  Hybrid organic-inorganic perovskites, such as methyl-ammonium lead tri-iodide
(MAPbI$_3$), are interesting candidates for efficient absorber materials in
next-generation solar cells, partly due to an unusual combination of low
exciton binding energy and strong optical absorption. Excitonic effects in this
material have been subject to debate both for experiment and theory, indicating
a need for better understanding of the screening mechanisms that act upon the
electron-hole interaction. Here we use cutting-edge first-principles
theoretical spectroscopy, based on density-functional and many-body
perturbation theory, to study atomic geometries, electronic structure, and
optical properties of three MAPbI$_3$ polymorphs and find good agreement with
earlier results and experiment. We then study the influence of free electrons
on the electron-hole interaction and show that this explains consistently
smaller exciton binding energies, compared to those in the material without
free electrons. Interestingly, we also find that the absorption line shape
strongly resembles that of the spectrum without free electrons up to high free
electron concentrations. We explain this unexpected behavior by formation of
Mahan excitons that dominate the absorption edge, making it robust against
free-electron induced changes observed in other semiconductors.
