The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig (1951) is known to be incomplete. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof, more directly than Adler (2013). We also extend Dantzig's game so that any max-min strategy gives either an optimal LP solution or shows that none exists.

翻译：零和博弈的最小最大定理可以轻松地从线性规划的强对偶定理中证明出来。对于反向推导，Dantzig（1951）提出的标准证明被认为是不完整的。我们解释并结合了关于非负变量线性方程组求解的经典定理，给出了一个比Adler（2013）更直接的正确替代证明。我们还扩展了Dantzig的博弈，使得任何最大最小策略都能给出最优线性规划解，或者表明解不存在。