To prove minimax for an arbitrary matrix $A$, first shift $A$ so that all entries are positive. Introduce an antisymmetric matrix built from $A$, for example $$ B= \begin{pmatrix} 0 & A & -1\\ -A^T & 0 & 1\\ 1 & -1 & 0 \end{pmatrix}. $$ An optimal strategy in the antisymmetric game $B$ yields optimal mixed strategies for both players in the original game $A$.

References

• [MFoGT, Section 2.8, Exercise 10(2)] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Deduce finite minimax from values of antisymmetric games.