Social Choice and Threshold Phenomena

Gil Kalai

Discussion Paper Series from The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem

Abstract: Arrow's theorem asserts that under certain conditions every non-dictatorial social choice function leads to nonrational social choice for some profiles. In other words, for the case of non-dictatorial social choice if we observe that the society prefers alternative A over B and alternative B over C we cannot deduce what its choice will be between B and C. Here we ask whether we can deduce anything from observing a sample of the society's choices on the society's choice in other cases? We prove that the answer is ``no'' for large societies for neutral and monotonic social choice function such that the society's choice is not typically determined by the choices of a few individuals. The proof is based on threshold properties of Boolean functions and on analysis of the social choice under some probabilistic assumptions on the profiles. A similar argument shows that under the same conditions for the social choice function but under certain other probabilistic assumptions on the profiles the social choice function will typically lead to rational choice for the society.

Pages: 22 pages
Date: 2001-11
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