An application of optimization theory to the study of equilibria for games: a survey

This contribution is a survey about potential games and their applications. In a potential game the information that is sufficient to determine Nash equilibria can be summarized in a single function on the strategy space: the potential function. We show that the potential function enable the application of optimization theory to the study of equilibria. Potential games and their generalizations are presented. Two special classes of games, namely team games and separable games, turn out to be potential games. Several properties satisfied by potential games are discussed and examples from concrete situations as congestion games, global emission games and facility location games are illustrated.     

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.     

不确定性下强Berge均衡的存在性

在已知不确定参数变化的范围下,研究了非合作博弈与广义非合作博弈的强Berge均衡的存在性,基于强Berge均衡与不确定性下非合作博弈的强Nash均衡的概念,给出了不确定参数下非合作博弈与广义非合作博弈的强Berge均衡的定义,并利用Fan-Glicksberg不动点定理证明其存在性,最后用算例验证其可行性.     

Values for Interior Operator Games

The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values.     

Reductions of interval noncooperative games

Noncooperative games of a finite number of persons with interval-valued payoff functions are considered. The concept of an equilibrium situation is introduced. A reduction of such games to deterministic noncooperative games is proposed. Properties of the reduced games are discussed. Interval antagonistic and bimatrix games are examined, and illustrative examples are considered.     

An amalgation of games

A special class of normal form games is the subject of the paper. Typically, the player set of the games in this class consists of two parties and the games are aggregations of conflicts between two players, one in each party. Two 2-person normal form games, closely related to the original game, are introduced and relations between the sets of (perfect) equilibria of these games and the original game are derived. Using the fact that the structure of the set of (perfect) equilibria of bimatrix games is known, the structure of the set of (perfect) equilibria of the original games is revealed. Two characteristic examples are treated in more detail.     

An average lexicographic value for cooperative games

For games with a non-empty core the Alexia value is introduced, a value which averages the lexicographic maxima of the core. It is seen that the Alexia value coincides with the Shapley value for convex games, and with the nucleolus for strongly compromise admissible games and big boss games. For simple flow games, clan games and compromise stable games an explicit expression and interpretation of the Alexia value is derived. Furthermore it is shown that the reverse Alexia value, defined by averaging the lexicographic minima of the core, coincides with the Alexia value for convex games and compromise stable games.     

Approximating the least core value and least core of cooperative games with supermodular costs

We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a 3-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time algorithm for computing a cost allocation in the 2-approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomial-time approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomial-time algorithms for computing the least core value as well as a least core cost allocation.     

Values of games with denumerably many players

In this paper axioms for values of games with denumerably many players are introduced and, on a certain space of games, a value is defined as a limit of values of finite games. Further, some relationships between the value that the topology on the space of games of bounded variation are investigated. It is also shown and the regular weighted majority games are members of the space on which the value is defined.     

Two characterizations of the consistent egalitarian solution and of the core on NTU games

In this paper, we provide two characterizations in the context of NTU games: of the consistent egalitarian solution and of the core. The first characterization is based on reduced games and contraction and expansion independence. The second one is based on reduced games and sub-reduced games.     

基于多维线性扩展的模糊联盟合作对策tau值性质与计算方法

研究模糊联盟合作对策tau值的计算方法及其性质. 利用多维线性扩展方法定义了模糊联盟合作对策的tau值, 证明了其存在性、唯一性等性质, 并推导出基于多维线性扩展凸模糊联盟合作对策tau值的计算公式. 研究结果发现, 基于多维线性扩展的模糊联盟合作对策tau值是对清晰联盟合作对策tau值的扩展, 而清晰联盟合作对策tau值仅是其特例. 特别地, 对于凸模糊联盟合作对策, 利用其tau值计算公式, 可进一步简化求解过程.     

多目标的随机结盟对策的ZS-值

本文研究的是多目标随机结盟对策的问题，是将单目标的随机结盟对策的ZS-值拓展到多目标的随机结盟对策上，同时考虑了局中人对不同目标的偏好程度，从而，给出了多目标随机结盟对策的ZS-值的定义，并讨论了该值的性质及定理。     

集合对策边缘解的公理化方法

本文将单调集合对策边缘解推广到一般集合对策上,利用三个公理,确定了集合对策边缘解的唯—性．在证明唯—性的过程中,引进了单调集合对策将集合对策进行分解．证明过程与Aarts等人的方法完全不一样,在某些地方吸收了Young的有关思想．     

Monotonic games are spanning network games

Spanning network games, which are a generalization of minimum cost spanning tree games, were introduced by Granot and Maschler (1991), who showed that these games are always monotonic. In this paper a subclass of spanning network games is introduced, namely simplex games, and it is shown that every monotonic game is a simplex game. Hence, the class of spanning network games coincides with the class of monotonic games.     

Proof of permutationally convexity of MCSF games

Granot and Huberman (1982) showed that minimum cost spanning tree (MCST) games are permutationally convex (PC). Recently, Rosenthal (1987) gave an extension of MCST games to minimum cost spanning forest (MCSF) games and showed these games have nonempty cores. In this note we show any MCSF game is a PC game.     

Survey on applications of semi-tensor product method in networked evolutionary games

Semi-tensor product (STP) method of matrices has received more and more attention from the communities of both engineering and economics in recent years. This paper presents a comprehensive survey on the applications of STP method in the theory of networked evolutionary games. In the beginning, some preliminary results on STP method are recalled. Then, the applications of STP method in many kinds of networked evolutionary games, such as general networked evolutionary games, networked evolutionary games with finite memories, networked evolutionary games defined on finite networks, and random networked evolutionary games, are reviewed. Finally, several research problems in the future are predicted.     

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.     

Submodularity and the evolution of Walrasian behavior

Vega-Redondo (1997) showed that imitation leads to the Walrasian outcome in Cournot Oligopoly. We generalize his result to aggregative quasi-submodular games. Examples are the Cournot Oligopoly, Bertrand games with differentiated complementary products, Common-Pool Resource games, Rent-Seeking games and generalized Nash-Demand games.I thank Jörg Oechssler, Rolf Tisljar, the editor, and an anonymous referee for helpful comments and suggestions. Financial support by the DFG is gratefully acknowledged.Submitted: September 2001 / Revised: January 2004     

Monotonicity of the core and value in dynamic cooperative games

We examine behavior of the core and value of certain classes of cooperative games in which a dynamic aspect is introduced. New players are added to the games while the underlying structure is held constant. This is done by considering games that satisfy properties like convexity, or games that are derived from optimization problems in which a player's addition can be defined naturally. For such games we give conditions regarding monotonicity of the core and value.