Economic sociologists have long established that trust problems give rise to market structures in which many exchanges take place within committed long-term relationships between specific trading parties. Returning to proven partners allows actors to mitigate the risk of being taken advantage of, and the preferential choice of partners who were reliable in the past provides incentives for honest behavior. We revisit the question of how trust problems affect exchange structures. Arguably, institutionalized reputations systems are a key source of trust in modern economies. They enable actors to learn from the experiences of others and thereby provide a basis for trust and trustworthiness that substitutes for the incentives in long-term relations. Do trust problems no longer shape exchange structures in the presence of technologically facilitated large-scale reputation systems? We argue that trust problems still have far-reaching implications for market structures, but that the key feature they bring about is no longer dyadic commitments. We show in a game-theoretic model and in a laboratory experiment that in the presence of reputation systems, trust problems instead lead to the emergence of high market concentration, with all buyers frequenting one or a few sellers and excluding many others. The few sellers who are lucky to serve the whole market additionally earn a sizeable premium for their good reputations.