Find a Nash equilibrium solver

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The solvers I know so far are designed only to allow payoffs as given numbers. But is there a solver allowing users to type payoffs as variables?

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Not that I am aware of. You could always write a script that generates instances of a parametrically defined game for an existing solver and then do a parameter sweep. For more than two players, even describing the set of equilibria for a given game (with all payoffs specified as actual numbers) is highly non-trivial, and given the state-of-the-art what you are asking for is rather unrealistic (except maybe for very small games).