Священная обитель разврата. Английская версия.

How is it correct to write about an already having been played game "Thank you for the game" or "thank you for a game". I seem "the", but am not sure, because some ones write "a"...

"The" is more correct, because you're thanking for a specific session of game, not just for the existence of a game :)

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Example: you can say "thank you for a game" to a game company, like Blizzard :) IMHO