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. $f(\bar A) > f(A)$ if $\bar A > A$ ; 2. $f(\tilde A) = f(A)$ if $\tilde A$ 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. $f(x) > x$ 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.