388 research outputs found
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
Approximations of Lovasz extensions and their induced interaction index
The Lovasz extension of a pseudo-Boolean function is
defined on each simplex of the standard triangulation of as the
unique affine function that interpolates at the
vertices of the simplex. Its degree is that of the unique multilinear
polynomial that expresses . In this paper we investigate the least squares
approximation problem of an arbitrary Lovasz extension by Lovasz
extensions of (at most) a specified degree. We derive explicit expressions of
these approximations. The corresponding approximation problem for
pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then
solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an
alternative definition of Banzhaf interaction index. Similarly we introduce a
new interaction index from approximations of and we present some of
its properties. It turns out that its corresponding power index identifies with
the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page
Measuring the interactions among variables of functions over the unit hypercube
By considering a least squares approximation of a given square integrable
function by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of . This definition extends the concept of Banzhaf interaction
index introduced in cooperative game theory. Our approach is partly inspired
from multilinear regression analysis, where interactions among the independent
variables are taken into consideration. We show that this interaction index has
appealing properties which naturally generalize the properties of the Banzhaf
interaction index. In particular, we interpret this index as an expected value
of the difference quotients of or, under certain natural conditions on ,
as an expected value of the derivatives of . These interpretations show a
strong analogy between the introduced interaction index and the overall
importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a
few applications of the interaction index
On the relationship between the Crescent Method and SUOWA operators
Producción CientíficaDifferent families of functions have been proposed in the literature with the purpose of simultaneously generalizing weighted means and OWA operators (see, for instance, WOWA and SUOWA operators). Recently, Jin, Mesiar, and Yager [L. Jin, R. Mesiar, and R. R. Yager, Melting probability measure with OWA operator to generate fuzzy measure: the Crescent Method, IEEE Transactions on Fuzzy Systems, in press] have introduced in the literature a new procedure called The Crescent Method to melt additive capacities with those of OWA operators. The main aim of this paper is to establish a relationship between the Crescent Method and SUOWA operators. From this relationship we give closed-form expressions for some well-known indices such
as the Shapley values, the veto and favor indices, and the k-conjunctiveness and k-disjunctiveness indices.Este trabajo forma parte del proyecto de investigación: MEC-FEDER Grant ECO2016-77900-P
The Consistent Value: Potentializability and Equivalent Relations on Fuzzy Games
Based on several equivalent relations, this paper is aimed to axiomatize the family of all solutions that admit a potential in the framework of fuzzy transferable-utility (TU) games. Further, several axiomatic results of the consistent value are also proposed by applying these equivalent relations
Symmetric approximations of pseudo-Boolean functions with applications to influence indexes
We introduce an index for measuring the influence of the k-th smallest
variable on a pseudo-Boolean function. This index is defined from a weighted
least squares approximation of the function by linear combinations of order
statistic functions. We give explicit expressions for both the index and the
approximation and discuss some properties of the index. Finally, we show that
this index subsumes the concept of system signature in engineering reliability
and that of cardinality index in decision making
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
Generalizations of weighted means and OWA operators by using unimodal weighting vectors
Producción CientíficaWeighted means and OWA operators are two families of functions well known in the literature. Given that both are specific cases of the Choquet integral, several procedures for constructing capacities that generalize simultaneously those of the
weighted means and the OWA operators have been suggested in recent years. In this paper we propose two methods that allow us to address the previous issue and that provide us with a wide variety of capacities when the weighting vector associated with
the OWA operator is unimodal.Este trabajo forma parte del proyecto de investigación: MEC-FEDER Grant ECO2016-77900-P
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
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