# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022098
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## Evolutionary game theory analysis of supply chain with fairness concerns of retailers

 1 Institute for Logistics and Emergency Management, School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, Sichuan, China 2 School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, Guizhou, China

*Corresponding author: Ying Dai

Received  November 2021 Revised  April 2022 Early access June 2022

This study focuses on evolutionary game analysis of supply chain (SC) strategies with fairness concerns. First, the equilibrium solutions under the eight combined strategies are obtained through the Stackelberg game. Subsequently, evolutionary game theory is applied to analyze the stability of these equilibrium solutions and to discuss the evolutionary stability strategy of the tripartite game. The study suggests that the effect of distributional fairness concerns on SC pricing differs from that of peer-induced fairness concerns in the short term. Implanting fairness concerns into SC pricing decisions is always detrimental to the supplier and not always beneficial to retailers. From a long-term perspective, only one evolutionarily stable equilibrium point exists. Retailers do not pay attention to the fairness of the channel profit distribution, and the supplier does not consider the fairness of retailers. The results help enterprises understand the influence of fairness preferences on pricing decisions and provide a reference for the efficient operation of SCs.

Citation: Yadong Shu, Ying Dai, Zujun Ma. Evolutionary game theory analysis of supply chain with fairness concerns of retailers. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022098
##### References:

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##### References:
SC Structure diagram with fairness concerns implanted
Tripartite Stackelberg game
The relation of $\pi_S^{l*}$ to $\lambda$
The relation of $\pi_S^{l*}$ to $\theta$
The relation of $\pi _{{R_1}}^{l*}$ to $\lambda$
The relation of $\pi _{{R_2}}^{l*}$ to $\lambda$
The relation of $\pi _{{R_2}}^{l*}$ to $\theta$
The relation of $\ u_{{R_i}}^{l*}$ to $\lambda$
The literature positioning of this paper
 Reference Number of participants Consider multiple fairness concerns Leader "actively" or "fairness" considers the fairness Stability analysis of equilibrium concerns of followers points Assarzadegan et al.[3] 2 $\times$ Passively $\surd$ Ho et al. [17] 3 $\surd$ Passively $\times$ Li et al. [29] 2 $\times$ Passively $\times$ Liu et al. [30] 3 $\surd$ Passively $\times$ Nie and Du[37] 3 $\surd$ Passively $\times$ Sharma[42] 2 $\times$ Passively $\times$ Shi and Zhu[43] 3 $\surd$ Passively $\times$ Yoshihara et al.[53] 3 $\times$ Passively $\times$ Zhang et al. [55] 2 $\times$ Passively $\times$ Zhang et al. [57] 3 $\times$ Passively $\times$ Zhang and Wang[58] 3 $\surd$ Passively $\times$ This paper 3 $\surd$ Passively and actively $\surd$
 Reference Number of participants Consider multiple fairness concerns Leader "actively" or "fairness" considers the fairness Stability analysis of equilibrium concerns of followers points Assarzadegan et al.[3] 2 $\times$ Passively $\surd$ Ho et al. [17] 3 $\surd$ Passively $\times$ Li et al. [29] 2 $\times$ Passively $\times$ Liu et al. [30] 3 $\surd$ Passively $\times$ Nie and Du[37] 3 $\surd$ Passively $\times$ Sharma[42] 2 $\times$ Passively $\times$ Shi and Zhu[43] 3 $\surd$ Passively $\times$ Yoshihara et al.[53] 3 $\times$ Passively $\times$ Zhang et al. [55] 2 $\times$ Passively $\times$ Zhang et al. [57] 3 $\times$ Passively $\times$ Zhang and Wang[58] 3 $\surd$ Passively $\times$ This paper 3 $\surd$ Passively and actively $\surd$
Notations description
 Parameter Meaning $S$ Supplier $R_{i}$ Retailer $i$, $i=1, 2$ $D_{i}$ Demand function of the retailer $R_{i}$, $D=D_{1}+D_{2}$ $Superscript$ "$\ast$" The optimal solution $\lambda_{R_{i}}$ Distributional fairness concerns coefficient of the retailer $R_{i}$, let $\lambda_{R_{1}}=\lambda_{R_{2}}$, and $0\leq \lambda \leq 1$ $\theta$ Peer-induced fairness concerns coefficient, and $0\leq \theta \leq 1$ Decision variables $p_{i}$ Selling price, and it is the retailer $R_{i}^{'}$s decision variable $\omega_{i}$ Wholesale price, and it is the supplier's decision variable Performance $\pi_{S}^{l}$ The profit of the supplier under model $l$, $l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\}$ $\pi_{R_{i}}^{l}$ The profit of the retailer $\pi_{R_{i}}$ under model $l$, $l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\}$ $\pi_{SJ}$ The supplier's profit in the $j$-th round of the game, $j=1, 2$ $\pi_{J}$ The profit of the SC in the $j$-th round of the game, $j=1, 2$ $\bar{\pi}_{R_{i}}$ Nash bargaining solution of the retailer $R_{i}$ $\bar{\pi}_{S}$ Nash bargaining solution of the supplier $u_{R_{i}}^{l}$ The utility of the retailer $R_{i}$ under model $l$, $l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\}$
 Parameter Meaning $S$ Supplier $R_{i}$ Retailer $i$, $i=1, 2$ $D_{i}$ Demand function of the retailer $R_{i}$, $D=D_{1}+D_{2}$ $Superscript$ "$\ast$" The optimal solution $\lambda_{R_{i}}$ Distributional fairness concerns coefficient of the retailer $R_{i}$, let $\lambda_{R_{1}}=\lambda_{R_{2}}$, and $0\leq \lambda \leq 1$ $\theta$ Peer-induced fairness concerns coefficient, and $0\leq \theta \leq 1$ Decision variables $p_{i}$ Selling price, and it is the retailer $R_{i}^{'}$s decision variable $\omega_{i}$ Wholesale price, and it is the supplier's decision variable Performance $\pi_{S}^{l}$ The profit of the supplier under model $l$, $l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\}$ $\pi_{R_{i}}^{l}$ The profit of the retailer $\pi_{R_{i}}$ under model $l$, $l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\}$ $\pi_{SJ}$ The supplier's profit in the $j$-th round of the game, $j=1, 2$ $\pi_{J}$ The profit of the SC in the $j$-th round of the game, $j=1, 2$ $\bar{\pi}_{R_{i}}$ Nash bargaining solution of the retailer $R_{i}$ $\bar{\pi}_{S}$ Nash bargaining solution of the supplier $u_{R_{i}}^{l}$ The utility of the retailer $R_{i}$ under model $l$, $l\in\left\{CCC, CCN, CNC, CNN, NCC, NCN, NNC, NNN\right\}$
The relation between the supplier's profits and distributional fairness concerns
 $\frac{\partial \pi_{S}^{CCC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{CCN\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{CNC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{CNN\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NCC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NCN\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NNC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NNN\ast}}{\partial \lambda}$ $-$ $-$ $-$ $-$ $-$ $-$ $-$ $/$
 $\frac{\partial \pi_{S}^{CCC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{CCN\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{CNC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{CNN\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NCC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NCN\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NNC\ast}}{\partial \lambda}$ $\frac{\partial \pi_{S}^{NNN\ast}}{\partial \lambda}$ $-$ $-$ $-$ $-$ $-$ $-$ $-$ $/$
The relation between the supplier's profits and peer-induced fairness concerns
 $\frac{{\partial\pi_S^{CCC*}}}{{\partial\theta}}$ $\frac{\partial \pi_{S}^{CCN\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{CNC\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{CNN\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NCC\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NCN\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NNC\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NNN\ast}}{\partial \theta}$ $+$ $+$ $+$ $+$ $+$ $/$ $+$ $/$
 $\frac{{\partial\pi_S^{CCC*}}}{{\partial\theta}}$ $\frac{\partial \pi_{S}^{CCN\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{CNC\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{CNN\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NCC\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NCN\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NNC\ast}}{\partial \theta}$ $\frac{\partial \pi_{S}^{NNN\ast}}{\partial \theta}$ $+$ $+$ $+$ $+$ $+$ $/$ $+$ $/$
The relation between the retailer ${R_1}$'s profits and distributional fairness concerns
 $\frac{{\partial \pi _{{R_1}}^{CCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{CCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{CNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{CNN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NNN*}}}{{\partial \lambda }}$ $+$ $+$ $+$ $+$ $-$ $-$ $/$ $/$
 $\frac{{\partial \pi _{{R_1}}^{CCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{CCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{CNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{CNN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_1}}^{NNN*}}}{{\partial \lambda }}$ $+$ $+$ $+$ $+$ $-$ $-$ $/$ $/$
The relation between the retailer ${R_2}$'s profits and distributional fairness concerns
 $\frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \lambda }}$ $+$ $+$ $+$ $+$ $-$ $/$ $-$ $/$
 $\frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \lambda }}$ $\frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \lambda }}$ $+$ $+$ $+$ $+$ $-$ $/$ $-$ $/$
The relation between the retailer ${R_2}$'s profits and peer-induced fairness concerns
 $\frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \theta}}$ $\frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \theta }}$ $+$ $+$ $+$ $+$ $-$ $/$ $-$ $/$
 $\frac{{\partial \pi _{{R_2}}^{CCC*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{CCN*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{CNC*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{CNN*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{NCC*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{NCN*}}}{{\partial \theta }}$ $\frac{{\partial \pi _{{R_2}}^{NNC*}}}{{\partial \theta}}$ $\frac{{\partial \pi _{{R_2}}^{NNN*}}}{{\partial \theta }}$ $+$ $+$ $+$ $+$ $-$ $/$ $-$ $/$
Equilibrium solutions under different strategy combination
 $w_1^{CCC*}$ $w_1^{CCN*}$ $w_1^{CNC*}$ $w_1^{CNN*}$ $w_1^{NCC*}$ $w_1^{NCN*}$ $w_1^{NNC*}$ $w_1^{NNN*}$ $416.7$ $416.7$ $416.7$ $416.7$ $500$ $500$ $500$ $500$ $w_2^{CCC*}$ $w_2^{CCN*}$ $w_2^{CNC*}$ $w_2^{CNN*}$ $w_2^{NCC*}$ $w_2^{NCN*}$ $w_2^{NNC*}$ $w_2^{NNN*}$ $437.5$ $437.5$ $437.5$ $437.5$ $500$ $500$ $500$ $500$ $p_1^{CCC*}$ $p_1^{CCN*}$ $p_1^{CNC*}$ $p_1^{CNN*}$ $p_1^{NCC*}$ $p_1^{NCN*}$ $p_1^{NNC*}$ $p_1^{NNN*}$ $750$ $750$ $708.3$ $708.3$ $800$ $800$ $750$ $750$ $p_2^{CCC*}$ $p_2^{CCN*}$ $p_2^{CNC*}$ $p_2^{CNN*}$ $p_2^{NCC*}$ $p_2^{NCN*}$ $p_2^{NNC*}$ $p_2^{NNN*}$ $750$ $718.7$ $750$ $718.7$ $785.7$ $750$ $785.7$ $750$
 $w_1^{CCC*}$ $w_1^{CCN*}$ $w_1^{CNC*}$ $w_1^{CNN*}$ $w_1^{NCC*}$ $w_1^{NCN*}$ $w_1^{NNC*}$ $w_1^{NNN*}$ $416.7$ $416.7$ $416.7$ $416.7$ $500$ $500$ $500$ $500$ $w_2^{CCC*}$ $w_2^{CCN*}$ $w_2^{CNC*}$ $w_2^{CNN*}$ $w_2^{NCC*}$ $w_2^{NCN*}$ $w_2^{NNC*}$ $w_2^{NNN*}$ $437.5$ $437.5$ $437.5$ $437.5$ $500$ $500$ $500$ $500$ $p_1^{CCC*}$ $p_1^{CCN*}$ $p_1^{CNC*}$ $p_1^{CNN*}$ $p_1^{NCC*}$ $p_1^{NCN*}$ $p_1^{NNC*}$ $p_1^{NNN*}$ $750$ $750$ $708.3$ $708.3$ $800$ $800$ $750$ $750$ $p_2^{CCC*}$ $p_2^{CCN*}$ $p_2^{CNC*}$ $p_2^{CNN*}$ $p_2^{NCC*}$ $p_2^{NCN*}$ $p_2^{NNC*}$ $p_2^{NNN*}$ $750$ $718.7$ $750$ $718.7$ $785.7$ $750$ $785.7$ $750$
Tripartite game benefit matrix
 Supplier Supplier $R_{1}$ Concerns ($y$) No concerns ($1-y$) Retailer $R_{2}$ Consideration ($x$) $A, I, q$ $b, T, r$ Concerns ($z$) No consideration ($1-x$) $c, k, s$ $d, I, B$ Consideration ($x$) $c, m, u$ $g, n, v$ (No concerns $1-z$) No consideration ($1-x$) $f, o, W$ $h, p, \varphi$
 Supplier Supplier $R_{1}$ Concerns ($y$) No concerns ($1-y$) Retailer $R_{2}$ Consideration ($x$) $A, I, q$ $b, T, r$ Concerns ($z$) No consideration ($1-x$) $c, k, s$ $d, I, B$ Consideration ($x$) $c, m, u$ $g, n, v$ (No concerns $1-z$) No consideration ($1-x$) $f, o, W$ $h, p, \varphi$
ESS stability analysis
 Equilibrium Points Eigenvalues Asymptotic Stability $\lambda_{1}$ $\lambda_{2}$ $\lambda_{3}$ $E_{1}(0, 0, 0)$ $\eta_{1}$ $\eta_{1}$ $\eta_{1}$ ESS $-$ $-$ $-$ $E_{2}(1, 0, 0)$ $-\eta_{1}$ $\alpha_{2}+\eta_{2}$ $\alpha_{3}+\eta_{3}$ Saddle-point $+$ $-$ $-$ $E_{3}(1, 1, 1)$ $-(\alpha_{1}+\theta_{1}+\eta_{1})$ $-(\alpha_{2}+\eta_{2})$ $-(\alpha_{3}+\eta_{3})$ Saddle-point $-$ $+$ $+$ $E_{4}(0, 1, 0)$ $\alpha_{1}+\eta_{1}$ $-\eta_{2}$ $\eta_{3}$ Saddle-point $+$ $+$ $-$ $E_{5}(0, 0, 1)$ $\theta_{1}+\eta_{1}$ $\eta_{2}$ $-\eta_{3}$ Saddle-point $Uncertain$ $-$ $+$ $E_{6}(1, 1, 0)$ $-(\alpha_{1}+\eta_{1})$ $-(\alpha_{2}+\eta_{2})$ $\alpha_{3}+\eta_{3}$ Saddle-point $-$ $+$ $-$ $E_{7}(1, 0, 1)$ $-(\theta_{1}+\eta_{1})$ $\alpha_{2}+\eta_{2}$ $-(\alpha_{3}+\eta_{3})$ Saddle-point $Uncertain$ $-$ $+$ $E_{8}(0, 1, 1)$ $\alpha_{1}+\theta_{1}+\eta_{1}$ $-\eta_{2}$ $-\eta_{3}$ Saddle-point $+$ $+$ $+$
 Equilibrium Points Eigenvalues Asymptotic Stability $\lambda_{1}$ $\lambda_{2}$ $\lambda_{3}$ $E_{1}(0, 0, 0)$ $\eta_{1}$ $\eta_{1}$ $\eta_{1}$ ESS $-$ $-$ $-$ $E_{2}(1, 0, 0)$ $-\eta_{1}$ $\alpha_{2}+\eta_{2}$ $\alpha_{3}+\eta_{3}$ Saddle-point $+$ $-$ $-$ $E_{3}(1, 1, 1)$ $-(\alpha_{1}+\theta_{1}+\eta_{1})$ $-(\alpha_{2}+\eta_{2})$ $-(\alpha_{3}+\eta_{3})$ Saddle-point $-$ $+$ $+$ $E_{4}(0, 1, 0)$ $\alpha_{1}+\eta_{1}$ $-\eta_{2}$ $\eta_{3}$ Saddle-point $+$ $+$ $-$ $E_{5}(0, 0, 1)$ $\theta_{1}+\eta_{1}$ $\eta_{2}$ $-\eta_{3}$ Saddle-point $Uncertain$ $-$ $+$ $E_{6}(1, 1, 0)$ $-(\alpha_{1}+\eta_{1})$ $-(\alpha_{2}+\eta_{2})$ $\alpha_{3}+\eta_{3}$ Saddle-point $-$ $+$ $-$ $E_{7}(1, 0, 1)$ $-(\theta_{1}+\eta_{1})$ $\alpha_{2}+\eta_{2}$ $-(\alpha_{3}+\eta_{3})$ Saddle-point $Uncertain$ $-$ $+$ $E_{8}(0, 1, 1)$ $\alpha_{1}+\theta_{1}+\eta_{1}$ $-\eta_{2}$ $-\eta_{3}$ Saddle-point $+$ $+$ $+$
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