A note on Condorcet's other paradox

We report in this note some results on the theoretical likelihood of Condorcet's Other Paradox in three alternative elections. This paradox occurs when we have a voting situation such that no Wheighted Scoring Rule (WSR) will select the Pairwise Majority Rule Winner as the WSR winner. We conclude from our study that actual observances of Condorcet's Other Paradox should be very rare events.Social Choice, Voting paradoxes, Scoring rules, Majority rule, Probability

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.

On the Likelihood of Dummy players in Weighted Majority Games

When the number of players is small in a weighted majority voting game, it can occur that one of the players has no influence on the result of the vote, in spite of a strictly positive weight. Such a player is called a “dummy” player in game theory. The purpose of this paper is to investigate the conditions that give rise to such a phenomenon and to compute its likelihood. It is shown that the probability of having a dummy player is surprisingly high and some paradoxical results are observed.Cooperative game theory, weighted voting games, dummy player, likelihood of voting paradoxes.

On the voting power of an alliance and the subsequent power of its members

Even, and in fact chiefly, if two or more players in a voting game have on a binary issue independent opinions, they may have interest to form a single voting alliance giving an average gain of influence for all of them. Here, assuming the usual independence of votes, we first study the alliance voting power and obtain new results in the so-called asymptotic limit for which the number of players is large enough and the alliance weight remains a small fraction of the total of the weights. Then, we propose to replace the voting game inside the alliance by a random game which allows new possibilities. The validity of the asymptotic limit and the possibility of new alliances are examined by considering the decision process in the Council of Ministers of the European Union.

On the Voting Power of an Alliance and the Subsequent Power of its Members

Even, and in fact chiefly, if two or more players in a voting gamehave on a binary issue independent opinions, they may haveinterest to form a single voting alliance giving an average gainof influence for all of them. Here, assuming the usualindependence of votes, we first study the alliance voting powerand obtain new results in the so-called asymptotic limit for whichthe number of players is large enough and the alliance weightremains a small fraction of the total of the weights. Then, wepropose to replace the voting game inside the alliance by a randomgame which allows new possibilities. The validity of theasymptotic limit and the possibility of new alliances are examinedby considering the decision process in the Council of Ministers ofthe European Union.Voting Power; Alliance

La détermination du nombre des délégués au sein des structures intercommunales : une application de l'indice de pouvoir de Banzhaf

Cet article propose une étude de la représentativité des communes au sein des différents établissements publics de coopération intercommunale (EPCI) bas-normands. Il utilise pour cela la notion d'indice de pouvoir (au sens de Banzhaf), outil de la théorie des jeux coopératifs qui permet de mesurer le pouvoir de vote dans le cas d'une décision prise à la majorité avec quota. Après une étude de la situation bas-normande, les auteurs s'intéressent à l'existence de coalitions préférentielles (qui nuancent le calcul a priori des indices de pouvoir) et à la recherche d'une distribution optimale par commune du nombre des délégués. Cette dernière permet de corriger les écarts de représentativité souvent constatés dans la réalité bas-normande.Indices de pouvoir, représentativité, intercommunalité, application.

The Likelihood of a Condorcet Winner in the Logrolling Setting

The purpose of this note is to compute the probability of logrolling for three different probabilistic cultures. The primary finding is that the restriction of preferences to be in accord with the condition of separable preferences creates enough additional structure among voters' preference rankings to create an increase in the likelihood that a Condorcet winner will exist with both IC and IAC-based scenarios

The Likelihood of a Condorcet Winner in the Logrolling Setting

The purpose of this note is to compute the probability of logrolling for three different probabilistic cultures. The primary finding is that the restriction of preferences to be in accord with the condition of separable preferences creates enough additional structure among voters' preference rankings to create an increase in the likelihood that a Condorcet winner will exist with both IC and IAC-based scenarios