What is the formal definition of game, as used in game theory?

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I am very much interested in a formal, rigorous definition of game, as used in the mathematical discipline of game theory. Now, I know that the philosopher Wittgenstein said that some words, like the word "game", can't be given an adequate definition, merely a family resemblance. However, I am still interested in such a formal definition, if one exists. Has anyone formally defined a game in a mathematical way, in some book or paper?