A Matrix Approach to the Modeling and Analysis of Networked Evolutionary Games With Time Delays

Using the semi-tensor product method, this paper investigates the modeling and analysis of networked evolutionary games(NEGs) with finite memories, and presents a number of new results. Firstly, a kind of algebraic expression is formulated for the networked evolutionary games with finite memories, based on which the behavior of the corresponding evolutionary game is analyzed. Secondly, under a proper assumption, the existence of Nash equilibrium of the given networked evolutionary games is proved and a free-type strategy sequence is designed for the convergence to the Nash equilibrium. Finally, an illustrative example is worked out to support the obtained new results.

Evolutionary games in a generalized Moran process with arbitrary selection strength and mutation

By using a generalized fitness-dependent Moran process, an evolutionary model for symmetric 2 × 2 games in a well-mixed population with a finite size is investigated. In the model, the individuals＇ payoff accumulating from games is mapped into fitness using an exponent function. Both selection strength β and mutation rate ε are considered. The process is an ergodic birth-death process. Based on the limit distribution of the process, we give the analysis results for which strategy will be favoured when s is small enough. The results depend on not only the payoff matrix of the game, but also on the population size. Especially, we prove that natural selection favours the strategy which is risk-dominant when the population size is large enough. For arbitrary β and ε values, the ＇Hawk-Dove＇ game and the ＇Coordinate＇ game are used to illustrate our model. We give the evolutionary stable strategy （ESS） of the games and compare the results with those of the replicator dynamics in the infinite population. The results are determined by simulation experiments.

Evolutionary Games in Two-Layer Networks with the Introduction of Dominant Strategy

We study evolutionary games in two-layer networks by introducing the correlation between two layers through the C-dominance or the D-dominance. We assume that individuals play prisoner＇s dilemma game （PDG） in one layer and snowdrift game （SDG） in the other. We explore the dependences of the fraction of the strategy cooperation in different layers on the game parameter and initial conditions. The results on two-layer square lattices show that, when cooperation is the dominant strategy, initial conditions strongly influence cooperation in the PDG layer while have no impact in the SDG layer. Moreover, in contrast to the result for PDG in single-layer square lattices, the parameter regime where cooperation could be maintained expands significantly in the PDG layer. We also investigate the effects of mutation and network topology. We find that different mutation rates do not change the cooperation behaviors. Moreover, similar behaviors on cooperation could be found in two-layer random networks.

Integrating the environmental factor into the strategy updating rule to promote cooperation in evolutionary games

Many previous studies have shown that the environment plays an important role for social individuals. In this paper, we integrate the environmental factor, which is defined as the average payoff of all a player＇s neighbours, with the standard Fermi updating rule by introducing a tunable parameter, w. It is found that the level of cooperation increases remarkably, and that the cooperators can better resist the invasion of defection with an increase in w. This interesting phenomenon is then explained from a microscopic view. In addition, the universality of this mechanism is also proved with the help of the small-world network and the random regular graph. This work may be helpful in understanding cooperation behaviour in species from unicellular organisms up to human beings.

Stability Analysis of Networked Evolutionary Games with Profile-Dependent Delays

This paper investigates the networked evolutionary games(NEGs)with profile-dependent delays,including modeling and stability analysis.Profile-dependent delay,which varies with the game profiles,slows the information transmission between participants.Firstly,the dynamics model is proposed for the profile-dependent delayed NEG,then the algebraic formulation is established using the algebraic state space approach.Secondly,the dynamic behavior of the game is discussed,involving general stability and evolutionarily stable profile analysis.Necessary and sufficient criteria are derived using the matrices,which can be easily verified by mathematical software.Finally,a numerical example is carried out to demonstrate the validity of the theoretical results.

Central environmental protection inspection and carbon emission reduction: A tripartite evolutionary game model from the perspective of carbon neutrality

Since the carbon neutrality target was proposed,many countries have been facing severe challenges to carbon emission reduction sustainably.This study is conducted using a tripartite evolutionary game model to explore the impact of the central environmental protection inspection(CEPI)on driving carbon emission reduction,and to study what factors influence the strategic choices of each party and how they interact with each other.The research results suggest that local governments and manufacturing enterprises would choose strategies that are beneficial to carbon reduction when CEPI increases.When the initial willingness of all parties increases 20%,50%—80%,the time spent for the whole system to achieve stability decreases from 100%,60%—30%.The evolutionary result of“thorough inspection,regulation implementation,low-carbon management”is the best strategy for the tripartite evolutionary game.Moreover,the smaller the cost and the larger the benefit,the greater the likelihood of the three-party game stability strategy appears.This study has important guiding significance for other developing countries to promote carbon emission reduction by environmental policy.

Evolutionary game dynamics of combining two different aspiration-driven update rules in structured populations

In evolutionary games,most studies on finite populations have focused on a single updating mechanism.However,given the differences in individual cognition,individuals may change their strategies according to different updating mechanisms.For this reason,we consider two different aspiration-driven updating mechanisms in structured populations:satisfied-stay unsatisfied shift(SSUS)and satisfied-cooperate unsatisfied defect(SCUD).To simulate the game player’s learning process,this paper improves the particle swarm optimization algorithm,which will be used to simulate the game player’s strategy selection,i.e.,population particle swarm optimization(PPSO)algorithms.We find that in the prisoner’s dilemma,the conditions that SSUS facilitates the evolution of cooperation do not enable cooperation to emerge.In contrast,SCUD conditions that promote the evolution of cooperation enable cooperation to emerge.In addition,the invasion of SCUD individuals helps promote cooperation among SSUS individuals.Simulated by the PPSO algorithm,the theoretical approximation results are found to be consistent with the trend of change in the simulation results.

An evolutionary game theory-based machine learning framework for predicting mandatory lane change decision

Mandatory lane change(MLC)is likely to cause traffic oscillations,which have a negative impact on traffic efficiency and safety.There is a rapid increase in research on mandatory lane change decision(MLCD)prediction,which can be categorized into physics-based models and machine-learning models.Both types of models have their advantages and disadvantages.To obtain a more advanced MLCD prediction method,this study proposes a hybrid architecture,which combines the Evolutionary Game Theory(EGT)based model(considering data efficient and interpretable)and the Machine Learning(ML)based model(considering high prediction accuracy)to model the mandatory lane change decision of multi-style drivers(i.e.EGTML framework).Therefore,EGT is utilized to introduce physical information,which can describe the progressive cooperative interactions between drivers and predict the decision-making of multi-style drivers.The generalization of the EGTML method is further validated using four machine learning models:ANN,RF,LightGBM,and XGBoost.The superiority of EGTML is demonstrated using real-world data(i.e.,Next Generation SIMulation,NGSIM).The results of sensitivity analysis show that the EGTML model outperforms the general ML model,especially when the data is sparse.

Some Analytical Properties of the Model for Stochastic Evolutionary Games in Finite Populations with Non-uniform Interaction Rate

Traditional evolutionary games assume uniform interaction rate, which means that the rate at which individuals meet and interact is independent of their strategies. But in some systems, especially biological systems, the players interact with each other discriminately. Taylor and Nowak (2006) were the first to establish the corresponding non-uniform interaction rate model by allowing the interaction rates to depend on strategies. Their model is based on replicator dynamics which assumes an infinite size population. But in reality, the number of individuals in the population is always finite, and there will be some random interference in the individuals' strategy selection process. Therefore, it is more practical to establish the corresponding stochastic evolutionary model in finite populations. In fact, the analysis of evolutionary games in a finite size population is more difficult. Just as Taylor and Nowak said in the outlook section of their paper, 'The analysis of non-uniform interaction rates should be extended to stochastic game dynamics of finite populations.' In this paper, we are exactly doing this work. We extend Taylor and Nowak's model from infinite to finite case, especially focusing on the influence of non-uniform connection characteristics on the evolutionary stable state of the system. We model the strategy evolutionary process of the population by a continuous ergodic Markov process. Based on the limit distribution of the process, we can give the evolutionary stable state of the system. We make a complete classification of the symmetric 2×2 games. For each case game, the corresponding limit distribution of the Markov-based process is given when noise intensity is small enough. In contrast with most literatures in evolutionary games using the simulation method, all our results obtained are analytical. Especially, in the dominant-case game, coexistence of the two strategies may become evolutionary stable states in our model. This result can be used to explain the emergence of cooperation in the Prisoner is Dilemma Games to some extent. Some specific examples are given to illustrate our results.

Evolution of Cooperation in Evolutionary Games for Heterogeneous Interactions

When a population structure is modelled as a square lattice,the cooperation may be improved for an evolutionary prisoner dilemma game or be inhibited for an evolutionary snowdrift game.In this work,we investigate cooperation in a population on a square lattice where the interaction among players contains both prisoner dilemma game and snowdrift game.The heterogeneity in interaction is introduced to the population in two different ways:the heterogenous character of interaction assigned to every player(HCP) or the heterogenous character of interaction assigned to every link between any two players(HCL).The resonant enhancement of cooperation in the case of HCP is observed while the resonant inhibition of cooperation in the case of HCL is prominent.The explanations on the enhancement or inhibition of cooperation are presented for these two cases.

Semi-tensor product approach to networked evolutionary games

In this paper a comprehensive introduction for modeling and control of networked evolutionary games （NEGs） via semi-tensor product （STP） approach is presented. First, we review the mathematical model of an NEG, which consists of three ingredients： network graph, fundamental network game, and strategy updating rule. Three kinds of network graphs are considered, which are i） undirected graph for symmetric games; ii） directed graph for asymmetric games, and iii） d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games （FEGs） are discussed, which are i） two strategies and symmetric （S-2）; ii） two strategies and asymmetric （A-2）; and iii） three strategies and symmetric （S-3）. Three strategy updating rules （SUR） are introduced, which are i） Unconditional Imitation （UI）; ii） Fermi Rule（FR）; iii） Myopic Best Response Adjustment Rule （MBRA）. First, we review the fundamental evolutionary equation （FEE） and use it to construct network profile dynamics （NPD）of NEGs. To show how the dynamics of an NEG can be modeled as a discrete time dynamics within an algebraic state space, the fundamental evolutionary equation （FEE） of each player is discussed. Using FEEs, the network strategy profile dynamics （NSPD） is built by providing efficient algorithms. Finally, we consider three more complicated NEGs： i） NEG with different length historical information, ii） NEG with multi-species, and iii） NEG with time-varying payoffs. In all the cases, formulas are provided to construct the corresponding NSPDs. Using these NSPDs, certain properties are explored. Examples are presented to demonstrate the model constructing method, analysis and control design technique, and to reveal certain dynamic behaviors of NEGs.

Dynamics and Stability of Potential Hyper-networked Evolutionary Games

This paper considers the modeling and convergence of hyper-networked evolutionary games （HNEGs）. In an HNEG the network graph is a hypergraph, which allows the fundamental network game to be a multi-player one. Using semi-tensor product of matrices and the fundamental evolutionary equation, the dynamics of an HNEG is obtained and we extend the results about the networked evolutionary games to show whether an HNEG is potential and how to calculate the potential. Then we propose a new strategy updating rule, called the cascading myopic best response adjustment rule （MBRAR）, and prove that under the cascading MBRAR the strategies of an HNEG will converge to a pure Nash equilibrium. An example is presented and discussed in detail to demonstrate the theoretical and numerical results.

Vector Space Structure of Finite Evolutionary Games and Its Application to Strategy Profile Convergence

A vector space structure is proposed for the set of finite games with fixed nmnbers of players and strategies for each players. Two statical equivalences are used to reduce tile dimension of finite games. Under the vector space structure the subspaces of exact and weighted potential games are investigated. Formulas are provided to calculate them. Then the finite evolutionary games （EGs） are considered. Strategy profile dynamics is obtained using different strategy updating rules （SURs）. Certain SURs, which assure the convergence of trajectories to pure Nash equilibriums, are investigated. Using the vector space structure, the projection of finite games to the subspace of exact （or weighted） potential games is considered, and a simple formula is given to calculate the projection. The convergence of near potential games to an c-equilibrium is studied. Further more, the Lyapunov function of EGs is defined and its application to the convergence of EGs is presented. Finally, the near potential function for an EG is defined, and it is proved that if the near potential function of an EG is a Lyapunov function, the EG will converge to a pure Nash equilibrium. Some examples are presented to illustrate the results.

Effects of the planarity and heterogeneity of networks on evolutionary two-player games

We study the effects of the planarity and heterogeneity of networks on evolutionary two-player symmetric games by considering four different kinds of networks, including two types of heterogeneous networks: the weighted planar stochastic lattice(a planar scale-free network) and the random uncorrelated scale-free network with the same degree distribution as the weighted planar stochastic lattice; and two types of homogeneous networks: the hexagonal lattice and the random regular network with the same degree k_0= 6 as the hexagonal lattice. Using extensive computer simulations, we found that both the planarity and heterogeneity of the network have a significant influence on the evolution of cooperation, either promotion or inhibition, depending not only on the specific kind of game(the Harmony, Snowdrift, Stag Hunt or Prisoner's Dilemma games), but also on the update rule(the Fermi, replicator or unconditional imitation rules).

Evolutionary privacy-preserving learning strategies for edge-based IoT data sharing schemes

The fast proliferation of edge devices for the Internet of Things(IoT)has led to massive volumes of data explosion.The generated data is collected and shared using edge-based IoT structures at a considerably high frequency.Thus,the data-sharing privacy exposure issue is increasingly intimidating when IoT devices make malicious requests for filching sensitive information from a cloud storage system through edge nodes.To address the identified issue,we present evolutionary privacy preservation learning strategies for an edge computing-based IoT data sharing scheme.In particular,we introduce evolutionary game theory and construct a payoff matrix to symbolize intercommunication between IoT devices and edge nodes,where IoT devices and edge nodes are two parties of the game.IoT devices may make malicious requests to achieve their goals of stealing privacy.Accordingly,edge nodes should deny malicious IoT device requests to prevent IoT data from being disclosed.They dynamically adjust their own strategies according to the opponent's strategy and finally maximize the payoffs.Built upon a developed application framework to illustrate the concrete data sharing architecture,a novel algorithm is proposed that can derive the optimal evolutionary learning strategy.Furthermore,we numerically simulate evolutionarily stable strategies,and the final results experimentally verify the correctness of the IoT data sharing privacy preservation scheme.Therefore,the proposed model can effectively defeat malicious invasion and protect sensitive information from leaking when IoT data is shared.

Evolutionary game-based optimization of green certificate-carbon emission right-electricity joint market for thermal-wind-photovoltaic power system

With the increasing proportion of renewable energy in the power market,the demands on government financial subsidies are gradually increasing.Thus,a joint green certificate-carbon emission right-electricity multi-market trading process is proposed to study the market-based strategy for renewable energy.Considering the commodity characteristics of green certificates and carbon emission rights,the dynamic cost models of green certificates and carbon rights are constructed based on the Rubinstein game and ladder pricing models.Furthermore,considering the irrational bidding behavior of energy suppliers in the actual electricity market,an evolutionary game based multi-market bidding optimization model is presented.Subsequently,it is solved using a composite differential evolutionary algorithm.Finally,the case study results reveal that the proposed model can increase profits and the consumption rate of renewable energy and reduce carbon emission.

Dynamic Evolutionary Game-based Modeling,Analysis and Performance Enhancement of Blockchain Channels

The recent development of channel technology has promised to reduce the transaction verification time in blockchain operations.When transactions are transmitted through the channels created by nodes,the nodes need to cooperate with each other.If one party refuses to do so,the channel is unstable.A stable channel is thus required.Because nodes may show uncooperative behavior,they may have a negative impact on the stability of such channels.In order to address this issue,this work proposes a dynamic evolutionary game model based on node behavior.This model considers various defense strategies'cost and attack success ratio under them.Nodes can dynamically adjust their strategies according to the behavior of attackers to achieve their effective defense.The equilibrium stability of the proposed model can be achieved.The proposed model can be applied to general channel networks.It is compared with two state-of-the-art blockchain channels:Lightning network and Spirit channels.The experimental results show that the proposed model can be used to improve a channel's stability and keep it in a good cooperative stable state.Thus its use enables a blockchain to enjoy higher transaction success ratio and lower transaction transmission delay than the use of its two peers.

Adaptive dynamic reconfiguration mechanism of unmanned swarm topology based on an evolutionary game

Autonomous cooperation of unmanned swarms is the research focus on“new combat forces”and“disruptive technologies”in military fields.The mechanism design is the fundamental way to realize autonomous cooperation.Facing the realistic requirements of a swarm network dynamic adjustment under the background of high dynamics and strong confrontation and aiming at the optimization of the coordination level,an adaptive dynamic reconfiguration mechanism of unmanned swarm topology based on an evolutionary game is designed.This paper analyzes military requirements and proposes the basic framework of autonomous cooperation of unmanned swarms,including the emergence of swarm intelligence,information network construction and collaborative mechanism design.Then,based on the framework,the adaptive dynamic reconfiguration mechanism is discussed in detail from two aspects:topology dynamics and strategy dynamics.Next,the unmanned swarms’community network is designed,and the network characteristics are analyzed.Moreover,the mechanism characteristics are analyzed by numerical simulation,focusing on the impact of key parameters,such as cost,benefit coefficient and adjustment rate on the level of swarm cooperation.Finally,the conclusion is made,which is expected to provide a theoretical reference and decision support for cooperative mode design and combat effectiveness generation of unmanned swarm operations.

EVOLUTIONARY GAME OF DYNAMIC KNOWLEDGE EXCHANGING IN KNOWLEDGE INTERACTION

Characteristics of knowledge exchanging behavior among individual agents in a knowledge dynamic interaction system are studied by using the game theory. An analytic model of evolutionary game of continuous dynamic knowledge interaction behavior is founded based on the structure of the evolutionary game chain. Possible evolution trends of the model are discussed. Finally, evolutionary stable strategies （ESSs） of knowledge transactions among individual agents in the knowledge network are identified by simulation data. Stable charicteristics of ESS in a continuous knowledge exchanging team help employee to communicate and grasp the dynamic regulation of shared knowledge.

Evolutionary Game Dynamics in a Fitness-Dependent Wright-Fisher Process with Noise

Evolutionary game dynamics in finite size populations can be described by a fitness-dependent Wright- Fisher process. We consider symmetric 2×2 games in a well-mixed population. In our model, two parameters to describe the level of player＇s rationality and noise intensity in environment are introduced. In contrast with the fixation probability method that used in a noiseless case, the introducing of the noise intensity parameter makes the process an ergodic Markov process and based on the limit distribution of the process, we can analysis the evolutionary stable strategy （ESS） of the games. We illustrate the effects of the two parameters on the ESS of games using the Prisoner＇s dilemma games （PDG） and the snowdrift games （SG）. We also compare the ESS of our model with that of the replicator dynamics in infinite size populations. The results are determined by simulation experiments.