On the problem of optimal fair exchange

TL;DR

This paper formulates the optimal fair exchange problem as an optimal transportation problem, proving existence, duality, and deriving a formula for the optimal value, especially in regular topological spaces.

Contribution

It establishes a theoretical framework connecting optimal exchange with transportation problems and provides new existence and duality results.

Findings

Proved existence of optimal solutions in regular topological spaces.

Established a duality theorem for the optimal exchange problem.

Derived a formula for the optimal value using the connection to transportation problems.

Abstract

We consider the problem of optimal exchange which can be formulated as a kind of optimal transportation problem. The existence of an optimal solution and a duality theorem for the optimal exchange problem are proved in case of completely regular topological spaces. We show the connection between the problem of optimal exchange and the optimal transportation problem with density constraints. With the use of this connection we obtain a formula for the optimal value in the problem of optimal exchange.