We propose a model of Kidney-Exchange that incorporates the main European institutional features. We assume that patients do not consider all compatible kidneys homogeneous and patients are endowed with reservation values over the minimal quality of the kidney they may receive. Under feasibility constraints, patients’ truthful revelation of reservation values is incompatible with constrained efficiency. In the light of this result, we introduce an alternative behavioral assumption on patients’ incentives. Patients choose their revelation strategies as to “protect” themselves from bad outcomes and use a lexicographic refinement of maximin strategies. In this environment, if exchanges are pairwise, then priority rules or rules that maximize a fixed ordering provide incentives for the patients to report their true reservation values. The positive result vanishes if larger exchanges are admitted.