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

Lattices and discrete methods in cooperative games and decisions

Questa tesi si pone l'obiettivo di presentare la teoria dei giochi, in particolare di quelli cooperativi, insieme alla teoria delle decisioni, inquadrandole formalmente in termini di matematica discreta. Si tratta di due campi dove l'indagine si origina idealmente da questioni applicative, e dove tuttavia sono sorti e sorgono problemi più tipicamente teorici che hanno interessato e interessano gli ambienti matematico e informatico. Anche se i contributi iniziali sono stati spesso formulati in ambito continuo e utilizzando strumenti tipici di teoria della misura, tuttavia oggi la scelta di modelli e metodi discreti appare la più idonea. L'idea generale è quindi quella di guardare fin da subito al complesso dei modelli e dei risultati che si intendono presentare attraverso la lente della teoria dei reticoli. Ciò consente di avere una visione globale più nitida e di riuscire agilmente ad intrecciare il discorso considerando congiuntamente la teoria dei giochi e quella delle decisioni. Quindi, dopo avere introdotto gli strumenti necessari, si considerano modelli e problemi con il fine preciso di analizzare dapprima risultati storici e solidi, proseguendo poi verso situazioni più recenti, più complesse e nelle quali i risultati raggiunti possono suscitare perplessità. Da ultimo, vengono presentate alcune questioni aperte ed associati spunti per la ricerca

Collaborative Models for Supply Networks Coordination and Healthcare Consolidation

This work discusses the collaboration framework among different members of two complex systems: supply networks and consolidated healthcare systems. Although existing literature advocates the notion of strategic partnership/cooperation in both supply networks and healthcare systems, there is a dearth of studies quantitatively analyzing the scope of cooperation among the members and its benefit on the global performance. Hence, the first part of this dissertation discusses about two-echelon supply networks and studies the coordination of buyers and suppliers for multi-period procurement process. Viewing the issue from the same angel, the second part studies the coordination framework of hospitals for consolidated healthcare service delivery. Realizing the dynamic nature of information flow and the conflicting objectives of members in supply networks, a two-tier coordination mechanism among buyers and suppliers is modeled. The process begins with the intelligent matching of buyers and suppliers based on the similarity of users profiles. Then, a coordination mechanism for long-term agreements among buyers and suppliers is proposed. The proposed mechanism introduces the importance of strategic buyers for suppliers in modeling and decision making process. To enhance the network utilization, we examine a further collaboration among suppliers where cooperation incurs both cost and benefit. Coalitional game theory is utilized to model suppliers\u27 coalition formation. The efficiency of the proposed approaches is evaluated through simulation studies. We then revisit the common issue, the co-existence of partnership and conflict objectives of members, for consolidated healthcare systems and study the coordination of hospitals such that there is a central referral system to facilitate patients transfer. We consider three main players including physicians, hospitals managers, and the referral system. As a consequence, the interaction within these players will shape the coordinating scheme to improve the overall system performance. To come up with the incentive scheme for physicians and aligning hospitals activities, we define a multi-objective mathematical model and obtain optimal transfer pattern. Using optimal solutions as a baseline, a cooperative game between physicians and the central referral system is defined to coordinate decisions toward system optimality. The efficiency of the proposed approach is examined via a case study

Global Coalitional Games

Global coalitional games are TU cooperative games intended to model situations where the worth of coalitions varies across different partitions of the players. Formally, they are real-valued functions whose domain is the direct product of the subset lattice and the lattice of partitions of a finite player set. Therefore, the dimension of the associated vector space grows dramatically fast with the cardinality of the player set, inducing flexibility as well as complexity. Accordingly, some reasonable restrictions that reduce such a dimension are considered. The solution concepts associated with the Shapley value and the core are studied for the general (i.e., unrestricted) case.lattice, lattice function, coalition, partition, Shapley value, core