Adv. Math. Econ. Volume 3, pp.97-124 (2001)

The Monge-Kantorovich problems and stochastic preference relations

Vladimir L. Levin
Central Economics and Mathematics Institute of Russian Academy of Sciences, 47 Nakhimovskii Prospect, 117418 Moscow, Russia

A method in mathematical utility theory based on the duality theorems for the general Monge-Kantorovich problem proves to be fruitful in various parts of mathematical economics. In the present survey we give further development of that method and study its applications to closed preference relations (resp. correspondences) on a topological space (resp. between two topological spaces) and to their convex stochastic extensions on the corresponding spaces of lotteries. Among other results, we prove characterization theorems:
- for a functionally closed preorder (Theorem 2.1);
- for the corresponding strong stochastic dominance (Theorems 2.2 and 3.1);
- for the convex stochastic extension of an arbitrary closed correspondence between two topological spaces (Theorem 4.1).

Radon measure, Monge-Kantorovich problem, functionally closed preorder, strong stochastic dominance, isotone function, utility function, closed preference relation, closed correspondence.