scholarly journals Remarks on the Afriat's Theorem and the Monge-Kantorovich Problem

2013 ◽  
Author(s):  
Olga Vladimirovna Kudryavtseva ◽  
Alexander Viktorovich Kolesnikov ◽  
Tigran Nagapetyan
Keyword(s):  
Afriat’S Theorem ◽  
Kantorovich Problem

scholarly journals Remarks on Afriat’s theorem and the Monge–Kantorovich problem

2013 ◽  
Vol 49 (6) ◽  
pp. 501-505 ◽  
Author(s):  
Alexander V. Kolesnikov ◽  
Olga V. Kudryavtseva ◽  
Tigran Nagapetyan
Keyword(s):  
Afriat’S Theorem ◽  
Kantorovich Problem

scholarly journals ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS

2021 ◽  
pp. 1-37
Author(s):  
Florian F. Gunsilius
Keyword(s):  
Convergence Rate ◽  
Poincaré Inequality ◽  
Optimal Transportation ◽  
Matching Theory ◽  
Economic Research ◽  
Poincare Inequality ◽  
Optimal Transportation Problem ◽  
Minimax Rate ◽  
The Cost ◽  
Kantorovich Problem

The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

scholarly journals On the matrix Monge–Kantorovich problem

2019 ◽  
Vol 31 (4) ◽  
pp. 574-600 ◽  
Author(s):  
YONGXIN CHEN ◽  
WILFRID GANGBO ◽  
TRYPHON T. GEORGIOU ◽  
ALLEN TANNENBAUM
Keyword(s):  
Euclidean Distance ◽  
Strong Duality ◽  
Wasserstein Distance ◽  
Positive Definite ◽  
Type Result ◽  
Density Matrices ◽  
Wirtinger Inequality ◽  
The Matrix ◽  
The Cost ◽  
Kantorovich Problem

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

On a Kantorovich Problem with a Density Constraint

2018 ◽  
Vol 104 (1-2) ◽  
pp. 39-47 ◽  
Author(s):  
A. N. Doledenok
Keyword(s):  
Kantorovich Problem

scholarly journals Revealed Preference in a Discrete Consumption Space

2013 ◽  
Vol 5 (1) ◽  
pp. 28-34 ◽  
Author(s):  
Matthew Polisson ◽  
John K.-H Quah
Keyword(s):  
Utility Function ◽  
Cost Efficiency ◽  
Revealed Preference ◽  
Efficiency Cost ◽  
Afriat’S Theorem ◽  
Continuous Quantities

We show that an agent maximizing some utility function on a discrete (as opposed to continuous) consumption space will obey the generalized axiom of revealed preference (GARP), so long as the agent obeys cost efficiency. Cost efficiency will hold if there is some good, outside the set of goods being studied by the modeler, that can be consumed by the agent in continuous quantities. An application of Afriat's Theorem then guarantees that there is a strictly increasing utility function on the discrete consumption space that rationalizes price and demand observations. (JEL D11)

scholarly journals From Nash to Cournot–Nash equilibria via the Monge–Kantorovich problem

2014 ◽  
Vol 372 (2028) ◽  
pp. 20130398 ◽  
Author(s):  
Adrien Blanchet ◽  
Guillaume Carlier
Keyword(s):  
Nash Equilibria ◽  
Transport Theory ◽  
Optimal Transport ◽  
Strategic Interactions ◽  
Complex Issue ◽  
Optimal Transport Theory ◽  
Kantorovich Problem

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of N player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article, we emphasize the role of optimal transport theory in (i) the passage from Nash to Cournot–Nash equilibria as the number of players tends to infinity and (ii) the analysis of Cournot–Nash equilibria.

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