The trade networks of the Roman Empire are among the most intensively researched large-scale market systems in antiquity, yet there is no consensus on the economic structure behind this vast network. The difficulty arises from data fragmentation and the lack of formal analytical methods. Here, the authors present a Bayesian analysis quantifying the extent to which four previously proposed hypotheses match the evidence for the market system in Roman olive oil. Results suggest that the size of economic agents involved in this network followed a power-law distribution, strongly indicating the presence of free market structures supplying olive oil to Rome. This new analysis offers an important tool to researchers exploring the impact of trade on the dynamics of past societies.