Soft cooperation systems and games (original) (raw)

The Shapley value of cooperative games under fuzzy settings: A survey

2014

We survey the recent developments in the studies of cooperative games under fuzzy environment. The basic problems of a cooperative game in both crisp and fuzzy contexts are to find how the coalitions form vis-á-vis how the coalitions distribute the worth. One of the fuzzification processes assumes that the coalitions thus formed are fuzzy in nature having only partial participations of the players. A second group of researchers fuzzify the worths of the coalitions while a few others assume that both the coalitions and the worths are fuzzy quantities. Among the various solution concepts of a cooperative game, the Shapley value is the most popular one-point solution concept which is characterized by a set of rational axioms. We confine our study to the developments of the Shapley value in fuzzy setting and try to highlight the respective characterizations.

Shapley Value of a Cooperative Game with Fuzzy Set of Feasible Coalitions

Cybernetics and Systems Analysis, 2017

The paper investigates Shapley value of a cooperative game with fuzzy set of feasible coalitions. It is shown that the set of its values is a type 2 fuzzy set (a fuzzy set whose membership function takes fuzzy values) of special type. Furthermore, the corresponding membership function is given. Elements of the support of this set are defined as particular Shapley values. The authors also propose the procedure of constructing these elements with maximum reliability of their membership and reliability of non-membership, not exceeding a given threshold.

Cooperative games with fuzzy coalitions and fuzzy characteristic functions

Fuzzy Sets and Systems, 2008

In this paper, an extension of a cooperative fuzzy game is proposed, which admits the representation of the rate of participation of every player in a coalition and also associates fuzziness with the value of the game. Games, subject to fuzzy coalitions as well as those pertaining to fuzzy characteristic functions or vague expectations are separately studied in the literature. We propose an extension of a fuzzy game with fuzzy coalitions and vague expectations together and obtain some interesting properties. It has been observed that most of the properties satisfied by a crisp game hold good in the fuzzy sense in this extension. A practical application of the proposed model in Investment Theory is being provided. Further, a Shapley function in the fuzzy sense has been proposed as a solution concept to this class of games. The notion of a fuzzy population monotonic allocation function (FPMAF) is defined and established that the proposed Shapley function is an FPMAF also.

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.

A Simplified Expression of Share Functions for Cooperative Games with Fuzzy Coalitions

Tatra Mountains Mathematical Publications

In this paper, we discuss the notion of Share functions for cooperative games with fuzzy coalitions or simply fuzzy cooperative games. We obtain the Share functions for some special classes of fuzzy games, namely the fuzzy games in proportional value form and the fuzzy games in Choquet integral form. The Shapley Share and Banzhaf Share functions for these classes are derived.

A new Shapley value for games with fuzzy coalitions

Fuzzy Sets and Systems, 2019

This paper deals with cooperative games over fuzzy coalitions. In these situations there is a continuous set of fuzzy coalitions instead of a finite set of them (as in the classical case), the unit square in an n-dimensional space. There exist in the literature two different extensions of the known Shapley value for crisp games to games with fuzzy coalitions: the crisp Shapley value and the diagonal value. The first value only uses a finite information in the set of fuzzy coalitions, the vertices of the square. While the second one uses a neighbourhood of the diagonal of the square. We propose a new extension of the Shapley value improving the crisp Shapley value for games with fuzzy coalitions. This new version uses the faces of the square, namely an infinity quantity of information. We analyze several properties of the new value, we endow it with an axiomatization and we study the behavior when it is applied to known fuzziness of crisp games.

Bi-cooperative games with fuzzy bi-coalitions

Fuzzy Sets and Systems

In this paper, we introduce the notion of a bi-cooperative game with fuzzy bi-coalitions and discuss the related properties. In real game theoretic decision making problems, many criteria concerning the formation of coalitions have bipolar motives. Our model tries to explore such bipolarity in fuzzy environment. The corresponding Shapley axioms are proposed. An explicit form of the Shapley value as a possible solution concept to a particular class of such games is also obtained. Our study is supplemented with an illustrative example.

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.

The value in games with restricted cooperation

2015

We consider cooperative games in which the cooperation among players is restricted by a set system, which outlines the set of feasible coalitions that actually can be formed by players in the game. In our setting, the structure of this set system is completely free, and the only restriction is that the empty set belongs to it. An extension of the Shapley value is provided as the sum of the dividends that players obtain in the game. In this general setting, we offer two axiomatic characterizations for the value: one by means of component efficiency and fairness, and the other one with efficiency and balanced contributions.

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.