Providing convincing explanations to accompany recommendationsis a key issue in decision-aiding. In the context of decisionsinvolving multiple criteria, the problem is made very difficultbecause the decision model itself may involve a complex process. Inthis paper, we investigate the following issue: when the preferentialinformation provided by the user is incomplete, is there a principledway to define what is a “simple” explanation for a recommendedchoice? We argue first that explanations may necessitate differentlevels of detail. Next, we show that even when a detailed explanationis necessary, it is possible to distinguish explanations of differentlevels of complexity. Our results rely on an original connection weestablish between the “mechanics” required to compute supportingcoalitions of criteria and the simplicity of the explanation.