A proof that given a pair of infinite metric spaces and a pair of respective finite mixed strategies on them, a separable game exists with bounded continuous payoff on their Cartesian product such that the given strategies constitute the unique solution of the game. If the spaces are identical, corresponding to any given finite mixture, a symmetric polynomial-like game can be obtained with bounded (skew-symmetric) continuous payoff so that the given strategy is the only optimal one. If the spaces are bounded subspaces of Euclidean n-space with sufficiently many cluster points in their closures, the payoff can be a polynomial and have the desired property.
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