Evolutionary Stable Strategies for Supply Chains: Selfishness, Fairness, and Altruism

Caichun Chai; Hailong Zhu; Zhangwei Feng

doi:10.21078/JSSI-2018-532-20

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Evolutionary Stable Strategies for Supply Chains: Selfishness, Fairness, and Altruism

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Veröffentlicht/Copyright: 10. Dezember 2018

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Aus der Zeitschrift Journal of Systems Science and Information Band 6 Heft 6

Abstract

The management strategies of a firm are inevitable affected by individual behavior preferences. The effect of individual preference on the evolutionary dynamics for supply chains is studied by employing replicator dynamics. Each firm has three behavior preferences: selfishness, fairness, and altruism. Firstly, the case that the strategy set of manufacturers and retailers including two pure strategies is considered and the effect of preference parameter on the equilibrium outcome in the short-term interaction is discussed. Secondly, the equilibrium state in the short-term is always disturbed because the change of the environment, firm’s structure, and so forth. Using the replicator dynamics, the evolutionary stable strategies of manufacturers and retailers in the long-term interaction are analyzed. Finally, the extend case that the strategy set of manufacturers and retailers include three pure strategies is investigated. These results are found that the strategy profile in which both manufacturer and retailer choose fairness or altruism, or one player chooses fair or altruistic strategy and the other player chooses selfish strategy may be evolutionary stable, the stability of these equilibria depends on the the preference parameters.

Keywords: stability; evolutionary stable strategy; equilibrium; preference

Supported by the National Natural Science Foundation of China (71371093), and the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (17KJB120006)

Acknowledgements

This paper has been completed under the guidance of Professor Tiaojun Xiao of Nanjing Univesity, and we are very grateful to him for his support.

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Appendix A

The proof of Table 1.

The profit of manufacturers is higher than that of retailers Based on Cui, et al.[24], the objective function of a retailer who cares about fairness is
maxpur=(p−w)(a−bp)−αrf[(w−c)−(p−w)](a−bp) s.t.p≤2w−c.
The second-order derivative of the objective function with respect to p is ∂²u_r/∂ p² = –2b[1+ (αrf)²]<0, i.e., u_r is a concave function of p. Solving the first-order condition ∂u_r/∂p = 0 for p under conditional on retailers’ disadvantageous inequality, we get the response function of retailers: p(w)=a+bw2b+αrf(w−c)2(1+αrf). Thus, conditional on disadvantageous inequality, we get the optimal price for retailers as follows:
p1(w)=a+bw2b+αrf(w−c)2(1+αrf),ifw>w1,2w−c,otherwise,
where, according to the condition that the retailers’ profits are lower than the manufacturers’ profits: p ≤ 2w-c. Solving a+bw2b+αrf(w−c)2(1+αrf) = 2w – c, we get w₁ = a(1+αrf)+bc(2+αrf)b(3+2αrf). The objective function of a manufacturer who cares about fairness concern is
maxwum(w)=(w−c)(a−bp(w))−βmf[(w−c)−(p(w)−w)](a−bp(w)) s.t.p(w)=a+bw2b+αrf(w−c)2(1+αrf),w>w1.
The second-order derivative of u_m(w) with respect to w is
d2um/dw2=−b(1+2αrf)[2(1+αrf)−βmf(3+2αrf)]2(1+αrf)2<0,
i.e., u_m(w) is a concave function of w. By indirect calculating, we get that when βmf<1−αrf−(αrf)23+2αrf, the equilibrium wholesale price of manufacturer is
wff=[2βmf(a+bc)−a−3bc](αrf)2+[(4βmf(a+bc)−4bc−2a]αrf+(2a+bc)βmf−a−bc4b(βmf−1)(αrf)2+2b(4βmf−3)αrf+b(3βmf−2);if elsewff=a+3bc4b.
When βa<1−αrf−(αrf)23+2αrf, the equilibrium profits of manufacturer and retailer are
πmff=(1−2βmf)(1+αrf−βmf)(1+αrf)2(a−bc)22b(1+2αrf)(2αrf−3βmf−2αrfβmf+2)2,πrff=(1+αrf−βmf)[5αrf−βmf−6αrfβmf−4(αrf)2+1](a−bc)24b(1+2αrf)(2αrf−3βmf−2αrfβmf+2)2,
respectively. If else πmff=πrff=(a−bc)28b.
The objective function of a selfish manufacturer is max_wπ_m(w) = (w–c)(a–bp(w)), s.t. p(w)=a+bw2b+αrf(w−c)2(1+αrf),w>w1. Similarly, the equilibrium prices of manufacturer and retailer are wsf=a(1+αrf)+bc(1+3αrf)2b(1+2αrf),psf=3a+bc4b, respectively. The equilibrium profits of manufacture and retailer are πmsf=(1+αrf)(a−bc)28b(1+2αrf),πrsf=(4αrf+1)(a−bc)216b(1+2αrf), respectively.
The profit of retailers is higher than that of manufacturers. The objective function of a retailer who cares about fairness is max_pu_2r = (p–w)(a-bp)– δrf [(p–w)–(w – c)](a - bp), s.t. p ≥ 2w – c. The second-order derivative of the objective function u_2r with respect to p is ∂²u_2r/∂p² = –2b[1– (δrf)²]<0, i.e., u_2r is a concave function of p. Solving the first-order condition ∂u_2r/∂p = 0 for p under conditional on retailers’ advantageous inequality, we obtain the response function of retailers p(w)=a+bw2b−δrf(w−c)2(1−δrf). Thus, the optimal price for retailers is
p2(w)=a+bw2b−δrf(w−c)2(1−δrf),ifw<w2,2w−c,otherwise,
where w2=bc(2−δrf)+a(1−δrf)b(3−2δrf), The optimal problem of a fairness concern manufacturer is max_pu_2m(w) = (w – c)(a – p(w))– γmf[(p(w)−w)−(w−c)](a−bp(w)),s.t.p(w)=a+bw2b−δrf(w−c)2(1−δrf),w<w2. There is no feasible solution in the feasible region, so the profit of manufacturer is always higher than that of the retailer.

Appendix B

The proof of Proposition 1. We only prove Case 1) of Proposition 1. According to the replicator dynamics (3), we get the Jacobian matrix

(1−2x)[πmsf−πmff+y(πmss+πmff−πmsf−πmfs)]x(1−x)(πmss+πmff−πmsf−πmfs)y(1−y)(πrss+πrff−πrsf−πrfs)(1−2y)[πrfs−πrff+x(πrss+πrff−πrsf−πrfs)].

Substituting x = 1, y = 0 into Jacobian matrix, we get that the Jacobian matrix at equilibrium point (1, 0) is (πmff−πmsf00πrss−πrsf). Hence the two eigenvalues of Jacobian matrix at (1, 0) are

λ1=πmff−πmsf=−(a−bc)2(1+αrf)(1+2αrf)(βmf)28b[2+2αrf(1−βmf)−3βmf]2<0,

λ2=πrss−πrsf=−(a−bc)2αrf8b(1+2αrf)<0, respectively. According to the stability theory of differential equation, we get that the equilibrium point (1, 0) is local asymptotically stable. Similarly, we can prove Case 2) in Proposition 1. Hence the Proposition 1 is true.

Appendix C

The proof of Table 3.

According to Appendix A, we find that the profit of the retailer is always less than the profit of the manufacturer. So we only consider the case that the profit of manufacturer is higher than the profit of the retailer in Appendix C.

When manufacturers care about altruism and retailers care about fairness. The objective of a retailer is max_pu_r = (p – w)(a – bp)– αrf [(w – c)-(p – w)](a – bp), s.t. p ≤ 2w – c. From the first-order condition, we get the optimal price of the retailer as follows: when w > w1,p3=a+bw2b+αrf(w−c)2(1+αrf); if else p₃ = 2w – c. So the objective of an altruistic manufacturer is max_wu_ma(w) = (w – c)(a–bp(w))+β_a(p(w)–w)(a–bp(w)), s.t. p(w) = a+bw2b+αrf(w−c)2(1+αrf),w>w₁. By the second-order derivative of the objective function with respective to w, d²u_ma/du² = −b(1+2αrf)[2(1+αrf)−βa]/[2(1+αrf)2]<0, we know the objective function is a concave function of w. Then when βa≤(1+αrf)(1−2αrf)2+3αrf+2(αrf)2, the equilibrium prices of the manufacturer and the retailer are
waf=a+bc−aβa+αrf(2a+4bc−2aβa)+(αrf)2[a+3bc−(1−bc)βa]b(1+2αrf)(2+2αrf−βa),paf=3a+bc−2aβa+αrf[3a+bc−(a−bc)βa]2b(2+2αrf−βa),
respectively; if else waf=a+3bc4b,paf=a+bc2b. Furthermore, when βa≤(1+αrf)(1−2αrf)2+3αrf+2(αrf)2, we get that the equilibrium profits of manufacturers and retailers are
πmaf=(a−bc)2(1+αrf)2(1−βa)(1+αrf+αrfβa)2b(1+2αrf)(2+2αrf−βa)2,πraf=(a−bc)2[1+4(αrf)2+αrf(5−βa)](1+αrf+αrfβa)4b(1+2αrf)(2+2αrf−βa)2,
respectively. If else πmaf=πraf=(a−bc)28b.
Manufacturers care about fairness and retailers are altruist The objective function of the retailer is max_pu_ra = (p – w)(a–bp)+α_a(w–c)(a–bp). The second-order derivative of the objective function with respect to p is ∂²u_ra/∂p² = –2b < 0, so the objective is a concave function of p. Hence, according to the first-order condition, we get that the response function of retailer is p(w)=a+bw2b−αa(w−c)2. Furthermore, we get that the optimal problem of manufacturers is maxwum(w)=(w−c)(a−bp(w))−βmf[(w−c)−(p(w)−w)](a−bp(w)),s.t.p(w)=a+bw2b−αa(w−c)2,w>(p+c)/2. The second-order derivative of the objective function with respect to w is d²u_m/dw² = –b(1–α_a)[2– βmf (3+β_a)]/2<0. So the objective function is a concave function of w. The equilibrium prices of manufacturers and retailers are
wfa=a+bc−2bcαa(1−βmf)−(2a+bc)βmf+bcαa2βmfb(1−αa)[2−(3+αa)βmf],pfa=3a+bc−[5a+bc+(a+bc)αa]βmf2b[2−(3+αa)βmf,
respectively.
Combing the results in Subsections 3.1–3.3, we get the results in Table 3. So Table 3 is true.

Received: 2017-11-08

Accepted: 2018-03-06

Published Online: 2018-12-10

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https://doi.org/10.21078/JSSI-2018-532-20

Schlagwörter für diesen Artikel

stability; evolutionary stable strategy; equilibrium; preference