This article contains the first detailed historical study of one of the new high-frequency trading (HFT) firms that have transformed many of the world’s financial markets. The study, of Automated Trading Desk (ATD), one of the earliest and most important such firms, focuses on how ATD’s algorithms predicted share price changes. The article argues that political-economic struggles are integral to the existence of some of the ‘pockets’ of predictable structure in the otherwise random movements of prices, to the availability of the data that allow algorithms to identify these pockets, and to the capacity of algorithms to use these predictions to trade profitably. The article also examines the role of HFT algorithms such as ATD’s in the epochal, fiercely contested shift in US share trading from ‘fixed-role’ markets towards ‘all-to-all’ markets.