  We study a tropical linear regression problem consisting in finding the best
approximation of a set of points by a tropical hyperplane. We establish a
strong duality theorem, showing that the value of this problem coincides with
the maximal radius of a Hilbert's ball included in a tropical polyhedron. We
also show that this regression problem is polynomial-time equivalent to mean
payoff games. We illustrate our results by solving an inverse problem from
auction theory. In this setting, a tropical hyperplane represents the set of
equilibrium prices. Tropical linear regression allows us to quantify the
distance of a market to the set of equilibria, and infer secret preferences of
a decision maker.
