Axiomatizations of the core on the universal domain and other natural domains

We prove that the core on the set of all transferable utility games with players contained in a universe of at least five members can be axiomatized by the zero inessential game property, covariance under strategic equivalence, anonymity, boundedness, the weak reduced game property, the converse reduced game property, and the reconfirmation property. These properties also characterize the core on certain subsets of games, e.g., on the set of totally balanced games, on the set of balanced games, and on the set of superadditive games. Suitable extensions of these properties yield an axiomatization of the core on sets of nontransferable utility games. Received September 1999/Final version December 2000     

On the use of binary decision diagrams for solving problems on simple games

Simple games are yes/no cooperative games which arise in many practical applications. Recently, we have used reduced ordered binary decision diagrams and quasi-reduced ordered binary decision diagrams (abbreviated as Robdds and Qobdds, respectively) for the representation of simple games and for the computation of some power indices. In the present paper, we continue this work. We show how further important computational problems on simple games can be solved using Qobdds, viz. the identification of some key players, the computation of the desirability relation on individuals, the test whether a simple game is proper and strong, respectively, and the computation of Qobdd-representations for the sets of all minimal winning coalitions, all shift-minimal winning coalitions and all blocking coalitions, respectively. Applications of these solutions include the computation of recent power indices based on shift-minimal winning coalitions and the test for linear separability of a directed simple game.     

Consistent and covariant solutions for TU games

One of the important properties characterizing cooperative game solutions is consistency. This notion establishes connections between the solution vectors of a cooperative game and those of its reduced game. The last one is obtained from the initial game by removing one or more players and by giving them the payoffs according to a specific principle (e.g. a proposed payoff vector). Consistency of a solution means that the restriction of a solution payoff vector of the initial game to any coalition belongs to the solution set of the corresponding reduced game. There are several definitions of the reduced games (cf., e.g., the survey of T. Driessen [2]) based on some intuitively acceptable characteristics. In the paper some natural properties of reduced games are formulated, and general forms of the reduced games possessing some of them are given. The efficient, anonymous, covariant TU cooperative game solutions satisfying the consistency property with respect to any reduced game are described.The research was supported by the NWO grant 047-008-010 which is gratefully acknowledgedReceived: October 2001     

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.     

On the reduced game property for and the axiomatization of theT-value of TU-games

Summary The τ-value is a one-point solution concept for transferable utility (TU-) games. The paper introduces a particular type of a reduced game. It is established that the τ-value possesses the reduced game property with respect to the reduced game presented. That is, there is no inconsistency in what the players of the reduced game receive-either in the original game or in the reduced game-according to the τ-value concept. In addition, the paper provides an axiomatic characterization of the τ-value in terms of the relevant reduced game property and standardness for two-person games.     

The core cover in relation to the nucleolus and the Weber set

The class of games for which the core coincides with the core cover (compromise stable games) is characterized. Moreover, an easy explicit formula for the nucleolus of this class of games is developed, using an approach based on bankruptcy problems. Also, the class of convex and compromise stable games is characterized. The relation between the core cover and the Weber set is studied and it is proved that under a weak condition their intersection is nonempty.     

双准均衡合作对策\tau值的公理化

介绍了能准确刻画现实生活中每个参与者有三种选择的双合作对策,在此基础上研究了双合作对策的\tau值,并对双准均衡合作对策的\tau值进行了公理化,其中双合作对策的上向量、间隙函数、让步向量的构造是刻画其\tau值的基础.     

The egalitarian solution and reduced game properties in convex games

The main purpose of this paper is to axiomatise the egalitarian solution of Dutta and Ray over the class of convex games. The main axioms used are the two reduced game properties due to Davis-Maschler and Hart-Mas Colell. The egalitarian solution is the only solution satisfying either of the two reduced game properties and agreeing with the egalitarian solution on two person games. Moreover, it is also shown that there is no solution satisfying symmetry, individual rationality and a monotonicity condition on two-person games and which simultaneously satisfies both the reduced game properties.     

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.     

A Study of the Generalized Pontryagin Test Example from the Theory of Differential Games

A generalization of L.S.Pontryagin’s test example from the theory of differential games is considered. The study is based on Pontryagin’s first direct method, which was developed for the constructive solution of linear pursuit–evasion differential games of kind.     

On the reduced game property and its converse

We investigate the relationship between two solutions, the core and the prekernel, and reduced games of coalitional games. An axiomatic characterization of these two solutions is obtained by means of the reduced game property and its converse.     

Consistent extensions and subsolutions of the core for the multichoice transferable-utility games

Abstract

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 multichoice transferable-utility games. First, we provide an example to illustrate that the core proposed by van den Nouweland et al. (Math Methods Oper. Res. 41:289–311, 1995) violates related consistency properties. Further, we propose the minimal consistent extensions of the core and the maximal consistent subsolutions of the core. We also provide an axiomatization based on related consistency properties and its converse.     

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     

On the computation of the nucleolus of a cooperative game

We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel. Received February 2000/Final version April 2001     

On the minimax principle and zero-sum stochastic differential games

The problem of prior and delayed commitment in zero-sum stochastic differential games is discussed. A new formulation and solution based on the delayed-commitment model is derived and its significant implications to stochastic games and control are considered.     

Oddness of the number of equilibrium points: A new proof

A new proof is offered for the theorem that, in “almost all” finite games, the number of equilibrium points isfinite andodd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those “exceptional” games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.     

Stable bargained equilibria for assignment games without side payments

We consider NTU assignment games, which are generalizations of two-sided markets. Matched pairs bargain over feasible allocations; the disagreement outcome is endogenuously determined, taking in account outside options which are based on the current payoff of other players. An allocation is in equilibrium if and only if each pair is in equilibrium (no player wishes to rebargain). The set of equilibria is not empty and it naturally generalizes the intersection of the core and prekernel of TU assignment games. A set with similar properties does not exist for general NTU games. The main source of technical difficulties is the relatively complicated structure of the core in NTU games. We make a strong use of reduced games and consistency requirements. We generalize also the results obtained by Rochford (1984) for TU assignment games.This is a part of my M.Sc. Thesis written at the Hebrew University, Jerusalem and at the University of Heidelberg. I am deeply indebted to my advisor, Prof. Bezalel Peleg. I wish also to thank Professors Michael Maschler, Avraham Neyman and Terje Lensberg for some helpful discussions, and to Prof. Werner Böge for his hospitality in Heidelberg. Finally, the comments of two anonymous referees greatly improved a preliminary version of this paper.     

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.     

基于Solidarity值的河流水资源分配

河流水资源分配问题可以抽象为图限制下合作对策解的模型.基于图限制下合作对策的Solidarity值对参与者的收益分配进行分析,构造了Solidarity值的结构,给出了该值的分支有效性等四个性质,并利用这四个性质刻画了Solidarity值的唯一性.最后通过一个简单的实例,证明了河流水资源分配问题中,基于Solidarity值的收益分配比Myerson值更优.     

The non-emptiness of the core of a partition function form game

The purpose of this paper is to provide a necessary and sufficient condition for the non-emptiness of the core for partition function form games. We generalize the Bondareva–Shapley condition to partition function form games and present the condition for the non-emptiness of “the pessimistic core”, and “the optimistic core”. The pessimistic (optimistic) core describes the stability in assuming that players in a deviating coalition anticipate the worst (best) reaction from the other players. In addition, we define two other notions of the core based on exogenous partitions. The balanced collections in partition function form games and some economic applications are also provided.