Price Dynamics, Social Choice, Voting Methods, Probability and Chaos

Abstract

An intriguing aspect of the mathematical social sciences is the discovery that so many things can go wrong! Mathematical models are developed to clarify, sharpen, and advance arguments originally put forth in a verbal form. Often these verbal arguments persuasively leave us with a sense of security and of orderliness; then often the mathematical formulation shatters this stability. (This is because the mathematics may uncover subtle combinatorics or hidden second and higher level interaction effects.) Perhaps the most dramatic illustration of this comes from the social choice literature with the well-known Arrow Impossibility Theorem [1]). In economics, the stability of the price adjustment model of tatonnement was threatened by Scarf’s example [2] where he showed that the tatonnement market forces of supply and demand could push the prices away from any equilibrium. Any remaining faith in this price story was destroyed by the work of Sonnenschein and others [3] which disclosed, as a corollary, that almost any type of price trajectory could occur.