1,311 research outputs found
The intrinsic dynamics of optimal transport
The question of which costs admit unique optimizers in the Monge-Kantorovich
problem of optimal transportation between arbitrary probability densities is
investigated. For smooth costs and densities on compact manifolds, the only
known examples for which the optimal solution is always unique require at least
one of the two underlying spaces to be homeomorphic to a sphere. We introduce a
(multivalued) dynamics which the transportation cost induces between the target
and source space, for which the presence or absence of a sufficiently large set
of periodic trajectories plays a role in determining whether or not optimal
transport is necessarily unique. This insight allows us to construct smooth
costs on a pair of compact manifolds with arbitrary topology, so that the
optimal transportation between any pair of probility densities is unique.Comment: 33 pages, 4 figure
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
Optimal transportation, topology and uniqueness
The Monge-Kantorovich transportation problem involves optimizing with respect
to a given a cost function. Uniqueness is a fundamental open question about
which little is known when the cost function is smooth and the landscapes
containing the goods to be transported possess (non-trivial) topology. This
question turns out to be closely linked to a delicate problem (# 111) of
Birkhoff [14]: give a necessary and sufficient condition on the support of a
joint probability to guarantee extremality among all measures which share its
marginals. Fifty years of progress on Birkhoff's question culminate in Hestir
and Williams' necessary condition which is nearly sufficient for extremality;
we relax their subtle measurability hypotheses separating necessity from
sufficiency slightly, yet demonstrate by example that to be sufficient
certainly requires some measurability. Their condition amounts to the vanishing
of the measure \gamma outside a countable alternating sequence of graphs and
antigraphs in which no two graphs (or two antigraphs) have domains that
overlap, and where the domain of each graph / antigraph in the sequence
contains the range of the succeeding antigraph (respectively, graph). Such
sequences are called numbered limb systems. We then explain how this
characterization can be used to resolve the uniqueness of Kantorovich solutions
for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
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