On the Chacteristic Numbers of Voting Games

This paper deals with the non-emptiness of the stability set for any proper voting game.We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles [6] for quota games.voting game, core, stability set

On the Chacteristic Numbers of Voting Games

This paper deals with the non-emptiness of the stability set for any proper voting game. We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles (1990) for quota games.voting game, core, stability set

Implementation of Social Choice Functions via Demanding Equilibria

We consider agents who do not have any information about others' preferences. In this situation they attempt to behave such as to maximize their chances to obtain their most preferred alternative. This defines a solution concept for games symmetrical to Barbera and Dutta's protective equilibrium, the demanding equilibrium. Necessary and sufficient conditions for self implementation in demanding equilibria (side) of social choice functions are provided.Implementation; Social Choice Function; Demanding Equilibrium; Voting

On the performance of the Shapley Shubik and Banzhaf power indices for the allocations of mandates

A classical problem in the power index literature is to design a voting mechanism such as the distribution of power of the different players is equal (or closer) to a pre established target. This tradition is especially popular when considering two tiers voting mechanisms: each player votes in his own jurisdiction to designate a delegate for the upper tier; and the question is to assign a certain number of mandates for each delegate according the population of the jurisdiction he or she represents. Unfortunately, there exist several measures of power, which in turn imply different distributions of the mandates for the same pre established target. The purposes of this paper are twofold: first, we calculate the probability that the two most important power indices, the Banzhaf index and the Shapley-Shubik index, lead to the same voting rule when the target is the same. Secondly, we determine which index on average comes closer to the pre established target.Banzhaf, Shapley-Shubik, power indices

Who benefits from the US withdrawal of the Kyoto Protocol? An application of the MMEA method to measure power

Since 1992, the international community is trying to arrive at a multilateral agreement on the reduction of emissions for greenhouse gases. A collective decision mechanism was adopted in 1997: An agreement is ratified if and only if it is approved by a coalition gathering more than 55 countries. Moreover, the ratifying industrialized countries - included in the Annex I of the Kyoto Protocol - must represent a total weight corresponding to at least 55% of the total CO2 emissions of the countries of the Annex I, taking the year 1990 as a reference point.One way to study the theoretical power distribution induced by this voting procedure is to compute the Banzhaf index for each country. Firstly, the results of the computation show that the power distribution is largely heterogeneous and benefits to the United-States. Secondly, we analyze the modifications generated by the European coalition scenario in order to prove that the European strategy to act as a single block counterbalanced the US leadership. Finally, we conclude that Japan and Russia benefited from the United States withdrawal in term of a priori decisional power.Power indices, environment, Kyoto Protocol, empirical game theory

On the Chacteristic Numbers of Voting Games

International audienceThis paper deals with the non-emptiness of the stability set for any proper voting game. We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles (1990) for quota games

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

The axiomatic characterizations of majority voting and scoring rules

The Arrovian framework of social choice theory is flexible enough to allow for a precise axiomatic study of the voting rules that are used in political elections, sport competitions or expert committees, etc. such as the majority rule or the scoring rules. The objective of this paper is to give an account of the results that have been obtained in this direction since 1951. We first present some basic conditions for a collective decision rule to be democratic. Next, we expound in detail two fundamental results: the characterization of the majority rule by May, and the axiomatization of the family of scoring rules by Young. Afterwards, using these results, some specific scoring rules, such as the plurality vote or the Borda count, have also been characterized. Some remarks on other directions of research and open issues conclude the paper.Le cadre arrowien de la théorie des choix collectifs est suffisamment flexible pour entreprendre une étude axiomatique précise des règles de vote qui sont communément utilisées dans des élections politiques, lors de compétitions sportives ou par des comités d'experts etc. comme le vote à la majorité ou les classements par points. L'objectif de cet article est de rendre compte des résultats qui ont été obtenus dans cette direction depuis 1951. Nous présentons d�abord les conditions qui garantissent qu'une règle de choix collectif est démocratique. Ensuite, nous exposons en détails deux résultats fondamentaux : la caractérisation de la règle de décision à la majorité par May, et l'axiomatisation de la famille des classements par points par Young. Par la suite, en utilisant ces résultats, des classements par points particuliers, comme le vote uninominal à un tour ou la méthode de Borda, ont aussi pu être axiomatisés. Quelques remarques sur d'autres voies de recherche et des questions ouvertes concluent l'article

La théorie des choix collectifs à portée de tous ! Commentaires sur quatre livres de vulgarisation de Donald Saari

D. Saari, "Geometry of voting", Studies in economic theory, Berlin-Heidelberg-New-York, Springer, 1994. D. Saari, "Basic geometry of voting", Berlin-Heidelberg-New-York, Springer, 1995. D. Saari, "Chaotic elections! A mathematician looks at voting", American Mathematical Society, 2001. D. Saari, "Decisions and elections, explaining the unexpected", Cambridge, Cambridge Universiy Press, 2001

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