Stability analysis of game models with fixed and stochastic delays

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Highlights

  • The three-strategy game model with continuous distributed time delay is analyzed.

  • The two-community three-strategy games with two delays is discussed.

  • The existence of Hopf bifurcation in replicator equation is proved.

  • The rock-paper-scissors game is simulated as an example.

Abstract

In this paper, we combine evolutionary game with dynamics, and discuss two kinds of game models with time delay. First, based on continuous distributed (kernel function) stochastic delay, we give the asymptotic stability condition of general three-strategy game model. Second, we study the replicator dynamics of two-community three-strategy game model with two fixed time delays (i.e. τ1 within the community and τ2 between different communities). Based on Hopf bifurcation theory, by calculating the critical conditions for the existence of bifurcation, the influence of bifurcation on the stability of equilibrium points is analyzed. Finally, we take rock-paper-scissors game as an example to verify the correctness of the theoretical results, when the delay τ meets certain conditions or doesn’t exceed the critical value, the stability of the system will not change.

Introduction

Evolutionary game theory combines evolutionary theory and game theory. It was not until Smith and Price [1] put forward the concept of evolutionarily stable strategy for the first time in their published article in 1973 that the formal birth of evolutionary game theory was marked. Taylor and Jonker [2] introduced the first game dynamics model - replicator dynamics, and thus investigated the relationship between static evolutionarily stable strategy and dynamic stability, which is the first and most important game dynamics, and also another breakthrough development of evolutionary game theory. Evolutionary game theory has been applied and promoted by many scholars. For instance, ecology is intrinsically related to evolution, [3], [4], [5] explored the eco-evolutionary dynamics of cooperation based on migration and stochastic imitation, under punishment and free space, and in the presence of policing respectively. Furthermore, Chen and Szolnoki [6] considered a feedback-evolving game to explore the utilization of common resources, and the conditions for the evolution of cooperation in a corrupt environment with corrupt enforcers and violators were also studied with the help of evolutionary dynamics [7]. In many practical cases, especially in biology and economics, the impact of action is not immediate, but only at some point in the future. Neural network system, immune system, epidemic model, van der Pol equation, Chen’s system, energy supply and demand system, energy conservation and emission reduction system, and financial system have all been discussed by scholars in the time delay case [8], [9], [10], [11], [12], [13], [14], [15], [16], and the research on the problem including time delay has attracted a large number of researchers’ attention. Therefore, it is of practical significance to introduce delay into the replicator equation to study practical problems.

At present, the research on delayed replicator equation can be roughly divided into fixed delay and stochastic delay. Many scholars have studied the replicator equation with fixed time delay. It was proposed by Tao and Wang [17] in 1997 that the influence of time delay effect was first proposed in the replicator dynamics, which studied continuous replicator systems. For discrete systems, Alboszta and Miȩkisz [18] studied the influence of time delay on the stability of evolutionarily stable strategies in social and biological game models, and proved different forms of time delays have different effects on the stability of mixed evolutionarily stable strategy. With the help of Hopf bifurcation theory, by calculating the critical conditions for the existence of Hopf bifurcation, Wettergren [19] studied the two-person and n-person games respectively and obtained the boundary conditions for the stability of the replicator dynamic system. The influence of time delay on the convergence of evolutionarily stable strategies under different types of evolutionary dynamics based on single population game has been discussed by Hamidou et al. [20]. In addition, Ryota [21] analyzed the stability of equilibrium point with time delay of heterogeneous distribution. Further, the effect of time delay was also discussed in public goods games by Szolnoki and Perc [22], who studied the evolution of cooperation under the influence brought by intermediate delays and long delays of goods. In order to reveal the influence of time delay on environmental change, Yan et al. [23] proposed a feedback-evolving game model, indicating that the coevolutionary dynamics of strategic interaction with environmental feedback was affected by the time delay of cooperative actions. All the above studies can show that the introduction of time delay factor is more practical, and it may have a certain impact on the stability of the system. However, all the game models discussed in the above literature focus on single population (i.e. homogeneous population), and there are few researches on multi-community game models with time delay. In real life, such as biological competition issues, traffic problems, enterprise competition problems, job hunting problems, often involve several different groups with different strategies, interaction happens between different groups, and also within the same group. For two-community models, Nesrine et al. [24] discussed three types of delay - strategy delay, space delay and space strategy delay, and obtained the influence of delay on the stability of Nash equilibrium in the game.

In addition, the occurrence of actions is not only affected by a specific moment or several specific moments in the past, but the time delay is often random or affected by all actions in a previous period of time. Therefore, it is necessary to consider the influence of stochastic delay with continuous distribution on replicator dynamic system. Many scholars have introduced the time delay of continuous distribution into the studied problems for discussion [25], [26], [27], [28], [29], [30]. In game theory, there are few literatures discussing the influence of continuous distributed delay on the stability of the replicator dynamic system and all of them are based on two-strategy games. Nesrine et al. [31] discussed the stability of a single population two - strategy replicator dynamic system with three kinds of continuous distributed delay (uniform distribution, exponential distribution and Erlang distribution). Zhong [32] analyzed the dynamic behavior of the replicator dynamics with bounded continuous distributed delay in the case of constant kernel function and exponential kernel function respectively, and carried out numerical simulation with hawk-dove game.

To sum up, this paper takes replicator dynamics as the research object on the basis of existing research. On the one hand, by increasing the number of strategies and considering the time delay of continuous distribution in the form of kernel function, we obtain the asymptotic stability conditions of the three-strategy game with weak kernel function and strong kernel function. On the other hand, considering the two-community three-strategy game, by increasing the number of strategies to enrich the network structure between the games, so as to get closer to the complex reality. With the help of Hopf bifurcation theory, the critical conditions for generating Hopf bifurcation are analyzed by taking the delays within community and between different communities as parameters, and the influence of bifurcation on system stability is also discussed.

Section snippets

Model building

In this section, we analyze a game model with continuous distributed delay. Considering a single-community three-strategy game with strategy {G,H,M}, (x1,x2,x3) is the proportion of choosing G,H,M, and its payoff matrix is:GHMGHM(a1b1c1a2b2c2a3b3c3).Based on evolutionary game theory, when there is no delay, let fxi(i=1,2,3) be the payoff of choosing G,H,M, thenfxi=aix1+bix2+cix3,i=1,2,3.In the presence of time delay, we first discuss the simple case. That is, the payoff of the players at time t

Model building

We consider a population of two communities where each individual comes from one of these two communities i=1,2. Each community has three strategies {Gi,Hi,Mi}, (x1,x2,x3) respectively represents the frequency with individuals in community 1 choose strategy {G1,H1,M1}, (y1,y2,y3) represents the frequency with individuals in community 2 choose strategy {G2,H2,M2}, and the probability of interaction in the same community is pi(0pi1)(i=1,2), in different community is 1pi. The payoff matrix is

Numerical simulation of rock-paper-scissors game

In this section, we take rock-paper-scissors game as an example to verify the correctness of Sections 2 and 3.

Example 1.  We consider the following game, its payoff matrix and replicator equation of continuous distributed delay are:RPSRPS(b111b111b),{x˙1=x1[(2b(1x1)x2(b+3))0p(τ)x1(tτ)dτ+((1x1)(3+b)2bx2)0p(τ)x2(tτ)dτ+(1x1)(1b)x2(1b)],x˙2=x2[((1x2)(b+3)2bx1)0p(τ)x1(tτ)dτ+(2b(1x2)x1(b3))0p(τ)x2(tτ)dτ+(1x2)(1b)x1(1b)].

As the interior equilibrium of this game is (13

Conclusions

In this paper, with the help of differential equation theory and bifurcation theory, we combine game theory with dynamics to consider two types of delay cases - continuous distributed delay and fixed delay, the conditions for asymptotic stability of the system under the influence of delay have been calculated. It can be seen from condition (a) and (b) in Section 2.2 that the stability of the interior equilibrium point of the replicator dynamic system with continuous distributed delay mainly

Acknowledgements

This work was supported by National Natural Science Foundation of China (No.12061020); Natural Fundation of Guizhou Province (QKH[2019]1123, QKHKY[2021]088, QKH-ZK[2021]331); Introduced Talent Foundation of Guizhou University (No:201811). We are very thankful to the anonymous reviewers for their insightful comments and suggestions, which helped us to improve the manuscript considerably and further open doors for future work.

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