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activity amount assert assignment problem assume basic solution called chapter choose column computation condition consider convex combination convex cone convex polytope COROLLARY corresponding define definition demand denote dual problem duality theorem economic edges entries equation example exchange matrix exists feasible vector finite cone flow follows Fundamental Theorem Game Theory give given hence independent subset inequalities integers intensity vector irreducible subsets LEMMA LIBRARIES STANFORD UNIVERSITY linear combination linear model linear programming linear programming problem linear space matrix game maximize maximum minimize mixed strategies n-vector nodes non-negative solution Note optimal strategies path pay-off matrix play positive precisely productive PROOF prove pure strategies rank real numbers saddle point saiavaan satisfy semi-positive solution sets of optimal simplex method solve STANFORD UNIVERSITY LIBRARIES strategy for player strategy x supply suppose symmetric game tableau theory unique verifies zero