The degree value for games with communication structure

A new value concept, called degree value, is proposed by employing the degree game induced by an original game for hypergraph communication situations (including graph communication situations). We provide an axiomatic characterization of the degree value for arbitrary hypergraph communication situations by applying component efficiency and balanced conference contributions, which is a natural extension of balanced link contributions introduced in Slikker (Int J Game Theory 33:505–514, 2005) for graph communication situations. By comparing the degree value with the position value and the Myerson value, it is verified that the degree value is a new allocation rule that differs from both the Myerson value and the position value, and the degree value highlights the important role of the degree of a player in hypergraph communication situations. Particularly, in a uniform hypergraph communication situation, where every conference contains the same number of players, we show that the degree value coincides with the position value.     

Consistency and the graph Banzhaf value for communication graph games

The graph Banzhaf value was introduced and axiomatically characterized by Alonso-Meijide and Fiestras-Janeiro (2006). In this paper we propose the reduced game and consistency of the graph Banzhaf value for communication situations. By establishing the relationship between the Harsanyi dividends of a coalition in a communication situation and the reduced communication situation, we provide a new axiomatization of the graph Banzhaf value by means of the axioms of consistency and standardness.     

On the convexity of communication games

A communication situation consists of a game and a communication graph. By introducing two different types of corresponding communication games, point games and arc games, the Myerson value and the position value of a communication situation were introduced. This paper investigates relations between convexity of the underlying game and the two communication games. In particular, assuming the underlying game to be convex, necessary and sufficient conditions on the communication graph are provided such that the communication games are convex. Moreover, under the same conditions, it is shown that the Myerson value and the posi tion value are in the core of the point game. Some remarks are made on superadditivity and balancedness.     

Decomposition of network communication games

Using network control structures, this paper introduces a general class of network communication games and studies their decomposition into unanimity games. We obtain a relation between the dividends in any network communication game and its underlying transferable utility game, which depends on the structure of the communication network. Moreover, we introduce a new class of network control values which contains both the Myerson value and the position value. The decomposition results are used to explicitly express these values in terms of dividends.     

Splitting graphs when calculating Myerson value for pure overhead games

A communication situation consists of a coalitional game and a graph, the nodes of the graph corresponding to the players of the game. To calculate the Myerson value for such situations, we obtain results which extend those well known for trees and cycle-complete graphs. On the other hand, in order to reduce the associated calculus for communication situations with a pure overhead game, the possibility of splitting the graph in several subgraphs is analyzed. For each fixed decomposition of the graph, a subspace of games compatible with this decomposition is given.     

The least square values and the shapley value for cooperative TU games

The main result proved in this paper is the fact that any Least Square Value is the Shapley value of a game obtained from the given game by rescaling. An Average per capita formula for Least Square Values, similar to the formula for the Shapley value (Dragan (1992)), will lead to this conclusion and allow a parallel computation for these values. The potential for the Least Square Values, a potential basis relative to Least Square Values and an approach similar to the one used for the Shapley value is allowing us to solve the Inverse problem for Least Square Values.     

A mean value for games with communication structures

The mean value is a new extension of the Shapley value for games with communication structure representable by a simple graph; only pairwise meetings can occur, although some of them might not be permitted. The new value is characterized by a set of axioms of which the one with the most far-reaching effect is an associated consistency property already used in various contexts. The mean value of an n-player unanimity game is the arithmetic average of the mean values of (n–1)-player unanimity games with connected support, which means games in which the deleted players are not articulation point of the considered graph.I wish to thank the anonymous referees for their helpful remarks. The usual disclaimer applies.Received: April 2002/Accepted: February 2004     

Inheritance of properties in communication situations

In this paper we consider cooperative games in which the possibilities for cooperation between the players are restricted because communication between the players is restricted. The bilateral communication possibilities are modeled by means of a (communication) graph. We are interested in how the communication restrictions influence the game. In particular, we investigate what conditions on the communication graph guarantee that certain appealing properties of the original game are inherited by the graph-restricted game, the game that arises once the communication restrictions are taken into account. We study inheritance of the following properties: average convexity, inclusion of the Shapley value in the core, inclusion of the Shapley values of a game and all its subgames in the corresponding cores, existence of a population monotonic allocation scheme, and the property that the extended Shapley value is a population monotonic allocation scheme. Received May 1998/Revised version January 2000     

A dynamic approach to the Shapley value based on associated games

We propose a dynamic process leading to the Shapley value of TU games or any solution satisfying Inessential Game (IG) and Continuity (CONT), based on a modified version of Hamiache's notion of an associated game. The authors are very grateful to William Thomson and two anonymous referees for valuable comments which much improve the paper. They provide better statements and proofs of several major results than the original ones.     

A new basis and the Shapley value

The purpose of this paper is to introduce a new basis of the set of all TU games. Shapley (1953) introduced the unanimity game in which cooperation of all players in a given coalition yields payoff. We introduce the commander game in which only one player in a given coalition yields payoff. The set of the commander games forms a basis and has two properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value. Second, the basis induces the null space of the Shapley value.     

Cooperative grey games and the grey Shapley value

This contribution is located in the common area of operational research and economics, with a close relation and joint future potential with optimization: game theory. We focus on collaborative game theory under uncertainty. This study is on a new class of cooperative games where the set of players is finite and the coalition values are interval grey numbers. An interesting solution concept, the grey Shapley value, is introduced and characterized with the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory. The paper ends with a conclusion and an outlook to future studies.     

A value for partially defined cooperative games

The Shapley value provides a method, which satisfies certain desirable axioms, of allocating benefits to the players of a cooperative game. When there aren players andn is large, the Shapley value requires a large amount of accounting because the number of coalitions grows exponentially withn. This paper proposes a modified value that shares some of the axiomatic properties of the Shapley value yet allows the consideration of games that are defined only for certain coalitions. Two different axiom systems are shown to determine the same modified value uniquely.     

The interval Shapley value: an axiomatization

The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory.     

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.     

超图结构上合作博弈的赋权Position值

在合作博弈的一般模型中总是假设所有联盟都能形成。不过,在实际中由于受到一些因素的制约,有些联盟是不能形成的。基于此,Myerson提出了具有图通讯结构的合作博弈。Myerson值和Position值是超图博弈上的两个重要分配规则。2005年,Slikker给出了在图博弈上Position值的公理化刻画。但超图博弈上Position值的公理化刻画一直悬而未决。本文通过引入“赋权平衡超边贡献公理”,并结合经典的“分支有效性”,提出了超图博弈上赋权Position值的公理化刻画。作为推论,解决了超图博弈上Position值的公理化刻画问题。     

Games on fuzzy communication structures with Choquet players

Myerson (1977) used graph-theoretic ideas to analyze cooperation structures in games. In his model, he considered the players in a cooperative game as vertices of a graph, which undirected edges defined their communication possibilities. He modified the initial games taking into account the graph and he established a fair allocation rule based on applying the Shapley value to the modified game. Now, we consider a fuzzy graph to introduce leveled communications. In this paper players play in a particular cooperative way: they are always interested first in the biggest feasible coalition and second in the greatest level (Choquet players). We propose a modified game for this situation and a rule of the Myerson kind.     

The Shapley value for the probability game

The main goal of this paper is to introduce the probability game. On one hand, we analyze the Shapley value by providing an axiomatic characterization. We propose the so-called independent fairness property, meaning that for any two players, the player with larger individual value gets a larger portion of the total benefit. On the other, we use the Shapley value for studying the profitability of merging two agents.     

A value for continuously-many-choice cooperative games

We extend a multi-choice cooperative game to a continuously-many-choice cooperative game. The set of all continuously-many-choice cooperative games is isomorphic to the set of all cooperative fuzzy games. A continuously-many-choice cooperative game and a cooperative fuzzy game have different physical interpretations. We define a value for the continuously-many-choice cooperative game and show that the value for the continuously-many-choice cooperative game has most properties as the traditional Shapley value does. Also, we give a probabilistic interpretation for the value. The probabilistic interpretation reveals some interesting properties of the value. Finally, we discuss the uniqueness of the value.     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

Mew mathematical properties of the semivalues of cooperative tu games *

The Semivalues were introduced by Dubey, Neiman and Weber (1981), as the values of TU games satisfying a set of axioms, precisely:linearity, symmetry, monotomcity and projection axioms. In this paper, a potential approach is used to prove a characterization of Semivalues as the unique values satisfying a new type of consistency relative to a new implicitly defined reduced game of Hart Mas Colell type and weighted standardness for all two person games. The definition of the reduced game requires some combinatorial results on the auxiliary game of the given game. As a byproduct, we derive from these results two other combinatorial properties of the Semivalues: (i) any Semivalue is the Shapley value of the auxiliary game of the given game, and (ii) any Semivalue satisfies the fairness principle introduced by Myerson (1977) as the principle of balanced contributions