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 convergence result of the PR dynamics is obtained in full generality, in the sense that it holds for Fisher markets with any quasi-linear utility functions. Conversely, the chaos results for the GA dynamics are established in the simplest possible setting of two firms and one good, and hold for a wide range of price functions with different demand elasticities. Our findings suggest that as multi-agent interactions grow larger, the ensuing market (instead of game-theoretic) conditions allow us to formally derive natural and stable learning protocols which converge to effective outcomes rather than being chaotic.