Discrete screening

We consider a principal who wishes to screen an agent with discrete types by offering a menu of discrete quantities and discrete transfers. We assume that the principal's valuation is discrete strictly concave and use a discrete first-order approach. We model the agent's cost types as non-integer, with integer types as a limit case. Our modeling of cost types allows us to replicate the typical constraint-simplification results and thus to emulate the well-treaded steps of screening under a continuum of contracts. We show that the solutions to the discrete F.O.C.s need not be unique even under discrete strict concavity, but we also show that there cannot be more than two optimal contract quantities for each type, and that-if there are two-they must be adjacent. Moreover, we can only ensure weak monotonicity of the quantities even if virtual costs are strictly monotone, unless we limit the "degree of concavity" of the principal's utility. Our discrete screening approach facilitates the use of rationalizability to solve the screening problem. We introduce a rationalizability notion featuring robustness with respect to an open set of beliefs over types called ∆-O Rationalizability, and show that the set of ∆-O rationalizable menus coincides with the set of usual optimal contracts-possibly augmented to include irrelevant contracts.