Zero-determinant strategies in iterated multi-strategy games

Self-serving,rational agents sometimes cooperate to their mutual benefit.The two-player iterated prisoner′s dilemma game is a model for including the emergence of cooperation.It is generally believed that there is no simple ultimatum strategy which a player can control the return of the other participants.The zero-determinant strategy in the iterated prisoner′s dilemma dramatically expands our understanding of the classic game by uncovering strategies that provide a unilateral advantage to sentient players pitted against unwitting opponents.However,strategies in the prisoner′s dilemma game are only two strategies.Are there these results for general multi-strategy games?To address this question,the paper develops a theory for zero-determinant strategies for multi-strategy games,with any number of strategies.The analytical results exhibit a similar yet different scenario to the case of two-strategy games.The results are also applied to the Snowdrift game,the Hawk-Dove game and the Chicken game.

Research on Children’s Game Activity Design Strategies Based on Embodied Cognition Theory

Edutainment,in the kindergarten education stage,emphasizes the game as the basic activity and combines the content of education with the form of the game,thus it also forms the educational method of gamification teaching.Through investigation and analysis,it is found that the current kindergarten game activity design has the problem of improper combination of educational content and game form.The current kindergarten game activity design has problems such as stereotypes,children’s lack of active learning opportunities in activities,teachers’insufficient theoretical understanding,inappropriate teacher guidance methods,and so on.Embodied cognition theory attaches importance to the important role of the body in the development of cognition,provides new guidance for classroom teaching,and opens up a new path for classroom teaching reform.Based on the perspective of embodied cognition theory,the concept of body and mind integration should be adhered to in kindergarten teaching with games as the basic activity,experiential teaching situation should be created,children’s subjective experience should be respected,and games and interactions should be designed to promote children’s physical and mental participation,thus laying a foundation for the realization of children’s individual freedom,autonomy,and all-round development.Therefore,this paper aims at the existing problems in the current kindergarten gamification teaching and discusses the design strategy of children’s game activities based on embodied cognition theory.

关于Banach空间值L^1S-games与随机变量阵列完全收敛性的几个注记

设B是一实可分的Banach空间,具有Radon-Nikodyn性质(RNP).{Xn,n≥1}是L1B中的序列,其子序列{Xs,s∈S}是一L1极限鞅.证明了{Xn,n≥1}是L1S-game的充分必要条件是{Xn,n≥1}在条件liminfnE‖XSn‖<∞下或条件∫{τ<∞}‖XSτ‖dP<∞,τ∈下依概率收敛,其中是由{Fn,n∈N}的停时组成的集合,Sn=inf{s∈S:n≤s},n∈N.这个结论推广与改进了Luu的相关结果.而行独立的B值随机变量阵列完全收敛性的两个结果则改进与推广了T.C.Hu等人的相应结果.

Strategic games on a hierarchical network model

Among complex network models,the hierarchical network model is the one most close to such real networks as world trade web,metabolic network,WWW,actor network,and so on.It has not only the property of power-law degree distribution,but also the scaling clustering coefficient property which Barabási-Albert(BA)model does not have.BA model is a model of network growth based on growth and preferential attachment,showing the scale-free degree distribution property.In this paper,we study the evolution of cooperation on a hierarchical network model,adopting the prisoner's dilemma(PD)game and snowdrift game(SG)as metaphors of the interplay between connected nodes.BA model provides a unifying framework for the emergence of cooperation.But interestingly,we found that on hierarchical model,there is no sign of cooperation for PD game,while the fre-quency of cooperation decreases as the common benefit decreases for SG.By comparing the scaling clustering coefficient prop-erties of the hierarchical network model with that of BA model,we found that the former amplifies the effect of hubs.Considering different performances of PD game and SG on complex network,we also found that common benefit leads to cooperation in the evolution.Thus our study may shed light on the emergence of cooperation in both natural and social environments.

Energy-Efficient Clustering Using Optimization with Locust Game Theory

Wireless sensor networks(WSNs)are made up of several sensors located in a specific area and powered by a finite amount of energy to gather environmental data.WSNs use sensor nodes(SNs)to collect and transmit data.However,the power supplied by the sensor network is restricted.Thus,SNs must store energy as often as to extend the lifespan of the network.In the proposed study,effective clustering and longer network lifetimes are achieved using mul-ti-swarm optimization(MSO)and game theory based on locust search(LS-II).In this research,MSO is used to improve the optimum routing,while the LS-II approach is employed to specify the number of cluster heads(CHs)and select the best ones.After the CHs are identified,the other sensor components are allo-cated to the closest CHs to them.A game theory-based energy-efficient clustering approach is applied to WSNs.Here each SN is considered a player in the game.The SN can implement beneficial methods for itself depending on the length of the idle listening time in the active phase and then determine to choose whether or not to rest.The proposed multi-swarm with energy-efficient game theory on locust search(MSGE-LS)efficiently selects CHs,minimizes energy consumption,and improves the lifetime of networks.The findings of this study indicate that the proposed MSGE-LS is an effective method because its result proves that it increases the number of clusters,average energy consumption,lifespan extension,reduction in average packet loss,and end-to-end delay.

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.

Cooperation influenced by the correlation degree of two-layered complex networks in evolutionary prisoner’s dilemma games

An evolutionary prisoner＇s dilemma game is investigated on two-layered complex networks respectively representing interaction and learning networks in one and two dimensions. A parameter q is introduced to denote the correlation degree between the two-layered networks. Using Monte Carlo simulations we studied the effects of the correlation degree on cooperative behaviour and found that the cooperator density nontrivially changes with q for different payoff parameter values depending on the detailed strategy updating and network dimension. An explanation for the obtained results is provided.

Polynomial Time Method for Solving Nash Equilibria of Zero-Sum Games

There are a few studies that focus on solution methods for finding a Nash equilibrium of zero-sum games. We discuss the use of Karmarkar’s interior point method to solve the Nash equilibrium problems of a zero-sum game, and prove that it is theoretically a polynomial time algorithm. We implement the Karmarkar method, and a preliminary computational result shows that it performs well for zero-sum games. We also mention an affine scaling method that would help us compute Nash equilibria of general zero-sum games effectively.

The Seventh National Games of the People's Republic of China

The Seveoth National Games opened on September 4,1993,in Beijing.Orer 7,000 athletes from China's different provinces,monicipalities,autonomous regions,the PLA(People's Liberation Army)and amateur sports associations participated in 43 compet-itive events,including track-and-field events,swimming,football,basketball,'weight lifting,sboting,boxing,cyeliog,wushu,weiqi,and table tennis.A total of 328 gold medals were given out.The Ministry of Posts and Telecomounications issued one stamp to commemorate the day.

Justifying the Choice and Use of a Game and a Song in My Lesson

Games and songs can bring a lot of pleasure to children. They are useful tools in children's language learning. As English teachers, we should know how to make full use of them to stimulate children's interest and promote their learning. In this essay, I will take one primary English lesson as an example to demonstrate my point. First I am going to talk about the advantages of using a game and a song, and then analyze the teaching or learning context and the activities in the lesson. At last I am going to focus on how and why to use the game and the song in the classroom. In a word, I am going to justify my choice and use of a game and a song in a revision lesson.

Zero-determinant strategy:An underway revolution in game theory

Repeated games describe situations where players interact with each other in a dynamic pattern and make decisions ac- cording to outcomes of previous stage games. Very recently, Press and Dyson have revealed a new class of zero-determinant （ZD） strategies for the repeated games, which can enforce a fixed linear relationship between expected payoffs of two play- ers, indicating that a smart player can control her unwitting co-player＇s payoff in a unilateral way [Proc. Acad. Natl. Sci. USA 109, 10409 （2012）]. The theory of ZD strategies provides a novel viewpoint to depict interactions among players, and fundamentally changes the research paradigm of game theory. In this brief survey, we first introduce the mathematical framework of ZD strategies, and review the properties and constrains of two specifications of ZD strategies, called pinning strategies and extortion strategies. Then we review some representative research progresses, including robustness analysis, cooperative ZD strategy analysis, and evolutionary stability analysis. Finally, we discuss some significant extensions to ZD strategies, including the multi-player ZD strategies, and ZD strategies under noise. Challenges in related research fields are also listed.

THE OPTIMAL CONTROL FOR PROMOTING THE COOPERATION IN EVOLUTION GAME GENERATED BY PRISONER'S DILEMMA

Natural selection opposes the evolution of cooperation unless specific mechanisms are at work in Prisoner＇s Dilemma. By taking advantage of the modern control theory, the controller design is discussed and the optimal control is designed for promoting cooperation based on the recent advances in mechanisms for the evolution of cooperation. Two con- trol strategies are proposed： compensation control strategy for the cooperator when playing against a defector and reward control strategy for cooperator when playing against a coop- erator. The feasibility and effectiveness of these control strategies for promoting cooperation in different stages are analyzed. The reward for cooperation can＇t prevent defection from being evolutionary stable strategy （ESS）. On the other hand, compensation for the coopera- tor can＇t prevent defection from emerging and sustaining. By considering the effect and the cost, an optimal control scheme with constraint on the admissible control set is put forward. By analyzing the special nonlinear system of replicator dynamics, the exact analytic solution of the optimal control scheme is obtained based on the maximum principle. Finally, the effectiveness of the proposed method is illustrated by examples.

Modeling and Analysis of Single Machine Scheduling Based on Noncooperative Game Theory

Considering the independent optimization requirement for each demander of modernmanufacture, we explore the application of noncooperative game in production scheduling research,and model scheduling problem as competition of machine resources among a group of selfish jobs.Each job has its own performance objective. For the single machine, multi-jobs and non-preemptivescheduling problem, a noncooperative game model is established. Based on the model, many prob-lems about Nash equilibrium solution, such as the existence, quantity, properties of solution space,performance of solution and algorithm are discussed. The results are tested by numerical example.

关于Schmidt's game的一个注记

Schmidt的(α,β)-game是由Schmidt所引入的研究分形集的一个非常有力的工具。在Schmidt's game意义下winning的集合具有满的Hausdorff维数,且具有可数交稳定性。已有的结论表明,存在(α,β)-winning的集合S,当α减小时不再是winning的。本文将通过数的展式构造(α,β)-winning的分形集S,证明当β增大时,此时S不再是(α,β)-winning的。

A 2-stage strategy updating rule promotes cooperation in the prisoner's dilemma game

In this study,we propose a spatial prisoner＇s dilemma game model with a 2-stage strategy updating rule,and focus on the cooperation behavior of the system.In the first stage,i.e.,the pre-learning stage,a focal player decides whether to update his strategy according to the pre-learning factor β and the payoff difference between himself and the average of his neighbors.If the player makes up his mind to update,he enters into the second stage,i.e.,the learning stage,and adopts a strategy of a randomly selected neighbor according to the standard Fermi updating rule.The simulation results show that the cooperation level has a non-trivial dependence on the pre-learning factor.Generally,the cooperation frequency decreases as the pre-learning factor increases;but a high cooperation level can be obtained in the intermediate region of- 3〈 β 〈-1.We then give some explanations via studying the co-action of pre-learning and learning.Our results may sharpen the understanding of the influence of the strategy updating rule on evolutionary games.

Adaptive co-evolution of strategies and network leading to optimal cooperation level in spatial prisoner's dilemma game

We study evolutionary prisoner＇s dilemma game on adaptive networks where a population of players co-evolves with their interaction networks. During the co-evolution process, interacted players with opposite strategies either rewire the link between them with probability p or update their strategies with probability 1 - p depending on their payoffs. Numerical simulation shows that the final network is either split into some disconnected communities whose players share the same strategy within each community or forms a single connected network in which all nodes are in the same strategy. Interestingly, the density of cooperators in the final state can be maximised in an intermediate range of p via the competition between time scale of the network dynamics and that of the node dynamics. Finally, the mean-field analysis helps to understand the results of numerical simulation. Our results may provide some insight into understanding the emergence of cooperation in the real situation where the individuals＇ behaviour and their relationship adaptively co-evolve.

Effects of Dimers on Cooperation in the Spatial Prisoner's Dilemma Game

We investigate the evolutionary prisoner＇s dilemma game in structured populations by introducing dimers, which are defined as that two players in each dimer always hold a same strategy. We find that influences of dimets on cooperation depend on the type of dimers and the population structure. For those dimers in which players interact with each other, the cooperation level increases with the number of dimers though the cooperation improvement level depends on the type of network structures. On the other hand, the dimers, in which there are not mutual interactions, will not do any good to the cooperation level in a single community, but interestingly, will improve the cooperation level in a population with two communities. We explore the relationship between dimers and self-interactions and find that the effects of dimers are similar to that of self-interactions. Also, we find that the dimers, which are established over two communities in a multi-community network, act as one type of interaction through which information between communities is communicated by the requirement that two players in a dimer hold a same strategy.

Effect of Strategy-Homogeneity on the Prisoner’s Dilemma Game in a Square Lattice

We investigate the effect of strategy-homogeneity on the prisoner’s dilemma game in a square lattice. Strategy-homogeneity means that the population contains at least one connected group in which individuals maintain the same strategy at each iteration and may update according to updating rule at next iteration. The simulation results show that the introduction of strategy-homogeneity increases the cooperation in the evolutionary stable state. For any value of temptation to defect, the density of cooperators in equilibrium state increases firstly and then decreases as the level of strategy-homogeneity increases constantly, and there exists an appropriate level of strategy-homogeneity, maximizing the density of cooperators. The results may be favorable for comprehending cooperative behaviors in societies composed of connected groups with coherent strategy.

The Game between FDI and the Beneficial Policies of Host Country

Foreign direct investment （FDI） has great function to the growth of the economy in various countries. The beneficial policies of the foreign capitals are the main means to various countries to absorb FDI. Utilizing this means appropriately and designing beneficial policies, which can make the host country get the maximized revenue of utilizing FDI, are the key to absorb FDI effectively. In this paper, we set up a game model between the level of the host country＇s beneficial policies and the scale of FDI, and analyze what the key factors that influence the level of host country＇s beneficial policies are and how to decide the level of beneficial policies according to these key factors.