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.

Effect of Noisy Fitness in Real-Time Strategy Games Player Behaviour Optimisation Using Evolutionary Algorithms

This paper investigates the performance and the results of an evolutionary algorithm （EA） specifically designed for evolving the decision engine of a program （which, in this context, is called bot） that plays Planet Wars. This game, which was chosen for the Google Artificial Intelligence Challenge in 2010, requires the bot to deal with multiple target planets, while achieving a certain degree of adaptability in order to defeat different opponents in different scenarios. The decision engine of the bot is initially based on a set of rules that have been defined after an empirical study, and a genetic algorithm （GA） is used for tuning the set of constants, weights and probabilities that those rules include, and therefore, the general behaviour of the bot. Then, the bot is supplied with the evolved decision engine and the results obtained when competing with other bots （a bot offered by Google as a sparring partner, and a scripted bot with a pre-established behaviour） are thoroughly analysed. The evaluation of the candidate solutions is based on the result of non-deterministic battles （and environmental interactions） against other bots, whose outcome depends on random draws as well as on the opponents＇ actions. Therefore, the proposed GA is dealing with a noisy fitness function. After analysing the effects of the noisy fitness, we conclude that tackling randomness via repeated combats and reevaluations reduces this effect and makes the GA a highly valuable approach for solving this problem.

Equilibrium Strategy of the Pursuit-Evasion Game in Three-Dimensional Space

The pursuit-evasion game models the strategic interaction among players, attracting attention in many realistic scenarios, such as missile guidance, unmanned aerial vehicles, and target defense. Existing studies mainly concentrate on the cooperative pursuit of multiple players in two-dimensional pursuit-evasion games. However, these approaches can hardly be applied to practical situations where players usually move in three-dimensional space with a three-degree-of-freedom control. In this paper,we make the first attempt to investigate the equilibrium strategy of the realistic pursuit-evasion game, in which the pursuer follows a three-degree-of-freedom control, and the evader moves freely. First, we describe the pursuer's three-degree-of-freedom control and the evader's relative coordinate. We then rigorously derive the equilibrium strategy by solving the retrogressive path equation according to the Hamilton-Jacobi-Bellman-Isaacs(HJBI) method, which divides the pursuit-evasion process into the navigation and acceleration phases. Besides, we analyze the maximum allowable speed for the pursuer to capture the evader successfully and provide the strategy with which the evader can escape when the pursuer's speed exceeds the threshold. We further conduct comparison tests with various unilateral deviations to verify that the proposed strategy forms a Nash equilibrium.

Air combat decision-making of multiple UCAVs based on constraint strategy games

Game theory can be applied to the air combat decision-making problem of multiple unmanned combat air vehicles(UCAVs).However,it is difficult to have satisfactory decision-making results completely relying on air combat situation information,because there is a lot of time-sensitive information in a complex air combat environment.In this paper,a constraint strategy game approach is developed to generate intelligent decision-making for multiple UCAVs in complex air combat environment with air combat situation information and time-sensitive information.Initially,a constraint strategy game is employed to model attack-defense decision-making problem in complex air combat environment.Then,an algorithm is proposed for solving the constraint strategy game based on linear programming and linear inequality(CSG-LL).Finally,an example is given to illustrate the effectiveness of the proposed approach.

Data network traffic analysis and optimization strategy of real-time power grid dynamic monitoring system for wide-frequency measurements

The application and development of a wide-area measurement system(WAMS)has enabled many applications and led to several requirements based on dynamic measurement data.Such data are transmitted as big data information flow.To ensure effective transmission of wide-frequency electrical information by the communication protocol of a WAMS,this study performs real-time traffic monitoring and analysis of the data network of a power information system,and establishes corresponding network optimization strategies to solve existing transmission problems.This study utilizes the traffic analysis results obtained using the current real-time dynamic monitoring system to design an optimization strategy,covering the optimization in three progressive levels:the underlying communication protocol,source data,and transmission process.Optimization of the system structure and scheduling optimization of data information are validated to be feasible and practical via tests.

Strategy dominance mechanism of autonomous collaboration in unmanned swarm within the framework of public goods game

The key advantage of unmanned swarm operation is its autonomous cooperation. How to improve the proportion of cooperators is one of the key issues of autonomous collaboration in unmanned swarm operations. This work proposes a strategy dominance mechanism of autonomous collaboration in unmanned swarm within the framework of public goods game. It starts with the requirement analysis of autonomous collaboration in unmanned swarm;and an aspiration-driven multiplayer evolutionary game model is established based on the requirement. Then the average abundance function and strategy dominance condition of the model are constructed by theoretical derivation. Furthermore, the evolutionary mechanism of parameter adjustment in swarm cooperation is revealed via simulation,and the influences of the multiplication factor r, aspiration levelα, threshold m and other parameters on the strategy dominance conditions were simulated for both linear and threshold public goods games(PGGs) to determine the strategy dominance characteristics;Finally, deliberate proposals are suggested to provide a meaningful exploration in the actual control of unmanned swarm cooperation.

Research on Optimal Configuration of Energy Storage in Wind-Solar Microgrid Considering Real-Time Electricity Price

Capacity allocation and energy management strategies for energy storage are critical to the safety and economical operation of microgrids.In this paper,an improved energymanagement strategy based on real-time electricity price combined with state of charge is proposed to optimize the economic operation of wind and solar microgrids,and the optimal allocation of energy storage capacity is carried out by using this strategy.Firstly,the structure and model of microgrid are analyzed,and the outputmodel of wind power,photovoltaic and energy storage is established.Then,considering the interactive power cost between the microgrid and the main grid and the charge-discharge penalty cost of energy storage,an optimization objective function is established,and an improved energy management strategy is proposed on this basis.Finally,a physicalmodel is built inMATLAB/Simulink for simulation verification,and the energy management strategy is compared and analyzed on sunny and rainy days.The initial configuration cost function of energy storage is added to optimize the allocation of energy storage capacity.The simulation results show that the improved energy management strategy can make the battery charge-discharge response to real-time electricity price and state of charge better than the traditional strategy on sunny or rainy days,reduce the interactive power cost between the microgrid system and the power grid.After analyzing the change of energy storage power with cost,we obtain the best energy storage capacity and energy storage power.

How do international agreement constraints affect marine litter management strategies?A research based on differential game

Based on differential game theory,the decision-making problem of two homogeneous countries facing transboundary marine litter governance is studied.On the basis of assuming that the input of marine litter is an exogenous variable,the focus is on reducing the accumulation of marine litter through cleanup and transfer processing by both parties.Considering the constant and increasing input of marine litter,in the framework of international agreement constraints,the analysis of the game behavior of the players in the marine litter governance under the open-loop strategy(in the case of agreement constraints)and the Markov strategy(in the case of no agreement constraints)was compared and analyzed.The research results show that when the direct pollution cost of marine litter is high enough,both sides of the game adopt an open-loop strategy that complies with the constraints of the agreement,which can reduce the accumulation of marine litter and improve the environmental quality.However,when there is a high initial accumulation of marine litter,the Markov strategy without protocol constraints will be better than the open-loop strategy.In the case that marine litter does not need to be transferred,there will be no difference between the two sides of the game adopting the Markov strategy and adopting the open-loop strategy on the equilibrium growth path.

On Topics in Quantum Games

This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.

Live-streaming selling strategies for competitive firms

The booming live-streaming commerce has significantly changed the traditional e-commerce model,thus attracting much attention from both industry and academia.In recent years,an increasing number of scholars have applied analytical models to explore live-streaming strategies for firms in different scenarios.However,the previous literature mainly considers monopolists,while in the real world,competition is not rare.To fill this gap between the literature and practical observations,this paper applies a game theoretical model to study live-streaming adoption and pricing strategy for firms under competitive environments.The results show that,for competitive firms,the equilibrium strategy depends on the relation between the commission rate and the intensity of the market expansion effect.Additionally,compared to the case in which no firm adopts live-streaming,competitive firms do not always benefit from the adoption of live-streaming selling.The paper also shows that competition plays a negative role in inducing a firm to adopt live-streaming.

An Incentive Mechanism for Federated Learning:A Continuous Zero-Determinant Strategy Approach

As a representative emerging machine learning technique, federated learning(FL) has gained considerable popularity for its special feature of “making data available but not visible”. However, potential problems remain, including privacy breaches, imbalances in payment, and inequitable distribution.These shortcomings let devices reluctantly contribute relevant data to, or even refuse to participate in FL. Therefore, in the application of FL, an important but also challenging issue is to motivate as many participants as possible to provide high-quality data to FL. In this paper, we propose an incentive mechanism for FL based on the continuous zero-determinant(CZD) strategies from the perspective of game theory. We first model the interaction between the server and the devices during the FL process as a continuous iterative game. We then apply the CZD strategies for two players and then multiple players to optimize the social welfare of FL, for which we prove that the server can keep social welfare at a high and stable level. Subsequently, we design an incentive mechanism based on the CZD strategies to attract devices to contribute all of their high-accuracy data to FL.Finally, we perform simulations to demonstrate that our proposed CZD-based incentive mechanism can indeed generate high and stable social welfare in FL.

A Mean-Field Game for a Forward-Backward Stochastic System With Partial Observation and Common Noise

This paper considers a linear-quadratic(LQ) meanfield game governed by a forward-backward stochastic system with partial observation and common noise,where a coupling structure enters state equations,cost functionals and observation equations.Firstly,to reduce the complexity of solving the meanfield game,a limiting control problem is introduced.By virtue of the decomposition approach,an admissible control set is proposed.Applying a filter technique and dimensional-expansion technique,a decentralized control strategy and a consistency condition system are derived,and the related solvability is also addressed.Secondly,we discuss an approximate Nash equilibrium property of the decentralized control strategy.Finally,we work out a financial problem with some numerical simulations.

Bi-matrix Games with Fuzzy Strategies and Fuzzy Payoffs and Their Mathematical Programming Equivalents

A fuzzy bi-matrix game（FBG）,namely a two-person non-zero-sum game with fuzzy strategies and fuzzy payoffs is proposed.We have defined and analyzed the optimal strategies of this FBG,and shown that it can be transformed into a corresponding fuzzy mathematical programming issue,for which a ranking function approach can be applied.In addition,optimal strategies of FBG for both Player I and Player II can be gotten.

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.

Influences of Network Delay on Quality of Experience for Soft Objects in Networked Real-Time Game with Haptic Sense

In this paper, we investigate the influences of network delay on QoE (Quality of Experience) such as the operability of haptic interface device and the fairness between players for soft objects in a networked real-time game subjectively and objectively. We handle a networked balloon bursting game in which two players burst balloons (i.e., soft objects) in a 3D virtual space by using haptic interface devices, and the players compete for the number of burst balloons. As a result, we find that the operability depends on the network delay from the local terminal to the other terminal, and the fairness is mainly dependent on the difference in network delay between the players’ terminals. We confirm that there exists a trade-off relationship between the operability and the fairness. We also see that the contribution of the fairness is larger than that of the operability to the comprehensive quality (i.e., the weighted sum of the operability and fairness). Assessment results further show that the output timing of terminals should be adjusted to the terminal which has the latest output timing to maintain the fairness when the difference in network delay between the terminals is large. In this way, the comprehensive quality at each terminal can be maintained as high as possible.

Game Analysis of Regulation for Online Prescription Drugs under Customer Feedback Mechanism

Objective To study the way to better regulate the online sales of prescription drugs,and to provide reference for the adjustment of relevant policies since the online sales of prescription drugs has become an inevitable trend.Methods A game model was constructed for the strategy choice of pharmaceutical e-commerce platform,customers and government departments based on differential game theory and Nash equilibrium game model to analyze the pure strategy Nash equilibrium,Nash equilibrium dominant strategy of each subject and the mixed strategy Nash equilibrium under different conditions.Besides,Matlab was used to carry out simulation analysis.Results and Conclusion The study shows that:(1)Improving the credibility of the government and reducing the cost of government regulation can not only make the pharmaceutical e-commerce platform operate with high quality,but also give greater play to government functions;(2)The greater the influence of social evaluation on pharmaceutical e-commerce platforms,the lower the cost of high-quality operation of pharmaceutical e-commerce platform,and the greater the probability of customer choosing real evaluation strategy;(3)The greater the customers’perception of potential risk,the greater the compensation,and the lower the cost of reporting.Then,the greater the probability that government departments will choose strict regulation.Finally,the model solution and simulation analysis are combined to provide countermeasures and suggestions for the safety regulation of online sales of prescription drugs.

Incentive Feedback Stackelberg Strategy in Mean-Field Type Stochastic Difference Games

This paper designs an incentive Stackelberg strategy for the discrete-time stochastic systems with mean-field terms.Sufficient conditions for the existence of such a design are suggested.Moreover,the incentive strategy is obtained as a feedback form including the deviation of the state and its mathematical expectation.Also,the stability analysis is involved.It is found that the stability can be guaranteed by the follower.In addition,the specific algorithm is proposed and its effectiveness is checked by two examples.

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.

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.

Evacuation of pedestrians from a hall by game strategy update

In this paper, a cellular automaton model considering game strategy update is proposed to study the pedestrian evac- uation in a hall. Pedestrians are classified into two categories, i.e., cooperators and defectors, and they walk to an exit according to their own strategy change. The conflicts that two or three pedestrians try to occupy the same site at the same time are investigated in the Game theory model. Based on it, the relationship between the pedestrian flow rate and the evacuation time as well as the variation of cooperative proportion against evacuation time is investigated from the different initial cooperative proportions under the influence of noise. The critical value of the noise is found when there is a small number of defectors in the initial time. Moreover, the influences of the initial cooperative proportion and strength of noise on evacuation are discussed. The results show that the lower the initial cooperative proportion as well as the bigger the strength of noise, the longer the time it takes for evacuation.