# Analysis and Design of Electoral Systems

### Michel L. Balinski

Ecole Polytechnique, Paris, France### Steven J. Brams

New York University, USA### Friedrich Pukelsheim

Universität Augsburg, Germany

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## Abstract

Electoral systems transform numbers into numbers. In order not to be blinded by the intricacies of any particular case, as interesting as particular cases may be, it is imperative to view theses transformations as functions, or as relations, and to study their properties and structure in their full generality. Promoting the mathematical foundations of the topic cannot be a goal in and for itself, but must reflect the practical needs defined by the problems themselves that are traditionally treated by such non-mathematical fields as political science, economics, social choice theory, constitutional law, etc. This diversity was well reflected in the expert fields represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under three headings: (1) voting schemes that aggregate many individual preference rankings of a given set of alternatives into a single global ranking; (2) proportional representation schemes that map vote counts (or population counts) into parliamentary representation; and (3) the determination of electoral districts that reasonably reflect geographical, political, and social structures.While mathematics at large provides the appropriate language to investigate these problems, the tools that are used draw on specific mathematical fields, including: \begin{itemize} \item discrete mathematics, in that preference rankings are (usually) partial orderings or permutations; \item combinatorial optimization, when finitely many units are assigned in a accordance with some optimality criterion; \item stochastics, where weights that are virtually continuous (proportions of votes) are to be approximated by weights that are intrinsically discrete (numbers of seats); and \item geometry, identifying the intricacies of voting with the symmetry structure of permutation groups, and with the geometry of the probability simplex. \end{itemize} The mathematical analysis of electoral systems has direct repercussions on current political issues. To mention but three, there is the question whether elections of the US president would be less likely to be disputed, and their outcomes considered more legitimate, were he to be elected directly. Another is the analysis of qualified majority rules in the EU Council of Ministers as stipulated in the draft constitution proposed by the European Convention 2003. A third relates to developing biproportional apportionment methods that simultaneously accommodate partisan and regional representation of the entire electorate. Somewhat unusual for an Oberwolfach conference was the rather broad mix of participants, representing the fields of mathematics, physics, economics, political science, and psychology. It was this breadth that virtually all participants experienced as challenging and fruitful. As always, the atmosphere was congenial for scientific discussions, and the Institute in its Schwarzwald setting made people feel creative by just being there \ldots~ particularly so, since the sky was blue and the sun was shining (most of the time). M.L. Balinski S.J. Brams F. Pukelsheim