The family of Vickrey-Clarke-Groves (VCG) mechanisms is arguably the most celebrated achievement in truthful mechanism design. However, VCG mechanisms have their limitations. They only apply to optimization problems with a utilitarian objective function, and their output should optimize the objective function. For many optimization problems, finding the optimal output is computationally intractable. If we apply VCG mechanisms to polynomialtime algorithms that approximate the optimal solution, the resulting mechanisms may no longer be truthful. In light of these limitations, it is useful to study whether we can design a truthful non-VCG payment scheme that is computationally tractable for a given output method O. In this paper, we focus our attention on binary demand games in which the agents’ only available actions are to take part in the a game or not to. For these problems, we prove that a truthful mechanism M = (O, P) exists (with proper payment method P) if and only if O satis...