On the power and importance of the mean preference in a mathematical model of democratic choice

The aggregation of individual preferences into a social preference ordering is one of the most fundamental of the problems in social science. The general problem is non-trivial even for a small group of individuals. Given plausible conditions which a social preference ordering should satsify, Arrow demonstrates in his classic work [1] that even when individual preferences satisfy reasonable conditions there need not exist a general social welfare function or preference ordering if the number of involved persons and alternatives under consideration is greater than or equal to three. Only in the special case of two alternatives is the device of decision by majority rule proven to provide a generally satisfactory social preference ordering for an arbitrary number of individuals.