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Is the game only compatible with Windows 8 and up? I have the latest directX installed on my computer (Windows 7- directX11) so I'm not sure what the problem is... the missing component error won't disappear.

If you have the lastest version of directX for your windows then there shouldn't be a problem. Did you make sure to keep the files together after you've extracted them in a new folder? The application file "Tyrania" or any of the other files inside the zip should always be together in a folder.