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.     

On the Semivalues and the Least Square Values Average Per Capita Formulas and Relationships

In this paper, it is shown that both the Semivalues and the Least Square Values of cooperative transferable utilities games can be expressed in terms of n^2 averages of values of the characteristic function of the game, by means of what we call the Average per capita formulas. Moreover, like the case of the Shapley value earlier considered, the terms of the formulas can be computed in parallel, and an algorithm is derived. From these results, it follows that each of the two values mentioned above are Shapley values of games easily obtained from the given game, and this fact gives another computational opportunity, as soon as the computation of the Shapley value is efficiently done.     

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     

The consistency principle and an axiomatization of the α-core

This paper examines the α-core of strategic games by means of the consistency principle. I provide a new definition of a reduced game for strategic games. And I define consistency (CONS) and two forms of converse consistency (COCONS and COCONS^*) under this definition of reduced games. Then I axiomatize the α-core for families of strategic games with bounded payoff functions by the axioms CONS, COCONS^*, weak Pareto optimality (WPO) and one person rationality (OPR). Furthermore, I show that these four axioms are logically independent. In proving this, I also axiomatize the α-individually rational solution by CONS, COCONS and OPR for the same families of games. Here the α-individually rational solution is a natural extension of the classical `maximin' solution. Received: June 1998/Final version: 6 July 2001     

A new look at the role of players’ weights in the weighted Shapley value

everal new families of semivalues for weighted n-person transferable utility games are axiomatically constructed and discussed under increasing collections of axioms, where the weighted Shapley value arises as the resulting one member family. A more general approach to such weighted games defined in the form of two components, a weight vector λ and a classical TU-game v, is provided. The proposed axiomatizations are done both in terms of λ and v. Several new axioms related to the weight vector λ are discussed, including the so-called “amalgamating payoffs” axiom, which characterizes the value of a weighted game in terms of another game with a smaller number of players. They allow for a new look at the role of players’ weights in the context of the weighted Shapley value for the model of weighted games, giving new properties of it. Besides, another simple formula for the weighted Shapley value is found and examples illustrating some surprising behavior of it in the context of players’ weights are given. The paper contains a wide discussion of the results obtained.     

A Shapley function on a class of cooperative fuzzy games

In this paper, we make a study of the Shapley values for cooperative fuzzy games, games with fuzzy coalitions, which admit the representation of rates of players' participation to each coalition. A Shapley function has been introduced by another author as a function which derives the Shapley value from a given pair of a fuzzy game and a fuzzy coalition. However, the previously proposed axioms of the Shapley function can be considered unnatural. Furthermore, the explicit form of the function has been given only on an unnatural class of fuzzy games. We introduce and investigate a more natural class of fuzzy games. Axioms of the Shapley function are renewed and an explicit form of the Shapley function on the natural class is given. We make sure that the obtained Shapley value for a fuzzy game in the natural class has several rational properties. Finally, an illustrative example is given.     

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.     

Economic lot-sizing games

In this paper we introduce a new class of OR games: economic lot-sizing (ELS) games. There are a number of retailers that have a known demand for a fixed number of periods. To satisfy demand the retailers order products at the same manufacturer. By placing joint orders instead of individual orders, costs can be reduced and a cooperative game arises. In this paper we show that ELS games are balanced. Furthermore, we show that two special classes of ELS games are concave.     

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.     

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.      

On cooperative games,inseparable by semivalues

Semivalues like the Shapley value and the Banzhaf value may assign the same payoff vector to different games. It is even possible that two games attain the same outcome for all semivalues. Due to the linearity of the semivalues, this exactly occurs in case the difference of the two games is an element of the kernel of each semivalue. The intersection of these kernels is called the shared kernel, and its game theoretic importance is that two games can be evaluated differently by semivalues if and only if their difference is not a shared kernel element. The shared kernel is a linear subspace of games. The corresponding linear equality system is provided so that one is able to check membership. The shared kernel is spanned by specific {–1,0,1}-valued games, referred to as shuffle games. We provide a basis with shuffle games, based on an a-priori given ordering of the players.     

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner’s dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.This research was partly funded by the College Engineering, Ohio State University.     

The prenucleolus and the prekernel for games with communication structures

It is well-known that the prekernel on the class of TU games is uniquely determined by non-emptiness, Pareto efficiency (EFF), covariance under strategic equivalence (COV), the equal treatment property, the reduced game property (RGP), and its converse. We show that the prekernel on the class of TU games restricted to the connected coalitions with respect to communication structures may be axiomatized by suitably generalized axioms. Moreover, it is shown that the prenucleolus, the unique solution concept on the class of TU games that satisfies singlevaluedness, COV, anonymity, and RGP, may be characterized by suitably generalized versions of these axioms together with a property that is called “independence of irrelevant connections”. This property requires that any element of the solution to a game with communication structure is an element of the solution to the game that allows unrestricted cooperation in all connected components, provided that each newly connected coalition is sufficiently charged, i.e., receives a sufficiently small worth. Both characterization results may be extended to games with conference structures.     

Submodularity of some classes of the combinatorial optimization games

Submodularity (or concavity) is considered as an important property in the field of cooperative game theory. In this article, we characterize submodular minimum coloring games and submodular minimum vertex cover games. These characterizations immediately show that it can be decided in polynomial time that the minimum coloring game or the minimum vertex cover game on a given graph is submodular or not. Related to these results, the Shapley values are also investigated.Supported by the Berlin-Zürich Joint Graduate Program Combinatorics, Geometry, and Computation (CGC), financed by ETH Zürich and the German Science Foundation (DFG).     

Set-valued nonlinear analogues of the shapley value

One of Shapley's classic axioms deals with the addition of the games. We consider the family of axiom systems depending on various operations over the set of cooperative games. The existence and the uniqueness of the correspondinga priori set-valued solutions are studied. The family of solutions (satisfying the proposed system of axioms) contains, as particular cases, the Shapley value and the -core. The modification of the proposed system of axioms is studied, prekernel being the maximal solution satisfying this modification.We would like to thank two anonymous referees for helpful comments.     

Biprobabilistic values for bicooperative games

The present paper introduces bicooperative games and develops some general values on the vector space of these games. First, we define biprobabilistic values for bicooperative games and observe in detail the axioms that characterize such values. Following the work of Weber [R.J. Weber, Probabilistic values for games, in: A.E. Roth (Ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley Cambridge University Press, Cambridge, 1988, pp. 101–119], these axioms are sequentially introduced observing the repercussions they have on the value expression. Moreover, compatible-order values are introduced and there is shown the relationship between these values and efficient values such that their components are biprobabilistic values.     

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.     

An axiomatization of the nucleolus

In this paper we give a set of axioms characterizing the nucleolus of a TU-game on the class of zero-monotonic games as well as on the class of balanced games. Among the axioms there are familiar ones like anonymity (ANN) and covariance (COV), a restriction of a known property, the restricted reduced game property (ResRGP) and a continuity axiom (LIM). Further we introduce another property of the nucleolus — the alternative reduced game property (AltRGP) — and show that this property together with the ones mentioned before characterizes the nucleolus almost completely on its definition set.     

Nonsymmetric variants of the prekernel and the prenucleolus

A solution on a class of TU games that satisfies the axioms of the pre-nucleolus or -kernel except the equal treatment property and is single valued for two-person games, is a nonsymmetric pre-nucleolus (NSPN) or -kernel (NSPK). We investigate the NSPKs and NSPNs and their relations to the positive prekernel and to the positive core. It turns out that any NSPK is a subsolution of the positive prekernel. Moreover, we show that an arbitrary NSPK, when applied to a TU game, intersects the set of preimputations whose dissatisfactions coincide with the dissatisfactions of an arbitrary element of any other NSPK applied to this game. This result also provides a new proof of sufficiency of the characterizing conditions for NSPKs introduced by Orshan (Non-symmetric prekernels, discussion paper 60. Center for Rationality, The Hebrew University of Jerusalem, 1994). Any NSPN belongs to ??its?? NSPK. Several classes of NSPNs are presented, all of them being subsolutions of the positive core. We show that any NSPN is a subsolution of the positive core provided that it satisfies the equal treatment property on an infinite subset of the universe of potential players. Moreover, we prove that, for any game whose prenucleolus is in its anticore, any NSPN coincides with the prenucleolus.