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Hybrid organic-inorganic perovskites, such as methyl-ammonium lead tri-iodide(MAPbI$_3$), are interesting candidates for efficient absorber materials innext-generation solar cells, partly due to an unusual combination of lowexciton binding energy and strong optical absorption. Excitonic effects in thismaterial have been subject to debate both for experiment and theory, indicatinga need for better understanding of the screening mechanisms that act upon theelectron-hole interaction. Here we use cutting-edge first-principlestheoretical spectroscopy, based on density-functional and many-bodyperturbation theory, to study atomic geometries, electronic structure, andoptical properties of three MAPbI$_3$ polymorphs and find good agreement withearlier results and experiment. We then study the influence of free electronson the electron-hole interaction and show that this explains consistentlysmaller exciton binding energies, compared to those in the material withoutfree electrons. Interestingly, we also find that the absorption line shapestrongly resembles that of the spectrum without free electrons up to high freeelectron concentrations. We explain this unexpected behavior by formation ofMahan excitons that dominate the absorption edge, making it robust againstfree-electron induced changes observed in other semiconductors.

Free-electron effects on optical absorption of hybrid perovskite CH$_3$NH$_3$PbI$_3$ from first principles