# On the convergence of no-regret learning in selfish routing

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Walid Krichene,
Benjamin Drighès,
Alexandre Bayen
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Proceedings of the 31st International Conference on Machine Learning, PMLR 32(2):163-171, 2014.

#### Abstract

We study the repeated, non-atomic routing game, in which selfish players make a sequence of routing decisions. We consider a model in which players use regret-minimizing algorithms as the learning mechanism, and study the resulting dynamics. We are concerned in particular with the convergence to the set of Nash equilibria of the routing game. No-regret learning algorithms are known to guarantee convergence of a subsequence of population strategies. We are concerned with convergence of the actual sequence. We show that convergence holds for a large class of online learning algorithms, inspired from the continuous-time replicator dynamics. In particular, the discounted Hedge algorithm is proved to belong to this class, which guarantees its convergence.

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