The simplified modified nucleolus of a cooperative TU-game

In the present paper, we introduce a new solution concept for TU-games, the simplified modified nucleolus or the SM-nucleolus. It is based on the idea of the modified nucleolus (the modiclus) and takes into account both the constructive power and the blocking power of a coalition. The SM-nucleolus inherits this convenient property from the modified nucleolus, but it avoids its high computational complexity. We prove that the SM-nucleolus of an arbitrary n-person TU-game coincides with the prenucleolus of a certain n-person constant-sum game, which is constructed as the average of the game and its dual. Some properties of the new solution are discussed. We show that the SM-nucleolus coincides with the Shapley value for three-person games. However, this does not hold for general n-person cooperative TU-games. To confirm this fact, a counter example is presented in the paper. On top of this, we give several examples that illustrate similarities and differences between the SM-nucleolus and well-known solution concepts for TU-games. Finally, the SM-nucleolus is applied to the weighted voting games.     

The consistent value for fuzzy games: alternative axiomatizations

In the framework of fuzzy transferable-utility games, we propose alternative reduced games by considering the players and their activity levels simultaneously. Further, we characterize the consistent value by means of related consistency and different properties.     

The least square nucleolus is a general nucleolus

This short note proves that the least square nucleolus (Ruiz et al. (1996)) and the lexicographical solution (Sakawa and Nishizaki (1994)) select the same imputation in each game with nonempty imputation set. As a consequence the least square nucleolus is a general nucleolus (Maschler et al. (1992)). Received: December 1998/Revised version: July 1999     

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 nucleolus is well-posed

The lexicographic order is not representable by a real-valued function, contrary to many other orders or preorders. So, standard tools and results for well-posed minimum problems cannot be used. We prove that under suitable hypotheses it is however possible to guarantee the well-posedness of a lexicographic minimum over a compact or convex set. This result allows us to prove that some game theoretical solution concepts, based on lexicographic order are well-posed: in particular, this is true for the nucleolus.     

Equal treatment, symmetry and Banzhaf value axiomatizations

The Banzhaf value is the only value satisfying the equal treatment, dummy player and marginal contributions conditions and neutrality of some linear operators on the spaces of games. Under some of these neutrality assumptions, equal treatment can be replaced by even weaker conditions. For linear values having the dummy player property, equal treatment is equivalent to symmetry. All these properties together imply the marginal contributions condition, but in some Banzhaf value axiomatizations marginal contributions cannot be a substitute for linearity. Received: December 1997/Revised version: May 2001     

Finite axiomatizations for existentially closed posets and semilattices

In this paper we exhibit axiomatizations for the theories of existentially closed posets and existentially closed semilattices. We do this by considering an infinite axiomatization which characterizes these structures in terms of embeddings of finite substructures, an axiomatization which exists for any locally finite universal class with a finite language and with the joint embedding and amalgamation properties. We then find particular finite subsets of these axioms which suffice to axiomatize both classes. Research supported by an NSERC Postdoctoral Fellowship. Research supported by NSERC Grant No. A7256.     

Fuzzy games: a complement-consistent solution,axiomatizations and dynamic approaches

By applying the supreme-utilities under fuzzy behavior, we propose a new solution on fuzzy games. In order to present the rationality for this solution, we adopt an extended reduction to provide related axiomatizations and dynamics process. Based on different viewpoint, we also define excess function to introduce alternative formulation and related dynamic process for this solution respectively.     

Bankruptcy and the per capita nucleolus: The claim-and-right rules family

This article describes the per capita nucleolus for bankruptcy games as a bankruptcy rule. This rule, called the clights rule, is based on the well-known constrained equal awards principle and it takes into account a vector of clights, a new term which is a blend of claims and rights. These clights only depend on the vector of claims while the height of the estate determines whether the clights should be interpreted as modified claims or as rights. It is shown that both the clights rule and the Aumann–Maschler rule can be captured within the family of so-called claim-and-right rules.     

Negation and partial axiomatizations of dependence and independence logic revisited

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit derivations for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms.     

The nucleolus of trees with revenues

Trees with revenues are a generalization of standard trees. In a tree with revenues, players have to pay for their connections to the root, but a player can also earn some revenue from being connected to the root. In this paper, we present an algorithm for calculating the nucleolus.     

The general nucleolus and the reduced game property

The nucleolus of a TU game is a solution concept whose main attraction is that it always resides in any nonempty -core. In this paper we generalize the nucleolus to an arbitrary pair (,F), where is a topological space andF is a finite set of real continuous functions whose domain is . For such pairs we also introduce the least core concept. We then characterize the nucleolus forclasses of such pairs by means of a set of axioms, one of which requires that it resides in the least core. It turns out that different classes require different axiomatic characterizations.One of the classes consists of TU-games in which several coalitions may be nonpermissible and, moreover, the space of imputations is required to be a certain generalized core. We call these gamestruncated games. For the class of truncated games, one of the axioms is a new kind ofreduced game property, in which consistency is achieved even if some coalitions leave the game, being promised the nucleolus payoffs. Finally, we extend Kohlberg's characterization of the nucleolus to the class of truncated games.     

Non-cooperative implementation of the nucleolus: The 3-player case

I present a non-cooperative bargaining game, in which responders may exit at any time and have endogenous outside options. When the order of proposers corresponds to the power that players have in the underlying coalitional function, the unique Markov perfect equilibrium outcome of the game is the prenucleolus. The result holds for 3-player superadditive games. An example shows that it cannot be extented to the same class of games forn players. The mechanism is inspired by the consistency property of the prenucleolus.I am grateful to Vijay Krishna, Andreu Mas-Colell, Eric Maskin, Amy Salsbury, and an anonymous referee for helpful comments and suggestions.     

The nucleolus in games with major and minor players

In this paper, we considern-person weighted majority games with major and minor players. We prove under certain conditions that if the minor players split into equal “mini-minor” players the nucleolus' components which correspond to the major players do not change.     

Equivalence theorem,consistency and axiomatizations of a multi-choice value

This paper is devoted to the study of solutions for multi-choice games which admit a potential, such as the potential associated with the extended Shapley value proposed by Hsiao and Raghavan (Int J Game Theory 21:301–302, 1992; Games Econ Behav 5:240–256, 1993). Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the extended Shapley value to an appropriately modified game. In the framework of multi-choice games, we also provide an extension of the reduced game introduced by Hart and Mas-Colell (Econometrica 57:589–614, 1989). Different from the works of Hsiao and Raghavan (1992, 1993), we provide two types of axiomatizations, one is the analogue of Myerson’s (Int J Game Theory 9:169–182, 1980) axiomatization of the Shapley value based on the property of balanced contributions. The other axiomatization is obtained by means of the property of consistency.     

The nucleolus and kernel of veto-rich transferable utility games

The process of computing the nucleolus of arbitrary transferable utility games is notoriously hard. A number of papers have appeared in which the nucleolus is computed by an algorithm in which either one or a huge number of huge linear programs have to be solved. We show that on the class of veto-rich games, the nucleolus is the unique kernel element. Veto-rich games are games in which one of the players is needed by coalitions in order to obtain a non-zero payoff. We then provide a fast algorithm which does not use linear programming techniques to compute the nucleolus of these games. Furthermore, we provide a few examples of economic situations which belong to the class of veto-rich games and which are treated in the literature.     

Convex decomposition of games and axiomatizations of the core and the D-core

This paper provides an axiomatic framework to compare the D-core (the set of undominated imputations) and the core of a cooperative game with transferable utility. Theorem 1 states that the D-core is the only solution satisfying projection consistency, reasonableness (from above), (*)-antimonotonicity, and modularity. Theorem 2 characterizes the core replacing (*)-antimonotonicity by antimonotonicity. Moreover, these axioms also characterize the core on the domain of convex games, totally balanced games, balanced games, and superadditive games.      

Network strength games: the core and the nucleolus

Mathematical Programming - The maximum number of edge-disjoint spanning trees in a network has been used as a measure of the strength of a network. It gives the number of disjoint ways that the...     

The kernel and the nucleolus of a product of simple games

The kernel and the nucleolus of a product of two simple games are given in terms of the kernels and the nucleoluses of the component games.     

A cooperative game of information trading: The core,the nucleolus and the kernel

A certain trade of the information about a technological innovation between the initial owner of the information andn identical producers is studied by means of a cooperative game theoretic approach. The information trading situation is modelled as a cooperative (n+1)-person game with side payments. The symmetrical strong -cores (including the core), the nucleolus and the kernel of the cooperative game model are determined. Interpretations of these game theoretic solutions and their implications for the information trading problem are given.