When is multidimensional screening a convex program?

TL;DR

This paper identifies a structural condition called non-negative cross-curvature that makes multidimensional screening problems convex, enabling better theoretical analysis and computational solutions for principal-agent models with aggregate preferences.

Contribution

It introduces the non-negative cross-curvature condition as necessary and sufficient for convexity in multidimensional screening, extending the theoretical framework and computational tractability.

Findings

Convexity of the principal's problem under non-negative cross-curvature.

Uniqueness and stability of optimal strategies are guaranteed.

Economic implications like high pricing strategies are analyzed.

Abstract

A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can choose only pure strategies, we identify a structural condition on the preference b(x,y) of agent type x for product type y -- and on the principal's costs c(y) -- which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal's problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal's optimum strategy -- and similarly of the strategy maximizing the expected welfare of the agents when the principal's profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang's necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.