Dynamic evolutionary games

To fill these gaps, we start by characterizing how the price strategy of supply chain members changes under different recycling channels based on nonlinear dynamics theory. Subsequently, in the collecting system, we investigate the dynamic changes of the system for used products on pricing decisions. The background of the model is given in this section.

The symbols and assumptions used in the following section are described as well. The closed-loop supply chain in this paper is composed of a manufacturer, a third-party online recycler, a third-party offline recycler, and consumers. Consumers can choose to return used products through the third-party online recycler, the third-party offline recycler, or the manufacturer. Besides, the third-party online recycler and the third-party offline recycler sale used products they collect from consumers to the manufacturer.

At the end, the manufacturer remanufactures used products and directly sales remanufactured products to consumers. The model is shown in Figure 1.

In order to simplify the analysis, we make these assumptions as follows: i When the quality level of the remanufactured product reaches the qualified standard, it can be put back into the market for sale. In practice, manufacturing companies do the same, such as refurbished machines sold by Apple. The market is large enough to consume enough products. Factors such as environment, service, and consumption level are not considered [ 49 — 51 ].

Because of the constraints of the market conditions, they cannot fully grasp the behaviors and decisions of other decision-makers and take a limited rational method to make decisions. In this section, we construct the corresponding dynamic model. The players make decisions based on the principle of bounded rationality. Also, the optimal solutions are derived. The functions of recycling quantity of the third-party recyclers and manufacturer are shown as follows:.

In this model, there are three channels, so we also assume. And because there is no study focusing on difference and degree of the price sensitivity coefficient between recycling channels, we assume that price sensitivity coefficients and price cross influence coefficients are invariable, that is, and.

According to the actual situation, in each recycling channel, the price of one channel has greater impact on the price of other channels, that is,. In this study, we can simplify the functions as follows:. The profit model can be given as follows:. But the high recycling price will weaken the price competitiveness of other channels and also cause vicious price competition, which violates the original intention of manufacturers, that is, expanding the market by using multiple recycling channel.

We put formula 2 into formulas 3 — 5. In this model, the third-party recycler and the manufacturer form Stackelberg game, so backward induction is used to analyze the model, and we consider the third-party recyclers first. The game process of the paper is organized as follows. The manufacturer makes simultaneous decisions based on its recycling price and wholesale price firstly.

The third-party online and offline recyclers make recycling price decisions according to the manufacturer. This study mainly aims to analyzing the complexity on the whole system.

According to formulas 6 and 7 , we can derive the following formula:. According to formula 9 , when , we obtain the optimal recovery price of the third-party online recycler and the third-party offline recycler, which is the optimal marginal profit function of the third-party recyclers. According to formula 10 , the optimal pricing expression of the third-party recyclers can be shown as follows:.

Substitute formula 11 into formula 8. We make the first derivative of the profit function of manufacturer equal to zero, and the functions are as follows:. According to formula 13 , when , the Nash equilibrium solution of this game can be obtained, which is the optimal marginal profit function of recycling price of used products collected by manufacturer to consumer, and recycling price of used products collected by manufacturer to the third-party, and the specific expression can be seen in Appendix.

Assuming that pricing game between the manufacturer and the third-party is not instantaneous, there is a repeated game pricing process. So, each participant is not a fully rational decision maker. Combining these conditions, we assume that the three participants in this model are both limited rational decision-makers, who cannot predict the market demand information accurately or completely control their pricing behavior.

In the periodic repeating game, the manufacturer dynamically adjusts the price. Based on the marginal profit of the decision of the previous period, the manufacturer makes current decision on the basis of forecast information.

The dynamic adjustment model of repeated game is. As a bounded rational economic offline recycler, the recycling players have long and prosperous business experience.

According to formula 15 , the pricing expression of the manufacturer in the repeated game can be obtained. If the marginal profit of the previous period is positive, the manufacturer will keep the strategy of the last period in the next period. Even if the third-party offline and online recycler raise their price to increase recycling quantity in the next period, the manufacturer will still not adjust strategy and keep the belief that his profit will be maximized.

When the manufacturer adjusts parameters in the next period, the whole system stability will be affected. In order to describe the dynamic change process accurately, we set some parameters for numerical simulation analysis.

In this paper, we set , , , , , , , , , , and then we can obtain. Let and , and we can get , , ,. According to the Jury stability criterion, we can get the stable region of and as follows. In the repeated game process of the adjustment, due to limited information, the manufacturer constantly adjusts the pricing strategy to get closer to the optimal price of profit maximization.

If the speed of price changing is too fast, the system will become instable and enter the chaotic state. In Figure 2 , we draw the stable region with the changes of and , then the reasonable ranges of parameters and can be got. In Figures 3 and 4 , we describe the changes of and with and. The system is steady at the beginning. With the increasing of and , the system becomes periodic bifurcation and enters chaotic state at the end.

Therefore, we can observe that, with the increase of price in recycling channel, the system is losing stability. In Figures 5 and 6 , we describe the changes of and with and. The third-party offline recycler and the third-party online recycler follow the manufacturer. In Figures 7 and 8 , we describe the changes of profits with and. When and are small, the profits from the three channels are stable.

This affects the profits of the three recycling channels, making their respective profits chaotic. Figures 3 — 8 show how the market changes during the price adjustment process. In summary, it is not difficult to see that price chaos will cause short-term irregular fluctuations in the market economy system. This is caused by the bounded rationality of decision-makers within the system.

Therefore, chaotic research can effectively reflect the overall changes in the market economy and help companies make timely decisions.

Combining Figures 5 and 6 , we can see from Figures 7 and 8 that the excessive adjustment speed of the system will affect the recovery price of the product, thus affecting the profit of each member of the system. When the recovery price enters into a chaotic state, the whole market will become disordered, which will have a certain impact on the decision-making of each member in the recovery system.

Next, we will discuss the Pareto solutions in a dynamic system, First, we set the initial variable. In Figures 9 — 11 , we describe the changes of Pareto solutions with , , and at different periods.

Figure 9 represents the Pareto solutions of the first period of the dynamic system, Figure 10 represents the fourth period, and Figure 11 represents the ninth period. In the initial period, when ranges from 0 to 0. Red, yellow, green, and blue are used to represent different regions. It can be seen from Figures 9 and 10 that the pareto solutions increase slowly in the four regions.

After the ninth period, the number of pareto solutions may increase and stabilize due to the emergence of chaos. According to the results, the decision maker can coordinate the adjustment speed in the recovery system and the retail price of the remanufactured product, so as to avoid the occurrence of the bad solution in the system and ensure the benefit of each member of the supply chain.

When the largest Lyapunov exponent is equal to zero, bifurcation happens. We draw the largest Lyapunov exponent when and , respectively, to determine whether the system enters chaos. When the largest Lyapunov exponent is negative, the system is in a stable state; however, when the largest Lyapunov exponent is positive, the system gradually enters the chaotic state.

The largest Lyapunov exponent of and each time corresponds to a cycle bifurcation point in Figures 3 and 4 when returning to the 0 axis. The largest Lyapunov exponents at and , respectively, are shown as follows. As shown in Figures 12 and 13 , when the largest Lyapunov exponent returns to 0 first, the system begins to enter the two times of the bifurcation point.

When the largest Lyapunov exponent returns to 0 for the third time, the system enters the chaotic state. Similarly, when the largest Lyapunov exponent revolves around zero axis, the system also enters the chaotic state.

The changes of relevant parameters also influence the stability of the whole system, so we verify the effect of the change of retailer price on the stability of the whole system. We make retailer price change from 0 to 80 and obtain the 3D stable region with changes of , , and as shown in Figure It can be seen from Figure 12 that when the value of retailer price exceeds a certain value, the stable region is getting smaller gradually.

We also set , , , , and draw the stable region with changes of and as shown in Figure In Figure 15 , we can get green border area, blue border area, red border area, purple border area, and yellow border area, respectively, when , , , , and.

Observing Figure 15 , we can draw the conclusion that is the same as what we can see in Figure It also can be concluded that an appropriate retailer price should be set to keep the stability of the system. According to the above results, we can clearly determine the stable region of the system.

This prevents the system from entering chaos and is conducive to the stability of the market. The findings can help policy makers implement management policies. The manufacturer is the leader in the closed-loop supply chain, who not only seeks to maximize profits, but also to maintain system stability.

In a word, after the objective factors in the market are determined, the price adjustment of the players in supply chain will have a great impact on the stability of the system. Players need to appropriately adjust their price to cope with the market competition, but once the volatility of the adjustment of players exceeds the reasonable threshold, the market will be in a chaotic state.

When the recycling market falls into the unstable state, it will be difficult for corporations to adjust price in a suitable state or obtain higher profit. According to we have stated, once the volatility of the adjustment of players exceeds the reasonable threshold, the market will be in a chaotic state, which is harmful to the whole system.

So, in this section, taking the characteristics of the whole decision-making process into consideration, we introduce the delay control method to control the chaos. The primary idea of the delay control method is to use feedback as an external input after a time delay and ultimately achieve stabilization of unstable periodic orbits of the chaotic attractor. The core idea of this method is taking part of the information of the output signal of the system into consideration.

That is, when making decision for the next period, we think over the decision in this period as well as the decision after a period of time at the same time.

Considering other periodic values and adding control factors, the delay control method can effectively eliminate the occurrence of system chaos [ 52 ]. Thus, the dynamic adjustment model can be shown as follows:. Formula 18 is the delay strategy equations of the system. Compared with the previous system model, formula 19 adds the difference between the forecast value of the next period and the current period value as the adjustment interval. Then, we set and draw bifurcation diagram with increasing as given in Figure We show the control of the parameter to the system in Figure 16 , where we can see that when is small, the system is in chaos, and with the control parameter increasing, the system goes into period doubling and becomes stable at the end.

In conclusion, adjusting decision-making method to get over market delay is a good way of system control. Manufacturers can take this method in practice. In the delay strategy, combined with the difference of and between the actual value in the current cycle and the predicted value in the next cycle, the manufacturers with bounded rationality adjust the value of and in the next cycle by setting certain control factors, so as to avoid large price adjustments and prevent the emergence of market chaos.

This can also help manufacturers better control their own recycling prices and make more accurate decisions. So, we can draw the conclusion that the efficacy is remarkable to control the chaos by the control parameter. To achieve this purpose, the manufacturer, the third-party offline, and online recycler should especially pay attention to the recycling process and services and control the price adjustment speed in the appropriate fields.

Consumers in the market should not only focus on the recycling price when choosing the recycling channel. In order to encourage companies to carry out larger-scale battery recycling and reduce the pollution of used batteries in society, the government often subsidizes enterprises in a variety of ways. There are two main forms of funding: one is direct cash subsidies, and the other is subsidies based on the number of products recovered by the company.

In order to complete the financial support for enterprises more conveniently, the current government often uses direct cash subsidies to complete the reward work for enterprises to recycle used batteries. This model does not take the recycling amount of enterprises as a reference for subsidies, and each enterprise is subsidized uniformly. Here, we set as the subsidy price. The profit model Model 1 is as follows:. The second method of subsidy is that the government subsidizes the company based on the amount of recycling.

Here, we set the subsidy amount per unit of recycled products as. The profit model Model 2 is as follows:. Next, we will discuss the changes in the profits of the three recycling entities under the direct subsidy and the price subsidy. We start with a simple adaptive rule, where after an encounter each … Expand.

View 2 excerpts, cites background. The replicator equation and other game dynamics. Medicine, Computer Science. Proceedings of the National Academy of Sciences. Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics. Journal of theoretical biology. Constrained evolutionary games by using a mixture of imitation dynamics.

State policy couple dynamics in evolutionary games. Mathematics, Computer Science. Evolutionary Game Theory. Evolutionary game theory developed as a means to predict the expected distribution of individual behaviors in a biological system with a single species that evolves under natural selection.

It has … Expand. Highly Influenced. View 6 excerpts, cites methods and background. Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics.

Symmetry and Collective Fluctuations in Evolutionary Games. In this monograph we bring together a conceptual treatment of evolutionary dynamics and a path-ensemble approach to non-equilibrium stochastic processes. Our framework is evolutionary game theory, in … Expand. View 3 excerpts, cites background. Transition matrix model for evolutionary game dynamics. Physical review. Dharini Hingu, K. Mallikarjuna Rao, A. About this book Introduction This contributed volume considers recent advances in dynamic games and their applications, based on presentations given at the 16th Symposium of the International Society of Dynamic Games, held July , , in Amsterdam.

Written by experts in their respective disciplines, these papers cover various aspects of dynamic game theory including differential games, evolutionary games, and stochastic games. They discuss theoretical developments, algorithmic methods, issues relating to lack of information, and applications in areas such as biological or economical competition, stability in communication networks, and maintenance decisions in an electricity market, just to name a few.

Advances in Dynamic and Evolutionary Games presents state-of-the-art research in a wide spectrum of areas. As such, it serves as a testament to the vitality and growth of the field of dynamic games and their applications. It will be of interest to an interdisciplinary audience of researchers, practitioners, and advanced graduate students.

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