We consider here an idealized infinite rectangular grid of roads with
a translationally symmetric O-D distribution. The total cost of travel
on all links approaching each junction is approximated by a quadratic
function of the four flows N, S, E, and W at that junction. If any of
the four eigenvalves of this quadratic form is negative, the system o
ptimal assignment problem is non-convex. If there are economies of sca
le (due possibly to construction costs) then all eigenvalues could be
negative and the optimal assignment will Lead to a hierarchical type o
f flow distribution (city streets, arterials, freeways, etc.). If cost
s arise only from congestion, however, it is possible that one or more
of the eigenvalues is negative particularly if the cost of travel N,
for example, is more sensitive to the flows E and/or W than to the flo
w N, or is more sensitive to the flow S than. N. If it is more sensiti
ve to the flow E-W an efficient assignment would seem to be one in whi
ch the space is divided into subregions such that in certain subregion
s traffic will be predominantly N or S and in other subregions it is p
redominately E or W. The optimal assignment is expected to be highly u
nstable to changes in the O-D distribution. If it is more sensitive to
the flow S, a user optimal assignment may be stable and translational
ly symmetric but not the system optimal. The conclusion is that a non-
convex assignment problem is not only a computational nightmare, but m
ay be inconsistent with social objective or impractical to implement.