Assessment of groups in a network organization based on the Shapley group value.

The focus of this paper is the assessment of groups of agents or units in a network organization. Given a social network, the relations between agents are modeled by means of a graph, and its functionality will be codified by means of a cooperative game. Building on previous work of Gomez et al. (2003) for the individual case, we propose a Myerson group value to evaluate the ability of each group of agents inside the social network to achieve the organization's goals. We analyze this centrality measure, and in particular we offer several decompositions that facilitate obtaining a precise interpretation of it.

Chemically tagging the Hyades Supercluster. A homogeneous sample of F6-K4 kinematically selected northern stars

Stellar kinematic groups are kinematical coherent groups of stars that might have a common origin. These groups are dispersed throughout the Galaxy over time by the tidal effects of both Galactic rotation and disc heating, although their chemical content remains unchanged. The aim of chemical tagging is to establish that the abundances of every element in the analysis are homogeneus among the members. We study the case of the Hyades Supercluster to compile a reliable list of members (FGK stars) based on our chemical tagging analysis. For a total of 61 stars from the Hyades Supercluster, stellar atmospheric parameters (T_eff, log g, ξ, and [Fe/H]) are determined using our code called StePar, which is based on the sensitivity to the stellar atmospheric parameters of the iron EWs measured in the spectra. We derive the chemical abundances of 20 elements and find that their [X/Fe] ratios are consistent with Galactic abundance trends reported in previous studies. The chemical tagging method is applied with a carefully developed differential abundance analysis of each candidate member of the Hyades Supercluster, using a well-known member of the Hyades cluster as a reference (vB 153). We find that only 28 stars (26 dwarfs and 2 giants) are members, i.e. that 46% of our candidates are members based on the differential abundance analysis. This result confirms that the Hyades Supercluster cannot originate solely from the Hyades cluster.

The scale of the universe, by Harlow Shapley ... and Heber D. Curtis.

Mode of access: Internet.

The Shapley value for shortest path games

In this paper shortest path games are considered. The transportation of a good in a network has costs and benet too. The problem is to divide the prot of the transportation among the players. Fragnelli et al (2000) introduce the class of shortest path games, which coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further four characterizations of the Shapley value (Shapley (1953)'s, Young (1985)'s, Chun (1989)'s, and van den Brink (2001)'s axiomatizations), and conclude that all the mentioned axiomatizations are valid for shortest path games. Fragnelli et al (2000)'s axioms are based on the graph behind the problem, in this paper we do not consider graph specic axioms, we take TU axioms only, that is, we consider all shortest path problems and we take the view of abstract decision maker who focuses rather on the abstract problem than on the concrete situations.

The Shapley value for airport and irrigation games

In this paper cost sharing problems are considered. We focus on problems given by rooted trees, we call these problems cost-tree problems, and on the induced transferable utility cooperative games, called irrigation games. A formal notion of irrigation games is introduced, and the characterization of the class of these games is provided. The well-known class of airport games Littlechild and Thompson (1977) is a subclass of irrigation games. The Shapley value Shapley (1953) is probably the most popular solution concept for transferable utility cooperative games. Dubey (1982) and Moulin and Shenker (1992) show respectively, that Shapley's Shapley (1953) and Young (1985)'s axiomatizations of the Shapley value are valid on the class of airport games. In this paper we show that Dubey (1982)'s and Moulin and Shenker (1992)'s results can be proved by applying Shapley (1953)'s and Young (1985)'s proofs, that is those results are direct consequences of Shapley (1953)'s and Young (1985)'s results. Furthermore, we extend Dubey (1982)'s and Moulin and Shenker (1992)'s results to the class of irrigation games, that is we provide two characterizations of the Shapley value for cost sharing problems given by rooted trees. We also note that for irrigation games the Shapley value is always stable, that is it is always in the core Gillies (1959).

The Shapley value for shortest path games

In this paper shortest path games are considered. The transportation of a good in a network has costs and benet too. The problem is to divide the prot of the transportation among the players. Fragnelli et al (2000) introduce the class of shortest path games, which coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further four characterizations of the Shapley value (Shapley (1953)'s, Young (1985)'s, Chun (1989)'s, and van den Brink (2001)'s axiomatizations), and conclude that all the mentioned axiomatizations are valid for shortest path games. Fragnelli et al (2000)'s axioms are based on the graph behind the problem, in this paper we do not consider graph specic axioms, we take TU axioms only, that is, we consider all shortest path problems and we take the view of abstract decision maker who focuses rather on the abstract problem than on the concrete situations.

On axiomatizations of the Shapley value for assignment games

We consider the problem of axiomatizing the Shapley value on the class of assignment games. We first show that several axiomatizations of the Shapley value on the class of all TU-games do not characterize this solution on the class of assignment games by providing alternative solutions that satisfy these axioms. However, when considering an assignment game as a communication graph game where the game is simply the assignment game and the graph is a corresponding bipartite graph buyers are connected with sellers only, we show that Myerson's component efficiency and fairness axioms do characterize the Shapley value on the class of assignment games. Moreover, these two axioms have a natural interpretation for assignment games. Component efficiency yields submarket efficiency stating that the sum of the payoffs of all players in a submarket equals the worth of that submarket, where a submarket is a set of buyers and sellers such that all buyers in this set have zero valuation for the goods offered by the sellers outside the set, and all buyers outside the set have zero valuations for the goods offered by sellers inside the set. Fairness of the graph game solution boils down to valuation fairness stating that only changing the valuation of one particular buyer for the good offered by a particular seller changes the payoffs of this buyer and seller by the same amount.

Young's axiomatization of the Shapley value - a new proof

We give a new proof of Young's characterization of the Shapley value. Moreover, as applications of the new proof, we show that Young's axiomatization of the Shapley value is valid on various well-known subclasses of TU games.

Pattern clustering using cooperative game theory

In this paper, we approach the classical problem of clustering using solution concepts from cooperative game theory such as Nucleolus and Shapley value. We formulate the problem of clustering as a characteristic form game and develop a novel algorithm DRAC (Density-Restricted Agglomerative Clustering) for clustering. With extensive experimentation on standard data sets, we compare the performance of DRAC with that of well known algorithms. We show an interesting result that four prominent solution concepts, Nucleolus, Shapley value, Gately point and \tau-value coincide for the defined characteristic form game. This vindicates the choice of the characteristic function of the clustering game and also provides strong intuitive foundation for our approach.

Novel Biobjective Clustering (BiGC) Based on Cooperative Game Theory

We propose a new approach to clustering. Our idea is to map cluster formation to coalition formation in cooperative games, and to use the Shapley value of the patterns to identify clusters and cluster representatives. We show that the underlying game is convex and this leads to an efficient biobjective clustering algorithm that we call BiGC. The algorithm yields high-quality clustering with respect to average point-to-center distance (potential) as well as average intracluster point-to-point distance (scatter). We demonstrate the superiority of BiGC over state-of-the-art clustering algorithms (including the center based and the multiobjective techniques) through a detailed experimentation using standard cluster validity criteria on several benchmark data sets. We also show that BiGC satisfies key clustering properties such as order independence, scale invariance, and richness.

Egalitarian distributions in coalitional models: The Lorenz criterion

The paper presents a framework where the most important single-valued solutions in the literature of TU games are jointly analyzed. The paper also suggests that similar frameworks may be useful for other coalitional models.

The coincidence of the kernel and nucleolus of a convex game: an alternative proof

In 1972, Maschler, Peleg and Shapley proved that in the class of convex the nucleolus and the kernel coincide. The only aim of this note is to provide a shorter, alternative proof of this result.

On the impact of independence of irrelevant alternatives: the case of two-person NTU games

On several classes of n-person NTU games that have at least one Shapley NTU value, Aumann characterized this solution by six axioms: Non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives (IIA). Each of the first five axioms is logically independent of the remaining axioms, and the logical independence of IIA is an open problem. We show that for n = 2 the first five axioms already characterize the Shapley NTU value, provided that the class of games is not further restricted. Moreover, we present an example of a solution that satisfies the first five axioms and violates IIA for two-person NTU games (N, V) with uniformly p-smooth V(N).

Characterizing distribution rules for cost sharing games

In noncooperative cost sharing games, individually strategic agents choose resources based on how the welfare (cost or revenue) generated at each resource (which depends on the set of agents that choose the resource) is distributed. The focus is on finding distribution rules that lead to stable allocations, which is formalized by the concept of Nash equilibrium, e.g., Shapley value (budget-balanced) and marginal contribution (not budget-balanced) rules.

Recent work that seeks to characterize the space of all such rules shows that the only budget-balanced distribution rules that guarantee equilibrium existence in all welfare sharing games are generalized weighted Shapley values (GWSVs), by exhibiting a specific 'worst-case' welfare function which requires that GWSV rules be used. Our work provides an exact characterization of the space of distribution rules (not necessarily budget-balanced) for any specific local welfare functions remains, for a general class of scalable and separable games with well-known applications, e.g., facility location, routing, network formation, and coverage games.

We show that all games conditioned on any fixed local welfare functions possess an equilibrium if and only if the distribution rules are equivalent to GWSV rules on some 'ground' welfare functions. Therefore, it is neither the existence of some worst-case welfare function, nor the restriction of budget-balance, which limits the design to GWSVs. Also, in order to guarantee equilibrium existence, it is necessary to work within the class of potential games, since GWSVs result in (weighted) potential games.

We also provide an alternative characterization—all games conditioned on any fixed local welfare functions possess an equilibrium if and only if the distribution rules are equivalent to generalized weighted marginal contribution (GWMC) rules on some 'ground' welfare functions. This result is due to a deeper fundamental connection between Shapley values and marginal contributions that our proofs expose—they are equivalent given a transformation connecting their ground welfare functions. (This connection leads to novel closed-form expressions for the GWSV potential function.) Since GWMCs are more tractable than GWSVs, a designer can tradeoff budget-balance with computational tractability in deciding which rule to implement.

Coalition and distribution in fuzzy dynamic supply chain systems

In this paper, two models of coalition and income's distribution in FSCS (fuzzy supply chain systems) are proposed based on the fuzzy set theory and fuzzy cooperative game theory. The fuzzy dynamic coalition choice's recursive equations are constructed in terms of sup-t composition of fuzzy relations, where t is a triangular norm. The existence of the fuzzy relations in FSCS is also proved. On the other hand, the approaches to ascertain the fuzzy coalition through the choice's recursive equations and distribute the fuzzy income in FSCS by the fuzzy Shapley values are also given. These models are discussed in two parts: the fuzzy dynamic coalition choice of different units in FSCS; the fuzzy income's distribution model among different participators in the same coalition. Furthermore, numerical examples are given aiming at illustrating these models., and the results show that these models are feasible and validity in FSCS.