Influence and interaction indexes in cooperative games: a unified least squares approach

The classical Banzhaf power and interaction indexes used in cooperative game theory appear naturally as leading coefficients in the standard least squares approximation of the game under consideration by a set function of a specified degree. We observe that this still holds true if we consider approximations by set functions depending on specified variables. We show that the Banzhaf influence index can also be obtained from this new approximation problem. Considering certain weighted versions of this approximation, we also introduce a class of weighted Banzhaf influence indexes and analyze their most important properties

Taxation and stability in cooperative games

Cooperative games are a useful framework for modeling multi-agent behavior in environments where agents must collaborate in order to complete tasks. Having jointly completed a task and generated revenue, agents need to agree on some reasonable method of sharing their profits. One particularly appealing family of payoff divisions is the core, which consists of all coalitionally rational (or, stable) payoff divisions. Unfortunately, it is often the case that the core of a game is empty, i.e. there is no payoff scheme guaranteeing each group of agents a total payoff higher than what they can get on their own. As stability is a highly attractive property, there have been various methods of achieving it proposed in the literature. One natural way of stabilizing a game is via taxation, i.e. reducing the value of some coalitions in order to decrease their bargaining power. Existing taxation methods include the ε-core, the least-core and several others. However, taxing coalitions is in general undesirable: one would not wish to overly tamper with a given coalitional game, or overly tax the agents. Thus, in this work we study minimal taxation policies, i.e. those minimizing the amount of tax required in order to stabilize a given game. We show that games that minimize the total tax are to some extent a linear approximation of the original games, and explore their properties. We demonstrate connections between the minimal tax and the cost of stability, and characterize the types of games for which it is possible to obtain a tax-minimizing policy using variants of notion of the ε-core, as well as those for which it is possible to do so using reliability extensions. Copyright © 2013, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved

Multicoalitional solutions

Documents de travail du Centre d'Economie de la Sorbonne 2013.62 - ISSN : 1955-611XThe paper proposes a new concept of solution for TU games, called multicoalitional solution, which makes sense in the context of production games, that is, where v(S) is the production or income per unit of time. By contrast to classical solutions where elements of the solution are payoff vectors, multicoalitional solutions give in addition an allocation time to each coalition, which permits to realize the payoff vector. We give two instances of such solutions, called the d-multicoalitional core and the c-multicoalitional core, and both arise as the strong Nash equilibrium of two games, where in the first utility per active unit of time is maximized, while in the second it is the utility per total unit of time. We show that the d-core (or aspiration core) of Benett, and the c-core of Guesnerie and Oddou are strongly related to the d-multicoalitional and c-multicoalitional cores, respectively, and that the latter ones can be seen as an implementation of the former ones in a noncooperative framework.Nous proposons dans ce papier un nouveau concept de solutions pour les jeux coopératifs à utilité transférable, que nous appelons solutions multicoalitionnelles. Ce type de solutions fera particulièrement sens pour modéliser des problèmes où la valeur d'un jeu correspondra à une quantité de biens produits par unité de temps. Contrairement au cadre classique où les vecteurs de paiements correspondent à un partage de la valeur de la grande coalition, les éléments d'une solution multicoalitionnelle indiqueront un vecteur de paiement et la durée d'activité de chaque coalition ayant permis d'engendrer ce vecteur de paiement. Nous étudions plus particulièrement deux exemples de ce type de solutions : le d-coeur multicoalitionnel et le c-coeur multicoalitionnel. Nous montrons que chacune de ces deux solutions correspond aux équilibres de Nash forts d'un jeu non coopératif très naturel pour modéliser une formation de coalitions. Une stratégie d'un joueur consiste à déclarer le temps qu'il souhaite passer dans chaque coalition et le paiement qu'il souhaite recevoir pour cela. Les équilibres de Nash forts du d-coeur multicoalitionnel sont issus du jeu ayant une utilité pour chaque joueur égale au gain par heure travaillée, tandis que les équilibres du c-coeur multicoalitionnel ayant une utilité pour chaque joueur égale au gain reçu à la fin du processus de formation coalitionnelle, c'est-à-dire au gain par heure vécue. Nous voyons enfin en quoi ces deux solutions sont fortement liées au concept de d-coeur (Bennett) et de c-coeur (Guesnerie et Oddou), et en quoi nous les implémentons dans un cadre non-coopératif

Taxation and Stability in Cooperative Games

ABSTRACT Cooperative games are a useful framework for modeling multiagent behavior in environments where agents must collaborate in order to complete tasks. Having jointly completed a task and generated revenue, agents need to agree on some reasonable method of sharing their profits. One particularly appealing family of payoff divisions is the core, which consists of all coalitionally rational (or, stable) payoff divisions. Unfortunately, it is often the case that the core of a game is empty, i.e. there is no payoff scheme guaranteeing each group of agents a total payoff higher than what they can get on their own. As stability is a highly attractive property, there have been various methods of achieving it proposed in the literature. One natural way of stabilizing a game is via taxation, i.e. reducing the value of some coalitions in order to decrease their bargaining power. Existing taxation methods include the ε-core, the least-core and several others. However, taxing coalitions is in general undesirable: one would not wish to overly tamper with a given coalitional game, or overly tax the agents. Thus, in this work we study minimal taxation policies, i.e. those minimizing the amount of tax required in order to stabilize a given game. We show that games that minimize the total tax are to some extent a linear approximation of the original games, and explore their properties. We demonstrate connections between the minimal tax and the cost of stability, and characterize the types of games for which it is possible to obtain a tax-minimizing policy using variants of notion of the ε-core, as well as those for which it is possible to do so using reliability extensions

Preserving coalitional rationality for non-balanced games

International audienceIn cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the k-additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most k, and is never empty for k ≥ 2. The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The k-additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal negotiation set. The idea is to select elements of the k-additive core mimimizing the total amount given to coalitions of size greater than 1. Thus the minimum negotiation set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We give a method of computing the minimum bargaining set, and lastly indicate how to eventually get classical solutions from general ones

Preserving coalitional rationality for non-balanced games

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2012.22 - ISSN : 1955-611X - Version réviséeIn cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the k-additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most k, and is never ampty for k ≥ 2. The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The k-additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal bargaining set. The idea is to select elements of the k-additive core minimizing the total amount given to coalitions of size greater than 1. Thus the minimum bargaining set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We introduce also the notion of unstable coalition, and show how to find them using the minimum bargaining set. Lastly, we give a method of computing the minimum bargaining set.En théorie des jeux coopératifs, le coeur est l'une des solutions fondamentales, car il définit un ensemble de partages rationnels pour les joueurs. Cependant, pour les jeux non équilibrés, le coeur est vide, et d'autres concepts de solutions ont dû être créés. Nous proposons l'utilisation de solutions générales, c'est-à-dire, de solutions distribuant la valeur totale du jeu aux groupes plutôt qu'aux individus. En particulier, le coeur k-additif, proposé par Grabisch et Miranda est une solution générale qui n'est jamais vide pour k ≥ 2. L'extension de coeur proposé par Bejan et Gomez peut aussi être vu comme une solution générale donnant un montant aux individus et à la grande coalition. Le coeur k-additif donnant un ensemble non borné et étant de ce fait difficile à utiliser en pratique, nous proposons un sous-ensemble du coeur k-additif que nous appelons l'ensemble de négociation minimum. L'idée est de sélectionner les éléments du coeur k-additif minimisant le montant total donné aux coalitions de tailles supérieures à 1. Ainsi, l'ensemble de négociation minimum se réduit au coeur quand celui-ci est non vide. Nous étudions cet ensemble, donnons des propriétés et des axiomatisations, ainsi que ses relations avec l'extension du coeur de Bejan et Gomez. Nous introduisons également la notion de coalitions instables et montrons comment les trouver en utilisant l'ensemble de négociation minimum. Enfin nous donnons une méthode pour calculer l'ensemble de négociation minimum