54 research outputs found
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
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
The Nucleolus, the Kernel, and the Bargaining Set: An Update
One of David Schmeidler’s many important contributions in his distinguished career was the introduction of the nucleolus, one of the central single-valued solution concepts in cooperative game theory. This paper is an updated survey on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. As a first approach to these concepts, we refer the reader to the great survey by Maschler (1992); see also the relevant chapters in Peleg and Sudholter (2003). Building on the notes of four lectures on the nucleolus and the kernel delivered by one of the authors at the Hebrew University of
Jerusalem in 1999, we have updated Maschler’s survey by adding more recent contributions to the literature. Following a similar structure, we have also added a new section that covers the bargaining set.
The nucleolus has a number of desirable properties, including nonemptiness, uniqueness, core selection, and consistency. The first way to understand it is based on an egalitarian principle among coalitions. However, by going over the axioms that characterize it, what comes across as important is its connection with coalitional stability, as formalized in the notion of the core. Indeed, if one likes a single-valued version of core stability that always yields a prediction, one should consider the nucleolus as a recommendation. The kernel, which contains the nucleolus, is based on the idea of “bilateral equilibrium” for every pair of players. And the bargaining set, which contains the
kernel, checks for the credibility of objections coming from coalitions. In this paper, section 2 presents preliminaries, section 3 is devoted to the nucleolus, section 4 to the kernel, and section 5 to the bargaining set.Iñarra acknowledges research support from the Spanish Government grant ECO2015-67519-P, and
Shimomura from Grant-in-Aid for Scientific Research (A)18H03641 and (C)19K01558
Game Theory: The Language of Social Science?
The present paper tries in a largely non-technical way to discuss the aim, the basic notions and methods as well as the limits of game theory under the aspect of providing a general modelling method or language for social sciences.
Compromise in cooperative game and the VIKOR method
Five approaches in conflict resolution are distinguished, based on cooperativeness and aggressiveness in resolving conflict. Compromise based on cooperativeness is emphasized here as a solution in conflict resolution. Cooperative game theory oriented towards aiding the conflict resolution is considered and the compromise value for TU(transferable utility)-game is presented. The method VIKOR could be applied to determine compromise solution of a multicriteria decision making problem with noncommensurable and conflicting criteria. Compromise is considered as an intermediate state between conflicting objectives or criteria reached by mutual concession. The applicability of the cooperative game theory and the VIKOR method for conflict resolution is illustrated
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
Contributions to Game Theory and Management. Vol. III. Collected papers presented on the Third International Conference Game Theory and Management.
The collection contains papers accepted for the Third International Conference Game Theory and Management (June 24-26, 2009, St. Petersburg University, St. Petersburg, Russia). The presented papers belong to the field of game theory and its applications to management. The volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments.
- …