Illustrating difficulty to find dominant strategies: series of simple one-shot two-players games
Start with the following matrix.
$$ \begin{matrix}1 &amp;&amp; 0 &amp;&amp; 0\\
1 &amp;&amp; 1 &amp;&amp; 0\\
1 &amp;&amp; 1 &amp;&amp; 1  \end{matrix}$$

You will choose a $1$ in the matrix and change it to a $0$. The other player will choose a $0$ and change it to a $1$. Your goal is for the resulting matrix to have an even determinant. The other player's goal is for the resulting matrix to have an odd determinant.

Hover over the box to find out the dominant strategy:


   Change the $1$ in the bottom left corner to a $0$.