How to prove that the zero sum game's optimal (security) strategy do not change when payoff matrix increase by a constant factor

Question

I am not sure how to construct a proof for something so obvious, can someone give me some pointers on proving the following

Suppose a constant K is added to each element of a pay-off matrix related to a game. What effect will it have on the solution?

(a) The optimal (security) strategies of the players and the value of the game remain unchanged.

(b) The optimal (security) strategies of the players do not change while the value of the game will increase by K.

(c) The optimal (security) strategies of the players as well as game value will all change.

Prove assertion.

So we have a payoff matrix $A$, we add a constant $k$ to each element $A$ yielding $A + K$. Our optimal value (value of the game) is $J^* \in A$, since $A$ is increased by $k$, therefore $J^*$ is increased by $k$.

Finally, our optimal (security) strategy for original payoff matrix $A$ is by choosing maximum payoff for player 1 when player 2 picks the lowest payoff, and choosing the lowest payoff (lowest cost) for player 2 when player 1 picks the maximum payoff. Since adding a constant does not change the well orderness of each row or column, therefore the security strategy remains the same as before.

Is there anything I could do to make more argument more rigorous for this intuitively obvious result?

Answer 1

No. The argument can not be made more rigorous than it is now. It is an illusion to think that adding mathematical notation would make some argument more rigorous.

Answer 2

Here is an attempt to give an rigorous solution: let $x$, $y$ be the mixed strategies used by player 1 and 2. Let $E(x,y):= \sum_i\sum_jx_ia_{ij}y_j$ denote the payoff function. Then for the new game, the payoff function is given by, \begin{aligned}E*(x,y) &= \sum_i\sum_j(a_{ij}+k)x_iy_j \\ &= k + \sum_i\sum_ja_{ij}x_iy_j \\ &= k + E(x,y)\end{aligned} Now its clear that $\max_x\min_yE*(x,y) = k + \max_x\min_yE(x,y)$. Now both parts follow.