Technology Category
Search and Optimization: Mixed Discrete/Continuous SearchMultiagent Systems: Mechanism DesignGame Theory and Economic Paradigms: Mechanism Design
Discrete Screening
🤖 AI Summary
This paper studies optimal screening mechanism design when both agent types and contracts are discrete, and the principal’s valuation function is discrete and strictly concave. To address the excessive restrictiveness of conventional monotonicity assumptions, we introduce the notion of “Δ-O rationalizability” and establish its robustness over open belief sets. Employing discrete first-order conditions, a virtual cost function, and constraint simplification techniques, we show that the optimal contract menu contains at most two adjacent contracts. Our approach dispenses with continuity assumptions, reproduces canonical simplification results from continuous models, and ensures solution uniqueness, weak monotonicity, and implementability. The key contribution is the first derivation—within a purely discrete framework—of an optimal menu structure fully consistent with that of continuous models, without requiring integer-valued types and applicable to arbitrary (including non-integer) type spaces.
Technology Category
Application Category
📝 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 extit{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 extit{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 extit{$Δ$-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.
Problem
Research questions and friction points this paper is trying to address.
Screens agents with discrete types using quantity-transfer menus
Analyzes non-unique solutions under discrete first-order conditions
Introduces robust rationalizability concept for optimal contract selection
Innovation
Methods, ideas, or system contributions that make the work stand out.
Discrete screening with non-integer cost types
First-order approach under discrete concavity
Δ-O Rationalizability for robust contract selection
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