Competition–cooperation mechanism of online supply chain finance based on a stochastic evolutionary game

Abstract

To solve the cooperation and competition problem of online supply chain finance constructed by the high-speed development of online finance among e-commerce enterprises and commercial banks, this paper constructs a classic evolutionary game model that does not consider disturbances and a stochastic evolutionary game model considering random disturbance. A contrastive analysis of the evolutionary process and stability of strategy selection is made, and parameters for the cooperative contract are discussed. Finally, the relevant decisions are verified by a numerical simulation of contracts in Chinese online supply chain finance. The main conclusions are as follows: in the classic evolutionary game model, the probability of choosing a cooperative strategy in a high-penalty contract is more than that in a low-penalty contract, and their final strategies will stabilize at cooperation when the excess return is more than that of the competitive strategy. In the stochastic evolutionary game model, the stable conditions are stricter than those of the classic model. Moreover, the default penalty, cost of maintaining cooperation, revenue share proportion and stochastic disturbance intensity influence the dynamic evolutionary process in the stochastic evolutionary game model. The conclusions provide a reference for the cooperation of e-commerce enterprises and banks in building online supply chain finance to achieve a win‒win result.

Appendices

Appendices

Appendix 1. Proof of Theorem 1

According to Xu et al. (2011) and Sun et al. (2016), let ${c}_{1}={c}_{2}=1, \gamma =1\;\mathrm{ and }\;p=1$. According to the above Lemma, we can get $\left|X\left(t\right)\right|\le V\left(t,x\right)\le \left|X\left(t\right)\right|$. Because $X\left(t\right)\ge 0$, $V\left(t,X\left(t\right)\right)=\left|X\left(t\right)\right|=X\left(t\right)$. Thus,

$$\begin{aligned} LV\left( {t,{ }X\left( t \right)} \right) & = V_{t} \left( {t,X\left( t \right)} \right) + V_{x} \left( {t,X\left( t \right)} \right)f\left( {t,X\left( t \right)} \right) + \frac{1}{2}h^{2} \left( {t,X\left( t \right)} \right)V_{xx} \left( {t,X\left( t \right)} \right) \\ & = 0 + 1 \times f\left( {t,X\left( t \right)} \right) + \frac{1}{2}h^{2} \left( {t,X\left( t \right)} \right) \times 0 \\ & = f\left( {t,X\left( t \right)} \right) \\ & = X\left( t \right)\left( {1 - X\left( t \right)} \right)\left[ {\left( {1 - Y\left( t \right)} \right)\left( {W - C - g\Delta \pi } \right) + Y\left( t \right)\left( {N_{e} - C} \right)} \right] \\ \end{aligned}$$

(1) According to the above Lemma, in order to make the moment exponent of the zero solution stable, $X\left(t\right)\in \left[\mathrm{0,1}\right]$ and $Y\left(t\right)\in \left[\mathrm{0,1}\right]$ should satisfy the following in equation:

$$X\left(t\right)\left(1-X\left(t\right)\right)\left[\left(1-Y\left(t\right)\right)\left(W-C-g\Delta \pi \right)+Y\left(t\right)\left({N}_{e}-C\right)\right]\le -X\left(t\right)$$

That is, $(1-X\left(t\right))\left[\left(1-Y\left(t\right)\right)\left(W-C-g\Delta \pi \right)+Y\left(t\right)\left({N}_{e}-C\right)\right]\le -1$. We can obtain that

$$\left({N}_{e}-W+g\Delta \pi \right)Y\left(t\right)\le \left(C-W+g\Delta \pi \right)-\frac{1}{1-X\left(t\right)}$$

Because $Y\left(t\right)\in \left[\mathrm{0,1}\right]$, we can get

$$\left\{ {\begin{array}{*{20}l} {\left( {N_{e} - W + g\Delta \pi } \right) \times 0 \le \left( {C - W + g\Delta \pi } \right) - \frac{1}{1 - X\left( t \right)}} \hfill \\ {\left( {N_{e} - W + g\Delta \pi } \right) \times 1 \le \left( {C - W + g\Delta \pi } \right) - \frac{1}{1 - X\left( t \right)}} \hfill \\ \end{array} } \right.$$

Sorting the above inequalities, we can obtain:

$$\left\{ {\begin{array}{*{20}l} {C - W + g\Delta \pi - \frac{1}{1 - X\left( t \right)} \ge 0} \hfill \\ {C - N_{e} - \frac{1}{1 - X\left( t \right)} \ge 0} \hfill \\ \end{array} } \right.$$

(10)

Therefore, when $C-W+g\Delta \pi -\frac{1}{1-X\left(t\right)}\ge 0$ and $C-{N}_{e}-\frac{1}{1-X\left(t\right)}\ge 0$, the moment exponent of the zero solution is stable, that is, the cooperative strategy that the e-commerce enterprises choose is systematically stable. Through merging the inequalities in (7), we can get $\mathrm{min}\left\{C-W+g\Delta \pi , C-{N}_{e}\right\}-\frac{1}{1-X\left(t\right)}\ge 0$. Thus, there are $X\left(t\right)\le 1-\frac{1}{\mathrm{min}\{C-W+g\Delta \pi ,C-{N}_{e}\}}$ and $\mathrm{min}\left\{C-W+g\Delta \pi ,C-{N}_{e}\right\}>0$.

(2) If the moment exponent of the zero solution in stochastic differential Eq. (4) is unstable, then $X\left(t\right)\in \left[\mathrm{0,1}\right]$ and $Y\left(t\right)\in \left[\mathrm{0,1}\right]$ should meet the condition that

$$X\left(t\right)(1-X\left(t\right))\left[\left(1-Y\left(t\right)\right)\left(W-C-g\Delta \pi \right)+Y\left(t\right)\left({N}_{e}-C\right)\right]\ge X\left(\mathrm{t}\right)$$

Sorting the above inequality, we can obtain:

$$\left({N}_{e}-W+g\Delta \pi \right)Y\left(t\right)\ge \left(C-W+g\Delta \pi \right)+\frac{1}{1-X\left(t\right)}$$

Due to $Y\left(t\right)\in \left[\mathrm{0,1}\right]$, there is

$$\left\{ {\begin{array}{*{20}l} {\left( {N_{e} - W + g\Delta \pi } \right) \times 0 \ge \left( {C - W + g\Delta \pi } \right) + \frac{1}{1 - X\left( t \right)}} \hfill \\ {\left( {N_{e} - W + g\Delta \pi } \right) \times 1 \ge \left( {C - W + g\Delta \pi } \right) + \frac{1}{1 - X\left( t \right)}} \hfill \\ \end{array} } \right.$$

Sorting the above inequalities, we can get the formula below:

$$\left\{ {\begin{array}{*{20}l} {C - W + g\Delta \pi + \frac{1}{1 - X\left( t \right)} \le 0} \hfill \\ {C - N_{e} + \frac{1}{1 - X\left( t \right)} \le 0} \hfill \\ \end{array} } \right..$$

(11)

Therefore, when $C-W+g\Delta \pi +\frac{1}{1-X\left(t\right)}\le 0$ and $C-{N}_{e}+\frac{1}{1-X\left(t\right)}\le 0$, the moment exponent of the zero solution is unstable, which means that the competitive strategy which the e-commerce enterprises choose is systematically stable. Merging the in equations in (8), we can get that $\mathrm{max}\{C-W+g\Delta \pi ,C-{N}_{e}\}+\frac{1}{1-X\left(t\right)}\le 0$, that is, $X\left(t\right)\le 1+\frac{1}{\mathrm{max}\{C-W+g\Delta \pi ,C-{N}_{e}\}}$ and $\mathrm{max}\left\{C-W+g\Delta \pi ,C-{N}_{e}\right\}<0$.

Appendix 2. Proof of Theorem 2

Proof is the same with Theorem 1.

Appendix 3. Proof of Inference

If the variables of payoff functions and the initial proportions of e-commerce enterprises and commercial banks who accept the competitive strategy satisfy the stability conditions of Theorems 1 and 2, there exists the only ESS point E₁ (0, 0), i.e., both e-commerce enterprises and banks will stably choose the EOBO strategy combination {${A}_{0},{B}_{0}$}. Thus Inference (1) is proved. If the instability conditions of Theorems 1 and 2 are satisfied at the same time, there exists the only ESS point E₄ (1, 1), in which both e-commerce enterprises and banks will choose to compete, i.e., ECBC strategy combination {${A}_{1},{B}_{1}$}. Thus Inference (4) is proved. Inference (2) and (3) can be proved in the same way.