When is multidimensional screening a convex program?
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 SpenceMirrlees singlecrossing) 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 nonnegative crosscurvature: it is also (i) necessary and sufficient to guarantee convexity of the set of bconvex 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 nonnegatively crosscurved models.
 Publication:

arXiv eprints
 Pub Date:
 December 2009
 DOI:
 10.48550/arXiv.0912.3033
 arXiv:
 arXiv:0912.3033
 Bibcode:
 2009arXiv0912.3033F
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Analysis of PDEs;
 91B24;
 90B50;
 90C25;
 49N30;
 58E17;
 35Q80
 EPrint:
 23 pages