Mechanism design with two alternatives in Quasi-linear environment

We study mechanism design in quasi-linear environments when there are two alternatives. We show that under a mild range condition, every implementable deterministic allocation rule is a generalized utility function maximizer. In unbounded domains, if we replace our range condition by an independence condition, then every implementable deterministic allocation rule is an affine maximizer. Our results extend Roberts' affine maximizer theorem (Roberts, 1979) to the case of two alternatives.