  We study the following repeated non-atomic routing game. In every round,
nature chooses a state in an i.i.d. manner according to a publicly known
distribution, which influences link latency functions. The system planner makes
private route recommendations to participating agents, which constitute a fixed
fraction, according to a publicly known signaling strategy. The participating
agents choose between obeying or not obeying the recommendation according to
cumulative regret of the participating agent population in the previous round.
The non-participating agents choose route according to myopic best response to
a calibrated forecast of the routing decisions of the participating agents. We
show that, for parallel networks, if the planner's signal strategy satisfies
the obedience condition, then, almost surely, the link flows are asymptotically
consistent with the Bayes correlated equilibrium induced by the signaling
strategy.
