Axiomatic Definition of the Value of a Matrix Game

Let a real function $f$, whose argument is a matrix $A$, satisfy the following axioms:
1. $f(\mathbf{\bar A})\geq(A)$ if $\mathbf{ \bar A}\geq A$;
2. $f(\mathbf{\bar A})=f(A)$ if $A$ differs from $A$ only by a row, which is dominated by others;
3. $f(-A^T)=-f(A)$, the index $T$ stands for transposition;
4. $f(x)\geq x$ for a real number $x$.
Then $f(A)$ is the game value function. Axioms $1$–$4$ are independent. Another similar set of axioms is given.