Weighted Nash equilibrium of incomplete-profile networked evolutionary games with multiple payoffs

This paper investigates the existence and convergence of weighted Nash equilibrium for incomplete-profile networked evolutionary games with multiple payoffs. First, the incomplete-profile networked evolutionary game under probabilistic myopic best response adjustment rule is transformed into an algebraic form based on the semi-tensor product of matrices. Second, a method for calculating weighted Nash equilibrium is presented, and the relationship between weighted Nash equilibrium and positive-probability fixed point is derived. Furthermore, a criterion is provided to verify whether the profiles in the feasible profile set can converge to the set of weighted Nash equilibriums with probability one. Finally, an illustrative example is given to support the new results obtained in this paper.     

Dynamics and stability for a class of evolutionary games with time delays in strategies一类具有时滞策略的演化博弈的动力学和稳定性

This paper investigates the modeling and stability of a class of finite evolutionary games with time delays in strategies. First, the evolutionary dynamics of a sequence of strategy profiles, named as the profile trajectory, is proposed to describe the strategy updating process of the evolutionary games with time delays. Using the semi-tensor product of matrices, the profile trajectory dynamics with two kinds of time delays are converted into their algebraic forms respectively. Then a sufficient condition is obtained to assure the stability of the delayed evolutionary potential games at a pure Nash equilibrium.     

系统直和博弈模型下的合作演化

演化博弈论是生物进化论与博弈论结合产生的理论,已成为研究合作演化行为的有力工具.本文研究了基于系统直和博弈模型下的合作演化行为.首先,利用复制者方程分析了双人双策略及三策略对称博弈的演化动力学过程.然后,以石头剪刀布模型和雪堆模型为基础,采用矩阵直和构建系统直和博弈模型,并将所构造的直和矩阵转化为含参数的系统总支付矩阵.随后,说明了这种方法可推广到n个博弈的情形.最后,利用MATLAB对系统直和博弈模型进行仿真模拟,从系统整体的角度分析合作演化.仿真结果表明,混合之后的系统直和博弈较单一博弈而言,合作策略的占比明显增加,且整个系统稳定性更好.这种合作演化机制呈现了全局互惠.     

Distributed Nash Equilibrium Seeking for General Networked Games With Bounded Disturbances

This paper is concerned with anti-disturbance Nash equilibrium seeking for games with partial information. First, reduced-order disturbance observer-based algorithms are proposed to achieve Nash equilibrium seeking for games with first-order and second-order players, respectively. In the developed algorithms, the observed disturbance values are included in control signals to eliminate the influence of disturbances, based on which a gradient-like optimization method is implemented for each player. Second, a signum function based distributed algorithm is proposed to attenuate disturbances for games with second-order integrator-type players. To be more specific, a signum function is involved in the proposed seeking strategy to dominate disturbances, based on which the feedback of the velocity-like states and the gradients of the functions associated with players achieves stabilization of system dynamics and optimization of players’ objective functions. Through Lyapunov stability analysis, it is proven that the players’ actions can approach a small region around the Nash equilibrium by utilizing disturbance observer-based strategies with appropriate control gains. Moreover, exponential (asymptotic) convergence can be achieved when the signum function based control strategy (with an adaptive control gain) is employed. The performance of the proposed algorithms is tested by utilizing an integrated simulation platform of virtual robot experimentation platform (V-REP) and MATLAB.      

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.     

Disturbance attenuation analysis of state feedback Nash strategy for two-player Linear Quadratic Sequential Games

There is limited formal mathematical analysis of one type of games — dynamic sequential games with large, or even infinitely large, planning horizons, from the point view of system controls. In this paper, we use a zero-sum game theoretical approach to address the disturbance attenuation analysis of state feedback Nash strategies for Dynamic Linear Quadratic Sequential Games (LQSGs) with uncertainties or disturbances. Based on the assumption that the disturbance will do the worst to the normal game players, we provide a simultaneous zero-sum game formulation between nature and each player, and a non-zero sum formulation among the players. For finite-horizon LQSGs, we first provide state feedback Nash strategies with optimal attenuation levels. Then we extend the approach to infinite-horizon LQSGs. We prove that the feedback system is Bounded Input Bounded Output (BIBO) stable with respect to the disturbances.     

Random Bimatrix Games Are Asymptotically Easy to Solve (A Simple Proof)

We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player’s payoffs are bounded and independent random variables, not necessarily identically distributed, but with almost common expectations. We show that the completely mixed uniform strategy profile, i.e., the combination of mixed strategies (one per player) where each player plays with equal probability each one of her available pure strategies, is with high probability a $\sqrt{\frac{\ln n}{n}}$ -Nash equilibrium and a $\sqrt{\frac{3\ln n}{n}}$ -well supported Nash equilibrium, where n is the number of pure strategies available to each player. This asserts that the completely mixed, uniform strategy profile is an almost Nash equilibrium for random bimatrix games, since it is, with high probability, an ?-well-supported Nash equilibrium where ? tends to zero as n tends to infinity.     

Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria

We consider a continuous-time form of repeated matrix games in which player strategies evolve in reaction to opponent actions. Players observe each other's actions, but do not have access to other player utilities. Strategy evolution may be of the best response sort, as in fictitious play, or a gradient update. Such mechanisms are known to not necessarily converge. We introduce a form of "dynamic" fictitious and gradient play strategy update mechanisms. These mechanisms use derivative action in processing opponent actions and, in some cases, can lead to behavior converging to Nash equilibria in previously nonconvergent situations. We analyze convergence in the case of exact and approximate derivative measurements of the dynamic update mechanisms. In the ideal case of exact derivative measurements, we show that convergence to Nash equilibrium can always be achieved. In the case of approximate derivative measurements, we derive a characterization of local convergence that shows how the dynamic update mechanisms can converge if the traditional static counterparts do not. We primarily discuss two player games, but also outline extensions to multiplayer games. We illustrate these methods with convergent simulations of the well known Shapley and Jordan counterexamples.     

基于纳什均衡的智能合约缺陷检测

本文研究区块链智能合约的缺陷检测问题,即检测合约中是否存在部分合约方无论选择什么动作,均无法避免损失的状态.将智能合约问题转换成合约状态迁移图上的博弈策略选择问题,提出了基于纳什均衡理论的合约缺陷自动检测方法,以及为了提高检测效率,提出了针对合约状态变迁图和博弈策略形式的一系列化简方法.最后,实现了一个智能合约建模、分...     

Equilibrium Trajectories in Dynamical Bimatrix Games with Average Integral Payoff Functionals

This paper considers models of evolutionary non-zero-sum games on the infinite time interval. Methods of differential game theory are used for the analysis of game interactions between two groups of participants. We assume that participants in these groups are controlled by signals for the behavior change. The payoffs of coalitions are defined as average integral functionals on the infinite horizon. We pose the design problem of a dynamical Nash equilibrium for the evolutionary game under consideration. The ideas and approaches of non-zero-sum differential games are employed for the determination of the Nash equilibrium solutions. The results derived in this paper involve the dynamic constructions and methods of evolutionary games. Much attention is focused on the formation of the dynamical Nash equilibrium with players strategies that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. An application of the minimax approach for constructing optimal control strategies generates dynamical Nash equilibrium trajectories yielding better results in comparison to static solutions and evolutionary models with the replicator dynamics. Finally, we make a comparison of the dynamical Nash equilibrium trajectories for evolutionary games with the average integral payoff functionals and the trajectories for evolutionary games with the global terminal payoff functionals on the infinite horizon.     

多组对策系统非劣Nash策略的最优均衡解算法

多组对策系统中求解组与组之间的非劣Nash策略至关重要.如何针对一般问题解析求出非劣Nash策略还没有有效的方法.本文阐述了一种利用组与组之间的非劣反应集构造求解非劣Nash策略的迭代算法.为此首先引进多组对策系统组内部合作对策的最优均衡值和最优均衡解的概念,然后通过证明最优均衡解是组内部隐含某一权重向量的合作对策的非劣解,得到求解合作对策的单目标规划问题.进一步说明在组内部该问题的解不仅是非劣解而且对所有局中人都优于不合作时的Nash平衡策略.最后给出了验证该算法有效性的一个实际例子.     

Matrix games with missing,interval, and ambiguous lottery payoffs of pure strategy profiles and compound strategy profiles

In a matrix game, the interactions among players are based on the assumption that each player has accurate information about the payoffs of their interactions and the other players are rationally self‐interested. As a result, the players should definitely take Nash equilibrium strategies. However, in real‐life, when choosing their optimal strategies, sometimes the players have to face missing, imprecise (i.e., interval), ambiguous lottery payoffs of pure strategy profiles and even compound strategy profile, which means that it is hard to determine a Nash equilibrium. To address this issue, in this paper we introduce a new solution concept, called ambiguous Nash equilibrium, which extends the concept of Nash equilibrium to the one that can handle these types of ambiguous payoff. Moreover, we will reveal some properties of matrix games of this kind. In particular, we show that a Nash equilibrium is a special case of ambiguous Nash equilibrium if the players have accurate information of each player's payoff sets. Finally, we give an example to illustrate how our approach deals with real‐life game theory problems.     

N人合作博弈的Nash及演化均衡稳定策略分析

在有限理性的基础上,对N人合作博弈的对称Nash均衡进行了分析,并引入演化博弈理论分析了参与人的演化均衡稳定策略,得到了不同策略选择下的均衡点。进而应用生物复制动态理论对离散时间及连续时间下的复制动态稳定集进行了研究。最后通过实例说明了该方法在博弈均衡选择上的有效性。     

A novel approximate method of computing extended Nash equilibria

In this paper, we propose a new method of computing an approximate Nash equilibrium with additional features. Existing algorithms often fail to produce an exact solution for games involving more than 3 players. Similarly, existing algorithms do not permit additional constraints on the problem. The principle idea of this paper involves proposing a methodology for computing approximate solutions through evolutionary computation. To do so, we first provide formal definitions of these problems and their approximate versions. Following which, we present the details of our solution. One of the most important advantages of the proposed solution is flexibility, which provides solutions to problems related to Nash equilibrium extensions. The proposed idea is tested on several types of games that vary with difficulty and size. All test sets are generated based on the well-known Gamut program. Additional comparisons with classical algorithms are also performed. Results indicate that Differential Evolution is capable of obtaining satisfactory solutions to large random and covariant games. The results also demonstrate that there is a high probability that even large games, in which a set of strategies with a non-zero probability of being chosen are very small, have a solution. The computation time depends mainly on the problem size, and the original Nash equilibrium problem is unaffected by additional modifications.     

Nash equilibria in risk-sensitive dynamic games

Dynamic games in which each player has an exponential cost criterion are referred to as risk-sensitive dynamic games. In this note, Nash equilibria are considered for such games. Feedback risk-sensitive Nash equilibrium solutions are derived for two-person discrete time linear-quadratic nonzero-sum games, both under complete state observation and shared partial observation     

Modelling and strategy optimisation for a kind of networked evolutionary games with memories under the bankruptcy mechanism

This paper investigates the evolutionary dynamic and strategy optimisation for a kind of networked evolutionary games whose strategy updating rules incorporate ‘bankruptcy’ mechanism, and the situation that each player's bankruptcy is due to the previous continuous low profits gaining from the game is considered. First, by using semi-tensor product of matrices method, the evolutionary dynamic of this kind of games is expressed as a higher order logical dynamic system and then converted into its algebraic form, based on which, the evolutionary dynamic of the given games can be discussed. Second, the strategy optimisation problem is investigated, and some free-type control sequences are designed to maximise the total payoff of the whole game. Finally, an illustrative example is given to show that our new results are very effective.     

On Equilibria in Quantitative Games with Reachability/Safety Objectives

In this paper, we study turn-based multiplayer quantitative non zero-sum games played on finite graphs with reachability objectives. In this framework each player aims at reaching his own goal as soon as possible. We focus on existence results for two solution concepts: Nash equilibrium and secure equilibrium. We prove the existence of finite-memory Nash (resp. secure) equilibria in n-player (resp. 2-player) games. For the case of Nash equilibria, we extend our result in two directions. First, we show that finite-memory Nash equilibria still exist when the model is enriched by allowing n-tuples of positive costs on edges (one cost by player). Secondly, we prove the existence of Nash equilibria in quantitative games with both reachability and safety objectives.     

Multi-objective design method based on evolution game and its application for suspension

Through research and bionics of biology survival mode, game players with competition, cooperation and self-adaptation capacity are introduced in the multi-objective design. The dynamic behavior and bounded rationality in game processes for players are considered according to Chinese saying “In success, commit oneself to the welfare of the society; in distress, maintain one’s own integrity”. An evolution rule, Poor-Competition-Rich-Cooperation (short for PCRC), is proposed. Then, the corresponding payoff functions of competition and cooperation behavior are established and a multi-objective design method based on evolution game is proposed. The calculation steps are as follows: 1) Taking the design objectives as different game players, and calculating factors of the design variables to objective and fuzzy clustering. The design variables are divided into multiple strategy subsets owned by each game player. 2) According to the evolution rule, each player determines its behavior and payoff function in this game round. 3) In their own strategy subsets, each game player takes their payoff as mono-objective for optimization. It gives the best strategy upon other players. And so the best strategies of all players conform the group strategy in this round. The final equilibrium solution is obtained through multi-round game based on convergence criterion. The validity and reliability of this method are shown by the results of an example of a tri-objective optimization design of passive suspension parameters.     

基于博弈论的SDN弹性控制机制研究

交换机在不同控制域之间迁移可以消除软件定义网络（SDN）中过载的控制器,过载控制器将部分交换机迁移到其邻居域需要判断迁移哪些交换机及目标控制器。首先将源目的控制器间的交互关系建模为两人非合作博弈,每个博弈参与者独立地选择策略并同时发布,通过设计合理的支付函数,该博弈存在纯策略纳什均衡。设计了一种基于非合作博弈的过载控制器消除算法(Overloaded Controller Eliminated Algorithm,OCEA),由于不同的博弈活动之间是独立进行的,因此OCEA可以合成为一种分布式的方法。最后,通过原型实现了GAME-SM,验证了算法的有效性。     

On the complexity of constrained Nash equilibria in graphical games

A widely accepted rational behavior for non-cooperative players is based on the notion of Nash equilibrium. Although the existence of a Nash equilibrium is guaranteed in the mixed framework (i.e., when players select their actions in a randomized manner) in many real-world applications the existence of “any” equilibrium is not enough. Rather, it is often desirable to single out equilibria satisfying some additional requirements (in order, for instance, to guarantee a minimum payoff to certain players), which we call constrained Nash equilibria.In this paper, a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games, where a player p may directly be interested in some of the other players only, called the neighbors of p. This setting is very useful for modeling large population games, where typically each player does not directly depend on all the players, and representing her utility function extensively is either inconvenient or infeasible.Based on this framework, the complexity of deciding the existence and of computing constrained equilibria is then investigated, in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players’ interactions. The analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies, i.e., when players are forced to deterministically choose the action to perform. In particular, for this latter case, restrictions on players’ interactions and on constraints are identified, that make the computation of Nash equilibria an easy problem, for which polynomial and highly-parallelizable algorithms are presented.