29,486 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
Regularity of optimal transport maps on multiple products of spheres
This article addresses regularity of optimal transport maps for cost="squared
distance" on Riemannian manifolds that are products of arbitrarily many round
spheres with arbitrary sizes and dimensions. Such manifolds are known to be
non-negatively cross-curved [KM2]. Under boundedness and non-vanishing
assumptions on the transfered source and target densities we show that optimal
maps stay away from the cut-locus (where the cost exhibits singularity), and
obtain injectivity and continuity of optimal maps. This together with the
result of Liu, Trudinger and Wang [LTW] also implies higher regularity
(C^{1,\alpha}/C^\infty) of optimal maps for more smooth (C^\alpha /C^\infty))
densities. These are the first global regularity results which we are aware of
concerning optimal maps on non-flat Riemannian manifolds which possess some
vanishing sectional curvatures. Moreover, such product manifolds have potential
relevance in statistics (see [S]) and in statistical mechanics (where the state
of a system consisting of many spins is classically modeled by a point in the
phase space obtained by taking many products of spheres). For the proof we
apply and extend the method developed in [FKM1], where we showed injectivity
and continuity of optimal maps on domains in R^n for smooth non-negatively
cross-curved cost. The major obstacle in the present paper is to deal with the
non-trivial cut-locus and the presence of flat directions.Comment: 35 pages, 4 figure
Entropy of the Randall-Sundrum brane world with the generalized uncertainty principle
By introducing the generalized uncertainty principle, we calculate the
entropy of the bulk scalar field on the Randall-Sundrum brane background
without any cutoff. We obtain the entropy of the massive scalar field
proportional to the horizon area. Here, we observe that the mass contribution
to the entropy exists in contrast to all previous results, which is independent
of the mass of the scalar field, of the usual black hole cases with the
generalized uncertainty principle.Comment: 12 pages. The improved version published in Phys. Rev.
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