A game theoretical perspective on the quantum probabilities associated with a GHZ state

In the standard approach to quantum games, players’ strategic moves are local unitary transformations on an entangled state that is subsequently measured. Players’ payoffs are then obtained as expected values of the entries in the payoff matrix of the classical game on a set of quantum probabilities obtained from the quantum measurement. In this paper, we approach quantum games from a diametrically opposite perspective. We consider a classical three-player symmetric game along with a known expression for a set of quantum probabilities relevant to a tripartite Einstein–Podolsky–Rosen (EPR) experiment that depends on three players’ directional choices in the experiment. We define the players’ strategic moves as their directional choices in an EPR setting and then express their payoff relations in the resulting quantum game in terms of their directional choices, the entries of the payoff matrix, and the quantum probability distribution relevant to the tripartite EPR experiment.     

Interactive strategy sets in multiple payoff games

In game theory, it is usually assumed that each player has only one payoff function and the strategy set of the game is composed of the topological product of individual players’ strategy sets. In real business and system design or control problems, however, players’ strategy sets may be interactive and each player may have more than one payoff function. This paper, investigates the more general situation of multiple payoff and multiple person games in a normal form. In this paper, each player has several payoff functions which are dominated by certain convex cones, and the feasible strategy set of each player may be interactive with those of the other players. This new model is applied to a classical example without requiring variational and quasi-variational inequalities, or point-to-set mappings.     

A Quantum Treatment of Public Goods Economics

Quantum generalizations of conventional games broaden the range of available strategies, which can help improve outcomes for the participants. With many players, such quantum games can involve entanglement among many states which is difficult to implement, especially if the states must be communicated over some distance. This paper describes a quantum approach to the economically significant n-player public goods game that requires only two-particle entanglement and is thus much easier to implement than more general quantum mechanisms. In spite of the large temptation to free ride on the efforts of others in the original game, two-particle entanglement is sufficient to give near optimal expected payoff when players use a simple mixed strategy for which no player can benefit by making different choices. This mechanism can also address some heterogeneous preferences among the players. PACS: 03.67-a; 02.50Le; 89.65.Gh     

Characterizing the Nash equilibria of a three-player Bayesian quantum game

Quantum games with incomplete information can be studied within a Bayesian framework. We consider a version of prisoner’s dilemma (PD) in this framework with three players and characterize the Nash equilibria. A variation of the standard PD game is set up with two types of the second prisoner and the first prisoner plays with them with probability p and \(1-p\), respectively. The Bayesian nature of the game manifests in the uncertainty that the first prisoner faces about his opponent’s type which is encoded either in a classical probability or in the amplitudes of a wave function. Here, we consider scenarios with asymmetric payoffs between the first and second prisoner for different values of the probability, p, and the entanglement. Our results indicate a class of Nash equilibria (NE) with rich structures, characterized by a phase relationship on the strategies of the players. The rich structure can be exploited by the referee to set up rules of the game to push the players toward a specific class of NE. These results provide a deeper insight into the quantum advantages of Bayesian games over their classical counterpart.     

Influence of initial conditions in 2\times 2 symmetric games

It is known that quantum game is characterized by the payoff matrix as well as initial states of the quantum objects used as carriers of information in a game. Further, the initial conditions of the quantum states influence the strategies adopted by the quantum players. In this paper, we identify the necessary condition on the initial states of quantum objects for converting symmetric games into potential games, in which the players acquire the same payoff matrix. The necessary condition to preserve the symmetric type and potential type of the game is found to be the same. The present work emphasizes the influence of the initial states in the quantization of games.     

Nonlocal sets of orthogonal multipartite product states with less members

Repeated quantum game theory addresses long-term relations among players who choose quantum strategies. In the conventional quantum game theory, single-round quantum games or at most finitely repeated games have been widely studied; however, less is known for infinitely repeated quantum games. Investigating infinitely repeated games is crucial since finitely repeated games do not much differ from single-round games. In this work, we establish the concept of general repeated quantum games and show the Quantum Folk Theorem, which claims that by iterating a game one can find an equilibrium strategy of the game and receive reward that is not obtained by a Nash equilibrium of the corresponding single-round quantum game. A significant difference between repeated quantum prisoner’s dilemma and repeated classical prisoner’s dilemma is that the classical Pareto optimal solution is not always an equilibrium of the repeated quantum game when entanglement is sufficiently strong. When entanglement is sufficiently strong and reward is small, mutual cooperation cannot be an equilibrium of the repeated quantum game. In addition, we present several concrete equilibrium strategies of the repeated quantum prisoner’s dilemma.

     

基于演化博弈的无线传感器网络节能分簇路由算法

考虑到无线传感器网络中节点在冲突环境下决策时具有有限理性,近年来学者引入博弈论解决传感网分簇路由中自私节点的簇头选举问题。以往经典博弈分簇路由算法要求知道所有参与者行动的完全信息,并假设节点完全理性,这对于资源有限的传感器节点不切实际。本文提出了一种基于演化博弈论的无线传感器网络节能分簇路由算法(EECEG),通过演化博弈复制动态方程证明存在演化稳定策略(ESS)。算法将所有节点模拟为自私的博弈参与者,参与者可决策宣称自己成为簇头候选者(D)或不成为候选者(ND)。所有参与者根据自身剩余能量、邻居节点个数等因素自私决策,通过观察和模仿对手进行演化,直到收益均衡。实验结果表明,EECEG协议可有效延长网络生命周期,均衡节点间能耗,同时使数据传输更高效。     

Three-player conflicting interest games and nonlocality

We outline the general construction of three-player games with incomplete information which fulfil the following conditions: (i) symmetry with respect to the permutations of players; (ii) the existence of an upper bound for total payoff resulting from Bell inequalities; (iii) the existence of both fair and unfair Nash equilibria saturating this bound. Conditions (i)–(iii) imply that we are dealing with conflicting interest games. An explicit example of such a game is given. A quantum counterpart of this game is considered. It is obtained by keeping the same utilities but replacing classical advisor by a quantum one. It is shown that the quantum game possesses only fair equilibria with strictly higher payoffs than in the classical case. This implies that quantum nonlocality can be used to resolve the conflict between the players.     

Fuzzy signaling game of deception between ant-inspired deceptive robots with interactive learning

In this study the robotic deception phenomenon is raised in the framework of a signaling game which utilizes fuzzy logic and game theory along with inspirations from nature. Accomplishing the fuzzy signaling strategy set for deceptive players serves as a great part of our contribution and on this aim, hierarchical fuzzy inference systems support receiver’s actions and sender’s ant-inspired deceptive signals (track and pheromone). In addition, special deceptive robots and visually-supported experimental environment are also provided. The fuzzy behavior of robots defines the strategy type of players. The final result of deception process depends on this strategy type which leads to proposing a payoff matrix in which each cell of mutual costs is defined with special supporting logic related to our deception game with pursuit–evasion applications. Furthermore, motivated by animal signaling, through applying mixed strategies on deceiver’s honesty level and rival’s trust level, the corresponding learning dynamics are investigated and the conceptual discussion put forward serves as a proof to the smart human-like behavior that occurs between the robots: the interactive learning. Simulation results show that robots are capable of interactive learning within deceptive interaction and finally change their strategies to adopt themselves to new situation occurred due to opponent’s strategy change. Because of repetitive change in strategies as a result of learning, the conditions of a persistent deception without breakdown holds for this game where deceiver can frequently benefit from deception without leaving rival to lose its trust totally. The change in strategy will happen after a short time needed to learn the new situation. In rival’s learning process, this short time, which we call the ignorance time, exactly is the period that deceiver can benefit from deception while its evil intends are still concealed. Moreover, in this study an algorithm is given for the proposed signaling game of deception and an illustrative experiment in the introduced experimental environment demonstrates the process of a successful deception. The paper also gives solution to the proposed game by analyzing mixed Nash equilibrium which turns out to be the interior center fixed point of the learning dynamics.     

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.     

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.     

Quantum games and quaternionic strategies

It is well-known that the phenomenon of entanglement plays a fundamental role in quantum game theory. Occasionally, games constructed via maximally entangled initial states (MEIS) will have new Nash equilibria yielding to the players higher payoffs than the ones they receive in the classical version of the game. When examining these new games for Nash equilibrium payoffs, a fundamental question arises; does a suitable choice of an MEIS improve the lot of the players? In this paper, we show that the answer to this question is yes for at least the case of a variant of the well-known two player, two strategy game of Chicken. To that end, we generalize Landsburg’s quaternionic representation of the payoff function of two player, two strategy maximally entangled states to games where the initial state is chosen arbitrarily from a circle of maximally entangled initial states and for the corresponding quantized games show the existence of superior Nash equilibrium payoffs when an MEIS is appropriately chosen.     

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.     

The interpretation of discontinuous state feedback control laws as nonanticipative control strategies in differential games

In differential games, one player chooses a feedback strategy to maximize a payoff. The other player counters by applying a minimizing open loop control. Classical notions of feedback strategies, based on state feedback control laws for which the corresponding closed loop dynamics uniquely define a state trajectory, are too restrictive for many problems, owing to the absence of minimizing classical feedback strategies or because consideration of classical feedback strategies fails to define, in a useful way, the value of the game. A number of feedback strategy concepts have been proposed to overcome this difficulty. That of Elliot and Kalton, according to which a feedback strategy is a nonanticipative mapping between control functions for the two players, has been widely taken up because it provides a value of the game which connects, via the Hamilton-Jacobi-Isaacs equation, with other fields of systems science. Heuristic analysis of specific games problems often points to discontinuous optimal feedback strategies. These cannot be regarded as classical feedback control strategies because the associated state trajectories are not in general unique. We give general conditions under which they can be interpreted as generalized feedback strategies in the sense of Elliot and Kalton.     

The Impact of Payoff Function and Local Interaction on the N-Player Iterated Prisoner's Dilemma

The N-player iterated prisoner's dilemma (NIPD) game has been widely used to study the evolution of cooperation in social, economic and biological systems. This paper studies the impact of different payoff functions and local interactions on the NIPD game. The evolutionary approach is used to evolve game-playing strategies starting from a population of random strategies. The different payoff functions used in our study describe different behaviors of cooperation and defection among a group of players. Local interaction introduces neighborhoods into the NIPD game. A player does not play against every other player in a group any more. He only interacts with his neighbors. We investigate the impact of neighborhood size on the evolution of cooperation in the NIPD game and the generalization ability of evolved strategies. Received 18 August 1999 / Revised 27 February 2000 / Accepted 15 May 2000     

时间尺度与选择倾向性协同作用下的演化博弈模型

针对合作行为的涌现与维持问题，基于演化博弈理论和网络理论，提出了一种促进合作的演化博弈模型。该模型同时将时间尺度、选择倾向性引入到演化博弈中。在初始化阶段，根据持有策略的时间尺度将个体分为两种类型：一种个体在每个时间步都进行策略更新；另一种个体在每一轮博弈后，以某种概率来决定是否进行策略更新。在策略更新阶段，模型用个体对周围邻居的贡献来表征他的声誉，并假设参与博弈的个体倾向于学习具有较好声誉邻居的策略。仿真实验结果表明，所提出的时间尺度与选择倾向性协同作用下的演化博弈模型中，合作行为能够在群体中维持；惰性个体的存在不利于合作的涌现，但是个体的非理性行为反而能够促进合作。     

Studying interval valued matrix games with fuzzy logic

Matrix games have been widely used in decision making systems. In practice, for the same strategies players take, the corresponding payoffs may be within certain ranges rather than exact values. To model such uncertainty in matrix games, we consider interval-valued game matrices in this paper; and extend the results of classical strictly determined matrix games to fuzzily determined interval matrix games. Finally, we give an initial investigation into mixed strategies for such games. This work is partially supported by the NSF grant CCF-0202042.     

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.     

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.     

三方多策略式博弈系统的长期演化稳定均衡特性研究

演化博弈论(EGT)基于有限理性假设且更加贴近现实,近年来已在众多领域得到了初步应用.基于此,本文关注一类较为常见的三方多策略式演化博弈系统,尝试通过理论分析总结其长期演化稳定均衡(ESE)特性,并进行仿真验证研究.首先,研究了一般情形下的三方两策略对称与非对称演化博弈系统;然后,将其扩展到更复杂的三方三策略非对称演化博弈类型,并对其长期ESE特性进行了理论分析与动态仿真验证;进一步,对通用三方n-策略(n 1)非对称演化博弈的建模思路进行了阐述与总结,给出其收敛迭代的计算方法.研究过程中详细定义了各类演化博弈模型的相对净支付(RNP)参数.实验结果表明可通过一些外部因素适当调整RNP参数使各类系统朝着期望的长期ESE状态自发收敛.最后,进行了实例验证.本文研究模型、方法和所得结论具有一定普适性,旨在丰富EGT研究,尤其是三方多策略演化博弈问题研究,并为相关领域非完全理性人参与的行为决策问题研究提供一些思路和理论参考.