The value in games with restricted cooperation (original) (raw)

Axiomatization of values of cooperative games using a fairness property

Applicationes Mathematicae, 2005

We propose new systems of axioms which characterize four types of values of cooperative games: the Banzhaf value, the Deegan-Packel value, the least square prenucleolus and the least square nucleolus. The common element used in these axiomatizations is a fairness property. It requires that if to a cooperative game we add another game in which two given players are symmetric, then their payoffs change by the same amount. In our analysis we will use an idea applied by R. van den Brink (2001) to obtain an axiomatic characterization of the Shapley value.

Axiomatizations of two types of Shapley values for games on union closed systems

Economic Theory, 2011

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication.

THE SHAPLEY-SOLIDARITY VALUE FOR GAMES WITH A COALITION STRUCTURE

International Game Theory Review, 2013

A value for games with a coalition structure is introduced, where the rules guiding cooperation among the members of the same coalition are different from the interaction rules among coalitions. In particular, players inside a coalition exhibit a greater degree of solidarity than they are willing to use with players outside their coalition. The Shapley value is therefore used to compute the aggregate payoffs for the coalitions, and the solidarity value to obtain the payoffs for the players inside each coalition.

Solidarity induced by group contributions: the MI$$^k$$-value for transferable utility games

Operational Research

The most popular values in cooperative games with transferable utilities are perhaps the Shapley and the Shapley like values which are based on the notion of players' marginal productivity. The equal division rule on the other hand, is based on egalitarianism where resource is equally divided among players, no matter how productive they are. However none of these values explicitly discuss players' multilateral interactions with peers in deciding to form coalitions and generate worths. In this paper we study the effect of multilateral interactions of a player that accounts for her contributions with her peers not only at an individual level but also on a group level. Based on this idea, we propose a value called the MI k-value and characterize it by the axioms of linearity, anonymity, efficiency and a new axiom: the axiom of MN k-player. An MN kplayer is one whose average marginal contribution due to her multilateral interactions upto level k is zero and can be seen as a generalization of the standard null player axiom of the Shapley value. We have shown that the MI k-value on a variable player set is asymptotically close to the equal division rule. Thus our value realizes solidarity among players by incorporating both their individual and group contributions.

The axiomatic approach to three values in games with coalition structure

European Journal of Operational Research, 2009

We study three values for 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.

A value for cooperative games with a coalition structure!

Discussion Papers in Economic Behaviour, 2011

A value for games with a coalition structure is introduced, where the rules guiding the cooperation among the members of the same coalition are di¤erent from the interaction rules among coalitions. In particular, players inside a coalition exhibit a greater degree of solidarity than they are willing to use with players outside their coalition. The Shapley value [Shapley, 1953] is therefore used to compute the aggregate payo¤s of the coalitions, and the Solidarity value [Nowak and Radzik, 1994] to obtain the payo¤s of the players inside each coalition.

Axiomatisation of the Shapley value and power index for bi-cooperative games

2006

Bi-cooperative games have been introduced by Bilbao as a generalization of classical cooperative games, where each player can participate positively to the game (defender), negatively (defeater), or do not participate (abstentionist). In a voting situation (simple games), they coincide with ternary voting game of Felsenthal and Mochover, where each voter can vote in favor, against or abstain. In this paper, we propose a definition of value or solution concept for bi-cooperative games, close to the Shapley value, and we give an interpretation of this value in the framework of (ternary) simple games, in the spirit of Shapley-Shubik, using the notion of swing. Lastly, we compare our definition with the one of Felsenthal and Machover, based on the notion of ternary roll-call, and the Shapley value of multi-choice games proposed by Hsiao and Ragahavan.

Values for two-stage games: Another view of the Shapley axioms

International Journal of Game Theory, 1990

This short study reports an application of the Shapley value axioms to a new concept of "two-stage games." In these games, the formation of a coalition in the first stage entities its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some "two-stage majority games" (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's "),-transfer value" for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for "multi-stage games."

Three additive solutions of cooperative games with a priori unions

Applicationes Mathematicae, 2003

We analyze axiomatic properties of three types of additive solutions of cooperative games with a priori unions structure. One of these is the Banzhaf value with a priori unions introduced by G. Owen (1981), which has not been axiomatically characterized as yet. Generalizing Owen's approach and the constructions discussed by J. Deegan and E. W. Packel (1979) and L. M. Ruiz, F. Valenciano and J. M. Zarzuelo (1996) we define and study two other solutions. These are the Deegan-Packel value with a priori unions and the least square prenucleolus with a priori unions. Each of known cooperative game solutions is usually constructed by means of different methods with specific assumptions. In this paper we investigate a modification of three types of such solutions. The first of these solutions, the Banzhaf value of a player, was introduced by J. F. Banzhaf III (1965). It describes the average profit for a coalition after co-opting the player. Numerous applications of this concept are now known in the social and economic practice, because the relevant formulas represent a good instrument to investigate the power of participants in collective decision processes. In 1981 G. Owen constructed a modification of this notion-the Banzhaf value with a priori unions. The main assumption of this model is a partition of the set of players into nonempty disjoint subsets called a priori unions or precoalitions. The Banzhaf value with a priori unions was constructed on the basis of the "normal" Banzhaf value. E. Lehrer (1988) suggested the first axiomatization of the Banzhaf value. It is the unique solution with the following properties: dummy player, equal treatment, amalgamation and additivity. An axiomatization theorem for the

Axiomatizations of the Shapley value for games on augmenting systems

European Journal of Operational Research, 2009

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed another model for cooperative games defined on lattice structures. We introduce a combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. In this framework, the Shapley value of games on augmenting systems is introduced and two axiomatizations of this value are showed.