ${t^{N_M}}^* = {t^{bN_M}}^*$ ${A_{1m}^{N_M}}^* = {A_{1m}^{bN_M}}^*$ ${A_{1r}^{N_M}}^* \leq {A_{1r}^{bN_M}}^*$ NM ${A_1^{N_R}}^* = {A_1^{bN_R}}^*$ ${t^{N_R}}^* = {t^{bN_R}}^*$ ${A_{1m}^{N_R}}^* = {A_{1m}^{bN_R}}^*$ ${A_{1r}^{N_R}}^* \leq {A_{1r}^{bN_R}}^*$
Effects of store brand's introduction on profits
 Games $\pi_m$ $\pi_r$ Comparisons Conditions SM ${\pi_m^{S_M}}^* \leq {\pi_m^{bS_M}}^*$ ${\pi_r^{S_M}}^* < (\geq) {\pi_r^{bS_M}}^*$ $a_2 < (\geq) C_1$ SR ${\pi_m^{S_R}}^* \leq {\pi_m^{bS_R}}^*$ ${\pi_r^{S_R}}^* < (\geq) {\pi_r^{bS_R}}^*$ $a_2 < (\geq) C_2$ NM ${\pi_m^{N_M}}^* \leq {\pi_m^{bN_M}}^*$ ${\pi_r^{N_M}}^* < (\geq) {\pi_r^{bN_M}}^*$ $a_2 < (\geq) C_3$ NM ${\pi_m^{N_R}}^* \leq {\pi_m^{bN_R}}^*$ ${\pi_r^{N_R}}^* < (\geq) {\pi_r^{bN_R}}^*$ $a_2 < (\geq) C_4$
 Games $\pi_m$ $\pi_r$ Comparisons Conditions SM ${\pi_m^{S_M}}^* \leq {\pi_m^{bS_M}}^*$ ${\pi_r^{S_M}}^* < (\geq) {\pi_r^{bS_M}}^*$ $a_2 < (\geq) C_1$ SR ${\pi_m^{S_R}}^* \leq {\pi_m^{bS_R}}^*$ ${\pi_r^{S_R}}^* < (\geq) {\pi_r^{bS_R}}^*$ $a_2 < (\geq) C_2$ NM ${\pi_m^{N_M}}^* \leq {\pi_m^{bN_M}}^*$ ${\pi_r^{N_M}}^* < (\geq) {\pi_r^{bN_M}}^*$ $a_2 < (\geq) C_3$ NM ${\pi_m^{N_R}}^* \leq {\pi_m^{bN_R}}^*$ ${\pi_r^{N_R}}^* < (\geq) {\pi_r^{bN_R}}^*$ $a_2 < (\geq) C_4$
Equilibrium solutions under four games without direct channel
 Equilibria SM game SR game NM game NR game ${A_0^c}^*$ $\frac{(\rho_1\theta_1)^2}{4}$ $\frac{(\rho_1\theta_1)^2}{4}$ $\frac{(\rho_1\theta_1)^2}{4}$ $\frac{(\rho_1\theta_1)^2}{4}$ ${A_1^c}^*$ $\frac{(2B_4+B_5)^2}{16}$ $\frac{(2B_5+B_4)^2}{16}$ $\frac{B_5^2}{4}$ $\frac{B_4^2}{4}$ ${A_2^c}^*$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ ${t^c}^*$ \left\{\begin{aligned}&\frac{2B_4-B_5}{2B_4+B_5}, &q_1>1/2\\&0, &q_1<1/2\end{aligned}\right. $\frac{2B_4}{2B_5+B_4}$ 0 1
 Equilibria SM game SR game NM game NR game ${A_0^c}^*$ $\frac{(\rho_1\theta_1)^2}{4}$ $\frac{(\rho_1\theta_1)^2}{4}$ $\frac{(\rho_1\theta_1)^2}{4}$ $\frac{(\rho_1\theta_1)^2}{4}$ ${A_1^c}^*$ $\frac{(2B_4+B_5)^2}{16}$ $\frac{(2B_5+B_4)^2}{16}$ $\frac{B_5^2}{4}$ $\frac{B_4^2}{4}$ ${A_2^c}^*$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ $\frac{(\rho_3\varphi_2-\rho_2\varphi_1)^2}{4}$ ${t^c}^*$ \left\{\begin{aligned}&\frac{2B_4-B_5}{2B_4+B_5}, &q_1>1/2\\&0, &q_1<1/2\end{aligned}\right. $\frac{2B_4}{2B_5+B_4}$ 0 1
Effects of direct channel's introduction on advertising decisions
 Game type Cooperative advertising Comparisons Conditions SM ${A_1^{S_M}}^* < {A_1^{cS_M}}^*$ $C_5\lambda < \rho_0\beta_0$ ${A_1^{S_M}}^* \geq {A_1^{cS_M}}^*$ $C_5\lambda \geq \rho_0\beta_0$ SR ${A_1^{S_R}}^* < {A_1^{cS_R}}^*$ $C_6\lambda < \rho_0\beta_0$ ${A_1^{S_R}}^* \geq {A_1^{cS_R}}^*$ $C_6\lambda \geq \rho_0\beta_0$ NM ${A_1^{N_M}}^* \leq {A_1^{cN_M}}^*$ - - - NR ${A_1^{N_R}}^* < {A_1^{cN_R}}^*$ $C_7\lambda < \rho_0\beta_0$ ${A_1^{N_R}}^* \geq {A_1^{cN_R}}^*$ $C_7\lambda \geq \rho_0\beta_0$
 Game type Cooperative advertising Comparisons Conditions SM ${A_1^{S_M}}^* < {A_1^{cS_M}}^*$ $C_5\lambda < \rho_0\beta_0$ ${A_1^{S_M}}^* \geq {A_1^{cS_M}}^*$ $C_5\lambda \geq \rho_0\beta_0$ SR ${A_1^{S_R}}^* < {A_1^{cS_R}}^*$ $C_6\lambda < \rho_0\beta_0$ ${A_1^{S_R}}^* \geq {A_1^{cS_R}}^*$ $C_6\lambda \geq \rho_0\beta_0$ NM ${A_1^{N_M}}^* \leq {A_1^{cN_M}}^*$ - - - NR ${A_1^{N_R}}^* < {A_1^{cN_R}}^*$ $C_7\lambda < \rho_0\beta_0$ ${A_1^{N_R}}^* \geq {A_1^{cN_R}}^*$ $C_7\lambda \geq \rho_0\beta_0$
Effects of direct channel's introduction on profits
 Game type Manufacturer's profits Retailer's Profits Comparisons Conditions Comparisons Conditions SM ${\pi_m^{S_M}}^* < {\pi_m^{cS_M}}^*$ $0 \leq a_0 < C_8$ ${\pi_r^{S_M}}^* < {\pi_r^{cS_M}}^*$ $Z_1 < 0$ ${\pi_m^{S_M}}^* \geq {\pi_m^{cS_M}}^*$ $a_0 \geq C_8$ ${\pi_r^{S_M}}^* \geq {\pi_r^{cS_M}}^*$ $Z_1 \geq 0$ SR ${\pi_m^{S_R}}^* < {\pi_m^{cS_R}}^*$ $0 \leq a_0 < C_9$ ${\pi_r^{S_R}}^* < {\pi_r^{cS_R}}^*$ $Z_2 < 0$ ${\pi_m^{S_R}}^* \geq {\pi_m^{cS_R}}^*$ $a_0 \geq C_9$ ${\pi_r^{S_R}}^* \geq {\pi_r^{cS_R}}^*$ $Z_2 \geq 0$ NM ${\pi_m^{N_M}}^* < {\pi_m^{cN_M}}^*$ $0 \leq a_0 < C_{10}$ ${\pi_r^{N_M}}^* < {\pi_r^{cN_M}}^*$ $Z_3 < 0$ ${\pi_m^{N_M}}^* \geq {\pi_m^{cN_M}}^*$ $a_0 \geq C_{10}$ ${\pi_r^{N_M}}^* \geq {\pi_r^{cN_M}}^*$ $Z_3 \geq 0$ NR ${\pi_m^{N_R}}^* < {\pi_m^{cN_R}}^*$ $0 \leq a_0 < C_{11}$ ${\pi_r^{N_R}}^* < {\pi_r^{cN_R}}^*$ $Z_4 < 0$ ${\pi_m^{N_R}}^* \geq {\pi_m^{cN_R}}^*$ $a_0 \geq C_{11}$ ${\pi_r^{N_R}}^* \geq {\pi_r^{cN_R}}^*$ $Z_4 \geq 0$
 Game type Manufacturer's profits Retailer's Profits Comparisons Conditions Comparisons Conditions SM ${\pi_m^{S_M}}^* < {\pi_m^{cS_M}}^*$ $0 \leq a_0 < C_8$ ${\pi_r^{S_M}}^* < {\pi_r^{cS_M}}^*$ $Z_1 < 0$ ${\pi_m^{S_M}}^* \geq {\pi_m^{cS_M}}^*$ $a_0 \geq C_8$ ${\pi_r^{S_M}}^* \geq {\pi_r^{cS_M}}^*$ $Z_1 \geq 0$ SR ${\pi_m^{S_R}}^* < {\pi_m^{cS_R}}^*$ $0 \leq a_0 < C_9$ ${\pi_r^{S_R}}^* < {\pi_r^{cS_R}}^*$ $Z_2 < 0$ ${\pi_m^{S_R}}^* \geq {\pi_m^{cS_R}}^*$ $a_0 \geq C_9$ ${\pi_r^{S_R}}^* \geq {\pi_r^{cS_R}}^*$ $Z_2 \geq 0$ NM ${\pi_m^{N_M}}^* < {\pi_m^{cN_M}}^*$ $0 \leq a_0 < C_{10}$ ${\pi_r^{N_M}}^* < {\pi_r^{cN_M}}^*$ $Z_3 < 0$ ${\pi_m^{N_M}}^* \geq {\pi_m^{cN_M}}^*$ $a_0 \geq C_{10}$ ${\pi_r^{N_M}}^* \geq {\pi_r^{cN_M}}^*$ $Z_3 \geq 0$ NR ${\pi_m^{N_R}}^* < {\pi_m^{cN_R}}^*$ $0 \leq a_0 < C_{11}$ ${\pi_r^{N_R}}^* < {\pi_r^{cN_R}}^*$ $Z_4 < 0$ ${\pi_m^{N_R}}^* \geq {\pi_m^{cN_R}}^*$ $a_0 \geq C_{11}$ ${\pi_r^{N_R}}^* \geq {\pi_r^{cN_R}}^*$ $Z_4 \geq 0$
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