Since its foundation by Arrow in his seminal contribution (Arrow, 1963), one of the main merit of social choice theory has been to provide a coherent framework for the analysis and comparison of different voting rules. First, many normative requirements about voting rules can be expressed precisely in this framework. Then it is possible to check whether a given voting rule satisfies a given property. Ideally, this type of analysis may lead to the axiomatic characterization of a voting rule. At last the propensity of situations for which a voting rule fails to satisfy a condition can be evaluated.
Peter Fishburn's contributions to this research program have been extremely important. For example, he proposed many new normative conditions for the analysis of voting rules (see in particular Fishburn, 1974, 1977; Fishburn & Brams, 1983), and developed axiomatic analysis for binary voting (Fishburn, 1973) and approval voting (Fishburn, 1978). Together with Gehrlein, he launched an important research program on the probabilistic analysis of voting rules. After Guilbauld's paper (Guilbauld, 1952), the use of probability models in voting was limited to the evaluation of the majority voting paradox under the assumption that each voter would pick his preference independently from the others from a uniform distribution. This assumption, today called the Impartial Culture assumption, puts an equal weight on each profile. Fishburn and Gehrlein developed the use of probabilistic models in two directions. First, to analyze the occurrence of Condorcet cycles, they proposed in Gehrlein and Fishburn (1976) a new probability assumption, the Impartial Anonymous Culture assumption, which assumes that each anonymous profile is equally likely to appear. Secondly, they applied these two probability models to a wider range of problems, the relationships between the scoring rules and the Condorcet principle being their favorite issue (see Fishburn & Gehrlein, 1976; Gehrlein & Fishburn, 1978a, 1978b). The results we will present in this paper are clearly a continuation of this research program, as we will compare voting rules suggested for the European Union on their propensity to fulfill a given property according to different probability assumptions.
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Feix, M.R., Lepelley, D., Merlin, V., Rouet, JL. (2009). On the Probability to Act in the European Union. In: Brams, S.J., Gehrlein, W.V., Roberts, F.S. (eds) The Mathematics of Preference, Choice and Order. Studies in Choice and Welfare. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79128-7_11
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