Fields Colloquium

 This is a featured talk

The principal-agent problem is an important paradigm in economic theory for studying the value of private information: the nonlinear pricing problem faced by a monopolist is one example; others include optimal taxation and auction design. Unidimensional versions of such problems have been solved in Mussa and Rosen (1978), and Nobel prize-winning work of Becker (1973), Mirrlees (1971), Myerson (1982), and Spence (1973).

For multidimensional spaces of consumers (i.e. agents) and products, Rochet and Chone (1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences are bilinear in the product and agent parameters. This optimization corresponds mathematically to a convexity-constrained obstacle problem whose endogenous obstacle has limited regularity. The solution is divided into multiple regions, according to the rank of the Hessian of the optimizer.

If the monopolists costs grow quadratically with the product type we show that a partially smooth free boundary delineates the region where it becomes efficient to customize products for individual buyers. We give the first complete solution of the problem on square domains, and discover new transitions from unbunched to targeted and from targeted to blunt bunching as market conditions become more and more favorable to the seller.

Based on works with (variously) Kelvin Shuangjian Zhang, Cale Rankin, and Lucas O'Brien:

Math. Models Methods Appl. Sci. 34 (2024) 2351-2394, J. Convex Anal. (Rockafellar 90 Issue), 32 (2) (2025) 579-584, arXiv 2303.04937, arxiv 2412.15505, and arXiv 2603.14100

CRM-PIMS-Fields 2026 Prize Lecture