Lets say you have a matrix game, where the matrix $A$ is the matrix, the column player can choose a column, the row player a row, and the row player pays the column player $A_{i,j}$.
Assume we want to find a random vector describing how the row and column player should play, this must then be a random vector. The row player has random vector x, the column player random vector y.
It is then argued that the column-player should solve the problem:
$\max_x \min_y y^TAx$ , where the requirement is that the vectors are nonegative and sum to 1.
The row player should solve:
$\min_y \max_x y^TAx$.
Bot these problems can be transformed to linear programming problems. Now comes my question: Why is do this give the optimal strategy for the columnplayer? When he solves the problem he assumes that the row player wants to minimize his loss, but what if the row player does not do that?
Can it be shown that $\max_x \min_y y^TAx$ will give a larger value then for instance: $\max_x y^{*T}Ax$, where $y^*$ is any strategy for the row player?
I mean we calculate expected values, so according to our calculations, the row player should earn in avarage $\max_x \min_y y^TAx$ per round in the long run. But here he assumes something about the row player, in his calculation he has assumed that the row player knows his strategy and has minimized his own loss according to this, but what if the row player does not do this?
You seem to have things mixed up. You are using the slightly less usual setup where $A$ denotes a payment from player 1, the row player, to player 2, the column player. Assume that $x$ represents a mixed strategy of the row player, and $y$ of the column player. With this in mind, the optimal strategies of player 1 solve the following problem:
$$\min_x \max_y \quad y^T Ax.$$
Here the row player is minimizing since he wants to pay less. The column player solves the following problem for her optimal strategies:
$$\max_y \min_x \quad y^T Ax.$$
She is maximizing the payment of the row player to her. Note that there is complete symmetry between the two players, and one can write, e.g.,
$$\max_y \min_x \quad y^T Ax \quad = \min_y \max_x \quad y^T (-A)x,$$
where $B=-A$ is the payoff matrix for player 2 in the (cost-version) bimatrix game $(A,B)$.
Whichever way it's written, with utilities as payoffs or costs, in terms of the utility for the player at hand or for the player's opponent, the optimization problem has the following interpretation:
Find a strategy $s$ that gives the best payoff guarantee given that the other player will be best responding against $s$. Note that you only do better if the other player does not best respond against such an optimal strategy. For example, in your cost-for-player-1-version, where he pays $A$, and his optimal strategies are given by $x$'s that solve
$$\min_x \max_y \quad y^T Ax,$$
the $\max_y$ is player 2 best responding and the $\min_x$ is player 1 choosing the best strategy in terms of payoff guarantee given this.