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alternating assertion AX g basic coefficient completes the proof complex numbers components consistent contains convex hull convex set corresponding Dantzig David Gale defined denote directed paths dual linear programs dual programs dual systems duality edge elements entries equated constraints equivalent exists face feasible vector finite number follows graph Hence hyperplane IEMMA implies incidence matrix integral intersection Lemma linear combination linear functionals linear inequalities linear subspace linearly independent Math Mathematics matrix game maximization problem maximum minimal minimax theorem mixed strategy nodes non-negative non-zero nonvacuous normed linear space optimal strategies optimal vectors pair of dual paper polyhedral convex cone polyhedron positive Princeton probability vector prove real numbers satisfies solution strict inequalities submodel subset Suppose systems of equated systems of linear Theory of Games tion variables vertex vertices X-space zero
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