On Ehrhart Polynomials and Probability Calculations in Voting Theory

In voting theory, analyzing how frequent is an event (e.g. a voting paradox) is, under some specific but widely used assumptions, equivalent to computing the exact number of integer solutions in a system of linear constraints. Recently, some algorithms for computing this number have been proposed in social choice literature by Huang and Chua [17] and by Gehrlein ([12, 14]). The purpose of this paper is threefold. Firstly, we want to do justice to Eug`ene Ehrhart, who, more than forty years ago, discovered the theoretical foundations of the above mentioned algorithms. Secondly, we present some efficient algorithms that have been recently developed by computer scientists, independently from voting theorists. Thirdly, we illustrate the use of these algorithms by providing some original results in voting theory.voting rules, manipulability, polytopes, lattice points, algorithms.

A formula for Nash equilibria in monotone singleton congestion games

This paper provides a simple formula describing all Nash equilibria in symmetric monotone singleton congestion games. Our approach also yields a new and short proof establishing the existence of a Nash equilibrium in this kind of congestion games without invoking the potential function or the nite improvement property.Singleton congestion games, Nash equilibria, Potential function, Finite improvement property

Nonsymmetric singleton congestion games: case of two resources

In this note we study the existence of Nash equilibria in nonsymmetric finite congestion games, complementing the results obtained by Milchtaich on monotone-decreasing congestion games. More specifically, we examine the case of two resources and we propose a simple method describing all Nash equilibria in this kind of congestion games. Additionally, we give a new and short proof establishing the existence of a Nash equilibrium in this type of games without invoking the potential function or the finite improvement property.Singleton congestion games, Nash equilibria, Potential function, Finite improvement property

Correlation, Partitioning and the Probability of Casting a Decisive Vote under the Majority Rule

The main purpose of this paper is to estimate the probability of casting a decisive vote under the majority rule for a class of random electorate models encompassing the celebrated Impartial Culture (IC) and Impartial Anonymous Culture (IAC) models. The emphasis is on the impact of correlation across votes on the order of magnitude of this event. Our proof techniques use arguments from probability theory on one hand and combinatorial and algorithmic tools for counting integer points inside convex polytopes on the other hand

On the stability of a scoring rules set under the IAC

A society facing a choice problem has also to choose the voting rule itself from a set of different possible voting rules. In such situations, the consequentialism property allows us to induce voters\u27 preferences on voting rules from preferences over alternatives. A voting rule employed to resolve the society\u27s choice problem is self-selective if it chooses itself when it is also used in choosing the voting rule. A voting rules set is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this paper a society which will make a choice from a set constituted by three alternatives {a, b, c} and a set of the three well-known scoring voting rules {Borda, Plurality, Antiplurality}. Under the Impartial Anonymous Culture assumption (IAC), we will derive a probability for the stability of this triplet of voting rules. We use Ehrhart polynomials in order to solve our problems. This method counts the number of lattice points inside a convex bounded polyhedron (polytope). We discuss briefly recent algorithmic solutions to this method and use it to determine the probability of stabillity of {Borda, Plurality, Antiplurality} set

The Probability of Casting a Decisive Vote: From IC to IAC trhough Ehrhart's Polynomials and Strong Mixing

The main purpose of this paper is to estimate the probability of casting a decisive vote for a class or random electorate models encompassing the celebrated IC and IAC models. The emphasis is on the impact of correlation across votes on the order of magnitude of this event. Our proof techniques use arguments from probability theory on one hand and the geometry of convex polytopes on the other hand

Contribution à la théorie du choix social : études axiomatiques et analyses probabilistes des classements par points

Thèse de doctorat de l'Université de Caen, mention sciences économiques, sous la direction du Pr. V. MERLI

Choix collectif et procédures de votes

National audienceno abstrac

Sukuk market development and Islamic banks’ capital ratios

This paper investigates the impact of Sukuk market development on Islamic banks’ capital ratios using a sample comprising 230 Islamic banks spanning the period 2005–2014. We characterize Islamic bank capital along multiple dimensions, namely: capital adequacy ratio, Tier 1 capital ratio, and capital-to-total assets ratio. We employ both the Prais-Winston technique and the system GMM estimator to tackle potential omitted variable bias, endogeneity, and simultaneity issues. The evidence shows that Sukuk market development has had a negative effect on capital ratios of Islamic banks. We argue that the development of Sukuk markets may have stimulated the competition between Islamic Banks, inducing them to hold lower capital ratios. Our results also show that trade openness and bank liquidity are positively and significantly related to capital ratios, while bank size and loan loss reserve ratio are negatively and significantly related to capital ratios, as expected

A formula for Nash equilibria in monotone singleton congestion games

International audienceThis paper provides a simple formula describing all Nash equilibria in monotone symmetric singleton congestion games. Our approach also yields a new and short proof establishing the existence of a Nash equilibrium in this kind of congestion games