A characterization of the average tree solution for tree games

For the class of tree games, a new solution called the average tree solution has been proposed recently. We provide a characterization of this solution. This characterization underlines an important difference, in terms of symmetric treatment of the agents, between the average tree solution and the Myerson value for the class of tree games.     

Cores and related solution concepts for multi-choice games

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. Cooperative games form a subclass of the class of multi-choice games.This paper extends some solution concepts for cooperative games to multi-choice games. In particular, the notions of core, dominance core and Weber set are extended. Relations between cores and dominance cores and between cores and Weber sets are extensively studied. A class of flow games is introduced and relations with non-negative games with non-empty cores are investigated.     

模糊联盟合作对策的平均树解

给出图对策中平均树解的拓展形式, 证明其是满足分支有效性和分支公平性的唯一解. 针对具有模糊联盟的图对策, 提出了一种模糊分配, 即模糊平均树解. 当模糊联盟图对策为完全图对策时, 模糊平均树解等于模糊Shapley值. 最后, 讨论了模糊平均树解与模糊联盟核心之间的关系, 并进行了实例论证.     

A constrained egalitarian solution for convex multi-choice games

This paper deals with a constrained egalitarian solution for convex multi-choice games named the d value. It is proved that the d value of a convex multi-choice game belongs to the precore, Lorenz dominates each other element of the precore and possesses a population monotonicity property regarding players’ participation levels. Furthermore, an axiomatic characterization is given where a specific consistency property plays an important role.     

The unit-level-core for multi-choice games: the replicated core for TU games

This note extends the solution concept of the core for traditional transferable-utility (TU) games to multi-choice TU games, which we name the unit-level-core. It turns out that the unit-level-core of a multi-choice TU game is a “replicated subset” of the core of a corresponding “replicated” TU game. We propose an extension of the theorem of Bondareva (Probl Kybern 10:119–139, 1963) and Shapley (Nav Res Logist Q 14:453–460, 1967) to multi-choice games. Also, we introduce the reduced games for multi-choice TU games and provide an axiomatization of the unit-level-core on multi-choice TU games by means of consistency and its converse.     

The multi-core,balancedness and axiomatizations for multi-choice games

This note extends the solution concept of the core for cooperative games to multi-choice games. We propose an extension of the theorem of Bondareva (Problemy Kybernetiki 10:119–139, 1963) and Shapley (Nav Res Logist Q 14:453–460, 1967) to multi-choice games. Also, we introduce a notion of reduced games for multi-choice games and provide an axiomatization of the core on multi-choice games by means of corresponding notion of consistency and its converse.     

The construction and characterization of egalitarian solutions for multi-choice NTU games

In this article we construct a procedure to define the egalitarian solutions in the context of multi-choice non-transferable utility (NTU) games. Also, we show that in the presence of other weak axioms the egalitarian solutions are the only monotonic ones.     

Monotonicity and dummy free property for multi-choice cooperative games

Given a coalition of ann-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, non-decreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.Partially funded by the NSF grant DMS-9024408     

Convex multi-choice games: Characterizations and monotonic allocation schemes

This paper focuses on new characterizations of convex multi-choice games using the notions of exactness and superadditivity. Furthermore, level-increase monotonic allocation schemes (limas) on the class of convex multi-choice games are introduced and studied. It turns out that each element of the Weber set of such a game is extendable to a limas, and the (total) Shapley value for multi-choice games generates a limas for each convex multi-choice game.     

On multi-choice TU games arising from replica of economies

In this paper we derive a multi-choice TU game from r-replica of exchange economy with continuous, concave and monetary utility functions, and prove that the cores of the games converge to a subset of the set of Edgeworth equilibria of exchange economy as r approaches to infinity. We prove that the dominance core of each balanced multi-choice TU game, where each player has identical activity level r, coincides with the dominance core of its corresponding r-replica of exchange economy. We also give an extension of the concept of the cover of the game proposed by Shapley and Shubik (J Econ Theory 1: 9-25, 1969) to multi-choice TU games and derive some sufficient conditions for the nonemptyness of the core of multi-choice TU game by using the relationship among replica economies, multi-choice TU games and their covers.     

Potential approach and characterizations of a Shapley value in multi-choice games

The main focus of this paper is on the restricted Shapley value for multi-choice games introduced by Derks and Peters [Derks, J., Peters, H., 1993. A Shapley value for games with restricted coalitions. International Journal of Game Theory 21, 351–360] and studied by Klijn et al. [Klijn, F., Slikker, M., Zazuelo, J., 1999. Characterizations of a multi-choice value. International Journal of Game Theory 28, 521–532]. We adopt several characterizations from TU game theory and reinterpret them in the framework of multi-choice games. We generalize the potential approach and show that this solution can be formulated as the vector of marginal contributions of a potential function. Also, we characterize the family of all solutions for multi-choice games that admit a potential. Further, a consistency result is reported.     

The Shapley value and average convex games

In this paper we reformulate the necessary and sufficient conditions for the Shapley value to lie in the core of the game. Two new classes of games, which strictly include convex games, are introduced: average convex games and partially average convex games. Partially average convex games, which need not be superadditive, include average convex games. The Shapley value of a game for both classes is in the core. Some Cobb Douglas production games with increasing returns to scale turn out to be average convex games. The paper concludes with a comparison between the new classes of games introduced and some previous extensions of the convexity notion.The authors thank G. Owen, S. Tijs, and J. Ostroy and two anonymous referees of the International Journal of Game Theory for their comments and suggestions. The usual disclamer applies. We are grateful to the Universidad del Pais Vasco-EHU (grant UPV 209.321-H053/90) and the Ministry of Education and Science of Spain (CICYT grant PB900654) for providing reseach support.     

Rooted-tree solutions for tree games

In this paper, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We introduce natural extensions of the average (rooted)-tree solution (see [Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for cycle free games. Games and Economic Behavior 62, 77–92]): the marginalist tree solutions and the random tree solutions. We provide an axiomatic characterization of each of these sets of solutions. By the way, we obtain a new characterization of the average tree solution.     

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.     

Successive approximations for average reward Markov games

This paper considers two-person zero-sum Markov games with finitely many states and actions with the criterion of average reward per unit time. Two special situations are treated and it is shown that in both cases the method of successive approximations yields anε-band for the value of the game as well as stationaryε-optimal strategies. In the first case all underlying Markov chains of pure stationary optimal strategies are assumed to be unichained. In the second case it is assumed that the functional equation Uv=v+ge has a solution.     

Consistent extensions and subsolutions of the core of multi-choice NTU games

We extend the reduced games introduced by Davis and Maschler (Naval Res Log Q 12:223–259, 1965) and Moulin (J Econ Theory 36:120–148, 1985) to multi-choice non-transferable utility games and define two related properties of consistency. We also show that the core proposed by Hwang and Li (Math Methods Oper Res 61:33–40, 2005) violates these two consistency properties. In order to investigate how seriously it violates these two consistency properties, we provide consistent extensions and consistent subsolutions of the core.     

图博弈的过程比例解

本文对具有图结构合作博弈（图博弈）进行了研究,采用比例原则和过程化分配方法,定义了比例分配过程,并对其性质进行了分析。随后,针对比例分配过程的超有效情况,运用等比例妥协的方式给出满足有效性的过程比例解,并研究了稳定性。最后,将比例分配过程与过程比例解应用到破产问题中,得到图博弈过程比例解与破产问题比例规则等价的结论。     

多选择NTU对策核心的一个非空条件及公理化

提出了\pi-均衡多选择NTU对策的概念,证明了\pi-均衡多选择NTU对策的核心非空, 定义了多选择NTU对策的非水平性质和缩减对策,给出了相容性和逆相容性等概念. 用个体合理性、单人合理性、相容性和逆相容性对非水平多选择NTU对策的核心进行了公理化.     

A solution for noncooperative games

In this paper, we study solutions of strict noncooperative games that are played just once. The players are not allowed to communicate with each other. The main ingredient of our theory is the concept of rationalizing a set of strategies for each player of a game. We state an axiom based on this concept that every solution of a noncooperative game is required to satisfy. Strong Nash solvability is shown to be a sufficient condition for the rationalizing set to exist, but it is not necessary. Also, Nash solvability is neither necessary nor sufficient for the existence of the rationalizing set of a game. For a game with no solution (in our sense), a player is assumed to recourse to a standard of behavior. Some standards of behavior are examined and discussed.This work was sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS-75-17385-A01. The author is grateful to J. C. Harsanyi for his comments and to S. M. Robinson for suggesting the problem.