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26.11.2018 | Original Paper

The theoretical Shapley–Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system

Social Choice and Welfare
Olivier de Mouzon, Thibault Laurent, Michel Le Breton, Dominique Lepelley
Wichtige Hinweise

Electronic supplementary material

The online version of this article (https://​doi.​org/​10.​1007/​s00355-018-1162-0) contains supplementary material, which is available to authorized users.
The numerical analysis of the question formulated in our paper is only sketched in Sect. 3. For a complete exposition of our method and conclusions including all the details, the reader is refered to the supplementary material presented in the online appendix. We thank the guest editors, Marc Fleurbaey and Kotaro Suzumura, and the referee for their encouragements as well as all the participants of the 13th Social Choice and Welfare conference and of the 13th meeting of the “Ingénieurs Statisticiens Toulousains”, for their comments and suggestions. We would like to express our deepest gratitude to Kazuya Kikuchi and Issofa Moyouwou for the very insightful comments and suggestions contained in the scientific correspondence that we had with them on the topic covered by this paper. Out of that, Kikuchi (2018) provides a short, elegant and assumption-free proof of the convergence to 0 of the IAC probability of an election inversion. Of course, any remaining errors are ours. Michel Le Breton and Dominique Lepelley would like to express their immense respect and gratitude to Ken Arrow to whom this special issue is dedicated. Ken Arrow was interested in many topics (not to say all) and majority voting (the topic of the current paper) was among them. The first (1951) edition of “Social Choice and Individual Values ” contains discussions of majority voting and more generally of the application of his theory to elections. For instance, on page 6, he writes “...The chief relevant point here is that virtually every particular scheme proposed for election from single-member constituencies has been shown to have certain arbitrary features. The problem of choosing one among a number of candidates for a single position, such as the Presidency of the United clearly of the same character as choosing one out of a number of alternative social policies” . The addition that appears in the second (1963) edition contains historical remarks including a discussion of the contribution of Condorcet to the theory of voting. The French translation appears in 1974. It contains a foreword, dated August 10, 1974, intended for the French readers. It reminds, among other things, the important work of the French precursors on voting methods and it is concluded as follows: “J’espère que cette traduction invitera quelques universitaires francais à choisir ce domaine de recherche ; les spécialistes ont besoin d’eux” . We have accepted that invitation and have never regretted, during our all professional life, to have done so.


In this article, we evaluate asymptotically the probability \(\phi \left( n\right) \) of an election inversion in a toy symmetric version of the US presidential electoral system. The novelty of this paper, in contrast to all the existing theoretical literature, is to assume that votes are drawn from an IAC (Impartial Anonymous Culture)/Shapley–Shubik probability model. Through the use of numerical methods, it is conjectured, that \(\sqrt{n}\) \( \phi \left( n\right) \) converges to 0.1309 when n (the size of the electorate in one district) tends to infinity. It is also demonstrated that \( \phi \left( n\right) =o\left( \sqrt{\frac{ln(n)^{3}}{n}}\right) \) and \(\phi \left( n\right) =\Omega \left( \frac{1}{\sqrt{n}}\right) \).

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