Cores of non-atomic market games

We study the cores of non-atomic market games, a class of transferable utility cooperative games introduced by Aumann and Shapley (Values of non-atomic games, 1974), and, more in general, of those games that admit a na-continuous and concave extension to the set of ideal coalitions, studied by Einy et al. (Int J Game Theory 28:1–14, 1999). We show that the core of such games is norm compact and some related results. We also give a Multiple Priors interpretation of some of our findings.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Duality and optimality conditions for generalized equilibrium problems involving DC functions

We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667–678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.     

Efficiency and equilibria in games of optimal derivative design

In this paper, optimal derivative design when multiple firms compete for heterogenous customers is studied. Ties in the agents’ best responses generate discontinuous payoffs. Efficient tie-breaking rules are considered: In a first step, the model presented by Carlier et al. (Math Financ Econ 1:57–80, 2007) is extended, and results of Page and Monteiro (J Math Econ 39:63–109, 2003, J Econ Theory 134:566–575, 2007, Econ Theory 34:503–524, 2008) are used to prove the existence of (mixed-strategies) Nash equilibria. In a second step, the case of risk minimizing firms is studied. Socially efficient allocations are introduced, and their existence is proved. In particular, the entropic risk measure is considered.     

Dynamic realization games in newsvendor inventory centralization

Consider a set N of n (> 1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et al. 2000, Games Econ Behav 31:26–49; Muller et al. 2002, Games Econ Behav 38:118–126). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror 2005, IIE Trans Scheduling Logistics 37:93–107). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a, Int J Game Theor 31:341–351). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer’s rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players’ demands at different stages.     

Deconvolution: a wavelet frame approach

This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b) as wavelet frame based high resolution image reconstruction methods. We first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation, which completes the formulation given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). This formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties. These missing coefficients are recovered iteratively together with a built-in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating. This approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). However, an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers. This paper is to establish the theoretical foundation of this wavelet frame approach. In particular, a proof of convergence, an analysis of the stability of algorithms and a study of the minimization property of solutions are given.     

Equivalence theorem,consistency and axiomatizations of a multi-choice value

This paper is devoted to the study of solutions for multi-choice games which admit a potential, such as the potential associated with the extended Shapley value proposed by Hsiao and Raghavan (Int J Game Theory 21:301–302, 1992; Games Econ Behav 5:240–256, 1993). Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the extended Shapley value to an appropriately modified game. In the framework of multi-choice games, we also provide an extension of the reduced game introduced by Hart and Mas-Colell (Econometrica 57:589–614, 1989). Different from the works of Hsiao and Raghavan (1992, 1993), we provide two types of axiomatizations, one is the analogue of Myerson’s (Int J Game Theory 9:169–182, 1980) axiomatization of the Shapley value based on the property of balanced contributions. The other axiomatization is obtained by means of the property of consistency.     

The gamma-core and coalition formation

This paper reinterprets the γ-core (Chander and Tulkens in Int Tax Pub Financ 2:279–293, 1995; in Int J Game Theory 26:379–401, 1997) and justifies it as well as its prediction that the efficient coalition structure is stable in terms of the coalition formation theory. The problem of coalition formation is formulated as an infinitely repeated game in which the players must choose whether to cooperate or not. It is shown that a certain equilibrium of this game corresponds to the γ-core assumption that when a coalition forms the remaining players form singletons, and that the grand coalition is an equilibrium coalition structure. An earlier version of this paper was presented at the conference on Game Theory and Its Applications held in Mumbai in 2003 and was subsequently circulated as CORE Discussion Paper 2003/46.     

An NTU value under complement reduced game

The paper follows Kalai and Samet’s (Econometrica 53:307–327, 1985) construction to define a possible extension of the equal allocation of nonseparable costs (EANSC) to games without transferable utilities. We offer a characterization of the EANSC based on the axiom of independence of irrelevant monotonic transformation [Chang and Hwang (Int J Game Theory 32:253–271, 2003) introduced this axiom, and they named it independence of irrelevant expansions. The referee points out that the terminology is inappropriate. So, we change the name.] introduced by Chang and Hwang (Int J Game Theory 32:253–271, 2003) as well as on several familiar axioms. Specifically, we show that the EANSC is the only solution to satisfy Pareto optimality, translation covariance, anonymity, TU-bilateral consistency (or TU-converse consistency), aggregate monotonicity, and independence of irrelevant monotonic transformation.     

No end of the world in 2012 for 4OR

The purpose of this note is to give the readers of 4OR information on the state of the journal and its future, abiding by a triennial tradition which started with Bouyssou et al. (4OR 1(1):1–6, 2003) and continued with Bouyssou et al. (4OR 4(1):1–9, 2006) and Bouyssou et al. (4OR Q J Oper Res 7(1):15, 2009). In the 3 years that have passed since the last editorial note, three volumes of the journal have been published, each containing four issues: vol. 7 (2009), vol. 8 (2010) and vol. 9 (2011).     

Random marginal and random removal values

We propose two variations of the non-cooperative bargaining model for games in coalitional form, introduced by Hart and Mas-Colell (Econometrica 64:357–380, 1996a). These strategic games implement, in the limit, two new NTU-values: the random marginal and the random removal values. Their main characteristic is that they always select a unique payoff allocation in NTU-games. The random marginal value coincides with the Consistent NTU-value (Maschler and Owen in Int J Game Theory 18:389–407, 1989) for hyperplane games, and with the Shapley value for TU games (Shapley in In: Contributions to the theory of Games II. Princeton University Press, Princeton, pp 307–317, 1953). The random removal value coincides with the solidarity value (Nowak and Radzik in Int J Game Theory 23:43–48, 1994) in TU-games. In large games we show that, in the special class of market games, the random marginal value coincides with the Shapley NTU-value (Shapley in In: La Décision. Editions du CNRS, Paris, 1969), and that the random removal value coincides with the equal split value.      

A Bilinear Pseudodifferential Calculus

In this paper we construct a new class of bilinear pseudodifferential operators which contains both the Coifman-Meyer class as well as the non-translation invariant class closely related both to the bilinear Hilbert transform and previously studied in Bényi et al. (J. Geom. Anal. 16(3):431–453, 2006), Bényi et al. (J. Anal. Math., 2009), Bernicot (Anal. PDE 1:1–27, 2008) as well as the bilinear Marcinkiewicz class studied in Grafakos and Kalton (Stud. Math. 146(2):115–156, 2001). We prove boundedness on Sobolev spaces for these operators as well as establish a symbolic calculus that exhibits the nice behavior of our new class under transposition and composition with linear operators.     

Counterexamples to some triality and tri-duality results

The aim of this paper is to show that the results on triality and tri-duality in Gao (J Glob Optim 17:127–160, 2000; J Glob Optim 29:377–399, 2004; J Glob Optim 35:131–143, 2006; Encyclopedia of optimization, 2nd edn. Springer, New York, pp 822–828, 2009) and Gao et al. (J Glob Optim 45:473–497, 2009) are false. To prove this we provide simple counterexamples.     

Feynman’s Operational Calculi: Disentangling Away from the Origin

In this paper we develop a method of forming functions of noncommuting operators (or disentangling) using functions that are not necessarily analytic at the origin in ℂ ⁿ . The method of disentangling follows Feynman’s heuristic rules from in (Feynman in Phys. Rev. 84:18–128, 1951) a mathematically rigorous fashion, generalizing the work of Jefferies and Johnson and the present author in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001). In fact, the work in (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) allow only functions analytic in a polydisk centered at the origin in ℂ ⁿ while the method introduced in this paper enable functions that are not analytic at the origin to be used. It is shown that the disentangling formalism introduced here reduces to that of (Jefferies and Johnson in Russ. J. Math. 8:153–181, 2001) and (Jefferies et al. in J. Korean Math. Soc. 38:193–226, 2001) under the appropriate assumptions. A basic commutativity theorem is also established.     

An Index Theorem for Band-Dominated Operators with Slowly Oscillating Coefficients (after Deundyak and Shteinberg)

We provide a proof of an index theorem for band-dominated operators with slowly oscillating coefficients. The statement is essentially the same as the main result of the announcement of Deundyak and Shteinberg (Funct Anal Appl 19(4):321–323, 1985), but our methods are very different from those hinted at there. The index theorem we prove can also be seen as a partial generalization to higher dimensions of the main result of the article of Rabinovich et al. (Integr Equ Oper Theory 49:221–238, 2004).     

Fuzzy identification problem for continuous extremal fuzzy dynamic system

This work deals with the problems of the Continuous Extremal Fuzzy Dynamic System (CEFDS) optimization and briefly discusses the results developed by Sirbiladze (Int J Gen Syst 34(2):107–138, 2005a; 34(2):139–167, 2005b; 34(2):169–198, 2005c; 35(4):435–459, 2006a; 35(5):529–554, 2006b; 36(1): 19–58, 2007; New Math Nat Comput 4(1):41–60, 2008a; Mat Zametki, 83(3):439–460, 2008b). The basic properties of extended extremal fuzzy measures and Sugeno’s type integrals are considered and several variants of their representation are given. Values of extended extremal conditional fuzzy measures are defined as a levels of expert knowledge reflections of CEFDS states in the fuzzy time intervals. The notions of extremal fuzzy time moments and intervals are introduced and their monotone algebraic structures that form the most important part of the fuzzy instrument of modeling extremal fuzzy dynamic systems are discussed. A new approach in modeling of CEFDS is developed. Applying the results of Sirbiladze (Int J Gen Syst 34(2) 107–138, 2005a; 34(2):139–167, 2005b), fuzzy processes with possibilistic uncertainty, the source of which are expert knowledge reflections on the states on CEFDS in extremal fuzzy time intervals, are constructed (Sirbiladze in Int J Gen Syst 34(2):169–198, 2005c). The dynamics of CEFDS’s is described. Questions of the ergodicity of CEFDS are considered. A fuzzy-integral representation of a continuous extremal fuzzy process is given. Based on the fuzzy-integral model, a method and an algorithm are developed for identifying the transition operator of CEFDS. The CEFDS transition operator is restored by means of expert data with possibilistic uncertainty, the source of which is expert knowledge reflections on the states of CEFDS in the extremal fuzzy time intervals. The regularization condition for obtaining quasi-optimal estimator of the transition operator is represented by the theorems. The corresponding calculating algorithm is provided. The results obtained are illustrated by an example in the case of a finite set of CEFDS states.     

Some new common fixed point theorems in fuzzy metric spaces

The purpose of this paper is to prove some new common fixed point theorems in (GV)-fuzzy metric spaces. While proving our results, we utilize the idea of compatibility due to Jungck (Int J Math Math Sci 9:771–779, 1986) together with subsequentially continuity due to Bouhadjera and Godet-Thobie (arXiv: 0906.3159v1 [math.FA] 17 Jun 2009) respectively (also alternately reciprocal continuity due to Pant (Bull Calcutta Math Soc 90:281–286, 1998) together with subcompatibility due to Bouhadjera and Godet-Thobie (arXiv:0906.3159v1 [math.FA] 17 Jun 2009) as patterned in Imdad et al. (doi:) wherein conditions on completeness (or closedness) of the underlying space (or subspaces) together with conditions on continuity in respect of any one of the involved maps are relaxed. Our results substantially generalize and improve a multitude of relevant common fixed point theorems of the existing literature in metric as well as fuzzy metric spaces which include some relevant results due to Imdad et al. (J Appl Math Inform 26:591–603, 2008), Mihet (doi:), Mishra (Tamkang J Math 39(4):309–316, 2008), Singh (Fuzzy Sets Syst 115:471–475, 2000) and several others.     

Crisp and fuzzy <Emphasis Type="Italic">k</Emphasis>-means clustering algorithms for multivariate functional data

Functional data analysis, as proposed by Ramsay (Psychometrika 47:379–396, 1982), has recently attracted many researchers. The most popular approach taken in recent studies of functional data has been the extension of statistical methods for the analysis of usual data to that of functional data (e.g., Ramsay and Silverman in Functional data Analysis Springer, Berlin Heidelberg New York, 1997, Applied functional data analysis: methods and case studies. Springer, Berlin Heidelberg New York, 2002; Mizuta in Proceedings of the tenth Japan and Korea Joint Conference of Statistics, pp 77–82, 2000; Shimokawa et al. in Japan J Appl Stat 29:27–39, 2000). In addition, several methods for clustering functional data have been proposed (Abraham et al. in Scand J Stat 30:581–595, 2003; Gareth and Catherine in J Am Stat Assoc 98:397–408, 2003; Tarpey and kinateder in J Classif 20:93–114, 2003; Rossi et al. in Proceedings of European Symposium on Artificial Neural Networks pp 305–312, 2004). Furthermore, Tokushige et al. (J Jpn Soc Comput Stat 15:319–326, 2002) defined several dissimilarities between functions for the case of functional data. In this paper, we extend existing crisp and fuzzy k-means clustering algorithms to the analysis of multivariate functional data. In particular, we consider the dissimilarity between functions as a function. Furthermore, cluster centers and memberships, which are defined as functions, are determined at the minimum of a certain target function by using a calculus-of-variations approach.     

A characterization of convex TU games by means of the Mas-Colell bargaining set (à la Shimomura)

Within the class of zero-monotonic games, we prove that a cooperative game with transferable utility is convex if and only if the core of the game coincides with the Mas-Colell bargaining set (à la Shimomura, in Int J Game Theory 26:283–302, 1997).     

Processing second-order stochastic dominance models using cutting-plane representations

Second-order stochastic dominance (SSD) is widely recognised as an important decision criterion in portfolio selection. Unfortunately, stochastic dominance models are known to be very demanding from a computational point of view. In this paper we consider two classes of models which use SSD as a choice criterion. The first, proposed by Dentcheva and Ruszczyński (J Bank Finance 30:433–451, 2006), uses a SSD constraint, which can be expressed as integrated chance constraints (ICCs). The second, proposed by Roman et al. (Math Program, Ser B 108:541–569, 2006) uses SSD through a multi-objective formulation with CVaR objectives. Cutting plane representations and algorithms were proposed by Klein Haneveld and Van der Vlerk (Comput Manage Sci 3:245–269, 2006) for ICCs, and by Künzi-Bay and Mayer (Comput Manage Sci 3:3–27, 2006) for CVaR minimization. These concepts are taken into consideration to propose representations and solution methods for the above class of SSD based models. We describe a cutting plane based solution algorithm and outline implementation details. A computational study is presented, which demonstrates the effectiveness and the scale-up properties of the solution algorithm, as applied to the SSD model of Roman et al. (Math Program, Ser B 108:541–569, 2006).