Learning in Markets: Greed Leads to Chaos but Following the Price is Right

TL;DR

This paper compares learning dynamics in different market models, showing that gradient ascent can lead to chaos in some cases, while a proportional response protocol reliably converges to market equilibrium.

Contribution

It provides a formal proof of chaos in gradient ascent dynamics in Cournot competitions and introduces a convergent proportional response protocol for Fisher markets.

Findings

Gradient ascent can be Li-Yorke chaotic with small step sizes in Cournot models.

Proportional response protocol converges to market equilibrium in Fisher markets.

Chaos occurs even in simple two-firm, one-good settings.

Abstract

We study learning dynamics in distributed production economies such as blockchain mining, peer-to-peer file sharing and crowdsourcing. These economies can be modelled as multi-product Cournot competitions or all-pay auctions (Tullock contests) when individual firms have market power, or as Fisher markets with quasi-linear utilities when every firm has negligible influence on market outcomes. In the former case, we provide a formal proof that Gradient Ascent (GA) can be Li-Yorke chaotic for a step size as small as $\Theta(1/n)$, where $n$ is the number of firms. In stark contrast, for the Fisher market case, we derive a Proportional Response (PR) protocol that converges to market equilibrium. The positive results on the convergence of the PR dynamics are obtained in full generality, in the sense that they hold for Fisher markets with \emph{any} quasi-linear utility functions. Conversely, the chaos results for the GA dynamics are established even in the simplest possible setting of two firms and one good, and they hold for a wide range of price functions with different demand elasticities. Our findings suggest that by considering multi-agent interactions from a market rather than a game-theoretic perspective, we can formally derive natural learning protocols which are stable and converge to effective outcomes rather than being chaotic.