# 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:

$$\begin{array}{|c|c|c|} \hline & 1 & 2 \\ \hline U & (0,0) & (0,0)\\ \hline L& (0,0)& (0,0)\\ \hline \end{array}$$

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.

• You should probably clarify/define what you mean by: simple game and non-trivial mixed equilibrium. – Herr K. Sep 30 at 20:36
• This question might be closed as unclear, but it doesn't look like homework to me. – Michael Greinecker Sep 30 at 21:01