Games on fuzzy communication structures with Choquet players

doi:10.1016/j.ejor.2010.06.014

European Journal of Operational Research

Volume 207, Issue 2, 1 December 2010, Pages 836-847

https://doi.org/10.1016/j.ejor.2010.06.014 Get rights and content

Abstract

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.

Section snippets

Preliminary

A transferable utility cooperative game is a pair (N, v) where N is a finite set and $v : 2^{N} \to R$ is a function with v(∅) = 0. Cooperative games was introduced by von Neumann and Morgensten (1944). The elements of N = {1, 2, … , n} are called players, the subsets S ⊆ N coalitions and v(S) is the worth (benefits) of S. A game (N, v) is superadditive iff for all S, T ⊂ N with S ∩ T = ∅ happen v(S ∪ T) ⩾ v(S) + v(T). In a superadditive game disjoint groups of players always prefer to join in a bigger coalition. We are only

Fuzzy graphs

We introduce now some concepts about fuzzy graphs for which the reader can see Mordeson and Nair (2000). In this paper we use the operators ∧, ∨ as the minimum and the maximum respectively.

Let N be a finite set. A fuzzy set in N is an application τ: N → [0, 1]. We denote by [0, 1]^N the family of fuzzy sets in N. If τ ∈ [0, 1]^N then a fuzzy relation over τ is ρ ∈ [0, 1]^N×N such that ρ(i, j) ⩽ τ(i) ∧ τ(j) for all i, j ∈ N. The fuzzy relation ρ over τ is reflexive if ρ(i, i) = τ(i) for each i ∈ N, and it is symmetric

Fuzzy communication structures

Let (N, v) be a superadditive cooperative game. We consider a fuzzy graph $ρ \in [0, 1]_{0}^{N \times N}$ which changes the initial situation. The number ρ(i, i) is interpreted as the real level of involvement of player i ∈ N in the game and ρ(i, j) represents the maximum level which the link (i, j) can be used. Therefore, we understand a fuzzy communication structure over N as a fuzzy graph without loops over N. For each fuzzy communication structure ρ over N we would take the different possibilities of communication

Particular cases

Returning to the Choquet extension (4) introduced by Tsurumi et al. (2001), we are going to establish the relationship between this extension and the fuzzy Myerson value.

Definition 7

A fuzzy graph $ρ \in [0, 1]_{0}^{N \times N}$ is complete by links iff it satisfies that ρ(i, j) = ρ(i, i) ∧ ρ(j, j) for all i, j ∈ N.

Each link in ρ complete by links can be used at maximum permitted from its vertices. Hence, we can identify the complete by links fuzzy communication structures with the fuzzy coalitions in N by the relation ρ(i, i) = τ(i) for

Acknowledgments

This research has been partially supported by the Spanish Ministry of Education an Science and the European Regional Development Funf, under Grant SEJ2006-00706, and by the FQM237 Grant of the Andalusian Government.

References (7)

M. Tsurumi et al.
A Shapley function on a class of cooperative fuzzy games
European Journal of Operational Research
(2001)
J.P. Aubin
Cooperative fuzzy games
Mathematics of Operation Research
(1981)
T.S.H. Driessen
Cooperative Games Solutions and Applications
(1988)

There are more references available in the full text version of this article.

Cited by (25)

Sharing profits in formal fuzzy contexts
2023, Fuzzy Sets and Systems
Citation Excerpt :
Also, the Shapley value is analyzed for cooperative games with additional fuzzy information about the agents. So, fuzzy communication situations [18], proximity relations among players [11], cohesion indices [12], fuzzy permission structures [13] or fuzzy authorization structures [14] have been studied. In all of them, fuzzy definitions of existing crisp structures were used.
Cooperative game theory is concerned with situations where a group of agents coordinate their actions to get a common benefit. An allocation rule for these situations is a way to share the common benefit among the agents. The search for a fair allocation rule may depend on the information one has about these agents. A formal context represents information about certain attributes of a set of objects in a table, and they have been used in the literature to describe information about the agents in a game. More recently, formal contexts are extended to the fuzzy setting. Now in this paper we establish a methodology to share the profits of a group of agents that have some information about them collected in a fuzzy formal context when those benefits depend on a set of attributes.
Measuring power in coalitional games with friends, enemies and allies
2022, Artificial Intelligence
We extend the well-known model of graph-restricted games due to Myerson to signed graphs. In our model, it is possible to explicitly define not only that some players are friends (as in Myerson's model) but also that some other players are enemies. As such our games can express a wider range of situations, e.g., animosities between political parties. We define the value for signed graph games using the axiomatic approach that closely follows the celebrated characterization of the Myerson value. Furthermore, we propose an algorithm for computing an arbitrary semivalue, including the extension of the Myerson value proposed by us. We also develop a pseudo-polynomial algorithm for power indices in weighted voting games for signed graphs with bounded treewidth. Moreover, we consider signed graph games with a priori defined alliances (unions) between players and propose algorithms to compute the extension of the Owen value to this setting.
The cg-position value for games on fuzzy communication structures
2018, Fuzzy Sets and Systems
Citation Excerpt :
Following this way, the uncertainty about the existence of the communications among the players can be extended. Recently, Jiménez-Losada et al. [22] introduced fuzzy graphs to analyze communication among players. Fuzzy graphs allow leveling the links between being feasible or not, and they also allow considering membership levels for the players.
A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. The best known of these allocations is the Shapley value. A communication structure defines the feasible bilateral communication relationships among the agents in a cooperative situation. Some solutions incorporating this information have been defined from the Shapley value: the Myerson value, the position value, etc. Later fuzzy communication structures were introduced. In a fuzzy communication structure the membership of the players and the relations among them are leveled. Several ways of defining the Myerson value for games on fuzzy communication structure were proposed, one of them is the Choquet by graphs (cg) version. Now we study in this work the cg-position value and its calculation. The cg-position value is defined as a solution for games with fuzzy communication structure which considers the bilateral communications as players. So, the Shapley value is applied for a new game (the link game) over the fuzzy sets of links in the fuzzy communication structure and the profit obtained for each link is allocated between both players in the link. As we see in our examples and results the cg-position value is more concerned with the graphical position of the players and their communications than the other cg-values. In this paper we also introduce a procedure to compute exactly the position value, avoiding to calculate the characteristic function of the link game for all coalitions. This procedure is used to determine the cg-position value. Finally we compare the new value with other cg-values in an applied example about the power of the groups in the European Parliament.
Cooperation among agents with a proximity relation
2016, European Journal of Operational Research
Citation Excerpt :
Tsurumi, Tanino, and Inuiguchi (2001) used the Choquet integral for fuzzy sets in order to define a Shapley-type solution of a family of games with fuzzy coalitions. Jiménez-Losada, Fernández, Ordóñez, and Grabisch (2010) Jiménez-Losada, Fernández, and Ordóñez (2013); Gallego, Fernández, Jiménez-Losada, and Ordóñez (2014) and Gallardo, Jiménez, Jiménez-Losada, and Lebrón (2014) proposed to use fuzzy structures as an additional information for a cooperative game. Particularly, they considered a fuzzy graph to study fuzzy communication structures in the Myerson way.
A cooperative game consists of a set of players and a characteristic function determining the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate, regardless of the actions that the other players take. The relationships of closeness among the players should modify the bargaining among them and therefore their payoffs. The first models that have studied this closeness used a priori unions or undirected graphs. In the a priori union model a partition of the big coalition is supposed. Each element of the partition represents a group of players with the same interests. The groups negotiate among them to form the grand coalition and later, inside each one, players bargain among them. Now we propose to use proximity relations to represent leveled closeness of the interests among the players and extending the a priori unions model.
Values of games with weighted graphs
2015, European Journal of Operational Research
In this paper we deal with TU games in which cooperation is restricted by means of a weighted network. We admit several interpretations for the weight of a link: capacity of the communication channel, flow across it, intimacy or intensity in the relation, distance between both incident nodes/players, cost of building or maintaining the communication link or even probability of the relation (as in Calvo, Lasaga, and van den Noweland, 1999). Then, according to the different interpretations, we introduce several point solutions for these restricted games in a way parallel to the familiar environment of Myerson. Finally, we characterize these values in terms of the (adapted) component efficiency, fairness and balanced contributions properties and we analyze the extent to which they satisfy a link/weight monotonicity property.
A Banzhaf value for games with fuzzy communication structure: Computing the power of the political groups in the European Parliament
2014, Fuzzy Sets and Systems
Citation Excerpt :
Following this way, the uncertainty about the existence of the communications among the players can be extended to the uncertainty about these communications. Recently, Jiménez-Losada et al. [8] introduced fuzzy graphs to analyze communication among players. Fuzzy graphs allow leveling the links between being feasible or not, and they also allow considering membership levels for the players.
In 2013, Jiménez-Losada et al. introduced several extensions of the Myerson value for games with fuzzy communication structure. In a fuzzy communication structure the membership of the players and the relations among them are leveled. Now we study a Banzhaf value for these situations. The Myerson model is followed to define the fuzzy graph Banzhaf value taking as base point the Choquet integral. We propose an axiomatization for this value introducing leveled amalgamation of players. An algorithm to calculate this value is provided and its complexity is studied. Finally we show an applied example computing by this fuzzy value the power of the groups in the European Parliament.

View all citing articles on Scopus

View full text

Stochastics and StatisticsGames on fuzzy communication structures with Choquet players

Abstract

Section snippets

Preliminary

Fuzzy graphs

Fuzzy communication structures

Particular cases

Acknowledgments

European Journal of Operational Research

Cooperative fuzzy games

Mathematics of Operation Research

Cooperative Games Solutions and Applications

Stochastics and Statistics
Games on fuzzy communication structures with Choquet players