DEA production games with fuzzy output prices

In DEA production models the technology is assumed to be implicit in the input-output data given by a set of recorded observations. DEA production games assess the benefits to different firms of pooling their resources and sharing their technology. The crisp version of this type of problems has been studied in the literature and methods to obtain stable solutions have been proposed. However, no solution approach exists when there is uncertainty in the unit output prices, a situation that can clearly occur in practice. This paper extends DEA production games to the case of fuzzy unit output prices. In that scenario the total revenue is uncertain and therefore the corresponding allocation among the players is also necessarily uncertain. A core-like solution concept is introduced for these fuzzy games, the Preference Least Core. The computational burden of obtaining allocations of the fuzzy total profit reached through cooperation that belong to the preference least core is high. However, the results presented in the paper permit us to compute the fuzzy total revenue obtained by the grand coalition and a fuzzy allocation in the preference least core by solving a single linear programming model. The application of the proposed approach is illustrated with the analysis of two cooperative production situations originated by data sets from the literature.     

The fuzzy core and Shapley function for dynamic fuzzy games on matroids

In this paper, the generalized forms of the fuzzy core and the Shapley function for dynamic fuzzy games on matroids are given. An equivalent form of the fuzzy core is researched. In order to better understand the fuzzy core and the Shapley function for dynamic fuzzy games on matroids, we pay more attention to study three kinds of dynamic fuzzy games on matroids, which are named as fuzzy games with multilinear extension form, with proportional value and with Choquet integral form, respectively. Meantime, the relationship between the fuzzy core and the Shapley function for dynamic fuzzy games on matroids is researched, which coincides with the crisp case.     

On the non-emptiness of the fuzzy core

The seminal contribution of Debreu and Scarf (Int Econ Rev 4:235–246, 1963) connects the two concepts of core and competitive equilibrium in exchange economies. In effect, their core-equilibrium equivalence result states that, when the set of economic agents is replicated, the set of core allocations of the replica economy shrinks to the set of competitive allocations. Florenzano (J Math Anal Appl 153:18–36, 1990) defines the fuzzy core as the set of allocations which cannot be blocked by any coalition with an arbitrary rate of participation and then shows the asymptotic limit of cores of replica economies coincides with the fuzzy core. In this note, we provide an elementary proof of the non-emptiness of the fuzzy core for an exchange economy. Hence, in motivation, our result is similar to the contribution of Vohra (On Scarf’s theorem on the non-emptiness of the core: a direct proof through Kakutani’s fixed point theorem. Brown University Working Paper, 1987) and Shapley and Vohra (Econ Theory 1:108–116, 1991) for the core. Unlike the classical Debreu–Scarf limit theorem (Debreu and Scarf in Int Econ Rev 4:235–246, 1963) and its numerous extensions our result does not require any asymptotic intersection—or limit—of the set of core allocations of replica economies. The author would like to thank the Netherlands Organisation for Scientific Research (NWO) for financial support.     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

Fuzzy cores and fuzzy balancedness

We study the relation between the fuzzy core and balancedness for fuzzy games. For regular games, this relation has been studied by Bondareva (Problemy Kibernet 10:119–139, 1963) and Shapley (Naval Res Logist Q 14: 453–460, 1967). First, we gain insight in this relation when we analyse situations where the fuzzy game is continuous. Our main result shows that any fuzzy game has a non-empty core if and only if it is balanced. We also consider deposit games to illustrate the use of the main result.     

Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

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.     

Some results on cooperative interval games

Uncertainty is a daily presence in the real world. It affects our decision-making and may have influence on cooperation. On many occasions, uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e. payoffs lie in some intervals. A suitable game theoretic model to support decision-making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players’ payoffs. In this paper, the relations between some set-valued solution concepts using interval payoffs, namely the interval core, the interval dominance core, the square interval dominance core and the interval stable sets for cooperative interval games, are studied. It is shown that the interval core is the unique stable set on the class of convex interval games.     

Airport games: The core and its center

An approach to define a rule for an airport problem is to associate to each problem a cooperative game, an airport game, and using game theory to come out with a solution. In this paper, we study the rule that is the average of all the core allocations: the core-center (González-Díaz and Sánchez-Rodríguez, 2007). The structure of the core is exploited to derive insights on the core-center. First, we provide a decomposition of the core in terms of the cores of the downstream-subtraction reduced games. Then, we analyze the structure of the faces of the core of an airport game that correspond to the no-subsidy constraints to find that the faces of the core can be seen as new airport games, the face games, and that the core can be decomposed through the no-subsidy cones (those whose bases are the cores of the no-subsidy face games). As a consequence, we provide two methods for computing the core-center of an airport problem, both with interesting economic interpretations: one expresses the core-center as a ratio of the volume of the core of an airport game for which a player is cloned over the volume of the original core, the other defines a recursive algorithm to compute the core-center through the no-subsidy cones. Finally, we prove that the core-center is not only an intuitive appealing game-theoretic solution for the airport problem but it has also a good behavior with respect to the basic properties one expects an airport rule to satisfy. We examine some differences between the core-center and, arguably, the two more popular game theoretic solutions for airport problems: the Shapley value and the nucleolus.     

Convex fuzzy mapping with differentiability and its application in fuzzy optimization

The concepts of differentiability, convexity, generalized convexity and minimization of a fuzzy mapping are known in the literature. The purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using Buckley–Feuring approach for fuzzy differentiation and derive Karush–Kuhn–Tucker condition for the constrained fuzzy minimization problem.     

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.     

Capital allocation for portfolios with non-linear risk aggregation

Existing risk capital allocation methods, such as the Euler rule, work under the explicit assumption that portfolios are formed as linear combinations of random loss/profit variables, with the firm being able to choose the portfolio weights. This assumption is unrealistic in an insurance context, where arbitrary scaling of risks is generally not possible. Here, we model risks as being partially generated by Lévy processes, capturing the non-linear aggregation of risk. The model leads to non-homogeneous fuzzy games, for which the Euler rule is not applicable. For such games, we seek capital allocations that are in the core, that is, do not provide incentives for splitting portfolios. We show that the Euler rule of an auxiliary linearised fuzzy game (non-uniquely) satisfies the core property and, thus, provides a plausible and easily implemented capital allocation. In contrast, the Aumann–Shapley allocation does not generally belong to the core. For the non-homogeneous fuzzy games studied, Tasche’s (1999) criterion of suitability for performance measurement is adapted and it is shown that the proposed allocation method gives appropriate signals for improving the portfolio underwriting profit.     

Cooperative location games based on the minimum diameter spanning Steiner subgraph problem

In this paper we introduce and analyze new classes of cooperative games related to facility location models. The players are the customers (demand points) in the location problem and the characteristic value of a coalition is the cost of serving its members. Specifically, the cost in our games is the service diameter of the coalition.We study the existence of core allocations for these games, focusing on network spaces, i.e., finite metric spaces induced by undirected graphs and positive edge lengths.     

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.     

The Shapley function for fuzzy cooperative games with multilinear extension form

In this paper, the definition of the Shapley function for fuzzy cooperative games is given, which is obtained by extending the classical case. The specific expression of the Shapley function for fuzzy cooperative games with multilinear extension form is given, and its existence and uniqueness are discussed. Furthermore, the properties of the Shapley function are researched. Finally, the fuzzy core for this kind of game is defined, and the relationship between the fuzzy core and the Shapley function is shown.     

Resource location games

In this paper, we introduce and analyze resource location games. We show core non-emptiness by providing a set of intuitive core allocations, called Resource-Profit allocations. In addition, we present a sufficient condition for which the core and the set of Resource-Profit allocations coincide. Finally, we provide an example showing that when the sufficient condition is not satisfied, the coincidence is not guaranteed.     

Proper cores and dominance cores of fuzzy games

Many kinds of proper cores and dominance cores of fuzzy games are proposed in this paper. We also consider two similar concepts of payoff of a fuzzy coalition that are called the true payoff and quasi-payoff. The different concepts of proper cores and dominance cores will be proposed based on the true payoff and quasi-payoff of a fuzzy coalition. Some mild sufficient conditions are provided in this paper to guarantee the equalities of different proper cores and dominance cores.     

A unifying model for matrix-based pairing situations

We present a unifying framework for transferable utility coalitional games that are derived from a non-negative matrix in which every entry represents the value obtained by combining the corresponding row and column. We assume that every row and every column is associated with a player, and that every player is associated with at most one row and at most one column. The instances arising from this framework are called pairing games, and they encompass assignment games and permutation games as two polar cases. We show that the core of a pairing game is always non-empty by proving that the set of pairing games coincides with the set of permutation games. Then we exploit the wide range of situations comprised in our framework to investigate the relationship between pairing games that have different player sets, but are defined by the same underlying matrix. We show that the core and the set of extreme core allocations are immune to the merging of a row player with a column player. Moreover, the core is also immune to the reverse manipulation, i.e., to the splitting of a player into a row player and a column player. Other common solution concepts fail to be either merging-proof or splitting-proof in general.