Discrete Screening
Alejandro Francetich, Burkhard C. Schipper
TL;DR
This paper studies a principal–agent screening problem with discrete types, quantities, and transfers, modeling agent marginal costs as non-integer to replicate continuum insight while keeping contracts discrete. It develops a discrete first-order framework using forward/backward derivatives and a round-up rent from integer transfers to ensure strict incentives, showing that each type has at most two adjacent optimal quantities and that uniqueness is knife-edged. It demonstrates that log-concave beliefs guarantee monotone allocations only weakly unless the principal’s value function $v(q)$ is restricted to limit concavity, and it introduces $\Delta$-O rationalizability, a robustness-based belief-reduction notion that yields the same set of optimal menus as standard analysis (potentially augmented with irrelevant contracts). By connecting to Francetich and Schipper (2025), the paper positions $\Delta$-O rationalizability as a full-awareness benchmark for screening with unawareness, enabling tractable solutions without relying on equilibrium tie-breaking. Overall, the discrete-concave screening framework provides sharp predictions for contract design under finite, non-integer marginal costs and offers a robust solution concept for incomplete-information settings.
Abstract
We consider a principal who wishes to screen an agent with \emph{discrete} types by offering a menu of \emph{discrete} quantities and \emph{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 \textit{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 \textit{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 \textit{$Δ$-O Rationalizability}, and show that the set of $Δ$-O rationalizable menus coincides with the set of usual optimal contracts -- possibly augmented to include irrelevant contracts.