In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium? Where I can find a simple proof for this fact?
For example, a trivial bimatrix game with alternating move has the following payoff matrix:
\begin{array}{|c|c|c|}
\hline
&amp; 1 &amp; 2 \\ \hline
U &amp; (0,0)  &amp; (0,0)\\ \hline
 L&amp;   (0,0)&amp; (0,0)\\ \hline
\end{array}
Then all of the pure and mixed strategies are trivially the equilibrium strategies.
I guess that, if the game structural is so complicated that it becomes impossible for the players to solve the game, then this complete information game becomes effectively like an incomplete information game. But I am not sure how to rigorously describe this.
This statement is wrong. Consider Alternating Matching Pennies with imperfect information (the follower doesn't observe the leader's move). The strategic form of this game is just the classical (simultaneous-move) Matching Pennies Game and the unique NE has both players mixing.