In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences. It turns out that there are two extreme types of behavior. On one side, there are social welfare functions, such as the majority rule, that lead to stochastic stability of the outcome in terms of perturbations of individual preferences. We identify and study a class of social welfare functions that demonstrate an extremely different type of behavior which is completely chaotic: they lead to a uniform probability distribution on all possible social preference relations and, for every ε > 0, if a small fraction e of individuals change their preferences (randomly) the correlation between the resulting social preferences and the original ones tends to zero as the number of individuals in the society increases. This class includes natural multi-level majority rules.
Kalai, G. (2010). Noise sensitivity and chaos in social choice theory. In Bolyai Society Mathematical Studies (Vol. 20, pp. 173–212). https://doi.org/10.1007/978-3-642-13580-4_8