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A distributed voting scheme to maximize preferences

Peter Auer, Nicolò Cesa-Bianchi (2006)

RAIRO - Theoretical Informatics and Applications

We study the problem of designing a distributed voting scheme for electing a candidate that maximizes the preferences of a set of agents. We assume the preference of agent i for candidate j is a real number xi,j, and we do not make any assumptions on the mechanism generating these preferences. We show simple randomized voting schemes guaranteeing the election of a candidate whose expected total preference is nearly the highest among all candidates. The algorithms we consider are designed so that...

A note on the voting problem.

Miguel Angel Fiol Mora (1992)

Stochastica

Let v(n) be the minimum number of voters with transitive preferences which are needed to generate any strong preference pattern (ties not allowed) on n candidates. Let k = [log2n]. Then it is shown that v(n) ≤ n-k if n and k have different parity, and v(n) ≤ n-k+1 otherwise.

A tight quantitative version of Arrow’s impossibility theorem

Nathan Keller (2012)

Journal of the European Mathematical Society

The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any ϵ > 0 , there exists δ = δ ( ϵ ) such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive...

Banzhafův index a dvoustupňové hlasovací systémy

Antonín Slavík (2017)

Pokroky matematiky, fyziky a astronomie

V textu představíme tzv. Banzhafův index, který umožňuje kvantifikovat sílu voliče v předepsaném hlasovacím systému. Definice indexu je zcela elementární, podrobnější zkoumání jeho vlastností však vede k zajímavé a hlubší matematice. Výklad je ilustrován řadou konkrétních příkladů ze světa politiky; uvidíme, že díky Banzhafovu indexu se matematika dostala i na stránky novin.

Condorcet's theory of voting

H. P. Young (1990)

Mathématiques et Sciences Humaines

Condorcet believed that the purpose of voting is to make a choice that is “best” for society. According to his view, there is one choice that is objectively best, another that is second-best, and so forth. Unfortunately, voters sometimes make mistakes ; they misperceive what is best. In designing a voting rule, therefore, the objective should be to choose the alternative that is most likely to be best. Condorcet solved this problem using a form of maximum likelihood estimation. The procedure that...

Cooperación y defensa.

Francesc Carreras (1993)

Qüestiió

Se aplican conceptos y técnicas de la teoría de juegos cooperativos a problemas de decisión que afectan a la política de Defensa del país. El análisis permite evaluar las propuestas sobre procedimientos de votación cualificada presentadas al Consejo Europeo en la cumbre de Maastricht de diciembre de 1991. Se ponen así de manifiesto las implicaciones que supondría para la posición estratégica de España la inédita capacidad operativa concedida a la Comunidad por el tratado de unión política.

Different approaches to weighted voting systems based on preferential positions

Robert Bystrický (2012)

Kybernetika

Voting systems produce an aggregated result of the individual preferences of the voters. In many cases the aggregated collective preference – the result of the voting procedure – mirrors much more than anything else the characteristics of the voting systems. Preferential voting systems work most of the time with equidistant differences between the adjacent preferences of an individual voter. They produce, as voting systems usually do, some paradoxical results under special circumstances. However,...

Fair majorities in proportional voting

František Turnovec (2013)

Kybernetika

In parliaments elected by proportional systems the seats are allocated to the elected political parties roughly proportionally to the shares of votes for the party lists. Assuming that members of the parliament representing the same party are voting together, it has sense to require that distribution of the influence of the parties in parliamentary decision making is proportional to the distribution of seats. There exist measures (so called voting power indices) reflecting an ability of each party...

Le dernier mot de Condorcet sur les élections

Pierre Crépel (1990)

Mathématiques et Sciences Humaines

Nous reconstituons ici le dernier mémoire (inédit) de Condorcet sur les élections ; ce texte était éparpillé en désordre dans plusieurs volumes différents des recueils de manuscrits de la Bibliothèque de l'Institut. Seul le début du mémoire a été publié, dans le Journal d'Instruction Sociale en 1793. En comparant les différentes formes d'élections proposées par Condorcet à partir de l'Essai sur l'application de l'analyse (1785) jusqu'à la Terreur, nous pouvons suivre l'évolution de ses idées et...

More on the tournament equilibrium set

G. Laffond, J. F. Laslier, M. Le Breton (1993)

Mathématiques et Sciences Humaines

Schwartz (1990) proposed a new solution concept for choosing from a tournament ; called the Tournament Equilibrium Set. He stated four problems concerning this solution. In this paper we introduce further questions and demonstrate some logical relationship between these questions.

Note sur le calcul de la probabilité des paradoxes du vote

Sven Berg, Dominique Lepelley (1992)

Mathématiques et Sciences Humaines

De nombreux travaux se sont efforcés au cours des années récentes de calculer la probabilité des paradoxes ou des difficultés que la théorie des choix collectifs a mis en évidence. On passe en revue dans cette note les principaux modèles de calcul utilisés dans ces travaux. On applique en outre l'un des modèles présentés au calcul de la probabilité de quelques paradoxes bien connus de la théorie du vote.

Parameters in collective decision making models : estimation and sensitivity

Tom A. B. Snijders, Evelien P. H. Zeggelink, Frans N. Stokman (1997)

Mathématiques et Sciences Humaines

Simulation models for collective decision making are based on theoretical and empirical insight in the decision making process, but still contain a number of parameters of which the values are determined ad hoc. For the dynamic access model, some of such parameters are discussed, and it is proposed to extend the utility functions with a random term of which the variance also is an unknown parameter. These parameters can be estimated by fitting model predictions to data, where the predictions can...

Power analysis of voting by count and account

Midori Hirokawa, Milan Vlach (2006)

Kybernetika

Using players’ Shapley–Shubik power indices, Peleg [4] proved that voting by count and account is more egalitarian than voting by account. In this paper, we show that a stronger shift in power takes place when the voting power of players is measured by their Shapley–Shubik indices. Moreover, we prove that analogous power shifts also occur with respect to the absolute Banzhaf and the absolute Johnston power indices.

Préférences et rationalité stochastiques

Pascal Bouyaux (1990)

Mathématiques et Sciences Humaines

Le but de cet article est de procéder à une présentation pédagogique d'un concept étendu de rationalité, la rationalité stochastique. Dans une première partie, nous exposons le problème à l'aide d'un exemple simple et posons un ensemble de définitions préliminaires. Puis, dans une seconde partie, nous présentons le résultat fondamental de Falmagne (1978) s'appliquant aux situations de choix multiples ; l'approche ensembliste de cet auteur est formalisée à partir du concept de polynômes de Block-Marschak...

Probabilistic comparison of weighted majority rules

Daniel Berend, Luba Bromberg, Luba Sapir (2012)

Applicationes Mathematicae

This paper studies a bi-parametric family of decision rules, so-called restricted distinguished chairman rules, which contains several one-parameter classes of rules considered previously in the literature. Roughly speaking, these rules apply to a variety of situations where the original committee appoints a subcommittee. Moreover, the chairman of the subcommittee, who is supposed to be the most competent committee member, may have more voting power than other jurors. Under the assumption of exponentially...

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