17 research outputs found
Power measures derived from the sequential query process
We study a basic sequential model for the discovery of winning coalitions in
a simple game, well known from its use in defining the Shapley-Shubik power
index. We derive in a uniform way a family of measures of collective and
individual power in simple games, and show that, as for the Shapley-Shubik
index, they extend naturally to measures for TU-games. In particular, the
individual measures include all weighted semivalues.
We single out the simplest measure in our family for more investigation, as
it is new to the literature as far as we know. Although it is very different
from the Shapley value, it is closely related in several ways, and is the
natural analogue of the Shapley value under a nonstandard, but natural,
definition of simple game. We illustrate this new measure by calculating its
values on some standard examples.Comment: 13 pages, to appear in Mathematical Social Science
Semivalues: weighting coefficients and allocations on unanimity games
This is a post-peer-review, pre-copyedit version of an article published in Optimization letters. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11590-017-1224-8.Each semivalue, as a solution concept defined on cooperative games with a finite set of players, is univocally determined by weighting coefficients that apply to players’ marginal contributions. Taking into account that a semivalue induces semivalues on lower cardinalities, we prove that its weighting coefficients can be reconstructed from the last weighting coefficients of its induced semivalues. Moreover, we provide the conditions of a sequence of numbers in order to be the family of the last coefficients of any induced semivalues. As a consequence of this fact, we give two characterizations of each semivalue defined on cooperative games with a finite set of players: one, among all semivalues; another, among all solution concepts on cooperative games.Peer ReviewedPostprint (author's final draft
On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games
Weighted voting games are a family of cooperative games, typically used to
model voting situations where a number of agents (players) vote against or for
a proposal. In such games, a proposal is accepted if an appropriately weighted
sum of the votes exceeds a prespecified threshold. As the influence of a player
over the voting outcome is not in general proportional to her assigned weight,
various power indices have been proposed to measure each player's influence.
The inverse power index problem is the problem of designing a weighted voting
game that achieves a set of target influences according to a predefined power
index. In this work, we study the computational complexity of the inverse
problem when the power index belongs to the class of semivalues. We prove that
the inverse problem is computationally intractable for a broad family of
semivalues, including all regular semivalues. As a special case of our general
result, we establish computational hardness of the inverse problem for the
Banzhaf indices and the Shapley values, arguably the most popular power
indices.Comment: To appear in AAAI 201
ASSESSMENT OF VOTING SITUATIONS: THE PROBABILISTIC FOUNDATIONS
In this paper we revise the probabilistic foundations of the theory of the measurement of 'voting power' either as success or decisiveness. For an assessment of these features two inputs are claimed to be necessary: the voting procedure and the voters' behavior. We propose a simple model in which the voters' behavior is summarized by a probability distribution over all vote configurations. This basic model, at once simpler and more general that other probabilistic models, provides a clear conceptual common basis to reinterpret coherently from a unified point of view di.erent power indices and some related game theoretic notions, as well as a wider perspective for a dispassionate assessment of the power indices themselves, their merits and their limitations.Voting rules, voting power, decisiveness, success, power indices
Some properties for probabilistic and multinomial (probabilistic) values on cooperative games
This is an Accepted Manuscript of an article published by Taylor & Francis in Optimization on 18-02-2016, available online: http://www.tandfonline.com/10.1080/02331934.2016.1147035.We investigate the conditions for the coefficients of probabilistic and multinomial values of cooperative games necessary and/or sufficient in order to satisfy some properties, including marginal contributions, balanced contributions, desirability relation and null player exclusion property. Moreover, a similar analysis is conducted for transfer property of probabilistic power indices on the domain of simple games.Peer ReviewedPostprint (author's final draft
Least Square Approximations and Conic Values of Cooperative Games
URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2015.47 - ISSN : 1955-611XThe problem of least square approximation for set functions by set functions satisfying specified linear equality or inequality constraints is considered. The problem has important applications in the field of pseudo-Boolean functions, decision making and in cooperative game theory, where approximation by additive set functions yields so-called least square values. In fact, it is seem that every linear value for cooperative games arises from least square approximation. We provide a general approach and problem overview. In particular, we derive explicit formulas for solutions under mild constraints, which include and extend previous results in the literature.On considère le problème de l'approximation au sens des moindres carrés des fonctions d'ensemble par des fonctions d'ensemble satisfaisant des contraintes linéaires d'égalité ou d'inégalité. Le problème a des applications importantes dans le domaine des fonctions pseudo-Booléennes, la décision et la théorie des jeux coopératifs, où l'approximation par des jeux additifs mène à la notion de valeur aux moindres carrés. En fait, on voit que toute valeur linéaire pour les jeux coopératifs vient d'un problème d'approximation par les moindres carrés. Nous proposons une approche générale du problème. En particulier, nous obtenons des formules explicites pour les solutions sous des hypothèses faibles, qui incluent et étendent des résultats précédents de la littérature
The Burkill-Cesari Integral on Spaces of Absolutely Continuous Games
We prove that the Burkill-Cesari integral is a value on a subspace ofACand then discuss its continuity with respect to both theBVand the Lipschitz norm. We provide an example of value on a subspace ofACstrictly containingpNAas well as an existence result of a Lipschitz continuous value, different from Aumann and Shapley's one, on a subspace ofAC∞
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