I characterize the set of implementable allocation functions in the standard one dimensional mechanism design environment where the relationship between private information and payoffs is possibly non-monotone. The characterization is useful in two aspects. First it leads to a rather mild condition under which individual rationality follows directly from incentive compatibility. Second, it can be conveniently used to determine the implementability of allocation functions in certain novel applications. In particular I show that neither monotonicity of allocations, nor the monotone differences property on values is necessary for implementation. In an application, I study a buyer-seller relationship where the buyer’s value displays habit formation, which enters into his payoff through a commonly known parameter. Habit implies that the agent’s value is a nonmonotone function of his type and that monotone diferences condition can not be satisfied for all parameters. For a set of parameters, the seller-optimal mechanism is nonmonotone: the seller screens out low and high types.
