When is multidimensional screening a convex program?

Abstract

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 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.

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

“…Therefore we have the following well-posedness result (existence essentially follows from [10] and uniqueness from [19]; for the sake of completeness we give a short proof).…”

“…− b(., y(x)) is minimized at x so that if b is differentiable and u is differentiable at x (and in fact, there are well-known conditions on b which guarantee a priori that b-convex functions are differentiable at least almost everywhere, see section 3) then one obtains that ∇u(x) = ∂ x b(x, y(x)) which is a local necessary condition for (2.3) to hold. If we go one step further, as was done in the seminal work of Figalli, Kim and McCann [19], by assuming that the relation q = ∂ x b(x, y) can be inverted in the sense that q = ∂ x b(x, y) ⇐⇒ y = y b (x, q) for some map y b , then one can actually deduce y(x) from the knowledge of q(x) := ∇u(x) by the relation y(x) = y b (x, q(x)). Replacing…”

“…Note that inequality (3.4) is an equality when x ′ = x so, under assumption (A2), whenever u is differentiable at x, we have q(x) = ∇u(x). Thanks to assumption (A3), first outlined by Figalli, Kim and McCann [19], the constraint (…”

“…In a recent pathbreaking article, Figalli, Kim and McCann [19] identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra's iterated projection algorithm to handle the b-convexity constraint in the framework of [19]. Our method also turns out to be simple for convex envelope computations.…”

“…One then has to minimize a certain integral functional among b-convex functions; general existence results can be proved but not much more in general and this has been a serious limitation for understanding principal-agent models in several dimensions. It is only recently with the pathbreaking work of Figalli, Kim and McCann [19] that some conditions (intimately related to the conditions of Ma, Trudinger and Wang [24] for the regularity of optimal transport maps) were identified to make the principal-agent problem a convex program and, in the first place, the set of b-convex functions a convex set (see section 4 below for an elementary presentation in a special case). This raised the hope to use convex optimization algorithms to solve numerically some convex minimization problems posed over the set of b-convex functions, provided it is convex.…”