Ann Oper Res DOI 10.1007/s10479-014-1764-6
Editorial: Contributions to game theory and social choice Jean-François Caulier · Michel Grabisch · Agnieszka Rusinowska
© Springer Science+Business Media New York 2014
Game Theory is a mathematical framework dealing with models of conflict and cooperation among interdependent decision makers or agents. It is a central tool for economics and the social sciences, which poses challenging research questions in mathematics, control and optimization, and is applied across a wide variety of fields, including neuroscience, philosophy, and biology. SING7 was the seventh in the series of “Spain–Italy–Netherlands Meetings on Game Theory” and the first one organized in France. It was held in Paris from 18th till 20th of July 2011. This special volume on game theory is dedicated to this SING7 conference. However, the submission invitation was not restricted only to SING7 presenters, but was addressed to all researchers and practitioners from the related disciplines. All submissions followed the high standard refereeing process of the Annals of Operations Research. Ten out of nineteen submitted papers have finally been accepted for publication. They show the variety of research topics of interest to SING participants. Following the traditional concern of cooperative game theory, several papers propose and develop axiomatically solutions or allocation rules of a worth generated through cooperation, and they do so in some specific generalizations of the classical setting that supposes that players can organize themselves in independent coalitions. Nicolas Andjiga and Sébastien Courtin present and develop axiomatically a solution concept in a setting where players organize themselves in concomitant non-necessary disjoint coalitions whose union is the grand coalition, and a player may belong to several coalitions. The solution concept they develop generalizes in this setting the van den Brink and van der Laan share function for coalition structures (partitions). In the paper of Encarnación Algaba, Jesús Mario Bilbao, and René van den Brink, cooperation is restricted to coalitions that belong to union stable systems that encompass as special
J.-F. Caulier Centre d’Economie de la Sorbonne, Université Paris 1, Paris, France M. Grabisch (B) · A. Rusinowska Paris School of Economics, CNRS, Centre d’Economie de la Sorbonne, Université Paris 1, Paris, France e-mail:
[email protected]
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cases communication graphs and permission structures. The solution they propose axiomatically leans on the distribution of Harsanyi dividends and they show that it coincides with the Myerson value under an equal power measure condition, and with the position value for a specific subclass of union-stable systems. For the situation arising when players must form tree-connected coalitions with costly links such that the set of coalitions partitions the grand coalition, Gustavo Bergantiños and María Gómez-Rúa axiomatically tackle the solution problem ensuing in a spanning tree that has minimal cost. This solution brings the Owen value for games with a coalition structure when players minimize cost-spanning trees. Imma Curiel considers dynamic versions of the TU-games. Members of consecutive coalitions in the sequence can rearrange themselves. She shows that under a modified equal gain splitting rule property, the resulting division rule belongs to the core of the multiple sequence TU game. The last paper dealing with cooperative game theory is by Jesús Getán, Josep M. Izquierdo, Jesús Montes, and Carles Rafels. It generalizes the well-known result of Maschler et al. of coincidence of the bargaining set and the core for convex games, to the class of the so-called almost-convex games, that is, whose all proper subgames are convex. The paper by Boaz Golany, Noam Goldberg, and Uriel Rothblum concerns multiple resource allocation. It investigates the problem of allocating multiple defensive resources to protect multiple sites against possible attacks by an adversary, which is solved under the form of a two-person zero-sum game with piecewise linear utility functions and polyhedral action sets. Olivier Hudry studies the complexity of computing the median relation of a collection of preference relations, that is, the relation minimizing the remoteness to the preference relations of the collection. The remoteness can be defined in several ways, like the sum of distances, or the sum of pth power of the distances, or their minimum, maximum, etc. In most cases, it is shown that the computation is an NP-hard problem. Another topic of SING7 of particular interest concerns auctions. Gisèle Umbhauer considers almost common value auctions. She shows that Bikhchandani’s equilibria are not the only equilibria of the two-player second-price sealed-bid common value auction. By introducing discontinuities in the bids, she establishes a new family of perfect equilibria with some interesting properties. Estrella Alonso, Joaquin Sanchez-Soriano, and Juan Tejada consider simultaneous auctions of two commonly ranked objects. They introduce a parametric family of auction mechanisms which includes discriminatory-price auction, uniform-price auction, and Vickrey auction, and provide the unique Bayesian Nash equilibrium for each auction in this family. Jack Douglas Stecher and Mark van Atten are interested in advantages of switching from classical to constructive mathematics. They use a result from intuitionistic mathematics known as Brouwer’s continuity principle to resolve a puzzle in finance and accounting that is insoluble using classical mathematics.
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