  We study the newly discovered Pt phosphides $A$Pt$_3$P ($A$=Sr, Ca, La) [ T.
Takayama et al. Phys. Rev. Lett. 108, 237001 (2012)] using first-principles
calculations and Migdal-Eliashberg theory. Given the remarkable agreement with
the experiment, we exclude the charge-density wave scenario proposed by
previous first-principles calculations, and give conclusive answers concerning
the superconducting state in these materials. The pairing increases from La to
Ca and Sr due to changes in the electron-phonon matrix elements and
low-frequency phonons. Although we find that all three compounds are well
described by conventional s-wave superconductivity and spin-orbit coupling of
Pt plays a marginal role, we show that it could be possible to tune the
structure from centrosymmetric to noncentrosymmetric opening new perspectives
towards the understanding of unconventional superconductivity.
