# American Institute of Mathematical Sciences

January  2018, 14(1): 325-347. doi: 10.3934/jimo.2017049

## Is social responsibility for firms competing on quantity evolutionary stable?

 1 School of Management Science and Engineering, Nanjing University, Nanjing 210093, China 2 Institute of Game Behavior and Operations Management, Nanjing University of Finance and Economics, Nanjing 210046, China 3 Economics Department, University of California, Santa Cruz, CA 95064, USA

* Corresponding author: T. Xiao

Received  March 2016 Revised  October 2016 Published  June 2017

Fund Project: This study is funded by: (ⅰ) China National Funds for Distinguished Young Scientists under Grant 71425001; (ⅱ) National Natural Science Foundation of China under Grants 71371093 and 11301001

This paper studies the evolutionary stable strategies and preferences regarding corporate social responsibility of competing firms. Firms randomly compete with each other in pairs. Shareholder-oriented firms have no social responsibility concern, whereas a firm that is concerned with social responsibility is stakeholder-oriented. Each firm first picks one of two production strategies: shareholder-oriented or stakeholder-oriented, and then decides production quantity. We find that socially responsible firms have lower retail prices. The evolutionary stability of a strategy depends on product substitutability and the degree to which firms care about social responsibility. When product substitutability is relatively high, stakeholder-oriented strategy is the evolutionary stable strategy; if product substitutability is lower than a threshold, shareholder-oriented strategy is evolutionary stable; and with moderate product substitutability, both strategies are evolutionary stable.

Furthermore, we consider how the degree of social responsibility preference evolves according to the adaptive dynamics to continuously stable preference. We find that the non social responsibility concern behavior is not an evolutionary stable preference; there is a unique continuously stable degree of social responsibility preference. Furthermore, we find the evolutionary stability of shareholder-oriented and stakeholder-oriented depends on the initial distribution of firms' strategies under the continuously stable social responsibility preference.

Citation: Caichun Chai, Tiaojun Xiao, Eilin Francis. Is social responsibility for firms competing on quantity evolutionary stable?. Journal of Industrial & Management Optimization, 2018, 14 (1) : 325-347. doi: 10.3934/jimo.2017049
##### References:
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##### References:
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Evolutionary dynamics when $2(1-\sqrt{1-\alpha})\leq d^2$
Evolutionary dynamics when $(2-\alpha)(1-\sqrt{1-\alpha})\geq d^2$
Evolutionary dynamics when $(2-\alpha)(1-\sqrt{1-\alpha}) < d^2 < 2(1-\sqrt{1-\alpha})$
Evolution of firms' behavior
SR preference and the equilibrium behavior
The profit of a firm with others' SR factor $\beta$
Profit of a firm versus $\alpha$ ($\beta = 0.6$)
Profit of a firm versus $\alpha$ and $\beta$
The dynamics of firm's SR preference
The equilibrium expressions under given strategy for one-shot game
 Strategy Quantity ($q_1, q_2$) Price ($p_1, p_2$) Profit ($\pi_1, \pi_2$) $TT$ $\frac{a-c}{2-\alpha+d},$ $\frac{c(1+d)+a(1-\alpha)}{2+d-\alpha},$ $\frac{(a-c)^2(1-\alpha)}{(2+d-\alpha)^2},$ $\frac{a-c}{2-\alpha+d}$ $\frac{c(1+d)+a(1-\alpha)}{2+d-\alpha}$ $\frac{(a-c)^2(1-\alpha)}{(2+d-\alpha)^2}$ $HH$ $\frac{a-c}{2+d},$ $\frac{a+c(1+d)}{2+d},$ $\frac{(a-c)^2}{(2+d)^2},$ $\frac{a-c}{2+d}$ $\frac{a+c(1+d)}{2+d}$ $\frac{(a-c)^2}{(2+d)^2}$ $TH$ $\frac{(a-c)(2-d)}{4-d^2-2\alpha},$ $\frac{c[2-d^2+d(1-\alpha)]+a(2-d)(1-\alpha)}{4-d^2-2\alpha},$ $\frac{(a-c)^2(2-d)^2(1-\alpha)}{(4-d^2-2\alpha)^2},$ $\frac{(a-c)(2-d-\alpha)}{4-d^2-2\alpha}$ $\frac{a(2-d-\alpha)+c(2+d-d^2-\alpha)}{4-d^2-2\alpha}$ $\frac{(a-c)^2(2-d-\alpha)^2}{(4-d^2-2\alpha)^2}$ $HT$ $\frac{(a-c)(2-d-\alpha)}{4-d^2-2\alpha},$ $\frac{a(2-d-\alpha)+c(2+d-d^2-\alpha)}{4-d^2-2\alpha},$ $\frac{(a-c)^2(2-d-\alpha)^2}{(4-d^2-2\alpha)^2},$ $\frac{(a-c)(2-d)}{4-d^2-2\alpha}$ $\frac{c[2-d^2+d(1-\alpha)]+a(2-d)(1-\alpha)}{4-d^2-2\alpha}$ $\frac{(a-c)^2(2-d)^2(1-\alpha)}{(4-d^2-2\alpha)^2}$
 Strategy Quantity ($q_1, q_2$) Price ($p_1, p_2$) Profit ($\pi_1, \pi_2$) $TT$ $\frac{a-c}{2-\alpha+d},$ $\frac{c(1+d)+a(1-\alpha)}{2+d-\alpha},$ $\frac{(a-c)^2(1-\alpha)}{(2+d-\alpha)^2},$ $\frac{a-c}{2-\alpha+d}$ $\frac{c(1+d)+a(1-\alpha)}{2+d-\alpha}$ $\frac{(a-c)^2(1-\alpha)}{(2+d-\alpha)^2}$ $HH$ $\frac{a-c}{2+d},$ $\frac{a+c(1+d)}{2+d},$ $\frac{(a-c)^2}{(2+d)^2},$ $\frac{a-c}{2+d}$ $\frac{a+c(1+d)}{2+d}$ $\frac{(a-c)^2}{(2+d)^2}$ $TH$ $\frac{(a-c)(2-d)}{4-d^2-2\alpha},$ $\frac{c[2-d^2+d(1-\alpha)]+a(2-d)(1-\alpha)}{4-d^2-2\alpha},$ $\frac{(a-c)^2(2-d)^2(1-\alpha)}{(4-d^2-2\alpha)^2},$ $\frac{(a-c)(2-d-\alpha)}{4-d^2-2\alpha}$ $\frac{a(2-d-\alpha)+c(2+d-d^2-\alpha)}{4-d^2-2\alpha}$ $\frac{(a-c)^2(2-d-\alpha)^2}{(4-d^2-2\alpha)^2}$ $HT$ $\frac{(a-c)(2-d-\alpha)}{4-d^2-2\alpha},$ $\frac{a(2-d-\alpha)+c(2+d-d^2-\alpha)}{4-d^2-2\alpha},$ $\frac{(a-c)^2(2-d-\alpha)^2}{(4-d^2-2\alpha)^2},$ $\frac{(a-c)(2-d)}{4-d^2-2\alpha}$ $\frac{c[2-d^2+d(1-\alpha)]+a(2-d)(1-\alpha)}{4-d^2-2\alpha}$ $\frac{(a-c)^2(2-d)^2(1-\alpha)}{(4-d^2-2\alpha)^2}$
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