Template-Type: ReDIF-Paper 1.0
Author-Name: Alejandro Francetich
Author-Name-First: Alejandro
Author-Name-Last: Francetich
Author-Name: Burkhard C. Schipper
Author-Name-First: Burkhard C.
Author-Name-Last: Schipper
Author-Workplace-Name: Department of Economics, University of California Davis
Title: Discrete Screening
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{$\Delta$-O Rationalizability}, and show that the set of $\Delta$-O rationalizable menus coincides with the set of usual optimal contracts---possibly augmented to include irrelevant contracts.
Length: 27
File-URL: https://repec.dss.ucdavis.edu/files/i0akxmuet3xg07scyjufv30n8wbv/disconsceen6.pdf
File-Format: application/pdf
Number: 375
Classification-JEL: D82
KeyWords: Screening, discrete concave optimization, rationalizability, level-$k$ reasoning
Creation-Date: 20251023
Handle: RePEc:cda:wpaper:375