量子博弈:新方法与新策略

论述了量子博弈的研究现状和最新进展,首先介绍了量子博弈中的一些基本概念和主要模型,即量子翻硬币模型、Eisert量子博弈模型和量子Monty Hall模型;然后以这些模型为基础,讨论了当前量子博弈研究中的热点问题：多人量子博弈、状态纠缠、进化稳定策略和平衡态、退相干性等;最后,对量子博弈的未来发展方向做了一些展望。     

Differential games and optimal pursuit-evasion strategies

In this paper it is shown that variational techniques can be applied to solve differential games. Conditions for capture and for optimality are derived for a class of optimal pursuit-evasion problems. Results are used to demonstrate that the well-known proportional navigation law is actually an optimal intercept strategy.     

Mixed Stackelberg strategies in continuous-kernel games

It is shown that continuous-kernel nonzero-sum games with compact strategy spaces could admit both pure and mixed Stackelberg equilibrium solutions, if the cost function of each player is either nonquadratic or nonconvex in his own decision variable. In such a case, the mixed Stackelberg strategy will yield a lower average cost for the leader than the pure Stackelberg strategy. It is also verified that, if the cost functions of the players are quadratic and strictly convex, then only pure Stackelberg strategies can exist.     

Adiabatic quantum games and phase-transition-like behavior between optimal strategies

In this paper we propose a game of a single qubit whose strategies can be implemented adiabatically. In addition, we show how to implement the strategies of a quantum game through controlled adiabatic evolutions, where we analyze the payment of a quantum player for various situations of interest: (1) when the players receive distinct payments, (2) when the initial state is an arbitrary superposition, and (3) when the device that implements the strategy is inefficient. Through a graphical analysis, it is possible to notice that the curves that represent the gains of the players present a behavior similar to the curves that give rise to a phase transition in thermodynamics. These transitions are associated with optimal strategy changes and occur in the absence of entanglement and interaction between the players.     

Children’s choices and strategies in video games

It is important to develop an understanding of children’s engagement and choices in learning experiences outside of school as this has implications for their development and orientations to other learning environments. This mixed-methods study examines relationships between the genres of video games children choose to play and the learning strategies they employ to improve at these games. It also explores students’ motivations for playing the games they choose to play. One hundred eighteen fourth- and fifth-grade students participated in this study. Qualitative analyses of student responses resulted in a model for classifying motivation for game choices. Children primarily cite reasons that can be classified as psychological or cognitive reasons for choosing to play certain video games, and are motivated by the challenge and thinking required in the games. Analyses using Chi-square tests of association demonstrated significant relationships between video game genre and learning strategy used for two of the six learning strategies (p < .05). Children playing action games are more likely to use repetition to learn the game and children playing adventure games are more likely to use their imaginations to take on the role of the character in the game and think the way the character would to make decisions in the game. There were also several gender differences in learning preferences.     

Learning machiavellian strategies for manipulation in Stackelberg security games

This paper suggests a new approach for repeated Stackelberg security games (SSGs) based on manipulation. Manipulation is a strategy interpreted by the Machiavellianism social behavior theory, which consists on three main concepts: view, tactics, and immorality. The world is conceptualized by manipulators and manipulated (view). Players employ Machiavelli’s tactics and Machiavellian intelligence in order to manipulate attacker/defender situations. The immorality plays a fundamental role in these games, defenders are able to not be attached to a conventional moral in order to achieve their goals. We consider a security game model involving manipulating defenders and manipulated attackers engaged cooperatively in a Nash game and at the same time restricted by a Stackelberg game. The resulting game is non-cooperative bargaining game. The cooperation is represented by the Nash bargaining solution. We propose an analytical formula for solving the manipulation game, which arises as the maximum of the quotient of two Nash products. The role of the players in the Stackelberg security game are determined by the weights of the players for the Nash bargaining approach. We consider only a subgame perfect equilibrium where the solution of the manipulation game is a Strong Stackelberg Equilibrium (SSE). We employ a reinforcement learning (RL) approach for the implementation of the immorality. A numerical example related to developing a strategic schedule for the efficient use of resources for patrolling in a smart city is handled using a class of homogeneous, ergodic, controllable, and finite Markov chains for showing the usefulness of the method for security resource allocation.

     

Three-level incentive schemes using follower's strategies in differential games

We deal with three-level incentive differential games in which first and second leaders have access not only to slate information but also to information on follower's strategies. We derive sufficient conditions for three-level incentive schemes using information on follower's strategies in both linear and non-linear differential games, and show that three-level incentive schemes using information on follower's strategies depend on an initial state value.     

Equilibrium strategies in dynamic games with multi-levels of hierarchy

This paper considers noncooperative equilibria of three-player dynamic games with three levels of hierarchy in decision making. In this context, first a general definition of a hierarchical equilibrium solution is given, which also accounts for nonunique responses of the players who are not at the top of the hierarchy. Then, a general theorem is proven which provides a set of sufficient conditions for a triple of strategies to be in hierarchical equilibrium. When applied to linear-quadratic games, this theorem provides conditions under which there exists a linear one-step memory strategy for the player (say, $\text{J}$1) at the top of the hierarchy, which forces the other two players to act in such a way so as to jointly minimize the cost function of $\text{J}$1. Furthermore, there exists a linear one-step memory strategy for the second-level player (say, $\text{J}$2), which forces the remaining player to jointly minimize the cost function of $\text{J}$2 under the declared equilibrium strategy of $\text{J}$1. A numerical example included in the paper illustrates the results and the convergence property of the equilibrium strategies, as the number of stages in the game becomes arbitrarily large.     

Noise-perturbed quaternionic Mandelbrot sets

Quaternionic Mandelbrot sets (abbreviated as M sets) have been a focus on the research in high-dimensional fractals. This paper explores the topological structure and the fission evolution of the quaternionic M sets under noise perturbations as well as the boundaries of their regions of stability. Our experimental results indicate that the additive dynamic noise displaces the M sets from the origin position, and the multiplicative dynamic noise shrinks the regions of stability of the M sets in a certain proportion. In addition, the influence of the output noise is mainly on the inner structure of the regions of stability. The presence of such noises leads to a decrease of the box dimension of the M sets except for the additive dynamic noise. It can be concluded that the earlier-mentioned noises have a great impact on the M sets.     

Nash strategies for M-person differential games with mixed information structures

This paper is concerned with a class of M-person linear-quadratic nonzero-sum differential games in which a subset of the players have access to closed-loop (CL) information and the rest to open-loop (OL) information. The state equation contains an additive random perturbation term, inclusion of which has been shown to be necessary in order to obtain, a unique globally optimal Nash equilibrium solution. For each player with CL information, the optimal strategy is a linear function of the current and the initial states, and for each player with OL information, the optimal strategy is a linear function of the initial state.     

不完全信息下开发商Markov博弈均衡的最优策略

房地产开发商对于资源的争夺存在零和博弈的特点,在信息不对称条件下对开发商经济博弈模型的研究具有重要价值。该研究考虑到市场上存在开发商Cartel联盟的情形,以Markov博弈模型为核心,针对不完全信息下的Cartel联盟与竞争者的Markov博弈均衡进行研究,得到了博弈双方的最优策略及演算方法。最后,利用实例对该模型的有效性和可行性进行了说明。     

Quantum games: a review of the history,current state,and interpretation

We review both theoretical and experimental developments in the area of quantum games since the inception of the subject circa 1999. We will also offer a narrative on the controversy that surrounded the subject in its early days, and how this controversy has affected the development of the subject.     

Guaranteeing cost strategies for infinite horizon difference games with uncertain dynamics

The paper addresses the problem of determination of guaranteeing cost control strategies for discrete-time two-persons zero-sum non-linear games over in.nite horizon. In the proposed approach the objective functional is appropriately modi.ed in order to cope with the uncertainties, and a su.cient condition is given to ensure that a given state-feedback is a guaranteeing cost control. The results are applied also for linear systems with uncertainties of linear fractional structure to derive guaranteeing cost strategies for both players. It is shown that this approach can successfully be applied in this case, when the method of introducing .ctitious games as proposed in previous papers may come up against a di.culty. The results are illustrated by numerical examples.     

Guaranteeing cost strategies for linear quadratic differential games under uncertain dynamics

This paper deals with the design of closed loop strategies for a class of two players zero-sum linear quadratic differential games, where each player does not know exactly the state equation and model it through a system subject to norm-bounded uncertainties. The finite horizon and the infinite horizon problems are both solved: it turns out that the optimal strategies, guaranteeing to each player a given level of performance, require, to be evaluated, the solution of two scaled differential (algebraic in the infinite horizon case) Riccati equations. A numerical example illustrates an application of the proposed technique.     

Stability of quaternionic linear systems

The main goal of this paper is to characterize stability and bounded-input-bounded-output (BIBO)-stability of quaternionic dynamical systems. After defining the quaternion skew-field, algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated. Having established these results, the Smith and the Smith-McMillan forms of quaternionic matrices are introduced and studied. Finally, all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework.     

Real linear quaternionic differential operators

The renewed interest in searching for quaternionic deviations of standard (complex) quantum mechanics resulted, in the last years, in a better understanding of the quaternionic mathematical tools needed to solve quantum mechanical problems. In particular, a relevant progress has been achieved in solving eigenvalue problems and differential equations for quaternionic operators. The practical methods recently proposed to solve quaternionic and complex linear second-order differential equations with constant coefficients represent a fundamental starting point to discuss quaternionic potentials in quantum mechanics and study possible violations from complex theories. Nevertheless, only for a restricted class of real linear quaternionic differential operators (namely, symmetric operators) the solution of differential problems was given. In this paper, we study real linear quaternionic differential equations. The proposed resolution's method is based on the Jordan canonical form of (real linear) quaternionic matrices.