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.     

On three Shapley-like solutions for cooperative games with random payoffs

Three solution concepts for cooperative games with random payoffs are introduced. These are the marginal value, the dividend value and the selector value. Inspiration for their definitions comes from several equivalent formulations of the Shapley value for cooperative TU games. An example shows that the equivalence is not preserved since these solutions can all be different for cooperative games with random payoffs. Properties are studied and a characterization on a subclass of games is provided.2000 Mathematics Subject Classification Number: 91A12.The authors thank two anonymous referees and an associate editor for their helpful comments.This author acknowledges financial support from the Netherlands Organization for Scientific Research (NWO) through project 613-304-059.Received: October 2000     

Sustainable cooperation in multicriteria multistage games

We use the imputation distribution procedure approach to ensure sustainable cooperation in a multistage game with vector payoffs. In order to choose a particular Pareto optimal and time consistent strategy profile and the corresponding cooperative trajectory we suggest a refined leximin algorithm. Using this algorithm we design a characteristic function for a multistage multicriteria game. Furthermore, we provide sufficient conditions for strong time consistency of the core.     

An axiomatization of probabilistic Owen value for games with coalition structure

In the framework of games with coalition structure, we introduce probabilistic Owen value which is an extension of the Owen value and probabilistic Shapley value by considering the situation that not all priori unions are able to cooperate with others. Then we use five axioms of probabilistic efficiency, symmetric within coalitions, symmetric across coalitions applying to unanimity games, strong monotone property and linearity to axiomatize the value.     

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.     

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 fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

The selectope for cooperative games

The selectope of a cooperative transferable utility game is the convex hull of the payoff vectors obtained by assigning the Harsanyi dividends of the coalitions to members determined by so-called selectors. The selectope is studied from a set-theoretic point of view, as superset of the core and of the Weber set; and from a value-theoretic point of view, as containing weighted Shapley values, random order values, and sharing values. Received May 1997/Revised version September 1999     

Values and potential of games with cooperation structure

Games with cooperation structure are cooperative games with a family offeasible coalitions, that describes which coalitions can negotiate in the game. We study a model ofcooperation structure and the corresponding restricted game, in which the feasible coalitions are those belonging to apartition system. First, we study a recursive procedure for computing the Hart and Mas-Colell potential of these games and we develop the relation between the dividends of Harsanyi in the restricted game and the worths in the original game. The properties ofpartition convex geometries are used to obtain formulas for theShapley andBanzhaf values of the players in the restricted game in terms of the original gamev. Finally, we consider the Owen multilinear extension for the restricted game.The author is grateful to Paul Edelman, Ulrich Faigle and the referees for their comments and suggestions. The proof of Theorem 1 was proposed by the associate editor's referee.     

The Shapley value,the Proper Shapley value,and sharing rules for cooperative ventures

In this note, we discuss two solutions for cooperative transferable utility games, namely the Shapley value and the Proper Shapley value. We characterize positive Proper Shapley values by affine invariance and by an axiom that requires proportional allocation of the surplus according to the individual singleton worths in generalized joint venture games. As a counterpart, we show that affine invariance and an axiom that requires equal allocation of the surplus in generalized joint venture games characterize the Shapley value.     

A reformulated Shapley-like value for cooperative games with interval payoffs

In this paper, a new value for cooperative interval games is proposed which may remedy the disadvantages of the interval Shapley-like value and of the improved interval Shapley-like value introduced by Han et al. (2012). Moreover, it is shown that the reformulated interval value uniquely satisfies the properties of efficiency, indifference null player, symmetry, and additivity.     

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.     

Partially ordered cooperative games: extended core and Shapley value

In this paper we analyze cooperative games whose characteristic function takes values in a partially ordered linear space. Thus, the classical solution concepts in cooperative game theory have to be revisited and redefined: the core concept, Shapley–Bondareva theorem and the Shapley value are extended for this class of games. The classes of standard, vector-valued and stochastic cooperative games among others are particular cases of this general theory. The research of the authors is partially supported by Spanish DGICYT grant numbers MTM2004-0909, HA2003-0121, HI2003-0189, MTM2007-67433-C02-01, P06-FQM-01366.     

Allocation processes in cooperative games

This paper deals with a temporal aspect of cooperative games. A solution of the game is reached through an allocation process. At each stage of the allocation process of a cooperative game a budget of fixed size is distributed among the players. In the first part of this paper we study a type of process that, at any stage, endows the budget to a player whose contribution to the total welfare, according to some measurements, is maximal. It is shown that the empirical distribution of the budget induced by each process of the family converges to a least square value of the game, one such value being the Shapley value. Other allocation processes presented here converge to the core or to the least core. Received: January 2001/Revised: July 2002 I am grateful to the Associate Editor and to the two anonymous referees of International Journal of Game Theory. This research was partially supported by the Israel Science Foundation, grant no. 178/99     

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.     

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     

Minimal large sets for cooperative games

We analyze the concept of large set for a coalitional game v introduced by Martínez-de-Albéniz and Rafels (Int. J. Game Theory 33(1):107–114, 2004). We give some examples and identify some of these sets. The existence of such sets for any game is proved, and several properties of largeness are provided. We focus on the minimality of such sets and prove its existence using Zorn’s lemma. Institutional support from research grants (Generalitat de Catalunya) 2005SGR00984 and (Spanish Government and FEDER) SEJ2005-02443/ECON is gratefully acknowledged, and the support of the Barcelona Economics Program of CREA.