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    K-balanced games and capacities

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    In this paper, we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of game. Based on the concept of k-additivity, we define to so-called k-balanced games and the corresponding generalization of core, the k-additive core, whose elements are not directly imputations but k-additive games. We show that any game is k-balanced for a suitable choice of k, so that the corresponding k-additive core is not empty. For the games in the k-additive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be k-balanced. For the general case, it is shown that any game is either balanced or 2-balanced. Finally, we treat the special case of capacities.Coopertaive games, k-additivity, balanced games, capacities, core.

    K-balanced games and capacities

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    URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/CESFramDP2008.htmClassification JEL : C7, D7.Documents de travail du Centre d'Economie de la Sorbonne 2008.79 - ISSN : 1955-611XIn this paper, we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of game. Based on the concept of k-additivity, we define to so-called k-balanced games and the corresponding generalization of core, the k-additive core, whose elements are not directly imputations but k-additive games. We show that any game is k-balanced for a suitable choice of k, so that the corresponding k-additive core is not empty. For the games in the k-additive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be k-balanced. For the general case, it is shown that any game is either balanced or 2-balanced. Finally, we treat the special case of capacities.Nous présentons une généralisation du concept de jeu balancé pour les jeux finis. En se basant sur la notion de jeu k-additif, nous définissons les jeux k-balancés, et la généralisation correspondante du coeur, que nous appelons le coeur k-additif, dont les éléments ne sont pas directement des imputations mais des jeux k-aditifs. Nous définissons également les jeux k-balancés monotones et le coeur k-additif monotone. Nous montrons que tout jeu est k-balancé pour une certaine valeur de k. Cette valeur vaut 2 pour les jeux, mais est en général différente pour le coeur k-additif monotone. Nous proposons une procédure de partage pour produire une imputation à partir d'un jeu k-additif contenu dans le coeur k-additif

    Cores of Cooperative Games in Information Theory

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    Cores of cooperative games are ubiquitous in information theory, and arise most frequently in the characterization of fundamental limits in various scenarios involving multiple users. Examples include classical settings in network information theory such as Slepian-Wolf source coding and multiple access channels, classical settings in statistics such as robust hypothesis testing, and new settings at the intersection of networking and statistics such as distributed estimation problems for sensor networks. Cooperative game theory allows one to understand aspects of all of these problems from a fresh and unifying perspective that treats users as players in a game, sometimes leading to new insights. At the heart of these analyses are fundamental dualities that have been long studied in the context of cooperative games; for information theoretic purposes, these are dualities between information inequalities on the one hand and properties of rate, capacity or other resource allocation regions on the other.Comment: 12 pages, published at http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/318704 in EURASIP Journal on Wireless Communications and Networking, Special Issue on "Theory and Applications in Multiuser/Multiterminal Communications", April 200

    Equilibrium in Labor Markets with Few Firms

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    We study competition between firms in labor markets, following a combinatorial model suggested by Kelso and Crawford [1982]. In this model, each firm is trying to recruit workers by offering a higher salary than its competitors, and its production function defines the utility generated from any actual set of recruited workers. We define two natural classes of production functions for firms, where the first one is based on additive capacities (weights), and the second on the influence of workers in a social network. We then analyze the existence of pure subgame perfect equilibrium (PSPE) in the labor market and its properties. While neither class holds the gross substitutes condition, we show that in both classes the existence of PSPE is guaranteed under certain restrictions, and in particular when there are only two competing firms. As a corollary, there exists a Walrasian equilibrium in a corresponding combinatorial auction, where bidders' valuation functions belong to these classes. While a PSPE may not exist when there are more than two firms, we perform an empirical study of equilibrium outcomes for the case of weight-based games with three firms, which extend our analytical results. We then show that stability can in some cases be extended to coalitional stability, and study the distribution of profit between firms and their workers in weight-based games

    Processing Games with Restricted Capacities

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    This paper analyzes processing problems and related cooperative games.In a processing problem there is a finite set of jobs, each requiring a specific amount of effort to be completed, whose costs depend linearly on their completion times.There are no restrictions whatsoever on the processing schedule.The main feature of the model is a capacity restriction, i.e., there is a maximum amount of effort per time unit available for handling jobs.Assigning to each job a player and letting each player have an individual capacity for handling jobs, each coalition of cooperating players in fact faces a processing problem with the coalitional capacity being the sum of the individual capacities of the members.The corresponding processing game summarizes the minimal joint costs for every coalition.It turns out that processing games are totally balanced.An explicit core element is constructed.games;capacity;scheduling;cooperation;allocation

    The core of bicapacities and bipolar games

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    Bicooperative games generalize classical cooperative games in the sense that a player is allowed to play in favor or against some aim, besides non participation. Bicapacities are monotonic bicooperative games, they are useful in decision making where underlying scales are of bipolar nature, i.e., they distinguish between good/satisfactory values and bad/unsatisfactory ones. We propose here a more general framework to represent such situations, called bipolar game. We study the problem of finding the core of such games, i.e., theset of additive dominating games.fuzzy measure, bicapacity, cooperative game, bipolar scale,core

    Processing Games with Shared Interest

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    A generalization of processing problems with restricted capacities is introduced.In a processing problem there is a finite set of jobs, each requiring a specific amount of effort to be completed, whose costs depend linearly on their completion times.The new aspect is that players have interest in all jobs. The corresponding cooperative game of this generalization is proved to be totally balanced.Processing games;scheduling;core allocation

    Optimal Budget Allocation in Social Networks: Quality or Seeding

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    In this paper, we study a strategic model of marketing and product consumption in social networks. We consider two competing firms in a market providing two substitutable products with preset qualities. Agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for the consumptions. At some point in time, firms receive a limited budget which they can use to trigger a larger consumption of their products in the network. Firms have to decide between marginally improving the quality of their products and giving free offers to a chosen set of agents in the network in order to better facilitate spreading their products. We derive a simple threshold rule for the optimal allocation of the budget and describe the resulting Nash equilibrium. It is shown that the optimal allocation of the budget depends on the entire distribution of centralities in the network, quality of products and the model parameters. In particular, we show that in a graph with a higher number of agents with centralities above a certain threshold, firms spend more budget on seeding in the optimal allocation. Furthermore, if seeding budget is nonzero for a balanced graph, it will also be nonzero for any other graph, and if seeding budget is zero for a star graph, it will be zero for any other graph too. We also show that firms allocate more budget to quality improvement when their qualities are close, in order to distance themselves from the rival firm. However, as the gap between qualities widens, competition in qualities becomes less effective and firms spend more budget on seeding.Comment: 7 page

    Cost sharing of cooperating queues in a Jackson network

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    We consider networks of queues in which the independent operators of individual queues may cooperate to reduce the amount of waiting. More specifically, we focus on Jackson networks in which the total capacity of the servers can be redistributed over all queues in any desired way. If we associate a cost to waiting that is linear in the queue lengths, it is known how the operators should share the available service capacity to minimize the long run total cost. We answer the question whether or not (the operators of) the individual queues will indeed cooperate in this way, and if so, how they will share the cost in the new situation. One of the results is an explicit cost allocation that is beneficial for all operators. The approach used also works for other cost functions, such as the server utilization
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