Ordinality of a voting system and reduction of coalitional manipulability

For any non-trivial voting system, there exist manipulable situations: a coalition of voters, by casting an insincere ballot, may secure an outcome that is better from their point of view. In this paper, we investigate how it is possible to reduce the manipulability rate, which is the probability of such situations, under a probabilistic assumption on the population called culture. We prove that when electors are independent, for any non-ordinal voting system (i.e. requiring information that is not included in the orders of preferences, for example grades), there exists an ordinal voting system whose manipulability rate is at most as high and which meets some other desirable properties. Furthermore, this result is also true when voters are not independent but the culture is decomposable, a weaker condition that we define. Combining this result with Condorcification theorem from Durand et al. (2014) and Green-Armytage et al. (2014), we conclude that when searching for a voting system whose manipulability is minimal (in a large class of systems), one can restrict to ordinal voting systems meeting the Condorcet criterion.