  We apply the fermion commutation technique for composite bosons to
polariton-polariton scattering in semiconductor planar microcavities.
Derivations are presented in a simple and physically transparent fashion. A
procedure of orthogonolization of the initial and final two-exciton state
wavefunctions is used to calculate the effective scattering matrix elements and
the scattering rates. We show how the bosonic stimulation of the scattering
appears in this full fermionic approach whose equivalence to the bosonization
method is thus demonstrated in the regime of low exciton density. We find an
additional contribution to polariton-polariton scattering due to the exciton
oscillator strength saturation, which we analyze as well. We present a theory
of the polariton-polariton scattering with opposite spin orientations and show
that this scattering process takes place mainly via dark excitonic states.
Analytical estimations of the effective scattering amplitudes are given.
