An overlapping generations stochastic differential game

The paper presents a noncooperative stochastic differential game played by an infinite number of overlapping generations of players. The number and types of players in future generations are uncertain and the state dynamics are given by a stochastic differential equation. A Markovian Nash equilibrium is characterized by a verification theorem of the Hamilton-Jacobi-Bellman type. A resource extraction game is offered as an illustration.     

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.     

Differential games of <Emphasis Type="Italic">N</Emphasis> players in stochastic systems with controlled diffusion terms

Differential Games of N players defined by stochastic systems with controlled diffusion terms are considered. Necessary conditions of equilibrium strategy are obtained. These conditions are specified for the linear quadratic differential game of N players in terms of differential Riccati equation for program and positional equilibrium situations.     

Linear−quadratic optimal control and nonzero‐sum differential game of forward−backward stochastic system

An existence and uniqueness result for one kind of forward–backward stochastic differential equations with double dimensions was obtained under some monotonicity conditions. Then this result was applied to the linear‐quadratic stochastic optimal control and nonzero‐sum differential game of forward–backward stochastic system. The explicit forms of the optimal control and the Nash equilibrium point are obtained respectively. We note that our method is effective in studying the uniqueness of Nash equilibrium point. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Resilient control for wireless networked control systems under DoS attack via a hierarchical game

In this paper, the resilient control problem is investigated for a wireless networked control system (WNCS) under denial‐of‐service (DoS) attack via a hierarchical game approach. In the presence of a wireless network, a DoS attacker leads to extra packet dropout in the cyber layer of WNCS by launching interference power. A zero‐sum Markov game is exploited to model the interaction between the transmitter and the DoS attacker under dynamic network environment. Additionally, with the attack‐induced packet loss, an H_∞ minimax controller is designed in the physical layer by using a delta operator approach. Both value iteration and Q‐learning methods are used to solve the hierarchical game problem for the WNCS. The proposed method is applied to a load frequency control system to illustrate the effectiveness.     

Graphical Congestion Games

We consider congestion games with linear latency functions in which each player is aware only of a subset of all the other players. This is modeled by means of a social knowledge graph G in which nodes represent players and there is an edge from i to j if i knows j. Under the assumption that the payoff of each player is affected only by the strategies of the adjacent ones, we first give a complete characterization of the games possessing pure Nash equilibria. Namely, if the social graph G is undirected, the game is an exact potential game and thus isomorphic to a classical congestion game. As a consequence, it always converges and possesses Nash equilibria. On the other hand, if G is directed an equilibrium is not guaranteed to exist, but the game is always convergent and an equilibrium can be found in polynomial time if G is acyclic, even if finding the best equilibrium remains an intractable problem.     

The problem of optimal approach in locally Euclidean spaces

A differential game of optimal approach with simple motions when players move in locally Euclidean spaces is studied. The game-end moment is fixed, and the game payment is a distance between the pursuer and the evader at the game-end moment. The value of game is obtained in the explicit form for any initial positions of players. Moreover, the differential game of optimal approach for the denumerable number of pursuers and one evader in the Euclidean space is solved. All pursuers are controlled by one parameter.     

Model‐Free H∞ Control Design for Unknown Continuous‐Time Linear System Using Adaptive Dynamic Programming

In this paper, a new online model‐free adaptive dynamic programming algorithm is developed to solve the H_∞ control problem of the continuous‐time linear system with completely unknown system dynamics. Solving the game algebraic Riccati equation, commonly used in H_∞ state feedback control design, is often referred to as a two‐player differential game where one player tries to minimize the predefined performance index while the other tries to maximize it. Using data generated in real time along the system trajectories, this new method can solve online the game algebraic Riccati equation without requiring the full knowledge of system dynamics. A rigorous proof of convergence of the proposed algorithm is given. Finally, simulation studies on two examples demonstrate the effectiveness of the proposed method.     

基于博弈论的H2/H∞混合控制及其在汽车主动悬架中的应用

文章提出把H₂/H_∞混合控制问题抽象为两个对局者信息不完全情况下的非零和博弈模型.在构造2×2非零和博弈模型中把反映系统鲁棒性能通道和动态性能通道作为参加博弈的两方,以H_∞和H₂控制方案作为两种纯策略,基于纳什谈判解原理设计出求解H₂/H_∞混合控制问题纳什均衡点的一般算法.把该算法应用于汽车主动悬架设计出基于纳什均衡点的H₂/H_∞输出反馈控制器.使用MATLAB进行仿真,仿真结果表明主动悬架系统在保持鲁棒稳定性与获得优化的动态性能指标之间取得平衡.     

Equilibria problems on games: Complexity versus succinctness

We study the computational complexity of problems involving equilibria in strategic games and in perfect information extensive games when the number of players is large. We consider, among others, the problems of deciding the existence of a pure Nash equilibrium in strategic games or deciding the existence of a pure Nash or a subgame perfect Nash equilibrium with a given payoff in finite perfect information extensive games. We address the fundamental question of how can we represent a game with a large number of players? We propose three ways of representing a game with different degrees of succinctness for the components of the game. For perfect information extensive games we show that when the number of moves of each player is large and the input game is represented succinctly these problems are PSPACE-complete. In contraposition, when the game is described explicitly by means of its associated tree all these problems are decidable in polynomial time. For strategic games we show that the complexity of deciding the existence of a pure Nash equilibrium depends on the succinctness of the game representation and then on the size of the action sets. In particular we show that it is NP-complete, when the number of players is large and the number of actions for each player is constant, and that the problem is -complete when the number of players is a constant and the size of the action sets is exponential in the size of the game representation. Again when the game is described explicitly the problem is decidable in polynomial time.     

ε-Equilibrium in LQ differential games with bounded uncertain disturbances: robustness of standard strategies and new strategies with adaptation

A finite time multi-persons linear-quadratic differential game (LQDG) with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances and uncertainties, the standard feedback Nash strategies are shown to yield to an ε-(or quasi) Nash equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, is suggested. It is defined an “adaptation period” where all participants apply the standard feedback Nash strategies with the, so-called, “shifting signal” generated only by a known external exciting signal. Then, during the adaptation, the readjustment (or correction) of the control strategies is realized to estimate the effect of unknown periodic disturbances by the corresponding correction of the shifting vector. After that adaptation period, the complete standard strategies including the recalculated shifting signal are activated allowing the achievement of pure (ε?=?0) Nash equilibrium for the rest of the game. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.     

Nash equilibrium in differential games and the quasi-strategy formalism

We consider a two-player nonzero-sum differential game in the case where players use nonanticipative strategies. We define the Nash equilibrium in this case and obtain a characterization of Nash equilibrium strategies. We show that a Nash equilibrium solution can be approximately realized by control-with-guide strategies.     

N-person credibilistic strategic game

This paper enlarges the scope of fuzzy-payoff game to n-person form from the previous two-person form. Based on credibility theory, three credibilistic approaches are introduced to model the behaviors of players in different decision situations. Accordingly, three new definitions of Nash equilibrium are proposed for n-person credibilistic strategic game. Moreover, existence theorems are proved for further research into credibilistic equilibrium strategies. Finally, two numerical examples are given to illustrate the significance of credibilistic equilibria in practical strategic games.     

Games to induce specified equilibria

Media access protocols in wireless networks require each contending node to wait for a backoff time, chosen randomly from a fixed range, before attempting to transmit on a shared channel. However, nodes acting in their own selfish interest may not follow the protocol. In this paper, a static version of the problem is modeled as a strategic game played by non-cooperating, rational players (the nodes). The objective is to design a game which exhibits a unique, a priori mixed-strategy Nash equilibrium. In the context of the media access problem, the equilibrium of the game would correspond to nodes choosing backoff times randomly from a given range of values, according to the given distribution. We consider natural variations of the problems concerning the number of actions available to the players and show that it is possible to design such a game when there are at least two players that each have the largest number of possible actions among all players. In contrast, we show that if there are exactly two players with different number of actions available to them, then it becomes impossible to design a strategic game with a unique such Nash equilibrium.     

A Comparative Study of Game Theoretic and Evolutionary Models of Bargaining for Software Agents

Most of the existing work in the study of bargaining behavior uses techniques from game theory. Game theoretic models for bargaining assume that players are perfectly rational and that this rationality is common knowledge. However, the perfect rationality assumption does not hold for real-life bargaining scenarios with humans as players, since results from experimental economics show that humans find their way to the best strategy through trial and error, and not typically by means of rational deliberation. Such players are said to be boundedly rational. In playing a game against an opponent with bounded rationality, the most effective strategy of a player is not the equilibrium strategy but the one that is the best reply to the opponents strategy. The evolutionary model provides a means for studying the bargaining behaviour of boundedly rational players. This paper provides a comprehensive comparison of the game theoretic and evolutionary approaches to bargaining by examining their assumptions, goals, and limitations. We then study the implications of these differences from the perspective of the software agent developer.     

A Group Pursuit Game

A differential game of pursuit of an evader by m dynamic pursuers under simple motion is studied. The time of game completion is fixed. Pursuers’ controls obey integral constraints, whereas the evader control obeys either an integral constraint or a geometric constraint. A differential game with cost defined by the distance between the evader and his nearest pursuer at the game completion instant is studied. Optimal strategies for players are constructed and the game cost is determined.__________Translated from Avtomatika i Telemekhanika, No. 8, 2005, pp. 24–35.Original Russian Text Copyright © 2005 by Ibragimov.     

基于纳什均衡的D2D通信功率控制博弈算法

为了减轻D2D通信在资源复用模式下的互干扰问题,提升蜂窝网络均衡性能收益,提出一种基于纳什均衡的功率控制博弈算法。算法中将互干扰用户间的功率控制过程描述为静态博弈模型,用户之间根据最小化代价函数的博弈决策,通过多步迭代调节发射功率,使系统收敛至纳什均衡的优化状态。在用户代价函数设计中,综合考虑了能耗及传输速率影响,同时给出了博弈算法纳什均衡存在性以及收敛性的证明。仿真实验表明,在最优响应策略及能耗因子的有效约束下,互干扰用户更理智的选择发射功率,系统拥有较好均衡性收益的同时能耗进一步降低。     

Analysis of a simple capacity game

We study a two‐player two‐fare‐class static single‐period capacity allocation game with complete information. Nonnested (partitioned) booking limit policies are investigated in both noncooperative and cooperative situations. We show the existence of unique Nash equilibrium in the noncooperative situation. In the cooperative game, we analyze the cost saving of the two players and investigate the concavity of the objective function. For both noncooperative and cooperative settings, we assume the demands to be a truncated normal distribution and provide a comprehensive sensitivity analysis to discover the effects of unit revenue, rejection cost, and transfer rate on the equilibrium solution. Our numerical experiments show that the nonnested model can be a good approximation to the nested booking limit model. For the cooperative setting, we identify conditions that give rise to improvements in the total system revenue. Finally, under each game‐theoretic setting, we present the managerial implications of our solutions along with numerical examples.     

Who plays,how much,and why? Debunking the stereotypical gamer profile

Online games have exploded in popularity, but for many researchers access to players has been difficult. The study reported here is the first to collect a combination of survey and behavioral data with the cooperation of a major virtual world operator. In the current study, 7,000 players of the massively multiplayer online game (MMO) EverQuest 2 were surveyed about their offline characteristics, their motivations and their physical and mental health. These self‐report data were then combined with data on participants’ actual in‐game play behaviors, as collected by the game operator. Most of the results defy common stereotypes in surprising and interesting ways and have implications for communication theory and for future investigations of games.     

Why We Distort in Self‐Report: Predictors of Self‐Report Errors in Video Game Play

Using Cognitive Dissonance and Balance Theory, this study investigates factors that predict how and why MMO players inaccurately report their game playing time. It was hypothesized that players belonging to categories other than the stereotypical game player (e.g. younger, less educated, male) would be likely to underreport playing time. It was also hypothesized that those players who held less positive attitudes toward the game would be more likely to underreport their playing time. Comparing people's self‐reported weekly usage of an MMO, EverQuest II, with their actual average weekly usage of the game, data showed that age, education, lack of enjoyment playing the game, and lack of an online sense of community predicted greater levels of underreporting.