Topology and Social Choice

Abstract

Recent events in the global economy have caused many writers to argue that the market is driven by animal spirits, by irrational exuberance or specu- lation. At the same time, the economic downturn has apparently caused many voters in the United States, and other countries, to change their opininion about the the proper role of government. Unfortunately, there does not exist a general equilibrium (GE) model of the political economy, combining a formal model of the existence, and convergence to a price equilibrium, as well as an equilibrium model of political choice. One impediment to such a theory is the so-called chaos theorem (Saari, 1997) which suggests that existence of a polit- ical equilibrium is non-generic. This paper surveys the results in GE based on the theory of dynamical systems, emphasizing the role of structural stability. In this context it is natural to consider a preference …eld H for the society, combining economic …elds, associated with the preferred changes wrought by agents in the economic market place, together with …elds of preferred changes in the polity. A condition called half-openess of H is su¢cient to guarantee existence of a local direction gradient, d; for the society. The paper argues that instead of seeking equilibrium, it is natural to examine the structural stability of the ‡ow induced by the social preference …eld. Instead of focusing on equi- librium analysis, based on some version of the Brouwer …xed point theoerem, and thus on the assumption that the social world is topologically contractible, political economy could consider the nature of the dynamical path of change, and utilize notions from the qualitative theory of dynamical systems.