Artificial Intelligence (AI) systems are precious support for decision-making, with many applications also in the medical domain. The interaction between MDs and AI enjoys a renewed interest following the increased possibilities of deep learning devices. However, we still have limited evidence-based knowledge of the context, design, and psychological mechanisms that craft an optimal human-AI collaboration. In this multicentric study, 21 endoscopists reviewed 504 videos of lesions prospectively acquired from real colonoscopies. They were asked to provide an optical diagnosis with and without the assistance of an AI support system. Endoscopists were influenced by AI ([Formula: see text]), but not erratically: they followed the AI advice more when it was correct ([Formula: see text]) than incorrect ([Formula: see text]). Endoscopists achieved this outcome through a weighted integration of their and the AI opinions, considering the case-by-case estimations of the two reliabilities. This Bayesian-like rational behavior allowed the human-AI hybrid team to outperform both agents taken alone. We discuss the features of the human-AI interaction that determined this favorable outcome.