  We consider a dynamic traffic routing game over an urban road network
involving a large number of drivers in which each driver selecting a particular
route is subject to a penalty that is affine in the logarithm of the number of
drivers selecting the same route. We show that the mean-field approximation of
such a game leads to the so-called linearly solvable Markov decision process,
implying that its mean-field equilibrium (MFE) can be found simply by solving a
finite-dimensional linear system backward in time. Based on this backward-only
characterization, it is further shown that the obtained MFE has the notable
property of strong time-consistency. A connection between the obtained MFE and
a particular class of fictitious play is also discussed.
