On metrically conservative societies

In this paper, social choice theory is considered from the standpoint of social change. Various metrics (in a discrete setting) are introduced to measure changes in individual and collective preferences, and a society is said to be metrically conservative if social change does not exceed total individual changes. Arrow's IIA Axiom is found to be intimately related to a very restrictive metrical condition called metrical ultraconservatism. Strong characterization theorems are proved for metrically ultraconservative societies. A natural relaxation is the condition of metrical conservatism. We show that metrically conservative societies exist, and the number of possibilities can in fact grow exponentially with the population. But when the metrical condition is placed into the more specific socio-economic context of strict preference orderings, normative restrictions appear. One is the constitutional protection against the election of a dictator; another is the nonexistence of metrically conservative stable matchings, in the sense of Gale-Shapley. Some similar questions have been raised in continuous social choice theory, but the conclusions are quite different. We also consider the effect of an increasing population on the average rate of social change.The author is grateful to Dr. T.M. Tang for first drawing his attention to Arrow's General Possibility Theorem, to Professors Robert M. Anderson, Kenneth J. Arrow, and an anonymous referee for many valuable suggestions, and to Dr. W.Y. Poon for pointing out an important reference. This work was done when the author was at the University of California at Berkeley and formed part of his Berkeley Ph.D. dissertation. The views expressed here are the author's, and not necessarily those of AT&T Bell Laboratories.