An axiomatization of the Banzhaf value for cooperative games on antimatroids

Cooperative games on antimatroids are cooperative games in which coalition formation is restricted by a combinatorial structure which generalizes permission structures. These games group several well-known families of games which have important applications in economics and politics. The current paper establishes axioms that determine the restricted Banzhaf value for cooperative games on antimatroids. The set of given axioms generalizes the axiomatizations given for the Banzhaf permission values. We also give an axomatization of the restricted Banzhaf value for the smaller class of poset antimatroids. Finally, we apply the above results to auction situations.     

The axiomatic approach to three values in games with coalition structure

We present a unified framework for a broad class of values in transferable utility games with coalition structure, including the Owen coalitional value and two weighted versions with weights given by the size of the coalitions. We provide three axiomatic characterizations using the properties of Efficiency, Linearity, Independence of Null Coalitions, and Coordination, with two versions of Balanced Contributions inside a Coalition and Weighted Sharing in Unanimity Games, respectively.     

The Owen value values friendship

This paper presents two new axiomatizations of the Owen value for games with coalition structures. Two associated games are defined and a consistency axiom is required. The construction of the associated games presupposes that coalitions behave in an aggressive manner towards players who are not members of the same unions and in a friendly manner towards players that do belong to their unions. The consistency axiom necessitates the definition of only one associated game which is not a reduced game. Received: February 1999/Revised version: January 2000     

A coalition formation value for games in partition function form

The coalition formation problem in an economy with externalities can be adequately modeled by using games in partition function form (PFF games), proposed by Thrall and Lucas. If we suppose that forming the grand coalition generates the largest total surplus, a central question is how to allocate the worth of the grand coalition to each player, i.e., how to find an adequate solution concept, taking into account the whole process of coalition formation. We propose in this paper the original concepts of scenario-value, process-value and coalition formation value, which represent the average contribution of players in a scenario (a particular sequence of coalitions within a given coalition formation process), in a process (a sequence of partitions of the society), and in the whole (all processes being taken into account), respectively. We give also two axiomatizations of our coalition formation value.     

Owen coalitional value without additivity axiom

We show that the Owen value for TU games with coalition structure can be characterized without the additivity axiom similarly as it was done by Young for the Shapley value for general TU games. Our axiomatization via four axioms of efficiency, marginality, symmetry across coalitions, and symmetry within coalitions is obtained from the original Owen’s one by the replacement of additivity and null-player via marginality. We show that the alike axiomatization for the generalization of the Owen value suggested by Winter for games with level structure is valid as well. The research was supported by NWO (The Netherlands Organization for Scientific Research) grant NL-RF 047.017.017.     

A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks

Semivalues are solution concepts for cooperative games that assign to each player a weighted sum of his/her marginal contributions to the coalitions, where the weights only depend on the coalition size. The Shapley value and the Banzhaf value are semivalues. Mixed modified semivalues are solutions for cooperative games when we consider a priori coalition blocks in the player set. For all these solutions, a computational procedure is offered in this paper.     

A polynomial expression for the Owen value in the maintenance cost game

The class of maintenance cost games was introduced in 2000 to deal with a cost allocation problem arising in the reorganization of the railway system in Europe. The main application of maintenance cost games regards the allocation of the maintenance costs of a facility among the agents using it. To that aim it was first proposed to utilize the Shapley value, whose computation for maintenance cost games can be made in polynomial time. In this paper, we propose to model this cost allocation problem as a maintenance cost game with a priori unions and to use the Owen value as a cost allocation rule. Although the computation of the Owen value has exponential complexity in general, we provide an expression for the Owen value of a maintenance cost game with cubic polynomial complexity. We finish the paper with an illustrative example using data taken from the literature of railways management.     

Share functions for cooperative games with levels structure of cooperation

In a standard TU-game it is assumed that every subset of the player set N can form a coalition and earn its worth. One of the first models where restrictions in cooperation are considered is the one of games with coalition structure of Aumann and Drèze (1974). They assumed that the player set is partitioned into unions and that players can only cooperate within their own union. Owen (1977) introduced a value for games with coalition structure under the assumption that also the unions can cooperate among them. Winter (1989) extended this value to games with levels structure of cooperation, which consists of a game and a finite sequence of partitions defined on the player set, each of them being coarser than the previous one.     

Two hardness results for core stability in hedonic coalition formation games

We establish NP-completeness of two problems on core stable coalitions in hedonic games. In the first problem every player has only two acceptable coalitions in his preference list, and in the second problem the preference structures arise from the distances in an underlying metric space.     

An axiomatization of the disjunctive permission value for games with a permission structure

Players that participate in acooperative game with transferable utilities are assumed to be part of apermission structure being a hierarchical organization in which there are players that need permission from other players before they can cooperate. Thus a permission structure limits the possibilities of coalition formation. Various assumptions can be made about how a permission structure affects the cooperation possibilities. In this paper we consider thedisjunctive approach in which it is assumed that each player needs permission from at least one of his predecessors before he can act. We provide an axiomatic characterization of thedisjunctive permission value being theShapley value of a modified game in which we take account of the limited cooperation possibilities.     

Values of games with graph restricted communication and a priori unions

Two new values for transferable utility games with graph restricted communication and a priori unions are introduced and characterized. Moreover, a comparison between these and the Owen graph value is provided. These values are used to analyze the distribution of power in the Basque Parliament emerging from elections in April 2005.     

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     

A new axiomatization of the Shapley–solidarity value for games with a coalition structure

In this paper, we propose a new kind of players as a compromise between the null player and the A-null player. It turns out that the axiom requiring this kind of players to get zero-payoff together with the well-known axioms of efficiency, additivity, coalitional symmetry, and intra-coalitional symmetry characterize the Shapley–solidarity value. This way, the difference between the Shapely–solidarity value and the Owen value is pinpointed to just one axiom.     

The Harsanyi value for nontransferable utility games with restricted cooperation

A nontransferable utility (NTU) game assigns a set of feasible pay-off vectors to each coalition. In this article, we study NTU games in situations in which there are restrictions on coalition formation. These restrictions will be modelled through interior structures, which extend some of the structures considered in the literature on transferable utility games for modelling restricted cooperation, such as permission structures or antimatroids. The Harsanyi value for NTU games is extended to the set of NTU games with interior structure.     

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.     

The Shapley value for bicooperative games

The aim of the present paper is to study a one-point solution concept for bicooperative games. For these games introduced by Bilbao (Cooperative Games on Combinatorial Structures, 2000) , we define a one-point solution called the Shapley value, since this value can be interpreted in a similar way to the classical Shapley value for cooperative games. The main result of the paper is an axiomatic characterization of this value.     

Some properties for probabilistic and multinomial (probabilistic) values on cooperative games

We investigate the conditions for the coefficients of probabilistic and multinomial values of cooperative games necessary and/or sufficient in order to satisfy some properties, including marginal contributions, balanced contributions, desirability relation and null player exclusion property. Moreover, a similar analysis is conducted for transfer property of probabilistic power indices on the domain of simple games.     

An algorithm for computing the nucleolus of disjunctive non-negative additive games with an acyclic permission structure

A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. In this paper we consider non-negative additive games with an acyclic permission structure. For such a game we provide a polynomial time algorithm for computing the nucleolus of the induced restricted game. The algorithm is applied to a market situation where sellers can sell objects to buyers through a directed network of intermediaries.     

The modeling of coalition for environmental protection within co-utility framework

The game theoretic modelling of coalition for environmental protection within the framework of a new concept of co-utility is discussed. The co-utility concept contains mainly two elements. Firstly, agents can increase their payoffs by collaborating with each-other. Secondly, the outcome of collaboration is robust towards internal and external disturbances. The advantages of using of co-utility are two-fold. Primarily, the co-utility concept is broad and can serve as an umbrella concept in all applications where agents have a space for simultaneous improvement of payoffs. Secondly the co-utility concept can be associated with different stability concept such as myopic or farsighted stability. The myopic and farsighted co-utile sets are defined and their element-co-utile outcomes are found.