The transport problem for non-additive measures

Non-additive measures, also known as fuzzy measures, capacities, and monotonic games, are increasingly used in different fields. Applications have been built within computer science and artificial intelligence related to e.g., decision making, image processing, machine learning for both classification, and regression. Tools for measure identification have been built. In short, as non-additive measures are more general than additive ones (i.e., than probabilities), they have better modeling capabilities allowing to model situations and problems that cannot be modeled by the latter. See e.g., the application of non-additive measures and the Choquet integral to model both Ellsberg paradox and Allais paradox. Because of that, there is an increasing need to analyze non-additive measures. The optimal transport problem is a useful tool for probabilities. It is used to define the Wasserstein distance, and also to solve problems as data matching. In this work with tackle the problem of defining the optimal transport problem for non-additive measures. We consider that it is necessary to provide appropriate definitions with a flavour similar to the one for probabilities, but for non-additive measures. The solutions should generalize the classical ones. We provide definitions based on the Möbius transform, but also based on the (max,+)-transform that we consider that has some advantages. We will discuss in this paper the problems that arise to define the transport problem for non-additive measures, and provide ways to solve them. We include the definitions of the optimal transport problem, and prove some properties. We also conclude providing a definition of the Wasserstein distance for non-additive measures