Fuzzy approaches to the game of Chicken

Game theory deals with decision-making processes involving two or more parties, also known as players, with partly or completely conflicting interests. Decision-makers in a conflict must often make their decisions under risk and under unclear or fuzzy information. In this paper, two distinct fuzzy approaches are employed to investigate an extensively studied 2×2 game model-the game of Chicken. The first approach uses a fuzzy multicriteria decision analysis method to obtain optimal strategies for the players. It incorporates subjective factors into the decision-makers' objectives and aggregates objectives using a weight vector. The second approach applies the theory of fuzzy moves (TFM) to the game of Chicken. The theory of moves (TOM) is designed to bring a dynamic dimension to the classical theory of games by allowing decision-makers to look ahead for one or several steps so that they can make a better decision. TOM is the crisp counterpart of TFM, the approach we implement here to deal with games that include fuzzy and uncertain information. The application of fuzzy approaches to the game of Chicken demonstrates their effectiveness in manipulating subjective, uncertain, and fuzzy information and provides valuable insights into the strategic aspects of Chicken     

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.     

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.     

Interval-valued intuitionistic fuzzy matrix games based on Archimedean t-conorm and t-norm

Fuzzy game theory has been applied in many decision-making problems. The matrix game with interval-valued intuitionistic fuzzy numbers (IVIFNs) is investigated based on Archimedean t-conorm and t-norm. The existing matrix games with IVIFNs are all based on Algebraic t-conorm and t-norm, which are special cases of Archimedean t-conorm and t-norm. In this paper, the intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm are employed to aggregate the payoffs of players. To derive the solution of the matrix game with IVIFNs, several mathematical programming models are developed based on Archimedean t-conorm and t-norm. The proposed models can be transformed into a pair of primal–dual linear programming models, based on which, the solution of the matrix game with IVIFNs is obtained. It is proved that the theorems being valid in the exiting matrix game with IVIFNs are still true when the general aggregation operator is used in the proposed matrix game with IVIFNs. The proposed method is an extension of the existing ones and can provide more choices for players. An example is given to illustrate the validity and the applicability of the proposed method.     

A fuzzy membership filtering aided neural network based transmission loss allocation scheme using game theory

The present paper proposes development of a transmission loss allocation scheme in a deregulated environment using fuzzy memberships and supervised neural networks. This method can be effectively utilized in online applications where game theory based solutions, which otherwise produce acceptable results, cannot be utilized for prohibitive computation load. We propose a fuzzy membership based approach to filter data from a global database and create a local relevant database, for each transaction detail online, each time. A neural network is trained for each such local database formed and utilized for estimating loss allocations among players, for the transaction detail under consideration. The proposed method has been employed for an IEEE 14 bus system and the results of our proposed method have been shown to be sufficiently accurate, when compared to results obtained by using game theoretic approach.     

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.     

A Rough Set Approach to Estimating the Game Value and the Shapley Value from Data

A value of a game v is a function which to each coalition S assigns the value v(S) of this coalition, meaning the expected pay–off for players in that coalition. A classical approach of von Neumann and Morgenstern [6] had set some formal requirements on v which contemporary theories of value adhere to. A Shapley value of the game with a value v [14] is a functional Φ giving for each player p the value Φ_p(v) estimating the expected pay-off of the player p in the game. Game as well as conflict theory have been given recently much attention on the part of rough and fuzzy set communities [11,8,1,4,7,2]. In particular, problems of plausible strategies [1] in conflicts as well as problems related to Shapley's value [3,2] have been addressed.We confront here the problem of estimating a value as well as Shapley's value of a game from a partial data about the game. We apply to this end the rough set ideas of approximations, defining the lower and the upper value of the game and, respectively, the lower and upper Shapley value. We also define a notion of an exact coalition, on which both values coincide giving the true value of the game; we investigate the structure of the family of exact sets showing its closeness on complements, disjoint sums, and intersections of coalitions covering the set of players. This work sets open a new area of rough set applications in mining constructs from data. The construct mined in this case are values as well as Shapley values of games.     

基于相对相似度关系的三角模糊数型不确定多属性决策法

针对属性权重未知的三角模糊数型不确定多属性决策问题, 提出一种新的规范三角模糊数与决策方案的相对相似度定义和三角模糊数相对相似度关系理论; 借鉴合作博弈中可能度最大化算法提出一种基于三角模糊数相对相似度关系的属性权重确定方法; 利用备选方案对象在方案集中的总体相对相似度值大小选取最优对象并排序, 以此给出三角模糊数型不确定多属性决策的相对相似度关系算法, 最后通过算例分析表明了所提出算法的可行性和有效性.     

基于攻防博弈的网络防御决策方法研究综述

博弈论研究冲突对抗条件下最优决策问题,是网络空间安全的基础理论之一,能够为解决网络防御决策问题提供理论依据.提炼网络攻防所具备的目标对立、策略依存、关系非合作、信息不完备、动态演化和利益驱动6个方面博弈特征.在理性局中人假设和资源有限性假设的基础上,采用攻防局中人、攻防策略集、攻防动作集、攻防信息集和攻防收益形式化定义...     

On a game with perfect information and time-claiming alternatives

This paper considers a new model of multistage games with perfect information in which players can control decision-making time. At each stage of the game, players choose a certain alternative from a finite set of basic alternatives and also time necessary to realize this basic alternative. The payoffs of players depend on the game path defined by the chosen alternatives and also on the time to realize this path at each stage. We use the subgame-perfect ε-Nash equilibrium as the optimality principle of the model. This paper is a continuation of the earlier research [5].     

Fuzzy utility and equilibria

A decision maker is frequently confronted with fuzzy constraints, fuzzy utility maximization, and fuzziness about the state of competitors. In this paper we present a framework for fuzzy decision-making, using techniques from fuzzy logic, game theory, and micro-economics. In the first part, we study the rationality of fuzzy choice. We introduce fuzzy constraints, and show that this can easily be combined with maximizing a fuzzy utility. The second part of the paper analyzes games with uncertainty about the state of the competitors. We implement fuzzy Cournot adjustment, define equilibria, and study their stability. Finally, we show how a play progresses where the players have uncertainty about the state of the other players, and about their utility. For a likely procedure of utility maximization, the equilibria are the same as for the game without utility maximization.     

Altruistic behavior in a nonantagonistic positional differential game

We consider a two-person nonantagonistic positional differential game (NPDG) whose dynamics is described by an ordinary nonlinear vector differential equation. Constraints on values of players’ controls are geometric. Final time of the game is fixed. Payoff functionals of both players are terminal. The formalization of positional strategies in an NPDG is based on the formalization and results of the general theory of antagonistic positional differential games (APDGs) (see monographs by N.N. Krasovskii and A.I. Subbotin [3, 4]). Additionally, in the present paper we assume that each player, together with the usual, normal (nor), type of behavior aimed at maximizing his own functional, can use other behavior types introduced in [2, 5]. In particular, these may be altruistic (alt), aggressive (agg), and paradoxical (par) types. It is assumed that in the course of the game players can switch their behavior from one type to another. Using the possibility of such switches in a repeated bimatrix 2 × 2 game in [5, 6] allowed to obtain new solutions of this game. In the present paper, extension of this approach to NPDGs leads to a new formulation of the problem. In particular, of interest is the question of how players’ outcomes at Nash solutions are transformed. An urgent problem is minimizing the time of “abnormal” behavior while achieving a good result. The paper proposes a formalization of an NPDG with behavior types (NPDGwBT). It is assumed that in an NPDGwBT each player, simultaneously with choosing a positional strategy, chooses also his own indicator function defined on the whole game horizon and taking values in the set {normal, altruistic, aggressive, paradoxical}. The indicator function of a player shows the dynamics of changes in the behavior type demonstrated by the player. Thus, in this NPDGwBT each player controls the choice of a pair {positional strategy, indicator function}. We define the notion of a BT-solution of such a game. It is expected that using behavior types in the NPDGwBT which differ from the normal one (so-called abnormal types) in some cases may lead to more favorable outcomes for the players than in the NPDG. We consider two examples of an NPDGwBT with simple dynamics in the plane in each of which one player keeps to altruistic behavior type over some time period. It is shown that in the first example payoffs of both players increase on a BT-solution as compared to the game with the normal behavior type, and in the second example, the sum of players’ payoffs is increased.     

Coalitional game with fuzzy payoffs and credibilistic core

As an important branch of game theory, coalitional game deals with situations that involve cooperation among the players. This paper deals with this topic further by incorporating the fuzzy payoff information. Based on the credibility theory, we introduce two decision criteria to define the preferences of players, which leads to two definitions of credibilistic cores—the solution of coalitional game with fuzzy transferable payoffs. Meanwhile, we give a sufficient and necessary condition to ensure non-emptiness of the credibilistic cores. Finally, an example is provided for illustrating the usefulness of the theory developed in this paper.     

Extension of the ELECTRE method based on interval-valued fuzzy sets

Decision-making is the process of finding the best option among the feasible alternatives. In classical multiple criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy ELECTRE method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For the purpose of proving the validity of the proposed model, we present a numerical example and build a practical maintenance strategy selection problem.     

Static Bayesian games with finite fuzzy types and the existence of equilibrium

This paper studies a special game with incomplete information, in which the payoffs of the players are both random and fuzzy. Such a game is considered in the context of a Bayesian game with the uncertain types characterized as fuzzy variables. A static fuzzy Bayesian game is then introduced and the decision rules for players are given based on credibility theory. We further prove the existence of the equilibrium of the game. Finally, a Cournot competition model with fuzzy efficiency under asymmetric information is investigated as an application and some results are presented.     

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.     

三角模糊数型不确定多指标决策的可能度关系法

针对指标权重未知的三角模糊数型不确定多指标决策问题, 提出4 种新的三角模糊数比较可能度的等价定义, 并得到一些优良性质关系. 借鉴合作博弈中极大极小算法, 提出一种基于三角模糊数比较可能度关系的指标权重确定方法; 集结所有决策方案比较的可能度, 并对决策方案集进行最优判定和排序, 即可得到三角模糊数型不确定多指标决策的比较可能度关系法. 最后通过算例表明所提出算法的可行性和有效性.

     

基于两参照点的动态混合多属性群决策算法

针对方案属性值为三参数区间灰数与三角模糊数相混合的动态多属性群决策问题,提出一种基于前景理论和两参照点的动态解决方案.针对灰数与模糊数混合的状况,提出混合靶心模型;设置时间参照点,通过均值和平均发展速度,考察各方案之间动态发展情况;利用向量之间的夹角,考察专家个体决策与群决策之间的相似度,建立相应的专家权重调整模型;考虑决策问题发展过程中的未来多个阶段,采用熵权法确定时间权重,并通过算例验证所提出方法的可行性和实用性.     

连锁门店选址与配送中心选择联合决策的f-MIGP模型

建立了连锁门店选址和配送中心选择联合决策问题的模糊多目标混合整数规划模型．针对该模型的特殊结构。提出一种适用的求解策略：首先确定每个模糊目标的隶属度函数；然后将模糊多目标混合整数规划模型转化为等价的清晰多目标混合整数规划模型，通过最大最小算子求出目标值；最后借助于两阶段算法，求出问题的最优解．通过应用算例进一步说明了该模型的有效性和可行性．     

基于博弈论的SDN主控制器重选机制

针对软件定义网络（SDN）中单一控制器容易发生过载的问题,提出了一种基于博弈论的主控制器重选机制——博弈系统模型（GAME-SM）。首先,把资源限定下的交换机迁移问题等效为零和博弈最大化收益问题,并提出了GAME-SM机制;其次,设置控制器负载上下限阈值来确定博弈触发的条件,达到负载上限的控制器邀请邻近的其他控制器作为博弈者参与博弈;最后,基于零和博弈最大化每个参与者的收益来设计博弈策略,使用利用度的改变进行反复博弈来重选主控制器,最终实现整个系统的负载均衡。仿真结果表明该机制能够明显改善控制器负载平衡状况,且与静态分布式控制器（D-CNTL）相比,控制器响应时间减少了50%。