One of the most basic questions in economics concerns the effects of competition onnmarket prices. We show that the neglect of both fairness concerns and decision errors prevents ansatisfactory understanding of how competition affects prices. We conducted experiments whichndemonstrate that the introduction of even a very small amount of competition to a bilateralnexchange situation - by adding just one competitor - induces large behavioral changes amongnbuyers and sellers, causing large changes in market prices. Models that assume that all people arenself-interested and fully rational fail to explain these changes satisfactorily. In contrast, a modelnthat combines heterogeneous fairness concerns with decision errors predicts all comparative staticneffects of changes in competition correctly. Moreover, the combined model enables us to predictnthe entire distribution of prices in many different competitive situations remarkably well.