Applied Math Seminar

Date: April 28, 2025

Time: 4:00PM - 5:00PM

Location: Zoom

Speaker: Christoph Borgers, Tufts University, Medford, MA

Title: The yard-sale convergence theorem

Abstract: Suppose n identical agents engage in a sequence of trades. Each trade involves a random pair of agents, and as a result of the trade, one agent gains some amount of wealth, and the other loses the same amount. The total amount of wealth is conserved, call it W. The amount of wealth transferred is a small fraction of the poorer trading partner’s pre-trade wealth, so nobody ever goes bankrupt. The direction of wealth transfer is random with both possibilities equally likely. The yard-sale convergence theorem states that with probability 1, the wealth of one agent will converge to W, and the wealth of all others will therefore converge to 0. For short, randomness that is fair in expectation inescapably leads to total oligarchy. This was first observed by Anirban Chakraborti 25 years ago. It is an immediate consequence of the martingale convergence theorem. However, I will give a more elementary proof, using nothing heavier than the Borel-Cantelli lemma, of a stronger result. This is joint work with Claude Greengard.