  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.
