Screening with Discrete Types and Discrete Contracts

We study screening when both the agent's types and the contracts available to the principal are finite: Quantities and transfers are drawn from a finite integer grid, while marginal costs lie on a slightly off-set grid, with integer costs as a limit case. We compare two solution concepts, equilibrium and rationalizability. Since rationalizability does not feature an equilibrium tie-breaking convention, we analyze screening under weak and under strict incentives. We develop a discrete first-order approach and characterize the optimal menus for full-support log-concave beliefs. Optimal quantities need not be unique, but at most two adjacent quantities are optimal and multiplicity is a knife-edge case in beliefs. Our rationalizability solution concept, Delta-O rationalizability, restricts the principal's beliefs over types and requires her best replies to be robust to belief perturbations. Two rounds of elimination suffice, and the rationalizable and the equilibrium outcomes essentially coincide: The predictions of equilibrium survive without a common prior, while avoiding a tie-breaking convention.