32 research outputs found
When is multidimensional screening a convex program?
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.Comment: 23 page
Existence of solutions to principal-agent problems with adverse selection under minimal assumptions
We prove an existence result for the principal-agent problem with adverse
selection under general assumptions on preferences and allocation spaces.
Instead of assuming that the allocation space is finite-dimensional or compact,
we consider a more general coercivity condition which takes into account the
principal's cost and the agents' preferences. Our existence proof is simple and
flexible enough to adapt to partial participation models as well as to the case
of type-dependent budget constraints.Comment: 22 page
The Exchange Value Embedded In A Transport System
This paper shows that a well designed transport system has an embedded
exchange value by serving as a market for potential exchange between consumers.
Under suitable conditions, one can improve the welfare of consumers in the
system simply by allowing some exchange of goods between consumers during
transportation without incurring additional transportation costs. We propose an
explicit valuation formula to measure this exchange value for a given
compatible transport system. This value is always nonnegative and bounded from
above. Criteria based on transport structures, preferences and prices are
provided to determine the existence of a positive exchange value. Finally, we
study a new optimal transport problem with an objective taking into account of
both transportation cost and exchange value.Comment: 20 pages, 6 figure
A glimpse into the differential topology and geometry of optimal transport
This note exposes the differential topology and geometry underlying some of
the basic phenomena of optimal transportation. It surveys basic questions
concerning Monge maps and Kantorovich measures: existence and regularity of the
former, uniqueness of the latter, and estimates for the dimension of its
support, as well as the associated linear programming duality. It shows the
answers to these questions concern the differential geometry and topology of
the chosen transportation cost. It also establishes new connections --- some
heuristic and others rigorous --- based on the properties of the
cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page
Handling convexity-like constraints in variational problems
We provide a general framework to construct finite dimensional approximations
of the space of convex functions, which also applies to the space of c-convex
functions and to the space of support functions of convex bodies. We give
estimates of the distance between the approximation space and the admissible
set. This framework applies to the approximation of convex functions by
piecewise linear functions on a mesh of the domain and by other
finite-dimensional spaces such as tensor-product splines. We show how these
discretizations are well suited for the numerical solution of problems of
calculus of variations under convexity constraints. Our implementation relies
on proximal algorithms, and can be easily parallelized, thus making it
applicable to large scale problems in dimension two and three. We illustrate
the versatility and the efficiency of our approach on the numerical solution of
three problems in calculus of variation : 3D denoising, the principal agent
problem, and optimization within the class of convex bodies.Comment: 23 page