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additional variables artificial variable assume basic feasible solution basic main basic solutions basic variables become negative best strategy bottom row boxes chance moves choice choose coefficients column constraints convex decide decisions defined dual problem dual system Duality easily seen equal equations equilibrium points example fact final tableau find non-negative hence indicated inequalities instance ith month John Wiley latter Linear Programming problem Main Theorem mathematics maximin minimax minimizing player mixed strategies non-basic variables non-negative values non-zero-sum games Noughts and Crosses objective function obtain Operational Research opponent optimal strategy outcome P(yi parlour games pay-off table perfect information pivot possible pound notes primal system produced proof proved pure strategies remains saddle point scheme shadow costs shadow prices similarly Simplex Method smaller solving straight line tableau transformation Theory of Games unoccupied cells value zero wool Xn+s yards of type zero-sum game