I don't really understand minimax theorem. I have watched this video minimax in chess and I understand that by backward induction as first player I can find best outcome for me (minimize my loss = minimize opponent's gain) but I don't get that equation (got from wikipedia) bellow. Can someone explain me it or recommend some nice video?
$$ \max_{x\in X} \min_{y\in Y} f(x,y) = \min_{y\in Y} \max_{x\in X} f(x,y) $$
Consider the following two sequencial games:
Game A: Player 1 chooses $x$, announces it. Then Player 2 chooses $y$.
Game B: Player 2 chooses $y$, announces it. Then Player 1 chooses $x$.
For any given $x$ in game A, the problem of Player 2 is to minimize $f(x,y)$. So the payoff will be $g(x)=\min_y f(x,y)$. Among all these possible options, Player 1 chooses the one that maximizes $g(x)$, so the total payoff is $\max_x \min_y f(x,y)$.
With a similar argument, the payoff in Game B is $\min_y \max_x f(x,y)$.
Clearly, in Game A player 2 has an advantage relative to the simultaneous game and to Game B, so $\max\min f\leq \min\max f$.
The minimax theorem states (with appropriate assumptions of $f,x,y$) that they are equal, so it doesn't matter who announces his strategy first (or if they play at the same time) the result is the same. Hence, we can actually play the strategies we wanted to announce in the simultaneous game (these are called the optimal strategies) and get this value.