This question is not focussed on video games but games in general. I went to a boardgame trade fair yesterday and asked myself if there is a way to calculate the fairness of a game. Sure, some of them require a good portion of luck, but it might be possible to calculate if some character is overpowered. Especially in role-playing games and trading card games. How, for example, can the creators of "Magic: The Gathering" make sure that there isn't the "one card that beats them all", given the impressive number of available cards?
Answer
Yes, it's theoretically possible - that's a good part of the game theory which deals with this subject.
However, it's only rarely practical, and even then mostly just for games which don't involve a randomiser (Chess, Reversi, Go and so on). Combinatorial explosion ensures that the theoretical time needed for such proofs for more complex games like Magic the Gathering can easily be several orders of magnitude longer than the current age of the universe.
In the end, for any non-trivial game you'll likely have to abandon the notion of proving balance or fairness of a game and instead go with a combination of common sense, designer instincts, game system reuse and throughout testing.
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