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We introduce an extension of the Optimal Transport problem when multiplecosts are involved. Considering each cost as an agent, we aim to share equallybetween agents the work of transporting one distribution to another. To do so,we minimize the transportation cost of the agent who works the most. Anotherpoint of view is when the goal is to partition equitably goods between agentsaccording to their heterogeneous preferences. Here we aim to maximize theutility of the least advantaged agent. This is a fair division problem. LikeOptimal Transport, the problem can be cast as a linear optimization problem.When there is only one agent, we recover the Optimal Transport problem. Whentwo agents are considered, we are able to recover Integral Probability Metricsdefined by $\alpha$-H\"older functions, which include the widely-known Dudleymetric. To the best of our knowledge, this is the first time a link is givenbetween the Dudley metric and Optimal Transport. We provide an entropicregularization of that problem which leads to an alternative algorithm fasterthan the standard linear program.

Equitable and Optimal Transport with Multiple Agents