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Social Choice and Catastrophe

This paper surveys the results in general equilibrium theory based on dynamical models, and emphasizes the role of structural stability. In this context it is natural to consider a preference field H for the society, combining economic fields, associated with the preferred changes wrought by agents in the economic market place, together with fields of preferred changes in the polity. A condition called half-openness of H is sufficient to guarantee existence of a local direction gradient,d, for the society, and thus of a social choice. The condition is equivalent to a dual property that the dual co-cones of H have non-empty intersection. This result is analogous to Chichilnisky's necessary and sufficient condition for market equilibrium.We then consider models of belief aggregation similar to Condorcet's Jury theorem and mention Penn's Theorem on existence of a belief equilibrium. The half open property may fail for general dynamical processes and this feature is linked to the creation of belief cascades at the "edge of chaos. Some comments are addressed to the notion of "punctuated equilibrium" in evolutionary models in biology and language. Finally, Thom's notion of a structurally stable chreod is introduced. Even when a belief or preference field is locally chaotic because of a failure of half openness, it is possible to define a generalized social correspondence called the "heart." The heart is lower hemi-continuous and is characterized by a possible rapid expansion at a critical value of the parameters. It is suggest that events in the heart obey a power law distribution which is generally associated with "the edge of chaos."