  We present a study of the thermoelectric (Seebeck and Nernst) response in
heavily overdoped, non-superconducting La$_{1.67}$Sr$_{0.33}$CuO$_4$. In spite
of the electron-like curvature of the Fermi surface, the Seebeck coefficient is
positive at low temperatures. Such a feature, previously observed in copper,
silver, gold and lithium, is caused by a non-trivial energy dependence of the
scattering time. We argue that this feature implies a strong asymmetry between
the lifetime of occupied and unoccupied states along the zone diagonals and
such an electron-hole asymmetry impedes formation of Cooper pairs along the
nodal direction in the superconducting ground state emerging at lower doping
levels.
