240 research outputs found
Cooperative Games with Overlapping Coalitions
In the usual models of cooperative game theory, the outcome of a coalition
formation process is either the grand coalition or a coalition structure that
consists of disjoint coalitions. However, in many domains where coalitions are
associated with tasks, an agent may be involved in executing more than one
task, and thus may distribute his resources among several coalitions. To tackle
such scenarios, we introduce a model for cooperative games with overlapping
coalitions--or overlapping coalition formation (OCF) games. We then explore the
issue of stability in this setting. In particular, we introduce a notion of the
core, which generalizes the corresponding notion in the traditional
(non-overlapping) scenario. Then, under some quite general conditions, we
characterize the elements of the core, and show that any element of the core
maximizes the social welfare. We also introduce a concept of balancedness for
overlapping coalitional games, and use it to characterize coalition structures
that can be extended to elements of the core. Finally, we generalize the notion
of convexity to our setting, and show that under some natural assumptions
convex games have a non-empty core. Moreover, we introduce two alternative
notions of stability in OCF that allow a wider range of deviations, and explore
the relationships among the corresponding definitions of the core, as well as
the classic (non-overlapping) core and the Aubin core. We illustrate the
general properties of the three cores, and also study them from a computational
perspective, thus obtaining additional insights into their fundamental
structure
Fuzzy coalitional structures (alternatives)
The uncertainty of expectations and vagueness of the interests belong to
natural components of cooperative situations, in general. Therefore, some
kind of formalization of uncertainty and vagueness should be included in
realistic models of cooperative behaviour. This paper attempts to contribute
to the endeavour of designing a universal model of vagueness in cooperative
situations. Namely, some initial auxiliary steps toward the development of
such a model are described. We use the concept of fuzzy coalitions suggested
in [1], discuss the concepts of superadditivity and convexity, and introduce a
concept of the coalitional structure of fuzzy coalitions.
The first version of this paper [10] was presented at the Czech-Japan
Seminar in Valtice 2003. It was obvious that the roots of some open questions
can be found in the concept of superadditivity (with consequences on some
other related concepts), which deserve more attention. This version of the
paper extends the previous one by discussion of alternative approaches to
this topic
Cooperative games with overlapping coalitions
In the usual models of cooperative game theory, the outcome of a coalition formation process is either the grand coalition or a coalition structure that consists of disjoint coalitions. However, in many domains where coalitions are associated with tasks, an agent may be involved in executing more than one task, and thus may distribute his resources among several coalitions. To tackle such scenarios, we introduce a model for cooperative games with overlapping coalitions—or overlapping coalition formation (OCF) games. We then explore the issue of stability in this setting. In particular, we introduce a notion of the core, which generalizes the corresponding notion in the traditional (non-overlapping) scenario. Then, under some quite general conditions, we characterize the elements of the core, and show that any element of the core maximizes the social welfare. We also introduce a concept of balancedness for overlapping coalitional games, and use it to characterize coalition structures that can be extended to elements of the core. Finally, we generalize the notion of convexity to our setting, and show that under some natural assumptions convex games have a non-empty core. Moreover, we introduce two alternative notions of stability in OCF that allow a wider range of deviations, and explore the relationships among the corresponding definitions of the core, as well as the classic (non-overlapping) core and the Aubin core. We illustrate the general properties of the three cores, and also study them from a computational perspective, thus obtaining additional insights into their fundamental structure
Two approaches to fuzzification of payments in NTU coalitional game
There exist several possibilities of
fuzzification of a coalitional game. It is quite usual to fuzzify,
e.\,g., the concept of coalition, as it was done in [1].
Another possibility is to fuzzify the expected pay-offs, see [3,4]. The latter possibility is dealt even here. We suppose
that the coalitional and individual pay-offs are expected only
vaguely and this uncertainty on the "input" of the game rules is
reflected also by an uncertainty of the derived "output" concept
like superadditivity, core, convexity, and others. This method of
fuzzification is quite clear in the case of games with
transferable utility, see [6,3]. The not transferable utility
(NTU) games are mathematically rather more complex structures. The
pay-offs of coalitions are not isolated numbers but closed subsets
of n-dimensional real space. Then there potentially exist two
possible approaches to their fuzzification. Either, it is possible
to substitute these sets by fuzzy sets (see, e.g.[3,4]).
This approach is, may be, more sophisticated but it leads to some
serious difficulties regarding the domination of vectors from
fuzzy sets, the concept of superoptimum, and others. Or, it is
possible to fuzzify the whole class of (essentially deterministic)
NTU games and to represent the vagueness of particular properties
or components of NTU game by the vagueness of the choice of the
realized game (see [5]). This approach is, perhaps, less
sensitive regarding some subtile variations in the the fuzziness
of some properties but it enables to transfer the study of fuzzy
NTU coalitional games into the analysis of classes of
deterministic games. These deterministic games are already well
known, which fact significantly simplifies the demanded analytical
procedures.
This brief contribution aims to introduce formal specifications of
both approaches and to offer at least elementary comparison of
their properties
Additivities in fuzzy coalition games with side-payments
summary:The fuzzy coalition game theory brings a more realistic tools for the mathematical modelling of the negotiation process and its results. In this paper we limit our attention to the fuzzy extension of the simple model of coalition games with side-payments, and in the frame of this model we study one of the elementary concepts of the coalition game theory, namely its “additivities”, i. e., superadditivity, subadditivity and additivity in the strict sense. In the deterministic game theory these additivites indicate the structure of eventual cooperation, namely the extent of finally formed coalitions, if the cooperation is possible. The additivities in fuzzy coalition games play an analogous role. But the vagueness of the input data about the expected coalitional incomes leads to consequently vague validity of the superadditivity, subadditivity and additivity. In this paper we formulate the model of this vagueness depending on the fuzzy quantities describing the expected coalitional pay-offs, and we introduce some elementary results mostly determining the links between additivities in a deterministic coalition game and its fuzzy extensions
Weighted Banzhaf power and interaction indexes through weighted approximations of games
The Banzhaf power index was introduced in cooperative game theory to measure
the real power of players in a game. The Banzhaf interaction index was then
proposed to measure the interaction degree inside coalitions of players. It was
shown that the power and interaction indexes can be obtained as solutions of a
standard least squares approximation problem for pseudo-Boolean functions.
Considering certain weighted versions of this approximation problem, we define
a class of weighted interaction indexes that generalize the Banzhaf interaction
index. We show that these indexes define a subclass of the family of
probabilistic interaction indexes and study their most important properties.
Finally, we give an interpretation of the Banzhaf and Shapley interaction
indexes as centers of mass of this subclass of interaction indexes
Values of Games for Information Decomposition
The information decomposition problem requires an additive decomposition of
the mutual information between the input and target variables into nonnegative
terms. The recently introduced solution to this problem, Information
Attribution, involves the Shapley-style value measuring the influence of
predictors in the coalitional game associated with the joint probability
distribution of the input random vector and the target variable. Motivated by
the original problem, we consider a general setting of coalitional games where
the players form a boolean algebra, and the coalitions are the corresponding
down-sets. This enables us to study in detail various single-valued solution
concepts, called values. Namely, we focus on the classes of values that can
represent very general alternatives to the solution of the information
decomposition problem, such as random-order values or sharing values. We extend
the axiomatic characterization of some classes of values that were known only
for the standard coalitional games
Approximations of solution concepts of cooperative games
The computation of a solution concept of a cooperative game usually depends
on values of all coalitions. However, in some applications, values of some of
the coalitions might be unknown due to various reasons. We introduce a method
to approximate standard solution concepts based only on partial information
given by a so called incomplete game. We demonstrate the ideas on the class of
minimal incomplete games. Approximations are derived for different solution
concepts including the Shapley value, the nucleolus, or the core. We show
explicit formulas for approximations of some of the solution concepts and show
how the approximability differs based on additional information about the game
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