Valid operations to the value of a matrix game

Question

Suppose I have a 2-player zero-sum matrix game with the payoff matrix

$U = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

Then, I can express the value of the game as

$v = value(\begin{pmatrix} a & b \\ c & d \end{pmatrix})$.

How do I know which operations I am allowed to use on this equation? For example, I read that I can add a constant to the value and then to all elements of $U$:

$v = value(\begin{pmatrix} a & b \\ c & d \end{pmatrix}) \Leftrightarrow v + \delta = value(\begin{pmatrix} a + \delta & b + \delta \\ c + \delta & d + \delta \end{pmatrix})$

Or that I could do the same with multiplication:

$v = value(\begin{pmatrix} a & b \\ c & d \end{pmatrix}) \Leftrightarrow v \alpha = value(\begin{pmatrix} \alpha a & \alpha b \\ \alpha c & \alpha d \end{pmatrix})$

Why is that, and are there further valid operations I can apply to both sides to obtain equivalent equations?

Thanks in advance!

Answer 1

Nope! These are the only ones. This is because utility function is an affine multivariable function...