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Journal Articles SIAM Journal on Discrete Mathematics Year : 2023

Tropical linear regression and mean payoff games: or, how to measure the distance to equilibria

Marianne Akian
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Stéphane Gaubert
Yang Qi
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Omar Saadi
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Abstract

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.

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hal-03504875 , version 1 (29-12-2021)

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Marianne Akian, Stéphane Gaubert, Yang Qi, Omar Saadi. Tropical linear regression and mean payoff games: or, how to measure the distance to equilibria. SIAM Journal on Discrete Mathematics, 2023, 37 (2), pp.632-674. ⟨10.1137/21M1428297⟩. ⟨hal-03504875⟩
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