Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

基于模糊博弈视角的企业联盟“双循环”合作对策研究

企业合作在“双循环”新发展格局下呈现出更多的模糊特征,能否形成稳定的模糊合作格局及持续合作的收益再分配策略成为合作联盟的关注点。将广义模糊超量博弈和模糊凸博弈及其广义解集应用到“双循环”新发展格局下企业联盟合作与收益再分配中,提出最大广义模糊超量博弈模型及其广义模糊谈判集,并基于最大广义模糊超量博弈均衡性的视角,对模糊凸博弈下广义模糊谈判集与核心的等价性质进行论证。研究结果不仅满足了“双循环”新发展格局下合作企业以部分资源参与合作的意愿,及企业联盟模糊凸合作博弈下再分配方案等价性证明需求,而且实现了保留部分收益用于联盟再发展的策略。     

On the bargaining set, kernel and core of superadditive games

We prove that for superadditive games a necessary and sufficient condition for the bargaining set to coincide with the core is that the monotonic cover of the excess game induced by a payoff be balanced for each imputation in the bargaining set. We present some new results obtained by verifying this condition for specific classes of games. For N-zero-monotonic games we show that the same condition required at each kernel element is also necessary and sufficient for the kernel to be contained in the core. We also give examples showing that to maintain these characterizations, the respective assumptions on the games cannot be lifted. Received: March 1998/Revised version: December 1998     

直觉模糊支付合作对策的核仁解

本文主要研究支付值为直觉模糊集的合作对策问题及其模糊核仁解.首先定义了直觉模糊集的得分函数和精确函数,并给出其排序方法,得到基于直觉模糊集的合作对策模型和适合这种模型的相应定义,同时提出了直觉模糊核仁解的概念;其次运用新的排序方法将求核仁解的问题转化为求解双目标非线性规划问题;最后通过实例分析验证了该方法的可行性和有效性。     

An Approach to Fuzzy Noncooperative Nash Games

Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility function over a set of the player decision space. Each player attempts to maximize/minimize his/her own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends.In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and operation of future military operations.     

基于个人超出值的模糊联盟合作博弈最小二乘预核仁

本文给出了基于个人超出值的无限模糊联盟合作博弈最小二乘预核仁的求解模型,得到该模型的显式解析解,并研究该解的若干重要性质。证明了:本文给出的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的相等解(The equalizer solution),基于个人超出值的字典序解三者相等。进一步证明了:基于Owen线性多维扩展的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的经典合作博弈最小二乘预核仁相等。最后,通过数值实例说明本文提出的无限模糊联盟合作博弈求解模型的实用性与有效性。     

A value for continuously-many-choice cooperative games

We extend a multi-choice cooperative game to a continuously-many-choice cooperative game. The set of all continuously-many-choice cooperative games is isomorphic to the set of all cooperative fuzzy games. A continuously-many-choice cooperative game and a cooperative fuzzy game have different physical interpretations. We define a value for the continuously-many-choice cooperative game and show that the value for the continuously-many-choice cooperative game has most properties as the traditional Shapley value does. Also, we give a probabilistic interpretation for the value. The probabilistic interpretation reveals some interesting properties of the value. Finally, we discuss the uniqueness of the value.     

An adjustable approach to intuitionistic fuzzy soft sets based decision making

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making.     

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set. Received September 2000/Revised version June 2001     

直觉模糊联盟合作博弈的非线性规划模型

本文研究联盟是直觉模糊集的合作博弈。首先,给出直觉模糊联盟的定义,并根据Choquet积分的直觉模糊形式,得到直觉模糊联盟合作博弈的区间值特征函数,进一步证明直觉模糊联盟合作博弈的区间值特征函数具有超可加性、凸性、弱超可加性. 其次根据区间数的闵可夫斯基距离、区间数的排序及损失函数的定义,建立直觉模糊联盟合作博弈的非线性规划模型,并对其求解得到最优分配. 最后给出一个具体的事例说明本文所建立的模型的合理性和有效性。     

博弈联盟模糊收益的分配

提出了联盟模糊收益合理分配的一种新方法.首先,在模糊收益α截集上定义了α合理分配集,分析了该分配集与模糊收益Shapley值的关系.接着,给出了模糊收益的α合理Shapley分配函数,对其性质进行了讨论.然后,构造了模糊合理Shapley分配,证明其连续性,得到了联盟模糊收益与模糊合理Shapley分配具有包含关系的结论.     

Polytope Games

Starting from the definition of a bimatrix game, we restrict the pair of strategy sets jointly, not independently. Thus, we have a set , which is the set of all feasible strategy pairs. We pose the question of whether a Nash equilibrium exists, in that no player can obtain a higher payoff by deviating. We answer this question affirmatively for a very general case, imposing a minimum of conditions on the restricted sets and the payoff. Next, we concentrate on a special class of restricted games, the polytope bimatrix game, where the restrictions are linear and the payoff functions are bilinear. Further, we show how the polytope bimatrix game is a generalization of the bimatrix game. We give an algorithm for solving such a polytope bimatrix game; finally, we discuss refinements to the equilibrium point concept where we generalize results from the theory of bimatrix games.     

Structural satisfaction in simple games

A fundamental maxim for any theory of social behavior is that knowledge of the theory should not cause behavior that contradicts the theory's assertions. Although this maxim consistently has been heeded in the theory of noncooperative games, it largely has been ignored in solution theory for cooperative games. Solution theory, the central concern of this paper, seeks to identify a subset of the feasible outcomes of a cooperative game that are ‘stable’ results of competition among participants, each of whom attempts to bring about an outcome he favors, rather than to prescribe ‘fair’ outcomes that accord with a standard of equity. We show that learning by participants about the solution theory can cause the outcomes identified as stable by certain solution concepts to become unstable, and discover that an important distinction in this regard is whether the solution concept requires each element of the solution set to defend itself against alternatives rather than relying on other elements for its defense. Finally, we develop a concept of ‘solid’ solutions which have a special claim for stability.The unifying theme of this paper concerns the sense in which certain outcomes of a cooperative game may be regarded as stable, and the extent to which this stability requires that the players are ignorant of the theory. Although the issues raised here have implications for the theory of cooperative games in general, Section 1 establishes the focus of the analysis on collective decision games. Section 2 develops some general perspectives on solution theory which are used in Sections 3 and 4 to evaluate the Condorcet solution, the core, the robust proposals set, von Neumann- Morgenstern solutions and competitive solutions. Section 5 presents the concept of a solid solution and relates this idea to the solution concepts reviewed earlier. We demonstrate that in general a solution concept has a strong claim to stability only if it is solid. Finally, Section 6 concludes by indicating that the basic argument also can be applied to Aumann and Maschler's bargaining sets and, more generally, to solution theory for any cooperative game.     

A convex representation of totally balanced games

We analyze the least increment function, a convex function of n variables associated to an n-person cooperative game. Another convex representation of cooperative games, the indirect function, has previously been studied. At every point the least increment function is greater than or equal to the indirect function, and both functions coincide in the case of convex games, but an example shows that they do not necessarily coincide if the game is totally balanced but not convex. We prove that the least increment function of a game contains all the information of the game if and only if the game is totally balanced. We also give necessary and sufficient conditions for a function to be the least increment function of a game as well as an expression for the core of a game in terms of its least increment function.     

几种新的截集及其应用

本文从“邻域”和“重域”的观点出发引入了三种新的截集：下截集，下重截集和上重截集。我们给出这些截集的性质及相应的分解定理，表现定理和扩展原理，作为应用，我们讨论了两方面的内容；一是利用落影表现理论，将一个模糊集看作“重云”的落影，然后用下生截集和落影表现理论来定义模糊集的运算，得到了与“２”一致的结果。     

On the non-emptiness of the one-core and the bargaining set of committee games

We study the committee decision making process using game theory. Shenoy  [15] introduced two solution concepts: the one-core and the bargaining set, and showed that the one-core of a simple committee game is nonempty if there are at most four players. We extend this result by proving that whether the committee is simple or not, as far as there are less than five players, the one-core is nonempty. This result also holds for the bargaining set.     

截集形式的模糊粗糙集及其性质

用模糊集的截集构造了模糊集的粗糙集,给出了模糊粗糙集的更加严格的数学定义,证明了与文[1]中的等价性,并用新的定义给出模糊粗糙集的相应性质.     

L—Fuzzy集上的Fuzzy值函数的Fuzzy积分的收敛

本文引入了L-Fuzzy集合上的Fuzzy值函数关于Fuzzy值Fuzzy测度的Fuzzy值Fuzzy积分的概念,给出上述概念的几个等价定义,讨论其基本性质,得到一系列积分序列的收敛定理。     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.