  The problem of a car following a lead car driven with constant velocity is
considered. To derive the governing equations for the following car dynamics a
cost functional that ranks the outcomes of different driving strategies is
constructed, which applies to fairly general properties of the driver behavior.
Assuming the driver behavior to be rational, the existence of the Nash
equilibrium is proved. Rational driving is defined by supposing that a driver
corrects continuously the car motion to follow the optimal path minimizing the
cost functional. The corresponding car-following dynamics is described quite
generally by a boundary value problem based on the obtained extremal equations.
Linearization of these equations around the stationary state results in a
generalization of the widely used optimal velocity model. Moreover, it is shown
that there are conditions (the ``dense traffic'' limit) under which the
rational car dynamics comprises two stages, fast and slow. During the fast
stage a driver eliminates the velocity difference between the cars, the
subsequent slow stage optimizes the headway. In the ``dense traffic'' limit an
effective Hamiltonian description is constructed. This allows a more detailed
nonlinear analysis. Finally, the differences between rational and bounded
rational driver behavior are discussed. The latter, in particular, justifies
some basic assumptions used recently by the authors to construct a
car-following model lying beyond the frameworks of rationality.
