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We show that Nash equilibrium components are universal for the collection of connected polyhedral sets. More precisely for every polyhedral set we construct a so-called binary game-a game where all players have two pure strategies and a common utility function with values either zero or one-whose success set (the set of strategy profiles where the maximal payoff of one is indeed achieved) is homeomorphic to the given polyhedral set. Since compact semi-algebraic sets can be triangulated, a similar result follows for the collection of connected compact semi-algebraic sets.
We discuss implications of our results for the strategic stability of success sets, and use the results to construct a Nash component with index k for any fixed integer k.