This paper deals with the implementation of Social Choice Functions in fair multiagent decision problems. In such problems the determination of the best alternatives often relies on the maximization of a non-utilitarian Social Welfare Function so as to account for equity. However, in such decision processes, agents may have incentive to misreport their preferences to obtain more favorable choices. It is well known that, for Social Choice Functions based on the maximization of an affine aggregator of individual utilities, we can preclude any manipulation by introducing payments (VCG mechanisms). Unfortunately such truthful mechanisms do not exist for non-affine maximizers (Roberts' Theorem). For this reason, we introduce here a notion of “almost-truthfulness” and investigate the existence of payments enabling the elaboration of almost-truthful mechanisms for non-additive Social Welfare Functions such as Social Gini Evaluation Functions used in fair optimization.